AASHTO Design Studio — Bridge Engineering

Chapter 1

Bridge engineering fundamentals and design basis

Preliminary proportioning, life-cycle cost screening, and the LRFD reliability framework — load modifiers, factored demand versus factored resistance and the target reliability index.

Module 1AASHTO LRFD 2.5.2.6.3 / 2.5.2.6.2Chapter 1

Preliminary sizing, span-to-depth & life-cycle cost

Table 2.5.2.6.3-1 minimum depth, the L/800 live-load deflection limit, deck and girder quantities, and the present value of owner cost over the analysis period.

Span LftOut-to-out widthftDeck thicknessinNumber of girders—Superstructure typeAnnual owner cost$/yrAnalysis periodyrDiscount rate—

Span-to-depth screening from Table 2.5.2.6.3-1, the L/800 live-load deflection limit, deck quantities and a present-value comparison of owner cost.

Minimum overall depth

46.1 in

L/31

Girder spacing S

8.80 ft

Δ_allow = L/800

1.80 in

Deck concrete

138.5 yd³

Life-cycle PV

$1188073

Derivation — equation, substitution, result

Span-to-depth screening

\begin{aligned}&\;d_{\operatorname{min}} = \;\text{(L/D ratio)} \cdot L\\[4pt]&=\;0.032 \times 120.0 \,\text{ft} \times 12\\[4pt]&=\;\boxed{d_{\operatorname{min}} \approx 46.1 \,\text{in} \,\text{overall}}\end{aligned}

Steel section depth

\begin{aligned}&\;d_{steel} = \,\text{ratio}_{comp} \cdot L\\[4pt]&=\;0.027 \times 1440 \,\text{in}\\[4pt]&=\;\boxed{38.9 \,\text{in}}\end{aligned}

Girder spacing

\begin{aligned}&\;S = w / N_g\\[4pt]&=\;44.0 / 5\\[4pt]&=\;\boxed{8.80 \,\text{ft}}\end{aligned}

Live-load deflection limit

\begin{aligned}&\;\Delta _{allow} = L/800 (§2.5.2.6.2)\\[4pt]&=\;1440 \,\text{in} / 800\\[4pt]&=\;\boxed{1.80 \,\text{in}}\end{aligned}

Deck concrete quantity

\begin{aligned}&\;V = w\cdot L\cdot (t_s/12)/27\\[4pt]&=\;44.0 \times 120.0 \times 0.708 / 27\\[4pt]&=\;\boxed{138.5 \,\text{yd}^{3}}\end{aligned}

Slab load per girder

\begin{aligned}&\;w_{deck} = t_s\cdot S\cdot \gamma _c\\[4pt]&=\;0.708 \,\text{ft} \times 8.80 \,\text{ft} \times 0.150 \,\text{kcf}\\[4pt]&=\;\boxed{0.935 \,\text{klf} \rightarrow 0.935 \,\text{klf}}\end{aligned}

Life-cycle present value

\begin{aligned}&\;\mathrm{PV} = A[1 − (1+i)^−n]/i\\[4pt]&=\;A = \$40000, i = 0.030, n = 75 \,\text{yr}\\[4pt]&=\;\boxed{\$1188073}\end{aligned}

Detailing — plan, elevation and section

Section — composite

Plan — framing

Elevation

Constructability & detailing notes

Keep the girder depth constant along the span where possible — haunched webs add fabrication cost that rarely pays back below 200 ft spans.
Check shipping limits early: 12 ft depth and 150 ft length are practical highway limits for a single piece.
Girder spacing between 8 and 12 ft usually minimises total cost; wider spacing thickens the deck and increases the distribution factor.
Allow 2 in minimum haunch for construction tolerance and screed adjustment.

Check	Demand	Capacity / limit	Status
Girder spacing within the §4.6.2.2 approximate-equation range 3.5 – 16 ft	8.80 ft	3.5 – 16 ft	PASS
Deck thickness ≥ 7 in (§9.7.1.1)	8.50 in	7.0 in	PASS

Independent verification	Expected	Computed	Status
Depth ≈ 0.033L for a continuous composite steel girder	47.5 in		VERIFIED

Assumptions & basis of design

Depth ratios from AASHTO LRFD Table 2.5.2.6.3-1 for constant-depth superstructures.
Concrete unit weight 0.150 kcf; the deck quantity excludes haunches, barriers and the wearing surface.
The present-value screening is a uniform-series discounting of owner cost — not an AASHTO provision.

Use the screening depth as the starting trial section, then confirm with the Chapter 8 flexure and Chapter 4 deflection modules.

Module 2AASHTO LRFD 1.3.2 / 3.4.1Chapters 1 · 2

LRFD design basis — η, factored demand and reliability

Ductility, redundancy and importance modifiers combined into η, the Strength I factored demand against φR_n, and the reliability index β = (μ_R − μ_Q)/√(σ_R² + σ_Q²).

η_D ductility—η_R redundancy—η_I importance—Nominal resistance R_nkip·ftφ—DC effectkip·ftDW effectkip·ftLL + IM effectkip·ftMean resistance μ_Rkip·ftσ_Rkip·ftMean load μ_Qkip·ftσ_Qkip·ftPermanent-load case

Combines the ductility, redundancy and importance modifiers into η, applies the Strength I factors, and back-checks the reliability index.

η (load modifier)

1.050

Factored demand Q

5748.8

Factored resistance R_r

5400.0

Utilisation Q/R_r

1.065

Reliability index β

4.17

Derivation — equation, substitution, result

Load modifier

\begin{aligned}&\;\eta = \eta _D \eta _R \eta _I \ge 0.95 (Eq. 1.3.2.1-2)\\[4pt]&=\;1.00 \times 1.00 \times 1.05 = 1.050\\[4pt]&=\;\boxed{\eta = 1.050}\end{aligned}

Factored demand

\begin{aligned}&\;Q = \eta [\gamma _{DC}\cdot \mathrm{DC} + \gamma _{DW}\cdot \mathrm{DW} + 1.75(\mathrm{LL}+\mathrm{IM})]\\[4pt]&=\;1.050[1.25(1500.0) + 1.50(300.0) + 1.75(1800.0)]\\[4pt]&=\;\boxed{Q = 5748.8}\end{aligned}

Factored resistance

\begin{aligned}&\;R_r = \phi R_n (Eq. 1.3.2.1-1)\\[4pt]&=\;0.90 \times 6000.0\\[4pt]&=\;\boxed{R_r = 5400.0}\end{aligned}

Design inequality

\begin{aligned}&\;\Sum \eta _i \gamma _i Q_i \le \phi R_n\\[4pt]&=\;5748.8 \,\text{vs} 5400.0\\[4pt]&=\;\boxed{\,\text{utilisation} = 1.065}\end{aligned}

Reliability index

\begin{aligned}&\;\beta = (\mu _R − \mu _Q)/\sqrt{\sigma _R^{2} + \sigma _Q^{2}}\\[4pt]&=\;(6600.0 − 3200.0)/\sqrt{660.0^{2} + 480.0^{2}}\\[4pt]&=\;\boxed{\beta = 4.17}\end{aligned}

Probability of failure

\begin{aligned}&\;p_f \approx 10^(−\beta \cdot 0.4 − 0.5) \;\text{(approximate)}\\[4pt]&=\;\boxed{p_f \approx 6.82e-3}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchLRFD load-and-resistance basis — factored demand vs. factored resistance

η-modified factored load Q_u = 5749 vs. φR_n = 5400 (nominal R_n = 6000). Reliability index β = 4.17 against the Strength I target β ≈ 3.50.

Constructability & detailing notes

η is applied to the load side, never to the resistance — a common mark-up error in student calculations.
Record the assumed operational classification on the cover sheet; it drives both η_I and the seismic R factor.

Check	Demand	Capacity / limit	Status
Strength limit state Σηγ Q ≤ φR_n	5748.8	5400.0	REVIEW
Target reliability β_T = 3.5 (§C1.3.2.1)	4.17	3.50	PASS
η ≥ 0.95 for maximum load factors	1.050	0.95	PASS

Independent verification	Expected	Computed	Status
Target reliability index for Strength I	β ≈ 3.5	4.17	VERIFIED

Assumptions & basis of design

AASHTO LRFD Eq. 1.3.2.1-1 with Strength I permanent factors from Table 3.4.1-2 (γ_DC 1.25/0.90, γ_DW 1.50/0.65).
η_D, η_R and η_I each fall between 0.95 and 1.05; the minimum-load case inverts the product.
The reliability index uses the lognormal-free first-order form; the AASHTO calibration target for girders is β_T = 3.5 at Strength I.

Demand exceeds resistance by 6.5 % — increase R_n by that margin, or reduce DW by specifying no future overlay allowance where the owner permits.

Chapter 3

Loads and load combinations

The HL-93 model, the position that maximises its effect, and the load factors that turn nominal effects into factored demands.

Module 3AASHTO LRFD 3.6.1Chapter 3

HL-93 live-load envelope (simply supported)

Envelope of design truck / tandem + lane load with IM and multiple-presence factors.

HL-93 Live-Load Effect — Simply Supported Span

Computes the maximum midspan moment and end shear from the HL-93 envelope. AASHTO LRFD §3.6.1.2 / §3.6.2

Span L (ft)

Dyn. load allowance IM (%)

Number of loaded lanes

Multiple-presence m

Max midspan moment

3656.4

kip-ft per design lane × factors

M = m · n · [(1+IM)·max(M_truck, M_tandem) + M_lane]

M_truck (per lane): 1883.0 kip-ft
M_tandem (per lane): 1450.3 kip-ft
M_lane (per lane): 1152.0 kip-ft
Governs: Design truck

Max end shear

119.3

kip per design lane × factors

V = m · n · [(1+IM)·max(V_truck, V_tandem) + V_lane]

V_truck (per lane): 60.80 kip
V_tandem (per lane): 49.17 kip
V_lane (per lane): 38.40 kip
Governs: Design truck

M_lane = w · L² / 8 V_lane = w · L / 2 w = 0.64 klf

(derived)

w: design lane load intensity [klf]
L: simply supported span length [ft]
IM: dynamic load allowance (33% typ.) [-]
m: multiple-presence factor [-]

Verification

Order-of-magnitude check: for L = 120 ft, the HL-93 envelope (per lane, per AASHTO tables) gives M ≈ 2,090 kip-ft (unfactored, no IM, no m). With IM = 1.33 and m = 1.00 the moment should approach ≈ 2,780 kip-ft plus lane contribution. Compare to the value returned above.

What can go wrong

This calculator idealizes a simply supported span. For continuous spans, use influence surfaces and apply the truck + lane combination for positive moment and two trucks (min. 50 ft between axles) × 0.90 + two lanes for negative moment AASHTO LRFD §3.6.1.3.1. Consult refined analysis for skew and curved bridges.

Detailing sketchHL-93 loading on the span — envelope

Governing vehicle: Design truck for moment (1883 vs 1450 k-ft truck/tandem), Design truck for shear. Design M_LL+IM = 3656 k-ft, design V_LL+IM = 119.3 kip on a 120 ft span.

Module 4AASHTO LRFD 3.6.1.2 / 3.6.2Chapters 3 · 4

Design-truck absolute maximum moment (Barré)

Locates the critical axle position by Barré's theorem, then computes truck, tandem and lane moments and the governing HL-93 moment and shear for one lane on a simple span.

Span length LftLane load wklf1 + IM/100—

Barré's theorem locates the absolute maximum moment under the axle nearest the resultant — not at midspan.

Maximum truck moment (no IM)

1164.9 k-ft

Maximum tandem moment (no IM)

950.0 k-ft

Lane moment

512.0 k-ft

Governing M_LL+IM per lane

2061.3 k-ft

Governing V_LL+IM per lane

110.19 kip

Detailing sketchHL-93 truck on the span — envelope moment

Governing live-load moment M_LL+IM = 2061.30 k-ft per lane on a 80.0 ft span, lane load 0.64 klf.

Step	Equation	Substitution	Result
Axle resultant location	x̄ = ΣPᵢxᵢ / ΣPᵢ	(8·0 + 32·14 + 32·28) / 72	18.67 ft from the front axle
Offset resultant-to-critical axle	e = \|x̄ − x_crit\|	\|18.67 − 14\|	4.67 ft
Critical axle position (Barré)	a = L/2 − e/2	80.0/2 − 4.67/2	37.67 ft from the left support
Left reaction for that position	R = Σ Pᵢ(L − xᵢ)/L	8(56.33) + 32(42.33) + 32(28.33) / 80.0	33.90 kip
Maximum truck moment	M = R·a − Σ P_left(a − xᵢ)	33.90·37.67 − Σ…	1164.9 k-ft
Design tandem moment	M = R·a₁ with axles at L/2 ± 2 ft	25.00·38.00	950.0 k-ft
Design lane moment	M = wL²/8	0.64·80.0²/8	512.0 k-ft
Truck + IM	M_truck(1 + IM/100)	1164.9 × 1.33	1549.3 k-ft
Governing HL-93 moment	M_LL+IM = max(M_truck, M_tandem)(1+IM) + M_lane	max(1549.3, 1263.5) + 512.0	2061.3 k-ft/lane
Governing HL-93 shear at support	V = max(V_truck, V_tandem)(1+IM) + V_lane	max(63.60, 48.75)·1.33 + 25.60	110.19 kip/lane

The design truck governs moment on this 80 ft span (1549.3 k-ft vs 1263.5 k-ft for the tandem). The tandem typically governs below about 40 ft.

Assumptions & code basis

HL-93 design truck 8-32-32 kip at 14 ft axle spacings (rear spacing at its 14 ft minimum for maximum simple-span moment), AASHTO LRFD §3.6.1.2.2.
Design tandem: two 25 kip axles 4 ft apart, §3.6.1.2.3. Design lane load 0.64 klf over the full span, §3.6.1.2.4.
Absolute maximum moment located by Barré's theorem: the span centreline bisects the distance between the resultant of the axles on the span and the axle nearest it.
Dynamic load allowance IM applies to the truck or tandem only, never to the lane load (§3.6.2.1).
Result is one lane of live load on a simple span before the distribution factor and multiple-presence factor are applied.

Module 5AASHTO LRFD 3.6.1.3Chapters 3 · 4

Influence-line explorer

Slide the HL-93 truck to locate the critical loading position for moment and shear at any section.

Influence-Line Explorer — HL-93 Truck on a 120-ft Simply Supported Span

Drag the truck along the span and reposition the analysis section. AASHTO LRFD §3.6.1.2 — Design Vehicular Live Load

Truck lead-axle position (ft)30.00

Variable rear-axle spacing (ft)14.00

Section x / L0.50

Moment at x = 60.0 ft

2904.00

kip-ft (unfactored, per lane, IM not applied)

Truck axles1752.00

Design lane (0.64 klf)1152.00

Shear at x = 60.0 ft

-19.60

kip (unfactored, per lane, IM not applied)

Truck axles-29.20

Design lane (0.64 klf)9.60

How to use this

Set the section (x/L) where you want the maximum effect.
Slide the truck until the heaviest axles align under the peak ordinate.
For shear at the end, sweep the truck toward the support; the lane load ordinate is largest just past the cut.
Change the variable spacing (14 ft governs for most spans < 100 ft; 30 ft can govern for shear at far support on long spans).

Module 6AASHTO LRFD 3.4.1Chapters 3 · 4

Load-combination generator

Applies the Strength, Service, and Fatigue load factors to your DC, DW, and LL+IM effects, with maximum and minimum permanent-load cases.

M_DC1 — non-compositek-ftM_DC2 — composite (barriers)k-ftM_DW effectk-ftM_LL+IM (distributed)k-ftLoad modifier η = η_D η_R η_ISpan L (diagram only)ft

Q = Σ η γᵢ Qᵢ. DC1 acts on the bare girder, DC2 on the composite section; both carry γ_DC. Maximum load factors govern where the effect adds; minimum factors govern where a permanent load relieves the effect (uplift, overturning, retaining walls).

M_DC = M_DC1 + M_DC2

900 k-ft

M_DW

140 k-ft

Factored M_u (Strength I)

3435 k-ft

Detailing sketchSimple-span moment diagram — DC1, DC2, DW and LL+IM

M_DC1 = 620 k-ft (girder + slab + haunch on the non-composite section) · M_DC2 = 280 k-ft (barriers on the composite section) · M_DW = 140 k-ft · M_LL+IM = 1200 k-ft (distributed) · factored M_u = 3435 k-ft.

Combination	γ_DC	γ_DW	γ_LL	Q (max)	Q (min)
Strength I	1.25	1.50	1.75	3435	3001
Strength II (permit)	1.25	1.50	1.35	2955	2521
Strength IV (dead governs)	1.50	1.50	0.00	1560	901
Service I	1.00	1.00	1.00	2240	2240
Service III (PS tension)	1.00	1.00	0.80	2000	2000
Fatigue I	0.00	0.00	1.75	2100	2100

Module 7AASHTO LRFD 3.6.1.1.1 / 3.6.1.1.2 / 3.6.2Chapter 3

Design lanes, multiple presence & dynamic allowance

N_L = INT(w/12), the Table 3.6.1.1.2-1 multiple-presence factor m, and IM for decks, fatigue, buried components and all other components with the fill-depth reduction.

Clear roadway widthftLanes loaded—ComponentDepth of fillftOne-lane reactionkip

Number of design lanes, multiple-presence factor and the dynamic load allowance for the component you are designing.

Design lanes N_L

3

Multiple presence m

0.85

IM

33.0 %

Live-load effect

217.1 kip

Derivation — equation, substitution, result

Number of design lanes

\begin{aligned}&\;N_L = \mathrm{INT}(w/12) (§3.6.1.1.1)\\[4pt]&=\;\mathrm{INT}(40.0/12)\\[4pt]&=\;\boxed{N_L = 3}\end{aligned}

Multiple presence

\begin{aligned}&\;m \operatorname{from} Table 3.6.1.1.2-1\\[4pt]&=\;3 \,\text{loaded} \,\text{lane}(s)\\[4pt]&=\;\boxed{m = 0.85}\end{aligned}

Dynamic load allowance

\begin{aligned}&\;\mathrm{IM} \operatorname{from} Table 3.6.2.1-1\\[4pt]&=\;\,\text{other}\\[4pt]&=\;\boxed{\mathrm{IM} = 33.0 \%}\end{aligned}

Applied to truck/tandem only

\begin{aligned}&\;\mathrm{LL}+\mathrm{IM} = m\cdot N\cdot [R_{truck}(1+\mathrm{IM}/100) + R_{lane}]\\[4pt]&=\;\boxed{}\end{aligned}

Factored one-component effect

\begin{aligned}&\;R = R_{1lane} \cdot N \cdot m \cdot (1 + \mathrm{IM}/100)\\[4pt]&=\;64.0 \times 3 \times 0.85 \times 1.330\\[4pt]&=\;\boxed{217.1 \,\text{kip}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchDesign lanes, multiple presence and dynamic load allowance

Clear roadway 40.0 ft ⇒ N_L = 3 design lane(s) at 200.00 ft each. Multiple-presence factor m = 0.85 governs when fewer than 3 lanes are loaded. IM = 33 % applied to the truck/tandem for the other case (not to the lane load).

Constructability & detailing notes

The multiple-presence factor is already embedded in the approximate distribution-factor equations — never apply it twice.
Fatigue design uses one truck in one lane with m = 1.0 and IM = 15 %.

Check	Demand	Capacity / limit	Status
Roadway wide enough for the assumed loaded lanes	3 lanes	3 lanes	PASS
m not applied with the approximate distribution equations (already embedded)Use m only with the lever rule, the rigid cross-section method or a refined analysis.	—	—	PASS

Independent verification	Expected	Computed	Status
N_L = INT(w/12)	3	3	VERIFIED

Assumptions & basis of design

Design lane width 12 ft with the 10-ft loaded strip placed to produce the extreme effect (§3.6.1.3.1).
IM applies to the design truck and tandem only, and not to fatigue-limit centrifugal or braking forces.
For 20 ft ≤ w < 24 ft, AASHTO requires two design lanes each equal to half the roadway width.

For the fatigue limit state use one lane with m = 1.00 (§3.6.1.4.3b) — the 1.20 single-lane factor is removed.

Module 8AASHTO LRFD 3.6.4 / 3.6.3Chapter 3

Braking (BR) and centrifugal (CE) forces

The governing of 25 % of axle weights and 5 % of truck-plus-lane, the centrifugal factor C = f v²/(gR), the 6 ft application height, and the overturning couple delivered to the substructure.

Loaded lengthftLanes contributing—Multiple presence m—Design truck weightkipLane loadklfDesign speedmphCurve radiusftForce height above deckftDeck to bearingft

BR is the larger of 25 % of the axle weights or 5 % of truck plus lane; CE uses the f = 4/3 factor for the truck only.

Braking force BR

36.00 kip

Centrifugal force CE

32.35 kip

Governing horizontal

36.00 kip

braking governs

Substructure moment

468.0 kip·ft

Derivation — equation, substitution, result

Braking — 25 % of the axles

\begin{aligned}&\;\mathrm{BR}_{1} = 0.25\;\text{(design truck or tandem)}\\[4pt]&=\;0.25 \times 72.0 \,\text{kip}\\[4pt]&=\;\boxed{18.00 \,\text{kip}/\,\text{lane}}\end{aligned}

Braking — 5 % of truck + lane

\begin{aligned}&\;\mathrm{BR}_{2} = 0.05[\,\text{truck} + \,\text{lane}\cdot L]\\[4pt]&=\;0.05[72.0 + 0.64(200.0)]\\[4pt]&=\;\boxed{10.00 \,\text{kip}/\,\text{lane}}\end{aligned}

Governing braking force

\begin{aligned}&\;\mathrm{BR} = \operatorname{max}(\mathrm{BR}_{1}, \mathrm{BR}_{2}) \cdot N_{lanes} \cdot m (§3.6.4)\\[4pt]&=\;18.00 \times 2 \times 1.00\\[4pt]&=\;\boxed{\mathrm{BR} = 36.00 \,\text{kip}}\end{aligned}

Centrifugal coefficient

\begin{aligned}&\;C = f v^{2}/(gR), f = 4/3 \operatorname{at} \,\text{strength} (§3.6.3)\\[4pt]&=\;(4/3)(80.7 \,\text{ft}/s)^{2}/(32.2 \times 1200)\\[4pt]&=\;\boxed{C = 0.2246}\end{aligned}

Centrifugal force

\begin{aligned}&\;\mathrm{CE} = C \cdot W_{truck} \cdot N \cdot m\\[4pt]&=\;0.2246 \times 72.0 \times 2 \times 1.00\\[4pt]&=\;\boxed{\mathrm{CE} = 32.35 \,\text{kip}}\end{aligned}

Overturning couple

\begin{aligned}&\;M = F\;\text{(hCG + ddeck-brg)}; \mathrm{BR} \,\text{acts} 6 \,\text{ft} \,\text{above} \,\text{the} \,\text{deck}\\[4pt]&=\;\,\text{arm} = 13.00 \,\text{ft}\\[4pt]&=\;\boxed{M_{BR} = 468.0 \,\text{kip}\cdot \,\text{ft}, M_{CE} = 420.5 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchBraking (BR) and centrifugal (CE) force — bearing elevation

BR = 36.0 kip acts 6.0 ft above the deck, transferred through 7.0 ft to the bearings. CE = 32.4 kip from a curve radius R = 1200 ft adds a lateral, overturning couple at the same elevation.

Constructability & detailing notes

Braking force acts 6 ft above the deck in both directions — check the bearings and the anchor bolts for the reversal.
Centrifugal force adds an overturning couple that can unload the inside girder; verify uplift at the bearings.
Detail fixed and expansion bearings so the horizontal load path into the substructure is explicit on the plans.

Check	Demand	Capacity / limit	Status
Braking applied in both directions on all lanes headed the same wayLanes carrying traffic in one direction only — use m for the number of loaded lanes.	—	—	PASS
Centrifugal force acts 6 ft above the roadway (§3.6.3)	6.0 ft	6.0 ft	PASS

Independent verification	Expected	Computed	Status
BR ≥ 0.25 × axle weights	18.0 kip per lane		VERIFIED

Assumptions & basis of design

BR per §3.6.4 with the dynamic load allowance intentionally omitted; the force acts 6 ft above the roadway surface.
CE per §3.6.3 with f = 4/3 for all limit states except fatigue, where f = 1.0.
Both forces are transmitted through the bearings into the substructure — check the anchor bolts and the bearing shear capacity.

Braking governs: detail the fixed bearing and its anchorage for this force, and check the pier for the resulting base moment.

Chapter 4

Section properties and composite action

Neutral axis, transformed sections, effective flange width, and the permanent loads each stage of the section must carry.

Module 9AASHTO LRFD 4.6.2.6 / 6.10.1.1.1b / 5.4.2.4Chapter 4

Transformed composite section properties

Modular ratio, §4.6.2.6 effective flange width, the composite neutral axis and inertia at n and 3n, and the stage-by-stage section moduli that DC1, DC2/DW and LL+IM act on.

Span LftGirder spacing SftDeck thickness t_sinOverhangftHaunch depthinDeck f′_cksiSteel section depthinSteel area Ain²Steel inertia Iin⁴Steel centroid from bottominGirder position

Enter the bare-steel (or precast) girder properties; the module transforms the deck at n and 3n and reports the stage-by-stage section moduli.

Modular ratio n

7.96

Effective flange width b_eff

108.0 in

Non-composite I / S_b

20000 / 833 in⁴ / in³

Short-term composite I (n)

50535 in⁴

Long-term composite I (3n)

38763 in⁴

S_b short term / long term

1128 / 1051 in³

Detailing sketchComposite section — transformed layer stack

Effective flange width b_eff = 108.00 in; short-term composite I = 50535 in⁴.

Step	Equation	Substitution	Result
Deck modulus	E_c = 33,000 w_c^1.5 √f′_c	33,000 (0.145)^1.5 √4.00	3644 ksi
Modular ratio	n = E_s / E_c	29000 / 3644	7.96
Effective flange width	b_eff = min(S, L_eff/4)	min(108.0, 300.0)	108.0 in
Transformed deck area (short term)	A_tr = b_eff·t_s / n	108.0·8.00 / 7.96	108.57 in²
Composite neutral axis (n)	ȳ_b = Σ A_i y_i / Σ A_i	(48.00·24.00 + 108.57·54.00) / 156.57	44.80 in above the bottom flange
Composite inertia (n)	I = ΣI_i + ΣA_i d_i²	20000 + …	50535 in⁴
Section moduli (n)	S_b = I/ȳ_b, S_t = I/y_t	50535/44.80, 50535/3.20	S_b = 1128 in³, S_t = 15806 in³
Composite inertia (3n, long term)	Repeat with n′ = 3n	n′ = 23.87	I = 38763 in⁴, ȳ_b = 36.90 in
Deck top-fibre modulus	S_deck = n·I / y_deck	7.96·50535 / 13.20	30473 in³

Composite action increases the bottom-fibre section modulus from 833 in³ (steel alone, carrying DC1) to 1128 in³ (n, live load) — a factor of 1.35. Stack the stresses stage by stage: DC1 on the steel section, DC2 and DW on the 3n section, LL+IM on the n section.

Assumptions & code basis

Modular ratio n = E_s/E_c with E_c = 33,000 w_c^1.5 √f′_c (AASHTO LRFD Eq. 5.4.2.4-1), w_c = 0.145 kcf.
Effective flange width per §4.6.2.6.1: the tributary width, limited by L_eff/4 for interior girders (one-quarter effective span; L/8 each side for exterior).
Short-term (n) properties govern live load; long-term (3n) properties govern superimposed dead load DC2/DW to account for creep (§6.10.1.1.1b).
Concrete deck is assumed uncracked and fully composite in positive-moment regions; in negative-moment regions only the longitudinal deck reinforcement is normally considered effective.
Haunch depth is included in the geometry but its concrete area is conservatively neglected.

Module 10AASHTO LRFD 3.5.1 / 4.6.2.2.1 / 3.4.1Chapters 3 · 4

Girder dead loads — DC1, DC2 & DW

Non-composite DC1 (slab, haunch, girder, diaphragms, forms), composite DC2 (barriers/sidewalk) and DW (future wearing surface), with midspan moments, support shears and factored permanent effects.

Span length LftGirder spacing SftNumber of girders N_b—Slab thickness tₛinHaunch depthinHaunch / top-flange widthinGirder self weightk/ftDiaphragms (smeared)k/ftStay-in-place formsk/ftBarrier weight (each)k/ftNumber of barriers—Roadway width (FWS)ftFWS thicknessinFWS unit weight γ_wskcfConcrete unit weight γ_ckcf

DC1 = slab + haunch + girder + diaphragms + SIP forms (non-composite) · DC2 = barriers / sidewalk ÷ N_b (composite) · DW = FWS × roadway width ÷ N_b

w_DC1 (non-composite)

0.9883 kip/ft

slab 0.8000 + haunch 0.0333

w_DC2 (composite)

0.1800 kip/ft

barriers shared by all girders

w_DW

0.2100 kip/ft

future wearing surface

M_DC1

790.7 kip-ft

M_DC2

144.0 kip-ft

M_DW

168.0 kip-ft

V_DC1

39.5 kip

V_DC2

7.2 kip

V_DW

8.4 kip

Strength I permanent moment (max: 1.25DC + 1.50DW)

1420.3 kip-ft

min case 0.90DC + 0.65DW = 950.4 kip-ft

Service I permanent moment (1.0DC + 1.0DW)

1102.7 kip-ft

factored permanent shear = 71.0 kip

Detailing sketchSimple-span moment diagram — DC1, DC2, DW and LL+IM

M_DC1 = 791 k-ft (girder + slab + haunch on the non-composite section) · M_DC2 = 144 k-ft (barriers on the composite section) · M_DW = 168 k-ft · M_LL+IM = 0 k-ft (distributed) · factored M_u = 1420 k-ft.

Live load (HL-93) is added separately in Module 2, then distributed to the girder with the factors from Module 4. Only DC1 acts on the bare girder section; DC2 and DW act on the composite section, so their stresses use the composite section moduli.

Module 11AASHTO LRFD 6.10.2 / 6.10.6 / 6.10.8.2 / 6.9.4Chapters 4 · 8

Steel I-section stability & compression design

Cross-section proportion limits, web/flange classification, flange local buckling and lateral–torsional buckling with L_p / L_r, plus a compression-member slenderness check.

I-section

Web depth DinWeb thickness t_winComp. flange width b_fcinComp. flange thk t_fcinTens. flange width b_ftinTens. flange thk t_ftinYield F_yksiUnbraced length L_bftC_b—Factored moment M_uk-ft

Compression member

Area Ain²Radius of gyration rinEffective length factor K—Unbraced length LftAxial demand P_ukipPlate b/t—

λ_f = b_fc/2t_fc · L_p = 1.0 r_t √(E/F_yc) · L_r = π r_t √(E/F_yr) · F_nc = min(FLB, LTB) · P_n = 0.658^(P_o/P_e) P_o

D / t_w

108.0

limit 150 (§6.10.2.1.1)

b_fc / 2t_fc

8.00

λ_pf = 9.15, λ_rf = 16.97

Web class

non-compact

2D_c/t_w = 119.1

R_b (load shedding)

1.000

L_p

7.4 ft

bracing for full yield

L_r

27.8 ft

L_b = 20.0 ft → inelastic

F_nc (FLB)

50.00 ksi

F_nc (LTB)

40.76 ksi

F_nc governing

40.76 ksi

Lateral–torsional buckling (§6.10.8.2.3)

φM_n vs M_u

3900 ≥ 4200 k-ft

S_xc = 1148 in³ · utilization 1.08

Design verdict & redesign guidance

• LTB governs at L_b = 240.0 ft: brace at or below L_p = 89.0 in (7.4 ft) to develop the full yield moment.
• φM_n = 3900 k-ft < M_u = 4200 k-ft — increase S_xc by about 8%: try t_fc = 1.077 in or D = 56 in.

KL/r

86.1

limit 120 (§6.9.3)

P_e (Euler)

568 kip

inelastic buckling

φ_c P_n vs P_u

406 ≥ 300 kip

Compression-member verdict

• Column adequate: KL/r = 86, utilization 0.74.

Detailing sketchComposite steel plate girder — section, elevation and framing plan

F_nc = 40.76 ksi · φM_n = 3900 k-ft vs M_u = 4200 k-ft

Chapter 4

Live-load distribution

How much of a lane a single girder carries — the approximate equations plus the stiffness, skew, lever-rule and rigid cross-section checks that surround them.

Module 12AASHTO LRFD 4.6.2.2.2 / 4.6.2.2.3Chapters 4 · 7 · 8

Live-load distribution factors

Approximate distribution equations for concrete decks on steel or prestressed I-girders, interior and exterior.

Girder spacing SftSpan length LftSlab thickness tₛinLongitudinal stiffness Kgin⁴Overhang distance dₑft

Cross-section type (k): concrete deck on steel or prestressed-concrete I-girders. Valid for 3.5 ≤ S ≤ 16 ft, 20 ≤ L ≤ 240 ft, 4.5 ≤ tₛ ≤ 12 in.

Kg / (12·L·tₛ³) = 5.0863 · g_M,int = 0.075 + (S/9.5)^0.6(S/L)^0.2(Kg/12Ltₛ³)^0.1

Interior moment — one lane

0.531 lanes/girder

Interior moment — two+ lanes

0.745 lanes/girder

Interior shear — one lane

0.680 lanes/girder

Interior shear — two+ lanes

0.814 lanes/girder

Exterior moment (e = 0.77 + dₑ/9.1)

0.737

e = 0.990

Exterior shear (e = 0.6 + dₑ/10)

0.652

e = 0.800

Governing interior moment factor is 0.745. Exterior girders must also be checked by the lever rule and the rigid cross-section (special analysis) equation for beam-slab bridges with diaphragms.

Detailing sketchDeck cross-section — girder spacing and lane positions

6 girders at S = 8.00 ft spacing, deck thickness t_s = 8.00 in, overhang d_e = 2.00 ft. Interior moment DF ≈ 0.745 lanes/girder; exterior moment DF ≈ 0.737 lanes/girder.

Module 13AASHTO LRFD 4.6.2.2.1 / 4.6.2.2.2d–e / 4.6.2.2.3cChapter 4

K_g, skew corrections, lever rule & rigid cross-section

The longitudinal stiffness parameter K_g = n(I + Ae_g²), the §4.6.2.2.2e moment reduction and §4.6.2.2.3c shear increase for skew, and the two exterior-girder checks — lever rule and the rigid cross-section special analysis — with multiple-presence factors.

Beam modulus E_BksiDeck modulus E_DksiGirder area Ain²Girder inertia Iin⁴Centroid distance e_ginGirder spacing SftSpan LftSlab t_sinSkew angle θdegNumber of girders N_b—Loaded lanes N_L—Overhang distance d_eftWheel 1 offsetftWheel 2 offsetft

Wheel offsets are the distances from the exterior girder to each wheel of the loaded lanes, measured along the deck.

K_g

502964 in⁴

Skew factor — moment

1.0000

Skew factor — shear (obtuse corner)

1.1087

Lever-rule exterior DF

0.4444 lanes/girder

Rigid cross-section DF

1.0222 lanes/girder

Governing exterior DF

1.0222 lanes/girder

Detailing sketchGirder cross-section — plan of framing

Governing exterior distribution factor = 1.022 lanes/girder for 5 girders at S = 9.00 ft, skew θ = 30°.

Step	Equation	Substitution	Result
Modular ratio	n = E_B/E_D	29000/3644	7.958
Longitudinal stiffness parameter	K_g = n(I + A e_g²)	7.958(20000 + 48.00·30.00²)	502964 in⁴
Stiffness term	K_g/(12 L t_s³)	502964/(12·100.0·8.00³)	0.8186
Skew correction — moment	1 − c₁(tan θ)^1.5 ; c₁ = 0.25(K_g/12Lt_s³)^0.25 (S/L)^0.5	c₁ = 0.0713, θ = 30.0°	1.0000
Skew correction — shear	1.0 + 0.20 (K_g/12Lt_s³)^0.3 tan θ	1 + 0.20(0.8186)^0.3 tan 30.0°	1.1087
Lever rule reaction	R = Σ P_i x_i / S (P = 0.5 axle)	Σ(7.00, 1.00)/9.00	0.4444 lanes
Lever rule DF with m	g = m·R	1.00 × 0.4444	0.4444 lanes/girder
Rigid cross-section reaction	R = N_L/N_b + X_ext ΣE / Σx²	2/5 + 18.00·2·14.00/810.0	1.0222
Rigid cross-section DF with m	g = m·R	1.00 × 1.0222	1.0222 lanes/girder

Exterior-girder design must use the largest of the approximate e·g equation, the lever rule (0.4444) and the rigid cross-section value (1.0222). Multiply moment DFs by 1.0000 and the support shear of the exterior girder at the obtuse corner by 1.1087.

Assumptions & code basis

K_g = n(I + A e_g²) per AASHTO LRFD Eq. 4.6.2.2.1-1, with e_g the distance between the girder and deck centroids.
Skew correction for moment applies only for 30° < θ ≤ 60° (§4.6.2.2.2e); for θ ≤ 30° no reduction is taken and for θ > 60° a refined analysis is required.
Skew correction for support shear at the obtuse corner of the exterior girder, Table 4.6.2.2.3c-1, always ≥ 1.0 — the shear correction is an increase.
Lever rule assumes the deck is hinged over the interior girder; wheel loads are placed 2 ft from the barrier face and 6 ft apart (§3.6.1.3.1).
The rigid cross-section (special analysis) equation §C4.6.2.2.2d applies to beam-slab bridges with sufficiently stiff diaphragms or cross-frames acting as a unit.
Multiple-presence factors m (1.20 / 1.00 / 0.85 / 0.65) are included in the lever-rule and rigid-section results but are already embedded in the approximate DF equations.

Module 14AASHTO LRFD 6.10.1.1.1cChapter 4

Cracked negative-moment composite section

Stage-dependent section properties neglecting concrete in tension, with the longitudinal deck steel acting compositely and the fibre stresses for DC1, DC2/DW and LL+IM at the pier.

Steel depthinSteel areain²Steel inertiain⁴Steel centroid from bottominLongitudinal deck steelin²Deck steel above top of steelinM_DC1kip·ftM_DC2kip·ftM_LL+IMkip·ftF_yksi

Negative-moment region: the deck concrete is neglected and only the longitudinal deck steel acts with the girder.

Cracked ȳ

32.26 in

I_cr

45203 in⁴

S_b,cr

1401 in³

f_bottom

-32.49 ksi

f_top

-27.40 ksi

Deck reinforcement stress

-16.20 ksi

Derivation — equation, substitution, result

Cracked-section area

\begin{aligned}&\;A_{cr} = A_{steel} + A_s,\,\text{deck} \;\text{(deck concrete neglected, §6.10.1.1.1c)}\\[4pt]&=\;62.0 + 14.00\\[4pt]&=\;\boxed{76.00 \,\text{in}^{2}}\end{aligned}

Neutral axis

\begin{aligned}&\;ȳ = \Sum A_i y_i / \Sum A_i\\[4pt]&=\;[62.0(26.00) + 14.00(60.00)]/76.00\\[4pt]&=\;\boxed{ȳ = 32.26 \,\text{in} \operatorname{from} \,\text{the} \,\text{bottom}}\end{aligned}

Cracked inertia

\begin{aligned}&\;I_{cr} = \Sum I_o + \Sum A_i d_i^{2}\\[4pt]&=\;32000 + 2432 + 10771\\[4pt]&=\;\boxed{I_{cr} = 45203 \,\text{in}^{4}}\end{aligned}

Section moduli

\begin{aligned}&\;S = I/c\\[4pt]&=\;S_b,\,\text{cr} = 1401, S_t,\,\text{cr} = 2080, S_{deck} = 1630 \,\text{in}^{3}\\[4pt]&=\;\boxed{\,\text{stage}-\,\text{dependent}}\end{aligned}

Bottom-flange stress

\begin{aligned}&\;f_b = M_{DC1}/S_b + (M_{DC2} + M_{LL}+\mathrm{IM})/S_b,\,\text{cr}\\[4pt]&=\;-1400(12)/1231 + -2200(12)/1401\\[4pt]&=\;\boxed{-32.49 \,\text{ksi}}\end{aligned}

Top-flange stress

\begin{aligned}&\;f_t = M_{DC1}/S_t + (M_{DC2} + M_{LL}+\mathrm{IM})/S_t,\,\text{cr}\\[4pt]&=\;\boxed{-27.40 \,\text{ksi}}\end{aligned}

Deck-steel stress

\begin{aligned}&\;f_s = (M_{DC2} + M_{LL}+\mathrm{IM})/S_{deck}\\[4pt]&=\;\boxed{-16.20 \,\text{ksi}}\end{aligned}

Detailing — plan, elevation and section

Section — composite

Plan — framing

Elevation

Constructability & detailing notes

Two-thirds of the longitudinal deck steel goes in the top mat; place it before the negative-moment pour.
Stagger deck-steel splices and extend the steel past the dead-load inflection point plus the development length.
Shear studs are required through the negative-moment region even though the concrete is neglected for strength.

Check	Demand	Capacity / limit	Status
Bottom flange ≤ F_y	32.49 ksi	50 ksi	PASS
Top flange ≤ F_y	27.40 ksi	50 ksi	PASS
Deck longitudinal steel ≥ 1 % of the slab area (§6.10.1.7)Two-thirds in the top mat; #6 bars or smaller at ≤ 12 in.	—	—	PASS

Independent verification	Expected	Computed	Status
Cracked inertia is smaller than the short-term composite value	I_cr < I_composite	in⁴	VERIFIED

Assumptions & basis of design

Negative-moment region: concrete in tension is neglected and only the longitudinal deck reinforcement within the effective flange width acts compositely (§6.10.1.1.1c).
DC1 acts on the bare steel section; DC2 and LL+IM act on the cracked composite section.
Stresses are elastic first-order values — compare against the §6.10.8 lateral–torsional buckling resistance, not simply F_y, when the bottom flange is in compression.

Elastic stresses are within yield; continue with the §6.10.8 stability checks and the §6.10.1.7 minimum deck steel.

Chapter 5

Deck design

Equivalent strips, the three overhang design cases with yield-line barrier analysis, and the empirical (isotropic) design method.

Module 15AASHTO LRFD 4.6.2.1 / A4Chapters 1 · 5

Deck equivalent strip & Strength I demand

Equivalent strip widths for positive, negative, and overhang design, with factored slab moments and both reinforcement mats.

Girder spacing SftOverhang load distance XftSlab thickness tₛinWearing surfacepsff′_cksif_yksiTop coverinBottom coverin

+M strip (26.0 + 6.6S)

78.8 in

−M strip (48.0 + 3.0S)

72.0 in

Overhang strip (45.0 + 10.0X)

75.0 in

M_DC

0.64 k-ft/ft

M_DW

0.16 k-ft/ft

M_LL+IM (approx.)

6.48 k-ft/ft

Strength I M_u

12.38 k-ft/ft

Bottom mat (+M)

#4 @ 5.5 in

A_s,req 0.432 → A_s,prov 0.436 in²/ft

Top mat (−M)

#6 @ 10.0 in

A_s,req 0.515 → A_s,prov 0.528 in²/ft

Deck bottom mat (positive moment) — AASHTO LRFD §5.6.3 / §5.10

A_s required (strength)	0.43 in²/ft
A_s minimum (§5.6.3.3 / §5.10.6)	0.19 in²/ft
A_s design (governing)	0.43 in²/ft
Bar selection	#4 @ 5.5 in c/c
Bar size / area	#4 — A_b = 0.20 in²
Spacing (c/c)	5.5 in
A_s provided	0.44 in²/ft
A_s,prov / A_s,req	1.01
Stress-block depth a	0.64 in
Net tensile strain ε_t	0.0236
φ	0.900
φM_n (as detailed)	12.5 k-ft/ft
M_cr (§5.6.3.3)	5.5 k-ft/ft
Minimum-resistance demand min(M_cr, 1.33M_u)	5.5 k-ft/ft
Reinforcement ratio ρ	0.0054

PASS — Adequate — detail as scheduled

A_s,prov / A_s,req = 1.01; ε_t = 0.0236 (tension-controlled, φ = 0.90).

Deck top mat (negative moment) — AASHTO LRFD §5.6.3 / §5.10

A_s required (strength)	0.51 in²/ft
A_s minimum (§5.6.3.3 / §5.10.6)	0.24 in²/ft
A_s design (governing)	0.51 in²/ft
Bar selection	#6 @ 10.0 in c/c
Bar size / area	#6 — A_b = 0.44 in²
Spacing (c/c)	10.0 in
A_s provided	0.53 in²/ft
A_s,prov / A_s,req	1.03
Stress-block depth a	0.78 in
Net tensile strain ε_t	0.0140
φ	0.900
φM_n (as detailed)	11.4 k-ft/ft
M_cr (§5.6.3.3)	5.5 k-ft/ft
Minimum-resistance demand min(M_cr, 1.33M_u)	5.5 k-ft/ft
Reinforcement ratio ρ	0.0085

PASS — Adequate — detail as scheduled

A_s,prov / A_s,req = 1.03; ε_t = 0.0140 (tension-controlled, φ = 0.90).

Detailing sketchDeck slab — section, plan and longitudinal section

Top mat #4 @ 5.5 in (A_s = 0.436 in²/ft) for the negative strip moment over the girders; bottom mat #4 @ 7.0 in for the positive strip moment between girders. Required A_s = 0.432 in²/ft at M_u = 12.38 k-ft/ft. Distribution steel and 2 in clear top / 1 in clear bottom cover per AASHTO 5.10.1 and 9.7.3.

Dead-load moments use the continuous-slab approximation wS²/10. For final design use the Appendix A4 live-load moment table, which already includes multiple presence and dynamic load allowance.

Module 16AASHTO LRFD A13.3.1 / A13.4.1 / 4.6.2.1.3Chapter 5

Deck overhang — collision & wheel-load design

Yield-line barrier resistance R_w and critical length L_c, the collision moment delivered into the deck at Extreme Event II, the Strength I wheel-load case on the 45 + 10X strip, and the governing overhang steel.

Overhang lengthftDeck thicknessinBarrier base widthinBarrier heightftWearing surfacepsfTest-level force F_t (TL-4)kipDistribution length L_tftForce height HftWall cantilever M_ck-ft/ftWall bending M_wk-ftTop beam M_bk-ftDeck f′_cksif_yksi

M_c, M_w and M_b are the barrier's yield-line resistances about a horizontal axis (wall cantilever), a vertical axis (wall bending) and the beam at the top of the parapet.

Barrier resistance R_w

82.7 kip

Critical length L_c

6.98 ft

Strength I overhang moment

12.39 k-ft/ft

Extreme Event II overhang moment

8.81 k-ft/ft

Governing overhang moment

12.39 k-ft/ft

Required top steel A_s

0.173 in²/ft

Detailing sketchDeck overhang — section

Overhang L = 3.50 ft; governing overhang moment M_u = 12.39 k-ft/ft requires A_s = 0.173 in²/ft.

Step	Equation	Substitution	Result
Yield-line critical length	L_c = L_t/2 + √[(L_t/2)² + 8H(M_b + M_w)/M_c]	1.75 + √[3.06 + 8·2.70(0.0+18.0)/16.00]	6.98 ft
Barrier resistance	R_w = 2/(2L_c − L_t) · [8M_b + 8M_w + M_c L_c²/H]	2/(2·6.98 − 3.5)·[8·0.0 + 8·18.0 + 16.00·6.98²/2.70]	82.7 kip
Collision moment into the deck	M = R_w/(L_c + 2H)	82.7/(6.98 + 2·2.70)	6.68 k-ft/ft
Overhang dead load	M_DC = w t_s L²/2 + W_barrier·arm	0.1000·3.50²/2 + barrier	2.07 k-ft/ft
Overhang strip	E = 45 + 10X	45 + 10·1.25	57.5 in
Wheel-load moment	M_LL+IM = P(1+IM)X / (E/12)	16·1.33·1.25/(57.5/12)	5.55 k-ft/ft
Case 2 — Strength I	1.25DC + 1.50DW + 1.75(LL+IM)	1.25·2.07 + 1.50·0.06 + 1.75·5.55	12.39 k-ft/ft
Case 1 — Extreme Event II	1.00DC + 1.00DW + M_collision	2.07 + 0.06 + 6.68	8.81 k-ft/ft
Required overhang steel	A_s = 0.85f′_c b d/f_y [1 − √(1 − 2M_u/(0.85φ f′_c b d²))]	M_u = 12.39 k-ft/ft, d = 5.19 in	0.173 in²/ft

OKBarrier resistance ≥ test level force F_tdemand 54.0 kipcapacity 82.7 kipratio 0.65
OKGoverning caseStrength I vertical wheel load governs.

Provide at least 0.173 in²/ft of top transverse steel in the overhang, developed fully into the first bay. Strength I governs.

Assumptions & code basis

Three design cases per AASHTO LRFD §A13.4.1: (1) transverse collision at Extreme Event II, (2) vertical wheel load at Strength I, (3) the deck design case checked at the design section.
Yield-line barrier analysis §A13.3.1 gives R_w and the critical length L_c of the failure pattern for an interior wall segment.
The barrier must fail before the deck: the deck overhang is designed for the barrier resistance R_w, not for the nominal test force F_t.
Wheel load placed 1 ft from the barrier face; overhang strip width 45 + 10X in (Table 4.6.2.1.3-1).
Extreme Event II uses γ_p = 1.00 on permanent loads and φ = 1.00 for flexure.

Module 17AASHTO LRFD 9.7.2.4 / 9.7.2.5Chapter 5

Empirical (isotropic) deck design method

Screens all eleven §9.7.2.4 applicability conditions and returns the prescribed 0.27 / 0.18 in²/ft isotropic mat with spacing and end-zone rules when the method applies.

Effective length SftTotal slab depthinCore depth (between mats)inOverhangftf_yksiSkewdegNumber of girders—Composite with girders

The empirical method replaces flexural analysis with a prescribed isotropic mat — but only when every §9.7.2.4 condition is satisfied.

Method applicable

No

S / t_core

21.60

Bottom steel each way

0.27 in²/ft

Top steel each way

0.18 in²/ft

Suggested bars

#5 @ 12 in bottom, #4 @ 12 in top

≤ 18 in spacing

Detailing sketchEmpirical deck — core and mats

S/t_core = 21.60; bottom steel 0.27 in²/ft each way, top 0.18 in²/ft each way.

Step	Equation	Substitution	Result
Effective length	S (face-to-face for monolithic, distance between flange tips + flange overhang for steel)	9.00 ft	108.0 in
Core depth	t_core = t_s − top cover − bottom cover	8.00 − covers	5.00 in
Span-to-depth ratio	S/t_core (must be 6 – 18)	108.0/5.00	21.60
Bottom mat, each direction	A_s = 0.27 in²/ft	—	0.27 in²/ft
Top mat, each direction	A_s = 0.18 in²/ft	f_y = 60 ksi ≥ 60 ksi	0.18 in²/ft

OKSupporting components are steel and/or concrete beams
OKDeck is fully cast-in-place and water cured
OKDeck is of uniform depth (haunches excepted)
NGEffective length-to-depth ratio 6.0 ≤ S/t_core ≤ 18.0
OKCore depth ≥ 4.0 in
OKEffective length S ≤ 13.5 ft
OKTotal slab depth ≥ 7.0 in
OKOverhang ≥ 5.0 times slab depth (or 3.0 t_s with a composite barrier)
OKDeck is made composite with the supporting components
OKAt least three girder lines (N_b ≥ 3)
OKSkew ≤ 25° for the standard reinforcement layout (otherwise double the end-zone steel)

Conditions failed: Effective length-to-depth ratio 6.0 ≤ S/t_core ≤ 18.0. Use the traditional equivalent-strip design (§4.6.2.1) instead, or adjust the geometry until the conditions are met.

Assumptions & code basis

Empirical (isotropic) deck design per AASHTO LRFD §9.7.2 — applicable only when every design condition of §9.7.2.4 is met.
Prescribed reinforcement §9.7.2.5: 0.27 in²/ft in each bottom layer and 0.18 in²/ft in each top layer, placed in both directions, f_y ≥ 60 ksi.
Maximum bar spacing 18 in; the reinforcement is placed as close to the outside surfaces as cover allows.
The overhang and the barrier are NOT covered by the empirical method — design them by the strip method with the three §A13.4.1 cases.
For skews above 25° the end-zone transverse reinforcement in the two end panels is doubled.

Module 18AASHTO LRFD 9.7.3.2 / 5.10.6 / 5.6.7 / 5.5.3.2Chapter 5

Deck secondary steel, crack control & rebar fatigue

Distribution steel as a percentage of the main mat, shrinkage-and-temperature steel, the §5.6.7 crack-control spacing with γ_e, and the Δf ≤ 24 − 20 f_min/f_y fatigue check on straight bars.

Effective span SftDeck thicknessinMain steel providedin²/ftEffective depth dinCover to bar centroid d_cinService I momentkip·ft/ftFatigue M_maxkip·ft/ftFatigue M_minkip·ft/ftf′_cksif_yksiExposure factor γ_e—Main steel orientation

Distribution steel, shrinkage and temperature steel, the Class 1/2 crack-control spacing and the rebar fatigue stress range.

Distribution steel

0.415 in²/ft

67.0 % of the main steel

Shrinkage & temp steel

0.110 in²/ft per face

Service stress f_ss

27.44 ksi

Max bar spacing

9.29 in

Fatigue Δf

16.46 ksi

Derivation — equation, substitution, result

Distribution steel percentage

\begin{aligned}&\;\% = 220/\sqrt{S} \le 67 \% (§9.7.3.2)\\[4pt]&=\;\sqrt{S} = 3.000\\[4pt]&=\;\boxed{67.0 \%}\end{aligned}

Bottom distribution steel

\begin{aligned}&\;A_s,\,\text{dist} = \% \cdot A_s,\,\text{main}\\[4pt]&=\;0.670 \times 0.620\\[4pt]&=\;\boxed{0.415 \,\text{in}^{2}/\,\text{ft}}\end{aligned}

Shrinkage & temperature steel

\begin{aligned}&\;A_s \ge 1.30 b h /[2(b+h)f_y], 0.11 \le A_s \le 0.60 (§5.10.6)\\[4pt]&=\;1.30(12)(8.50)/[2(12+8.50)(60)]\\[4pt]&=\;\boxed{0.110 \,\text{in}^{2}/\,\text{ft} \operatorname{per} \,\text{face}}\end{aligned}

Cracked elastic lever arm

\begin{aligned}&\;k = \sqrt{(n\rho )^{2} + 2n\rho } − n\rho , j = 1 − k/3\\[4pt]&=\;n = 7.97, \rho = 0.00876\\[4pt]&=\;\boxed{k = 0.310, j = 0.897}\end{aligned}

Service steel stress

\begin{aligned}&\;f_{ss} = M_s/(A_s j d) \le 0.60 f_y\\[4pt]&=\;7.50(12)/(0.620 \times 0.897 \times 5.90)\\[4pt]&=\;\boxed{27.44 \,\text{ksi}}\end{aligned}

Crack-control spacing

\begin{aligned}&\;s \le 700\gamma _e/(\beta _s f_{ss}) − 2d_c (Eq. 5.6.7-1)\\[4pt]&=\;\beta _s = 1 + 2.00/[0.7(8.50 − 2.00)] = 1.440\\[4pt]&=\;\boxed{s_{\operatorname{max}} = 9.29 \,\text{in}}\end{aligned}

Fatigue stress range

\begin{aligned}&\;\Delta f = f_{\operatorname{max}} − f_{\operatorname{min}} \le (\Delta F)_{TH} = 26 − 22(f_{\operatorname{min}}/f_y) (§5.5.3.2)\\[4pt]&=\;12.81 − -3.66 = 16.46 \,\text{ksi} \,\text{vs} 27.34 \,\text{ksi}\\[4pt]&=\;\boxed{\,\text{satisfied}}\end{aligned}

Detailing — plan, elevation and section

Section — composite

Plan — framing

Elevation

Constructability & detailing notes

Distribution steel goes in the bottom mat, tied to the main bars — it is not a substitute for temperature steel in the top mat.
Use epoxy-coated or stainless bars in the top mat where deicing salts are used; repair coating damage before the pour.
Maintain 2.5 in top cover (2 in with an integral wearing surface) using chairs at 4 ft o.c. maximum.
Screed rails must bear on the girders, never on the reinforcing steel.

Check	Demand	Capacity / limit	Status
f_ss ≤ 0.60 f_y	27.44 ksi	36.0 ksi	PASS
Crack-control spacing achievable	bar spacing used	9.29 in	PASS
Rebar fatigue Δf ≤ (ΔF)_TH	16.46 ksi	27.34 ksi	PASS
Deck deflection Δ ≤ S/800 (optional, §2.5.2.6.2)Only mandatory for the owner-invoked deflection criterion.	—	—	PASS

Independent verification	Expected	Computed	Status
Distribution steel = 220/√S ≤ 67 % of the main steel	67.0 %		VERIFIED

Assumptions & basis of design

Straight, non-skewed deck with the primary reinforcement perpendicular to traffic; §9.7.3.2 bottom distribution steel.
Class 1 exposure γ_e = 1.00; use 0.75 for decks exposed to de-icing salts (§5.6.7).
Fatigue I with the fatigue truck at a 30 ft rear axle spacing and IM = 15 %.

Distribution, crack-control and fatigue requirements are all satisfied with the given mat.

Module 19AASHTO LRFD 9.8.3 / 6.6.1.2Chapters 5 · 8

Orthotropic steel deck — three-system fatigue

Superposition of the local rib-wall, rib-as-beam and global girder stress systems, the resulting stress range at the rib-to-deck weld, and the infinite / finite life prediction for the governing detail category.

Deck plate t_pinRib spacinginWheel load with IMkipTyre patch lengthinFloorbeam spacinginRib section modulusin³Global stress rangeksiDetail categoryADTT (single lane)trucks/dayDesign lifeyr

Superimposes the local plate, rib and global girder stress ranges, then checks infinite and finite life for the governing detail category.

f₁ deck plate

38.71 ksi

f₂ rib

29.08 ksi

Total Δf

71.78 ksi

(ΔF)_n

5.00 ksi

Predicted life

0 yr

Derivation — equation, substitution, result

Contact pressure

\begin{aligned}&\;w = P_{wheel}/\;\text{(patch²)}\\[4pt]&=\;21.0/(10.0^{2})\\[4pt]&=\;\boxed{0.2100 \,\text{ksi}}\end{aligned}

System 1 — deck-plate local bending

\begin{aligned}&\;M_{1} = w\cdot a^{2}/12 \;\text{(fixed–fixed strip)}, f_{1} = M_{1}/S_{plate}, S = t_p^{2}/6\\[4pt]&=\;a = 12.0 \,\text{in}, t_p = 0.625 \,\text{in}\\[4pt]&=\;\boxed{f_{1} = 38.71 \,\text{ksi}}\end{aligned}

System 2 — rib bending between floorbeams

\begin{aligned}&\;M_{2} \approx 0.20 P L_{fb} \;\text{(continuous rib)}, f_{2} = M_{2}/S_{rib}\\[4pt]&=\;0.20 \times 21.0 \times 180.0 / 26.0\\[4pt]&=\;\boxed{f_{2} = 29.08 \,\text{ksi}}\end{aligned}

System 3 — global girder action

\begin{aligned}&\;f_{3} \operatorname{from} \,\text{the} \,\text{girder} \,\text{section} \,\text{modulus}\\[4pt]&=\;\boxed{f_{3} = 4.00 \,\text{ksi}}\end{aligned}

Superimposed stress range

\begin{aligned}&\;\Delta f = f_{1} + f_{2} + f_{3}\\[4pt]&=\;38.71 + 29.08 + 4.00\\[4pt]&=\;\boxed{\Delta f = 71.78 \,\text{ksi}}\end{aligned}

Cycles

\begin{aligned}&\;N = \;\text{(ADTTSL)}(365)\;\text{(years)}(n)\\[4pt]&=\;2000 \times 365 \times 75\\[4pt]&=\;\boxed{N = 5.48e+7}\end{aligned}

Fatigue resistance

\begin{aligned}&\;(\Delta F)_n = (A/N)^{1/3} \ge ½(\Delta F)_{TH} (Eq. 6.6.1.2.5-1)\\[4pt]&=\;A = 4.40e+9 \,\text{ksi}^{3}, (\Delta F)_{TH} = 10.0 \,\text{ksi}\\[4pt]&=\;\boxed{(\Delta F)_n = 5.00 \,\text{ksi}}\end{aligned}

Predicted life

\begin{aligned}&\;Y = A/\;\text{(Δf³ · 365 · ADTTSL)}\\[4pt]&=\;\boxed{0 \,\text{years}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchOrthotropic deck plate — rib section and combined fatigue stress range

Deck plate t_p = 0.625 in over ribs at 12.0 in spacing. Wheel load 21.0 kip on a 10.0 in tyre patch produces local Δf = 67.8 ksi; combined with the global girder range 4.0 ksi gives total Δf = 71.8 ksi against the threshold (ΔF)_TH = 10.0 ksi.

Constructability & detailing notes

Rib-to-deck welds must achieve 80 % penetration with tight fit-up — the fatigue performance of the whole deck depends on it.
Use continuous ribs passing through slotted floorbeam webs and grind the cut-out radius smooth.
Specify a wearing surface (epoxy asphalt or polymer) that bonds and shares load with the deck plate.

Check	Demand	Capacity / limit	Status
Infinite life Δf ≤ (ΔF)_TH (Category C)	71.78 ksi	10.0 ksi	REVIEW
Finite life Δf ≤ (ΔF)_n	71.78 ksi	5.00 ksi	REVIEW
Deck plate ≥ 5/8 in (§9.8.3.7.2)	0.625 in	0.625 in	PASS
Rib-to-deck weld ≥ 80 % penetrationDetail per §9.8.3.7.3 to keep the rib-to-deck joint in Category C.	—	—	PASS

Independent verification	Expected	Computed	Status
Combined stress range below the constant-amplitude threshold			VERIFIED

Assumptions & basis of design

AASHTO LRFD §9.8.3 orthotropic deck design by the three-system superposition (local plate, rib/panel, global girder).
Deck-plate strip idealised as fixed–fixed between ribs; ribs idealised as continuous beams over the floorbeams with M ≈ 0.20 P L.
Fatigue I load factor 1.75 with the fatigue truck; single-lane ADTT_SL entered directly.

Thicken the deck plate (f₁ ∝ 1/t_p²) or close the rib spacing before enlarging the ribs — local plate bending dominates the rib-to-deck detail.

Chapter 6

Reinforced concrete — flexure, shear, torsion and interfaces

Strength design of concrete sections including the general (MCFT) shear procedure, torsion reinforcement, and shear transfer across cold joints.

Module 20AASHTO LRFD 5.6.3 / 5.5.4.2Chapters 5 · 6

Reinforced-concrete flexural capacity

Stress-block depth, net tensile strain, resistance factor, and the φM_n ≥ M_u check with bar selection.

Width binOverall depth hinEffective depth dinTrial tension steel A_sin²f′_cksif_yksiFactored demand M_uk-ftFactored shear V_ukip

a = A_s f_y / (0.85 f′_c b) · M_n = A_s f_y (d − a/2) · ε_t = 0.003 (d − c)/c

β₁

0.850

Stress-block depth a

5.88 in

Neutral axis c

6.92 in

Net tensile strain ε_t

0.0100

Tension-controlled

φ

0.900

M_n

541 k-ft

φM_n vs M_u

487 ≥ 400 k-ft

Section adequate for flexure.

Flexural reinforcement design — AASHTO LRFD §5.6.3 / §5.10

A_s required (strength)	3.22 in²
A_s minimum (§5.6.3.3 / §5.10.6)	0.70 in²
A_s design (governing)	3.22 in²
Bar selection	2 – #14
Bar size / area	#14 — A_b = 2.25 in²
Bars / layers	2 bars in 1 layer (clear spacing 5.1 in)
A_s provided	4.50 in²
A_s,prov / A_s,req	1.40
Stress-block depth a	6.62 in
Net tensile strain ε_t	0.0086
φ	0.900
φM_n (as detailed)	540.5 k-ft
M_cr (§5.6.3.3)	93.4 k-ft
Minimum-resistance demand min(M_cr, 1.33M_u)	93.4 k-ft
Reinforcement ratio ρ	0.0125

PASS — Adequate — detail as scheduled

A_s,prov / A_s,req = 1.40; ε_t = 0.0086 (tension-controlled, φ = 0.90).

Transverse reinforcement — AASHTO LRFD §5.7.2.5 / §5.7.3.3

Concrete contribution V_c	41.0 kip
Steel required V_s = V_u/φ − V_c	92.4 kip
Stirrup schedule	#4 2-leg @ 7.0 in c/c (A_v = 0.40 in²)
Maximum spacing s_max	21.6 in
φV_n = φ(V_c + V_s)	120.2 kip
Web-crushing limit 0.25 f′_c b_v d_v	324.0 kip

PASS — φV_n = 120.2 kip ≥ V_u = 120.0 kip.

Detailing sketchReinforced-concrete beam — section, elevation and bar plan

5 #9 bars (5.00 in² provided against A_s = 4.50 in² assumed), 2 rows, 2 in clear cover, #4 stirrups. Whitney block a = 6.62 in, c = 7.79 in, f_y = 60 ksi. φM_n = 540 k-ft vs M_u = 400 k-ft (AASHTO 5.6.3.2).

Module 21AASHTO LRFD 5.7.3.3 / 5.7.3.4.2 / 5.7.2.5Chapters 6 · 7

Sectional shear — general (MCFT) procedure

Longitudinal strain ε_s, β and θ from Eqs. 5.7.3.4.2-1..-3, V_c and V_s with cot θ, the web-crushing ceiling, minimum transverse steel and the required stirrup spacing with the §5.7.3.4.1 maximum.

V_u at the sectionkipM_u at the sectionk-ftAxial N_u (+ tension)kipWeb width b_vinEffective shear depth d_vinf′_cksif_y (stirrups)ksiMild tension steel A_sin²Strand area A_psin²Harped strand V_pkipStirrup area A_v (both legs)in²Trial spacing sin

The general procedure iterates on ε_s: β falls and θ rises as the section strains, so a heavily loaded web needs both more stirrups and a flatter truss angle.

ε_s

-4.000e-4

β

6.857

θ

27.60 deg

V_c

294.2 kip

V_s

229.5 kip

φV_n

507.3 kip

Design spacing

24.0 in

Detailing sketchMCFT shear — demand vs capacity

θ = 27.60° and β = 6.857 at ε_s = -4.000e-4.

Step	Equation	Substitution	Result
Longitudinal strain	ε_s = [\|M_u\|/d_v + 0.5N_u + \|V_u − V_p\| − A_ps f_po] / (E_s A_s + E_p A_ps)	[18000/60.00 + 0.5·0.0 + 260.0 − 869.4] / (29000·2.00 + 28500·4.60)	-4.000e-4
Minimum transverse steel	A_v,min = 0.0316 √f′_c b_v s / f_y	0.0316√8.00·8.00·12.00/60	0.143 in² (provided 0.400 in²)
β	β = 4.8/(1 + 750ε_s)	4.8/(1 + 750·-4.000e-4)	6.857
θ	θ = 29 + 3500 ε_s	29 + 3500·-4.000e-4	27.60°
Concrete contribution	V_c = 0.0316 β √f′_c b_v d_v	0.0316·6.857·√8.00·8.00·60.00	294.2 kip
Stirrup contribution	V_s = A_v f_y d_v cot θ / s	0.400·60·60.00·1.913/12.00	229.5 kip
Nominal shear	V_n = min(V_c + V_s + V_p, 0.25f′_c b_v d_v + V_p)	min(563.7, 1000.0)	563.7 kip
Factored resistance	φV_n ≥ V_u	0.90·563.7 vs 300.0	507.3 kip
Required stirrup spacing	s = A_v f_y d_v cot θ / (V_u/φ − V_c − V_p)	0.400·60·60.00·1.913/(333.3 − 294.2)	stirrups not required by strength
Maximum spacing	v_u < 0.125f′_c → s ≤ min(0.8d_v, 24 in); else min(0.4d_v, 12 in)	v_u = 0.625 ksi	24.00 in

OKShear strength φV_n ≥ V_udemand 300.0 kipcapacity 507.3 kipratio 0.59
OKWeb crushing V_n ≤ 0.25f′_c b_v d_v + V_pdemand 563.7 kipcapacity 1000.0 kip
OKMinimum transverse steeldemand 0.143 in²capacity 0.400 in²

Use 24.0 in stirrup spacing (governed by the maximum-spacing rule); φV_n = 507.3 kip ≥ V_u = 300.0 kip with θ = 27.6°.

Assumptions & code basis

General (MCFT) sectional procedure, AASHTO LRFD §5.7.3.4.2, with β and θ from Eqs. 5.7.3.4.2-1..-3.
Longitudinal strain ε_s taken at mid-depth of the tension reinforcement; negative values would require the concrete-tension refinement of Eq. 5.7.3.4.2-4 and are floored here.
d_v = max(0.9d_e, 0.72h, moment arm) must be supplied consistently with the flexural design.
Nominal shear capped at 0.25 f′_c b_v d_v + V_p (Eq. 5.7.3.3-2) to prevent web crushing.
Longitudinal reinforcement must also satisfy §5.7.3.5 — the tension-tie demand from shear.

Module 22AASHTO LRFD ACI 318-19 §22.7 · AASHTO §5.7.2.1 / 5.7.3.6Chapters 6 · 9

Torsion — closed ties & longitudinal A_ℓ

Threshold and cracking torsion, A_oh / p_h / A_o geometry, the combined shear-plus-torsion crushing check, the (A_v + 2A_t)/s tie schedule and the distributed longitudinal torsion steel.

Factored torsion T_uk-ftFactored shear V_ukipWeb width b_winOverall depth hinEffective depth dinf′_cksif_y = f_ytksiClear coverinClosed tie bar no.Strut angle θdeg

φT_th = φλ√f′_c A_cp²/p_cp · A_t/s = T_u/(φ2A_o f_yt cot θ) · A_ℓ = (A_t/s)p_h cot²θ · s ≤ p_h/8 ≤ 12″

φT_th

13.4 k-ft

torsion designed

T_cr

71.5 k-ft

A_oh / p_h

384 / 82.0 in², in

A_t/s (one leg)

0.0184 in²/in

(A_v+2A_t)/s req

0.0950 in²/in

Tie spacing

4.0 in

s_max = 10.3 in

A_ℓ required

1.673 in²

A_ℓ,min = 1.673

A_ℓ provided

1.760 in²

Crushing check

0.307 / 0.530 ksi

Torsion reinforcement schedule — ACI 318-19 §22.7 / §9.7.5 / §9.7.6.3

Closed ties	#4 closed ties @ 4.0 in c/c (2 legs)
Tie legs / hooks	2 legs, 135° hooks (§25.7.1.6)
Longitudinal torsion steel	4 – #6 distributed @ ≈ 20.5 in around the perimeter
Perimeter bar spacing	20.5 in (≤ 12 in required)
A_o = 0.85A_oh	327 in²
cot θ	1.000
V_c	67.4 kip

PASS — Provide closed #4 ties @ 4.0 in c/c (135° hooks, §25.7.1.6) plus 4 – #6 longitudinal bars distributed around the perimeter at ≤ 12 in (§9.7.5.1).

Detailing sketchReinforced-concrete beam — section, elevation and bar plan

4 #6 bars (1.76 in² provided against A_s = 1.76 in² assumed), 1 row, 2 in clear cover, #4 stirrups. Whitney block a = 0.00 in, c = 0.00 in, f_y = 60 ksi. φM_n = 0 k-ft vs M_u = 0 k-ft (AASHTO 5.6.3.2).

Module 23AASHTO LRFD 5.7.4Chapters 6 · 7

Interface (cold-joint) shear transfer

v_ui = V_u/(b_vi d_v), the c + μ(A_vf f_y + P_c) shear-friction resistance with the K₁f′_c and K₂ ceilings for each surface condition, minimum interface steel and the extended-stirrup spacing across the girder-to-deck joint.

V_u at the sectionkipd_vinInterface width b_viinDeck f′_cksif_yksiBar area crossing joint A_vfin²Spacing sinPermanent compression P_ckip/ftInterface condition

Interface shear governs the extended-stirrup spacing at the top of a precast girder far more often than the web shear does.

v_ui

0.2500 ksi

V_ui

60.00 kip/ft

φV_ni

82.08 kip/ft

A_vf required

0.200 in²/ft

Max spacing of extended stirrups

24.0 in

Detailing sketchInterface — girder / deck cold joint

Interface shear v_ui = 0.2500 ksi; required A_vf = 0.200 in²/ft at 12.00 in spacing (roughened surface).

Step	Equation	Substitution	Result
Interface shear stress	v_ui = V_u/(b_vi d_v)	300.0/(20.00·60.00)	0.2500 ksi
Interface shear per foot	V_ui = v_ui b_vi (12 in)	0.2500·20.00·12	60.00 kip/ft
Interface steel provided	A_vf/ft = A_vf(12)/s	0.400·12/12.00	0.400 in²/ft
Nominal interface resistance	V_ni = c A_cv + μ(A_vf f_y + P_c)	0.28·240.0 + 1(0.400·60 + 0.0)	91.20 kip/ft
Upper limits	V_ni ≤ K₁ f′_c A_cv and ≤ K₂ A_cv	0.3·4.00·240.0 = 288.0; 1.8·240.0 = 432.0	288.00 kip/ft
Factored resistance	φV_ni ≥ V_ui	0.9·91.20 vs 60.00	82.08 kip/ft
Required interface steel	A_vf = (V_ui/φ − cA_cv)/(μ f_y) ≥ 0.05A_cv/f_y	(66.67 − 67.20)/(1·60)	0.200 in²/ft
Required stirrup spacing across the joint	s = A_vf(12)/A_vf,req	0.400·12/0.200	24.00 in

OKInterface shear φV_ni ≥ V_uidemand 60.00 kip/ftcapacity 82.08 kip/ftratio 0.73
OKMinimum interface reinforcement 0.05A_cv/f_ydemand 0.200 in²/ftcapacity 0.400 in²/ft

Extend the girder stirrups into the deck at 24.0 in or closer and roughen the top flange to a 0.25 in amplitude; φV_ni = 82.08 kip/ft ≥ V_ui = 60.00 kip/ft.

Assumptions & code basis

Interface shear transfer (shear friction) per AASHTO LRFD §5.7.4 — the cold joint between a precast girder and a cast-in-place deck.
Surface condition: roughened; c = 0.28 ksi, μ = 1, K₁ = 0.3, K₂ = 1.8 ksi (§5.7.4.4).
Interface shear per unit length v_ui = V_u/(b_vi d_v); the check is performed on a 12 in length of interface.
Permanent net compressive force P_c across the interface may be taken as zero unless it is reliably present.
Minimum interface steel 0.05 A_cv/f_y may be waived when v_ui < 0.210 ksi and the surface is roughened to 0.25 in amplitude (§5.7.4.2).

Module 24AASHTO LRFD AASHTO §5.10.6 · ACI 318-19 §24.4Chapters 5 · 12

Shrinkage & temperature steel

The ACI ratio equation and the AASHTO per-face equation side by side, governing A_s, bar size and spacing against the 5h ≤ 18 in cap, provided ρ and the bar count for the strip.

Member thickness hinDesign strip width binf_yksiTrial bar no.Faces reinforced (1 or 2)

ρ = max(0.0018·60/f_y, 0.0014) · A_s = ρbh · A_s(AASHTO) = 1.30bh/[2(b+h)f_y] · s ≤ min(5h, 18″)

A_s (ACI)

0.194 in²

ρ = 0.00180

A_s (AASHTO/face)

0.110 in²/ft

A_s governing

0.194 in²

Spacing

18.0 in

s_max = 18 in

A_s provided

0.267 in²

ρ provided

0.00247

Shrinkage & temperature reinforcement schedule — ACI 318-19 §24.4 · AASHTO LRFD §5.10.6

Call-out	#4 @ 18.0 in c/c each face
Faces	Each face (top and bottom)
Bars over strip length	9 bars
Steel weight	72 lb per 12 ft strip
Max spacing	18 in

PASS — Provide #4 @ 18.0 in c/c each way, each face (max 18 in, §24.4.3.3). ρ_prov = 0.00247.

Detailing sketchSlab — shrinkage & temperature steel, section and plan

Strip 12 in wide × 9.0 in thick, #4 @ 18.0 in c/c each way, top and bottom.

Chapter 7

Prestressed concrete

Stress limits at every stage, immediate and time-dependent losses, service fibre stresses, and flexural resistance with bonded strand.

Module 25AASHTO LRFD 5.9.2.2 / 5.9.2.3Chapter 7

Prestressing stress limits — steel & concrete

Table 5.9.2.2-1 strand limits at jacking, transfer and service, with the §5.9.2.3 concrete compression and tension limits at release and in Service I / Service III, each screened against your fibre stresses.

f_puksiJacking stress f_pjksiStress after transfer f_ptksiEffective stress f_peksif′_ci at releaseksif′_c at 28 daysksiBottom fibre stressksiTop fibre stressksiExposure

Enter fibre stresses with compression negative and tension positive; the module screens every steel and concrete limit at transfer and at service.

f_py

243.0 ksi

Jacking limit

202.5 ksi

Transfer compression limit

3.600 ksi

Service III tension limit

0.5374 ksi

All limits satisfied

Yes

Detailing sketchPrestress stress limits — actual vs bound

All limits satisfied: Yes.

Step	Equation	Substitution	Result
Assumed yield strength	f_py = 0.90 f_pu (low relaxation)	0.90·270	243.0 ksi
Jacking limit	0.75 f_pu	0.75·270	202.5 ksi
Transfer limit	0.70 f_pu	0.70·270	189.0 ksi
Service steel limit	0.80 f_py	0.80·243.0	194.4 ksi
Transfer compression limit	0.60 f′_ci	0.60·6.00	3.600 ksi
Transfer tension limit	0.24 √f′_ci (bonded) / min(0.0948√f′_ci, 0.2) otherwise	0.24√6.00 = 0.5879	0.5879 ksi (unbonded 0.2000)
Service compression limits	0.45f′_c permanent / 0.60f′_c all loads	0.45·8.00, 0.60·8.00	3.600 / 4.800 ksi
Service III tension limit	0.19 √f′_c	0.19√8.00	0.5374 ksi

OKJacking stress f_pj ≤ 0.75f_pudemand 202.5 ksicapacity 202.5 ksi
OKStress immediately after transfer f_pt ≤ 0.70f_pudemand 185.0 ksicapacity 189.0 ksi
OKEffective stress at service f_pe ≤ 0.80f_pydemand 155.0 ksicapacity 194.4 ksi
OKConcrete compression at transfer ≤ 0.60f′_cidemand 2.900 ksicapacity 3.600 ksi
OKConcrete tension at transfer (bonded reinf.) ≤ 0.24√f′_cidemand 0.3500 ksicapacity 0.5879 ksi
OKService III tension ≤ 0.19√f′_cdemand 0.0000 ksicapacity 0.5374 ksi
OKService I compression (all loads) ≤ 0.60f′_cdemand 0.000 ksicapacity 4.800 ksi

All strand and concrete stress limits are satisfied at both transfer and service.

Assumptions & code basis

Steel stress limits from AASHTO LRFD Table 5.9.2.2-1 for low-relaxation strand: 0.75f_pu at jacking, 0.70f_pu immediately after transfer, 0.80f_py at service after all losses.
Concrete stress limits from §5.9.2.3: 0.60f′_ci compression at transfer, 0.24√f′_ci tension where bonded reinforcement is provided (0.0948√f′_ci ≤ 0.2 ksi otherwise).
Service limits §5.9.2.3.2: 0.45f′_c under permanent loads, 0.60φ_w f′_c under all loads, and 0.19√f′_c tension in Service III (0.0948√f′_c for corrosive exposure).
Fibre stresses must be entered with compression negative and tension positive.

Module 26AASHTO LRFD 5.9.3.2 / 5.9.3.3Chapter 7

Prestress losses — immediate & time-dependent

Friction and anchorage set for post-tensioning, elastic shortening from f_cgp, and the approximate long-term loss with γ_h and γ_st — totalled to f_pe with P_i and P_e.

Jacking stress f_pjksiStrand area A_psin²Gross area A_gin²Gross inertia I_gin⁴Strand eccentricity einGirder self-weight moment M_gk-ftf′_ciksif′_cksiRelative humidity H%Prestressing typeFriction coefficient μ/radWobble K/ftTotal angle change αradTendon length xftAnchor set Δin

Friction and anchorage-set terms apply to post-tensioning only; they are reported for reference when pretensioning is selected.

Elastic shortening Δf_pES

19.32 ksi

Friction Δf_pF

11.32 ksi

not applicable to pretensioning

Anchorage set Δf_pA

2.23 ksi

not applicable

Long-term Δf_pLT

21.41 ksi

Total loss

40.73 (20.1 %) ksi

f_pe after all losses

161.8 ksi

P_i at transfer / P_e at service

1121 / 990 kip

Detailing sketchPrestress losses — component breakdown

f_pe after all losses = 161.8 ksi.

Step	Equation	Substitution	Result
Concrete modulus at transfer	E_ci = 33,000 w^1.5 √f′_ci	33,000(0.145)^1.5 √6.00	4463 ksi
Concrete stress at strand centroid	f_cgp = P/A_g + P e²/I_g − M_g e/I_g	1.6158 + 2.2383 − 0.8283	3.0258 ksi
Elastic shortening	Δf_pES = (E_p/E_ci) f_cgp	(28500/4463)·3.0258	19.32 ksi
Friction loss (post-tensioned)	Δf_pF = f_pj[1 − e^−(Kx + μα)]	202.5[1 − e^−(0.0002·100.0 + 0.25·0.150)]	11.32 ksi
Anchorage set loss	L_set = √(ΔE_p/p), Δf_pA = 2p·L_set	L_set = √(0.375/12·28500/0.11644) = 9.6 ft	2.23 ksi
Humidity factor	γ_h = 1.7 − 0.01H	1.7 − 0.01·70	1.000
Strength factor	γ_st = 5/(1 + f′_ci)	5/(1 + 6.00)	0.714
Long-term loss	Δf_pLT = 10(f_pi A_ps/A_g)γ_h γ_st + 12γ_h γ_st + 2.4	10·1.462·1.000·0.714 + 12·1.000·0.714 + 2.4	21.41 ksi
Total loss	Δf_pT = Δf_pES + Δf_pLT	19.32 + 21.41	40.73 ksi (20.1 %)
Effective strand stress	f_pe = f_pj − Δf_pT	202.5 − 40.73	161.8 ksi
Prestress force	P_i = (f_pj − Δf_pES)A_ps ; P_e = f_pe A_ps	183.2·6.12 ; 161.8·6.12	P_i = 1121 kip, P_e = 990 kip

OKTotal loss within the usual 20–30 % bandratio 20.1 %
OKf_pe ≤ 0.80 f_py (= 0.72 f_pu)demand 161.8 ksicapacity 194.4 ksi

Design with P_e = 990 kip at service (f_pe = 161.8 ksi, 20.1 % total loss) and P_i = 1121 kip at transfer. Losses are in the normal range for a pretensioned girder.

Assumptions & code basis

Immediate losses: friction Δf_pF = f_pj[1 − e^−(Kx + μα)] and anchorage set (post-tensioned only), and elastic shortening Δf_pES = (E_p/E_ci) f_cgp (§5.9.3.2).
For pretensioned members the full (E_p/E_ci)f_cgp is used; for post-tensioned members with sequential stressing the average is taken as one-half.
Time-dependent losses by the approximate estimate of §5.9.3.3: Δf_pLT = 10 (f_pi A_ps/A_g) γ_h γ_st + 12 γ_h γ_st + Δf_pR, with γ_h = 1.7 − 0.01H, γ_st = 5/(1+f′_ci) and Δf_pR = 2.4 ksi for low-relaxation strand.
f_cgp is the concrete stress at the centroid of the prestressing force at transfer, including the girder self-weight moment relief.
The approximate method requires standard precast, pretensioned members with normal-weight concrete, average conditions, and a specified concrete strength at transfer; use the refined method of §5.9.3.4 for unusual members.

Module 27AASHTO LRFD 5.9.3.3Chapter 7

Prestress-loss estimator (approximate method)

Elastic shortening plus the approximate long-term loss with humidity and strength correction factors — the quick screening version.

f_puksiA_psin²f′_ci at releaseksif′_c at 28 daysksiRelative humidity H%Partial prestress ratio PPR

γ_h = 1.7 − 0.01H · γ_st = 5 / (1 + f′_ci) · Δf_pLT = 10 γ_h γ_st + 12 γ_h γ_st + Δf_pR

Jacking stress f_pj = 0.75 f_pu

202.5 ksi

γ_h

1.000

γ_st

0.769

Elastic shortening Δf_pES

18.2 ksi

Long-term Δf_pLT

19.3 ksi

Total loss

37.5 ksi

18.5 % of jacking stress

Effective prestress f_pe

165.0 ksi

Use with Service III tension and flexural checks.

Detailing sketchPrestress losses — breakdown and strand profile

Jacking stress f_pj = 202.5 ksi, elastic shortening Δf_pES = 18.2 ksi, long-term Δf_pLT = 19.3 ksi, effective prestress f_pe = 165.0 ksi.

Module 28AASHTO LRFD 5.9.2.3 / 5.6.3.3Chapter 7

Prestressed-girder service stresses & flexure

Transfer and Service III fibre stresses, cracking moment, and the minimum-reinforcement check for a prestressed I-girder.

Section area Ain²Section modulus Sb (bottom)in³Section modulus St (top)in³Strand eccentricity einInitial prestress PikipTotal prestress loss%Mdl (girder + deck)k-ftMsdl (superimposed)k-ftMLL+IMk-ftf′c (28-day)ksif′ci (release)ksiNoncomposite Sncin³

f = P/A ± Pe/S ∓ M/S · Mcr = Sb(fr + fcpe) − Md(Sb/Snc − 1) · fr = 0.24√f′c

Effective prestress Pe

1210 kip

Top stress at transfer

-0.735 ksi

Bottom stress at transfer

3.868 ksi

Bottom stress, Service III

2.341 ksi

Top stress, Service I

0.664 ksi

Cracking moment Mcr

5784 k-ft

Transfer compression

0.735 ≤ 3.000 ksi

Transfer tension

3.868 ≤ 0.200 ksi

Service III tension

2.341 ≤ 0.484 ksi

Service I compression

0.664 ≤ 2.925 ksi

Minimum reinforcement requires φMn ≥ min(1.33Mu, Mcr) = 4256 k-ft; verify against the section's computed φMn.

Detailing sketchPrestressed girder — strand pattern and fiber stress diagram

Strand centroid eccentricity e = 30.6 in below the section centroid. Top fiber f_t = 0.664 ksi (limit 2.925 ksi), bottom fiber f_b = 2.341 ksi (tension limit 0.484 ksi).

Module 29AASHTO LRFD 5.6.3.1 / 5.6.3.2 / 5.6.3.3Chapter 7

Flexural resistance with prestressing steel — f_ps and φM_n

Strand stress at nominal resistance f_ps = f_pu(1 − kc/d_p) with automatic rectangular / flanged equilibrium, φ from the net tensile strain, and the §5.6.3.3 minimum-reinforcement check against M_cr and 1.33M_u.

M_u (Strength I)k-ftEffective flange width binWeb width b_winFlange (deck) thickness h_find_p to strand centroidinComposite depth hinA_psin²f_puksiMild steel A_sin²f_yksiDeck f′_cksiComposite S_c (bottom)in³Non-composite S_ncin³Non-composite DL moment M_dnck-ftf_cpe (compression from P_e)ksi

Rectangular behaviour is assumed first; the module switches to flanged equilibrium automatically when a = β₁c exceeds the flange.

c

5.184 in

f_ps

264.4 ksi

M_n

9142.0 k-ft

φ

1.000

φM_n

9142.0 k-ft

Behaviour

Rectangular

Detailing sketchPrestressed flexure — strain and stress block

φM_n = 9142.0 k-ft vs M_u = 7500.0 k-ft; f_ps = 264.4 ksi.

Step	Equation	Substitution	Result
Strand k factor	k = 2(1.04 − f_py/f_pu)	2(1.04 − 243.0/270)	0.280
β₁	0.85 − 0.05(f′_c − 4) within 0.65–0.85	f′_c = 4.00 ksi	0.850
Neutral axis	c = [A_ps f_pu + A_s f_y − A′_s f′_y] / [0.85f′_c β₁ b + k A_ps f_pu/d_p]	A_ps = 6.120 in², f_pu = 270 ksi, d_p = 70.00 in	5.184 in (rectangular behaviour)
Stress block depth	a = β₁ c	0.850·5.184	4.407 in
Strand stress at nominal resistance	f_ps = f_pu(1 − k c/d_p)	270(1 − 0.280·5.184/70.00)	264.4 ksi
Nominal moment	M_n = A_ps f_ps(d_p − a/2) + A_s f_y(d_s − a/2)	6.120·264.4(70.00 − 2.203) + …	9142.0 k-ft
Net tensile strain and φ	ε_t = 0.003(d_t − c)/c ; φ = 0.75 + 0.25(ε_t − ε_cl)/(ε_tl − ε_cl)	ε_t = 0.03751	φ = 1.000
Factored resistance	φM_n	1.000·9142.0	9142.0 k-ft
Cracking moment	M_cr = γ₃[(γ₁ f_r + γ₂ f_cpe)S_c − M_dnc(S_c/S_nc − 1)]	f_r = 0.4800 ksi, S_c = 26000 in³	11232.6 k-ft
Minimum reinforcement	φM_n ≥ min(M_cr, 1.33M_u)	min(11232.6, 9975.0)	9975.0 k-ft

OKStrength φM_n ≥ M_udemand 7500.0 k-ftcapacity 9142.0 k-ftratio 0.82
NGMinimum reinforcement §5.6.3.3demand 9975.0 k-ftcapacity 9142.0 k-ft
OKDuctility ε_t ≥ 0.005 (tension controlled)demand 0.03751capacity 0.00500

Strength is adequate but the §5.6.3.3 minimum-reinforcement check fails — add mild steel or strand so φM_n ≥ 9975.0 k-ft.

Assumptions & code basis

Bonded strand: f_ps = f_pu(1 − k c/d_p) with k = 2(1.04 − f_py/f_pu) = 0.28 for low-relaxation strand (AASHTO LRFD Eq. 5.6.3.1.1-1).
Equilibrium of the rectangular stress block gives c; flanged behaviour is triggered automatically when a = β₁c exceeds the flange thickness.
Flexural resistance φM_n with φ from §5.5.4.2 varying between 0.75 (compression controlled) and 1.00 (tension controlled) for prestressed concrete.
Minimum reinforcement §5.6.3.3: φM_n ≥ min(M_cr, 1.33M_u) with M_cr = γ₃(γ₁ f_r + γ₂ f_cpe)S_c − M_dnc(S_c/S_nc − 1).
Compression steel A′_s is included in equilibrium but conservatively assumed to yield.

Module 30AASHTO LRFD 5.6.3.5.2 · PCI Design HandbookChapter 7

Prestress camber & deflection — PCI multipliers

Camber at release from the strand profile, self-weight deflection, and the PCI time-dependent multipliers at erection and final service with the net camber for screed setting.

SpanftP at transferkipEffective PkipMidspan eccentricityinEnd eccentricityinI girderin⁴I compositein⁴f′_ciksif′_cksiGirder self-weightklfDeck weightklfSuperimposed DLklfLive loadklfStrand profile

Time-dependent camber with PCI multipliers at release, erection and final — the numbers the fabricator and the screed crew actually need.

Camber at release

1.559 in

Camber at erection

2.806 in

Final net camber

-3.030 in

Live-load deflection

0.929 in

Δ_allow = L/800

1.650 in

Haunch build-up

3.53 in

Derivation — equation, substitution, result

Moduli at release and in service

\begin{aligned}&\;E_c = 1,820\sqrt{f'_c} (\,\text{ksi})\\[4pt]&=\;E_{ci} = 1,820\sqrt{6.00}, E_c = 1,820\sqrt{8.00}\\[4pt]&=\;\boxed{4458 / 5148 \,\text{ksi}}\end{aligned}

Camber from prestress

\begin{aligned}&\;\Delta _p = P_i e_c L^{2}/(8E_{ci} I_g) \operatorname{with} \,\text{the} \,\text{harped} \,\text{correction} −P_i(e_c−e_e)L^{2}/(12E_{ci} I_g)\\[4pt]&=\;1350 \times 22.00 \times 1320^{2}/(8 \times 4458 \times 260000)\\[4pt]&=\;\boxed{3.890 \,\text{in} ↑}\end{aligned}

Self-weight deflection

\begin{aligned}&\;\Delta _g = 5w_g L^{4}/(384 E_{ci} I_g)\\[4pt]&=\;w_g = 0.820 \,\text{klf}\\[4pt]&=\;\boxed{2.330 \,\text{in} ↓}\end{aligned}

Net camber at release

\begin{aligned}&\;\Delta _{rel} = \Delta _p − \Delta _g\\[4pt]&=\;3.890 − 2.330\\[4pt]&=\;\boxed{1.559 \,\text{in} ↑}\end{aligned}

Camber at erection

\begin{aligned}&\;\Delta _{erect} = 1.80 \Delta _{rel} \;\text{(PCI multiplier)}\\[4pt]&=\;\boxed{2.806 \,\text{in} ↑}\end{aligned}

Deck and superimposed dead load

\begin{aligned}&\;\Delta = 5wL^{4}/(384 E_c I); \,\text{deck} \,\text{on} I_g, \mathrm{SDL} \,\text{on} I_c\\[4pt]&=\;\Delta _{deck} = 2.092, \Delta _{SDL} = 0.258 \,\text{in}\\[4pt]&=\;\boxed{2.350 \,\text{in} ↓}\end{aligned}

Long-term net

\begin{aligned}&\;\Delta _{final} = 2.20\Delta _p,e − \;\text{(2.70Δg + 2.40Δdeck + 3.00ΔSDL)}\\[4pt]&=\;\boxed{-3.030 \,\text{in} ↓ (\,\text{sag})}\end{aligned}

Live-load deflection

\begin{aligned}&\;\Delta _{LL} = 5w_{LL} L^{4}/(384 E_c I_c) \le L/800\\[4pt]&=\;0.929 \,\text{in} \,\text{vs} 1.650 \,\text{in}\\[4pt]&=\;\boxed{\,\text{satisfied}}\end{aligned}

Haunch build-up

\begin{aligned}&\;\,\text{build}-\,\text{up} = |\,\text{sag}| + 0.5 \,\text{in} \,\text{minimum} \,\text{haunch}\\[4pt]&=\;\boxed{3.53 \,\text{in} \operatorname{at} \,\text{midspan}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchGirder camber growth and strand profile

Predicted camber at erection = 2.81 in, long-term final = -3.03 in over a span of 110 ft. Strand profile: harped, midspan eccentricity e_c = 22.0 in, end eccentricity e_e = 12.0 in.

Constructability & detailing notes

Publish the predicted camber at 28, 60 and 120 days — the haunch detail must absorb the difference between predicted and measured.
Girders stored longer than expected keep growing in camber; measure before setting the screed elevations.
Differential camber between adjacent girders is the usual cause of deck thickness problems; specify a maximum of 3/4 in.

Check	Demand	Capacity / limit	Status
Live-load deflection ≤ L/800 (§2.5.2.6.2)	0.929 in	1.650 in	PASS
Residual camber positive after all dead load	-3.030 in	≥ 0	REVIEW
Camber at erection within ±0.5 in of the assumed screed elevationsLarge cambers complicate deck screeding — consider debonding rather than harping.	—	—	PASS

Independent verification	Expected	Computed	Status
Net camber positive (upward) at erection	> 0 in	2.806	VERIFIED

Assumptions & basis of design

PCI Design Handbook multipliers: 1.80 at erection, 2.20/2.70/2.40/3.00 for the long-term components.
Prestress camber uses the gross section; the composite section resists SDL and live load.
Live-load deflection uses the distribution factor already embedded in w_LL, with all girders deflecting equally (§2.5.2.6.2).

The girder sags under full dead load — increase the drape or the strand count, or specify additional haunch build-up so the deck profile is preserved.

Chapters 8 – 11

Steel girders, splices, cross-frames and connections

Classification, plastic and yield moments, lateral–torsional buckling, web shear, splices and bearing stiffeners.

Module 31AASHTO LRFD 6.10.2 / 6.10.6 / 6.10.8.2Chapters 8 · 11

Steel girder flexural resistance & LTB

Web and flange classification, M_y and M_p, r_t with L_p and L_r, the three LTB zones with C_b, flange local buckling, φM_n and every §6.10.2 proportion limit — with the bracing or flange size needed when a check fails.

F_yksiWeb depth DinWeb thickness t_winCompression flange widthinCompression flange thicknessinTension flange widthinTension flange thicknessinUnbraced length L_bftMoment gradient C_b—M_uk-ft

Cross-frame spacing L_b is the single most powerful lever on a plate girder's flexural resistance — compare the result at L_p.

M_y

4784 k-ft

M_p

5867 k-ft

L_p / L_r

8.19 / 30.75 ft

LTB zone

Plastic / no LTB (L_b ≤ L_p)

φM_n

4784 k-ft

Utilization

1.25

Detailing sketchPlate girder — flange/web section

Plastic / no LTB (L_b ≤ L_p); φM_n = 4784 k-ft vs M_u = 6000 k-ft at L_b = 22.00 ft.

Step	Equation	Substitution	Result
Web slenderness	2D_c/t_w vs λ_pw = 3.76√(E/F_y), λ_rw = 5.7√(E/F_y)	2·27.00/0.500 = 108.00; λ_pw = 90.55, λ_rw = 137.27	Non-compact web
Compression-flange slenderness	λ_f = b_fc/2t_fc vs λ_pf = 0.38√(E/F_y)	16.00/(2·1.000) = 8.00; λ_pf = 9.15	Compact flange
Elastic section properties	I, S_xc = I/c_t, S_xt = I/c_b	I = 35333 in⁴, c_t = 30.77 in	S_xc = 1148 in³, S_xt = 1387 in³
Yield moment	M_y = F_y S	50·1148/12	4784 k-ft
Plastic moment	M_p = F_y Z_x	50·1408/12	5867 k-ft
Effective radius of gyration	r_t = b_fc/√[12(1 + D_c t_w/(3 b_fc t_fc))]	16.00/√[12(1 + 27.00·0.500/(3·16.00·1.000))]	4.080 in
Bracing limits	L_p = 1.0 r_t √(E/F_y) ; L_r = π r_t √(E/F_yr)	r_t = 4.080 in, F_yr = 35.0 ksi	L_p = 8.19 ft, L_r = 30.75 ft
LTB resistance	Plastic / no LTB (L_b ≤ L_p)	L_b = 22.00 ft, C_b = 1.00	F_nc(LTB) = 50.00 ksi
Flange local buckling resistance	F_nc = R_b R_h F_yc	λ_f = 8.00	50.00 ksi
Nominal flexural resistance	M_n = min(F_nc,LTB, F_nc,FLB)·S_xc (or M_p when compact and braced)	F_n = 50.00 ksi	4784 k-ft
Factored resistance	φ_f M_n ≥ M_u	1.00·4784 vs 6000	4784 k-ft

NGFlexural strength φM_n ≥ M_udemand 6000 k-ftcapacity 4784 k-ftratio 1.25
OKWeb proportion D/t_w ≤ 150demand 108.0capacity 150
OKFlange proportion b_f/2t_f ≤ 12demand 8.00capacity 12
OKFlange width b_f ≥ D/6demand 16.00capacity 9.00
OKFlange thickness t_f ≥ 1.1 t_wdemand 1.000capacity 0.550

Deficient by 1.25×. Either brace tighter — reducing L_b to L_p = 8.19 ft recovers the full F_y — thicken the compression flange to lower λ_f, or increase S_xc by about 25 %.

Assumptions & code basis

I-section flexural resistance per AASHTO LRFD §6.10.6 – §6.10.8 (LRFD, φ_f = 1.00).
Cross-section proportion limits of §6.10.2 must also be satisfied: D/t_w ≤ 150 (unstiffened), b_f/2t_f ≤ 12, b_f ≥ D/6, t_f ≥ 1.1t_w.
Web classification uses 2D_c/t_w against 3.76√(E/F_y) (compact) and 5.7√(E/F_y) (slender, load-shedding factor R_b applies).
Lateral–torsional buckling uses r_t = b_fc/√[12(1 + D_c t_w/(3 b_fc t_fc))] with L_p = 1.0 r_t √(E/F_y) and L_r = π r_t √(E/F_yr), F_yr = 0.7F_y.
Hybrid factor R_h and web load-shedding factor R_b are taken as 1.0; apply them explicitly for hybrid or slender-web girders.
Composite sections in positive flexure with a compact web may be governed instead by the §6.10.7.1 plastic-moment provisions and the D_p/D_t ductility limit.

Module 32AASHTO LRFD 6.10.9Chapters 8 · 11

Steel girder web shear resistance

Shear-buckling coefficient, ratio C, and the nominal shear resistance of a stiffened or unstiffened web.

Web depth DinWeb thickness t_winWeb yield F_ywksiStiffener spacing d_oinFactored shear V_ukip

V_p = 0.58 F_yw D t_w · k = 5 + 5/(d_o/D)² · V_n = C V_p (unstiffened / stiffened web)

D / t_w

120.0

Shear-buckling coefficient k

6.250

Ratio C

0.395

Plastic shear V_p

870 kip

V_n

344 kip

φ_v V_n (φ_v = 1.0)

344 kip

Shear check

344 ≥ 300 kip

Web adequate in shear.

Detailing sketchPlate-girder web panel — elevation and stiffener section

Web 60.0 in deep × 0.500 in thick, D/t_w = 120, transverse stiffeners at d_o = 120.0 in (d_o/D = 2.00). V_u = 300 kip vs φV_n = 344 kip (AASHTO 6.10.9).

Module 33AASHTO LRFD 6.13.2 / 6.13.6Chapters 8 · 11

Bolted field splice design

Shear, bearing, and slip resistance of a high-strength bolted splice against a factored demand.

Bolt diameter dinBolt gradeNumber of boltsShear planes NsHole typePlate thicknessinPlate FuksiEdge distance LcinSlip coefficient KsBolt spacinginFactored splice force Vukip

Rn,shear = 0.38 Ab Fub Ns · Rn,bearing = min(2.4dtFu, 1.2LctFu) · Rn,slip = Kh Ks Ns Pt

Bolt area Ab

0.601 in²

Shear resistance (splice)

702 kip

Bearing resistance (splice)

1310 kip

Slip resistance (splice)

412 kip

Governing capacity vs demand

702 ≥ 650 kip

Splice adequate for Strength I shear/bearing.

Detailing sketchBolted field splice — elevation and section through the plates

16 × 0.875 in Ø A325 bolts in 2 rows of 8, 3 in pitch, 1.75 in edge distance (AASHTO 6.13.2.6). Demand per bolt 650.0 kip vs R_n = 702.0 kip.

Slip resistance governs serviceability (Service II) checks and is not combined additively with the bearing-type strength resistance.

Module 34AASHTO LRFD AISC 360-22 §J10 · AASHTO §6.10.11Chapters 4 · 8

Web concentrated forces & bearing stiffeners

Web local yielding, crippling and compression buckling at reactions and point loads, then a full bearing-stiffener design — plate size, the 25t_w effective column, KL/r, F_cr, φP_n and the fillet weld.

Concentrated force R_ukipBearing length ℓ_binDepth dinWeb t_winFlange t_finFlange width b_fink_desinF_yksi

R_n = F_ywt_w(5k+ℓ_b) · R_n = 0.80t_w²[1+3(ℓ_b/d)(t_w/t_f)^1.5]√(EF_ywt_f/t_w) · R_n = 24t_w³√(EF_yw)/h

φR_n yielding

205.7 kip

φR_n crippling

123.1 kip

φR_n web buckling

52.0 kip

Governing

52.0 kip

Web compression buckling (§J10.5)

Stiffener plates

2 – PL 3.78″ × 0.500″

φP_n stiffener

385.0 kip

Concentrated-force limit states & bearing stiffener design — AISC 360-22 §J10 / §J7 / §E3

Clear web depth h	21.32 in
Governing limit state	Web compression buckling (§J10.5)
Stiffener call-out	2 – PL 3.78″ × 0.500″ bearing stiffeners, 3/16″ fillet both sides
Effective area A_eff	8.62 in²
KL / r	16.0 in / 1.576 in → 10.1
F_e / F_cr	2780.6 / 49.6 ksi
Weld	3/16 in fillet, both sides, full web depth

PASS — Provide a pair of bearing stiffeners 3.78 in × 0.500 in, fitted to the loaded flange, with 3/16 in fillet welds each side to the web (§J10.8).

Detailing sketchBearing stiffener — girder elevation and plate/weld section

End reaction R_u = 180 kip. Stiffener pair PL 3.78 × 0.500 in, fillet welds 3/16 in each side, full web depth.

Module 35AASHTO LRFD 6.10.2 / 2.5.2.6.3Chapters 8 · 11

Plate-girder trial sizing & proportion limits

Web depth from span-to-depth guidance, D/t_w and b_f/2t_f screening, the §6.10.2 proportion limits and the flange plate sizes needed to satisfy each one.

SpanftGirder spacingftNumber of girders—F_yksiDeck thicknessinContinuity

First-pass plate girder dimensions that satisfy the §6.10.2 proportion limits before any analysis is run.

Web D × t_w

60 × 0.4375 in

Top flange

15.0 × 0.7500 in

Bottom flange

15.0 × 1.0000 in

Total depth

61.75 in

Area

52.5 in²

Self weight

0.1786 klf

Derivation — equation, substitution, result

Web depth

\begin{aligned}&\;D \approx L/27.5 \;\text{(continuous)}\\[4pt]&=\;1680 \,\text{in} / 27.5\\[4pt]&=\;\boxed{D = 60 \,\text{in}}\end{aligned}

Web thickness

\begin{aligned}&\;t_w \approx D/150, \ge 7/16 \,\text{in}, D/t_w \le 150 (§6.10.2.1.1)\\[4pt]&=\;60/150 = 0.400\\[4pt]&=\;\boxed{t_w = 0.4375 \,\text{in} (D/t_w = 137.1)}\end{aligned}

Flange width

\begin{aligned}&\;b_f \approx D/4 \ge D/6 \operatorname{and} \ge 12 \,\text{in} (§6.10.2.2)\\[4pt]&=\;60/4\\[4pt]&=\;\boxed{b_f = 15.0 \,\text{in}}\end{aligned}

Flange thickness

\begin{aligned}&\;b_f/(2t_f) \le 12 \operatorname{and} t_f \ge 1.1 t_w (§6.10.2.2)\\[4pt]&=\;t_{fc} = 0.7500 \,\text{in}, t_{ft} = 1.0000 \,\text{in}\\[4pt]&=\;\boxed{\lambda _f = 10.00}\end{aligned}

Trial section

\begin{aligned}&\;A = D t_w + b_{fc} t_{fc} + b_{ft} t_{ft}\\[4pt]&=\;60(0.4375) + 15.0(0.7500) + 15.0(1.0000)\\[4pt]&=\;\boxed{A = 52.5 \,\text{in}^{2}, \,\text{total} \,\text{depth} 61.75 \,\text{in}}\end{aligned}

Self weight

\begin{aligned}&\;w = A(490 \,\text{pcf})/144\\[4pt]&=\;\boxed{0.1786 \,\text{klf}}\end{aligned}

Compactness screening

\begin{aligned}&\;\lambda _f \le 0.38\sqrt{E/F_{yc}}, 2D_{cp}/t_w \le 3.76\sqrt{E/F_{yc}}\\[4pt]&=\;\lambda _{pf} = 9.15, \lambda _{pw} = 90.6\\[4pt]&=\;\boxed{\,\text{non}-\,\text{compact} \,\text{flange}}\end{aligned}

Deck fit

\begin{aligned}&\;Girder \,\text{spacing} S \operatorname{with} \,\text{slab} t_s\\[4pt]&=\;S = 10.00 \,\text{ft}, 5 \,\text{girders}, t_s = 8.50 \,\text{in}\\[4pt]&=\;\boxed{\,\text{within} \,\text{the} §4.6.2.2 \,\text{range}}\end{aligned}

Detailing — plan, elevation and section

Section — composite

Plan — framing

Elevation

Constructability & detailing notes

Keep the web thickness constant for the whole girder line; changing it saves less steel than it costs in fabrication.
Flange transitions should occur at field splices or at least 10 ft from them, and the width change should be a 1:2.5 taper.
Minimum flange width of 12 in keeps the girder stable during shipping and erection.

Check	Demand	Capacity / limit	Status
D/t_w ≤ 150 without longitudinal stiffeners	137.1	150	PASS
b_f ≥ D/6 (§6.10.2.2-2)	15.0 in	10.0 in	PASS
t_f ≥ 1.1 t_w (§6.10.2.2-3)	0.7500 in	0.4813 in	PASS
b_f/(2t_f) ≤ 12 (§6.10.2.2-1)	10.00	12.0	PASS
Girder spacing 3.5 – 16 ft	10.00 ft	3.5 – 16 ft	PASS

Independent verification	Expected	Computed	Status
Web slenderness within the non-longitudinally-stiffened limit	D/t_w ≤ 150		VERIFIED

Assumptions & basis of design

Depth ratios from AASHTO LRFD Table 2.5.2.6.3-1 with plate sizes rounded to commercial increments (web depth to 3 in, plate thickness to 1/8 in).
Bottom flange is proportioned 25 % thicker than the top flange to reflect composite action in positive-moment regions.
Steel unit weight 490 pcf; the estimate excludes stiffeners, cross-frames and connection material (add roughly 12 – 18 %).

Take this section into the composite section-property, flexure and shear modules; iterate the flange plates until the utilisation lands near 0.90 – 0.98.

Module 36AASHTO LRFD Appendix D6.1 / 6.10.7.1.2Chapters 8 · 11

Composite plastic moment M_p and PNA location

Slab, flange and web plastic forces, Appendix D6.1 case selection (I, II, III) for the plastic neutral axis, M_p, and the D_p ≤ 0.42 D_t ductility screen.

Effective flange widthinSlab thicknessinDeck f′_cksiHaunchinTop flange widthinTop flange thicknessinWeb depthinWeb thicknessinBottom flange widthinBottom flange thicknessinF_ycksiF_ywksiF_ytksiM_u Strength Ikip·ft

Locates the plastic neutral axis by Appendix D6.1 case selection and screens ductility with D_p ≤ 0.42 D_t.

PNA case

II

D_p

10.95 in

0.42 D_t

30.66 in

M_p

12952 kip·ft

φ_f M_n

12952 kip·ft

Utilisation

0.695

Derivation — equation, substitution, result

Plastic forces

\begin{aligned}&\;P_s = 0.85f'_c b_{eff} t_s \cdot P_c = F_{yc} b_{fc} t_{fc} \cdot P_w = F_{yw} D t_w \cdot P_t = F_{yt} b_{ft} t_{ft}\\[4pt]&=\;P_s = 3121, P_c = 800, P_w = 1688, P_t = 1350 \,\text{kip}\\[4pt]&=\;\boxed{\,\text{forces} \,\text{set}}\end{aligned}

PNA case

\begin{aligned}&\;Compare P_t + P_w + P_c \operatorname{with} P_s, \,\text{then} P_t + P_w \operatorname{with} P_s + P_c \;\text{(Table D6.1-1)}\\[4pt]&=\;\boxed{\mathrm{II} — \mathrm{PNA} \,\text{in} \,\text{the} \,\text{top} \,\text{flange} \;\text{(§D6.1, Case II)}}\end{aligned}

PNA location

\begin{aligned}&\;Ȳ \operatorname{from} \,\text{the} \,\text{governing} \,\text{case} \,\text{equation}\\[4pt]&=\;\boxed{Ȳ = 0.45 \,\text{in}, D_p = 10.95 \,\text{in} \,\text{below} \,\text{the} \,\text{top} \operatorname{of} \,\text{the} \,\text{slab}}\end{aligned}

Plastic moment

\begin{aligned}&\;M_p = \Sum P_i d_i \,\text{taken} \,\text{about} \,\text{the} \mathrm{PNA}\\[4pt]&=\;\Sum = 155422 \,\text{kip}\cdot \,\text{in}\\[4pt]&=\;\boxed{M_p = 12952 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Total depth

\begin{aligned}&\;D_t = t_s + \,\text{haunch} + t_{fc} + D + t_{ft}\\[4pt]&=\;8.50 + 2.00 + 1.000 + 60.0 + 1.500\\[4pt]&=\;\boxed{D_t = 73.00 \,\text{in}}\end{aligned}

Ductility requirement

\begin{aligned}&\;D_p \le 0.42 D_t (Eq. 6.10.7.3-1)\\[4pt]&=\;10.95 \,\text{vs} 30.66 \,\text{in}\\[4pt]&=\;\boxed{\,\text{ductile} — M_n = M_p}\end{aligned}

Nominal flexural resistance

\begin{aligned}&\;M_n = M_p (§6.10.7.1.2)\\[4pt]&=\;\boxed{\phi _f M_n = 12952 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Detailing — plan, elevation and section

Section — composite

Plan — framing

Elevation

Constructability & detailing notes

M_p is only available if the section is compact and adequately braced during the deck pour — check the non-composite stage separately.
Ductility (D_p ≤ 0.42 D_t) is what allows the section to reach M_p; a deep slab with a shallow girder often fails it.

Check	Demand	Capacity / limit	Status
Ductility D_p ≤ 0.42 D_t (§6.10.7.3)	10.95 in	30.66 in	PASS
Flexure φ_f M_n ≥ M_u	9000 kip·ft	12952 kip·ft	PASS
Compact web 2D_cp/t_w ≤ 3.76√(E/F_yc)	0.0	90.6	PASS

Independent verification	Expected	Computed	Status
Ductility requirement D_p ≤ 0.42 D_t	≤ 0.42 D_t	10.95	VERIFIED

Assumptions & basis of design

AASHTO LRFD Appendix D6.1 plastic-moment cases for a composite section in positive flexure; concrete in tension neglected.
Longitudinal deck reinforcement is conservatively neglected in the positive-moment plastic force sum.
φ_f = 1.00 for flexure (§6.5.4.2); compact composite sections in positive flexure may reach M_p.

Section is compact and ductile — proceed to the §6.10.9 shear check and the §6.10.10 shear-connector design.

Module 37AASHTO LRFD 6.10.10.1 / 6.10.10.4Chapters 8 · 11

Shear connectors — strength and fatigue pitch

Stud resistance Q_r, the number of studs for the strength limit state between points of maximum moment and zero moment, and the Fatigue I / II pitch from Z_r and the range of horizontal shear flow.

Stud diameterinStud heightinStuds per row—Deck f′_cksiStud F_uksiDeck thicknessinTop flange thicknessinFatigue shear range V_srkip/inHorizontal force PkipADTT single lanetrucks/dayDesign lifeyr

Strength count from the plastic horizontal force and fatigue pitch from Z_r and the horizontal shear range.

Q_n per stud

36.08 kip

Q_r per stud

30.67 kip

Studs required (strength)

79

Z_r fatigue

2.105 kip

Governing pitch

3.5 in

Callout

3 — 0.875 in ø × 6.0 in studs @ 3.5 in

Derivation — equation, substitution, result

Stud area

\begin{aligned}&\;A_{sc} = \pi d^{2}/4\\[4pt]&=\;\pi (0.875)^{2}/4\\[4pt]&=\;\boxed{0.6013 \,\text{in}^{2}}\end{aligned}

Nominal stud strength

\begin{aligned}&\;Q_n = 0.5A_{sc}\sqrt{f'_c E_c} \le A_{sc} F_u (Eq. 6.10.10.4.3-1)\\[4pt]&=\;0.5(0.6013)\sqrt{4.00 \times 3640} \,\text{vs} 36.08\\[4pt]&=\;\boxed{Q_n = 36.08 \,\text{kip}}\end{aligned}

Factored strength

\begin{aligned}&\;Q_r = \phi _{sc} Q_n, \phi _{sc} = 0.85\\[4pt]&=\;\boxed{Q_r = 30.67 \,\text{kip}/\,\text{stud}}\end{aligned}

Strength-limit stud count

\begin{aligned}&\;n = P/Q_r (§6.10.10.4.1)\\[4pt]&=\;2400/30.67\\[4pt]&=\;\boxed{79 \,\text{studs} \,\text{between} \,\text{the} \operatorname{max}-\,\text{moment} \,\text{point} \operatorname{and} \,\text{the} \,\text{support}}\end{aligned}

Design cycles

\begin{aligned}&\;N = (365)\;\text{(years)}\;\text{(ADTTSL)}(n)\\[4pt]&=\;365 \times 75 \times 1500\\[4pt]&=\;\boxed{N = 4.11e+7}\end{aligned}

Fatigue shear resistance

\begin{aligned}&\;Z_r = \alpha d^{2} \ge 5.5d^{2}/2, \alpha = 34.5 − 4.28 \operatorname{log} N (Eq. 6.10.10.2-1/2)\\[4pt]&=\;\alpha = 0.00\\[4pt]&=\;\boxed{Z_r = 2.105 \,\text{kip}/\,\text{stud}}\end{aligned}

Fatigue pitch

\begin{aligned}&\;p \le n_{studs} Z_r / V_{sr} (Eq. 6.10.10.1.2-1)\\[4pt]&=\;3 \times 2.105 / 2.200\\[4pt]&=\;\boxed{p = 2.87 \,\text{in} \rightarrow \operatorname{use} 3.5 \,\text{in}}\end{aligned}

Detailing — plan, elevation and section

Section — composite

Plan — framing

Elevation

Constructability & detailing notes

Studs are welded in the shop through clean, dry flange surfaces; bend-test the first two of each shift and 1 % thereafter.
Keep 2 in clear cover over the stud head and 1 in clear between the stud and the deck bottom mat.
Vary the pitch in bands rather than continuously — the fabricator lays out from a table, not a curve.

Check	Demand	Capacity / limit	Status
Pitch 6d ≤ p ≤ 24 in (§6.10.10.1.2)	3.5 in	5.3 – 24 in	REVIEW
Stud height H ≥ 4d (§6.10.10.1.1)	6.0 in	3.5 in	PASS
Stud diameter ≤ 2.5 t_fc (§6.10.10.1.4)	0.875 in	2.500 in	PASS
Penetration ≥ 2 in into the deck with ≥ 2 in top covertop cover ≈ 2.5 in	6.0 in in a 8.5 in slab	5.5 – 6.5 in	PASS
Transverse spacing ≥ 4d and ≤ 8 t_sCheck the flange width can accommodate the row.	—	—	PASS

Independent verification	Expected	Computed	Status
Stud height-to-diameter ratio ≥ 4	≥ 4.0	6.86	VERIFIED

Assumptions & basis of design

AASHTO LRFD §6.10.10; φ_sc = 0.85; normal-weight concrete with E_c = 1,820√f′_c.
Fatigue pitch computed from the Fatigue I horizontal shear range V_sr = V_f Q/I at the section considered.
The strength check governs the total number of connectors; the fatigue check governs their spacing.

The required fatigue pitch is tighter than the 6d minimum — add a third or fourth stud per row rather than closing the pitch further.

Module 38AASHTO LRFD 6.6.1.2.3 / 6.6.1.2.5 / Table 6.6.1.2.3-1Chapters 8 · 18

Fatigue detail categories & remaining life

Category A–E′ constants, the Fatigue I and II load combinations, the single-lane ADTT_SL with cycles per truck passage, the (ΔF)_TH infinite-life screen and N = A/(Δf)³ finite life.

Detail categoryADTTtrucks/dayLanes available to trucks—Design lifeyrFatigue M_maxkip·ftFatigue M_minkip·ftSection modulusin³Cycles per truck n—Limit state

Fatigue I (infinite life) and Fatigue II (finite life) checks for AASHTO detail categories A through E′.

ADTT_SL

2550 trucks/day

Δf

10.50 ksi

(ΔF)_n

10.00 ksi

Utilisation

1.050

Estimated life

4 yr

Derivation — equation, substitution, result

Single-lane truck traffic

\begin{aligned}&\;\mathrm{ADTT}_{SL} = p \cdot \mathrm{ADTT} \;\text{(Table 3.6.1.4.2-1)}\\[4pt]&=\;p = 0.85 \operatorname{for} 2 \,\text{lane}(s)\\[4pt]&=\;\boxed{\mathrm{ADTT}_{SL} = 2550 \,\text{trucks}/\,\text{day}}\end{aligned}

Fatigue load factor

\begin{aligned}&\;Fatigue I: \gamma = 1.75 \;\text{(infinite life)}\\[4pt]&=\;\boxed{\gamma = 1.75}\end{aligned}

Factored stress range

\begin{aligned}&\;\Delta f = \gamma \;\text{(Mmax − Mmin)}/S\\[4pt]&=\;1.75(900.0 − -150.0)(12)/2100\\[4pt]&=\;\boxed{\Delta f = 10.50 \,\text{ksi}}\end{aligned}

Design cycles

\begin{aligned}&\;N = (365)\;\text{(years)}(n)\;\text{(ADTTSL)}\\[4pt]&=\;365 \times 75 \times 1.0 \times 2550\\[4pt]&=\;\boxed{N = 6.98e+7}\end{aligned}

Nominal fatigue resistance

\begin{aligned}&\;(\Delta F)_n = (\Delta F)_{TH}\\[4pt]&=\;A = 4.40e+9, (\Delta F)_{TH} = 10.0 \,\text{ksi}\\[4pt]&=\;\boxed{(\Delta F)_n = 10.00 \,\text{ksi}}\end{aligned}

Estimated life

\begin{aligned}&\;Y = A/\;\text{(Δf³ · 365 · n · ADTTSL)}\\[4pt]&=\;\boxed{4 \,\text{years}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchFatigue detail category C — S-N resistance curve

Category C: γ(Δf) = 10.50 ksi plotted at N = 69806250 cycles against the finite-life resistance (ΔF)_n = 10.00 ksi and the constant-amplitude threshold (ΔF)_TH = 10.00 ksi. Point falls above the resistance curve — review the detail.

Constructability & detailing notes

Category E′ details (long cover plates on thick flanges) should be designed out, not checked — weld the plate away entirely if you can.
Distortion-induced fatigue is not covered by these categories: weld connection plates to both flanges (§6.6.1.3).
Grind weld toes and specify a peened or ground transition where a category improvement is needed.

Check	Demand	Capacity / limit	Status
γ(Δf) ≤ (ΔF)_n — Category C	10.50 ksi	10.00 ksi	REVIEW
Infinite-life screening Δf ≤ (ΔF)_TH	10.50 ksi	10.0 ksi	REVIEW
Fracture-critical member? apply CVN and redundancy provisions§6.6.2 requires refined analysis and enhanced inspection for FCMs.	—	—	PASS

Independent verification	Expected	Computed	Status
Stress range below the resistance	10.00		VERIFIED

Assumptions & basis of design

AASHTO LRFD §6.6.1.2 detail categories with the fatigue truck (constant 30 ft rear axle spacing) and IM = 15 %.
Fatigue I gives infinite life when Δf ≤ (ΔF)_TH; Fatigue II gives the finite-life resistance from the S-N curve.
Cycles per truck passage n from Table 6.6.1.2.5-2 (typically 1.0 for spans > 40 ft, 2.0 for shorter spans and cantilevers).

Upgrade the detail — moving from Category C to the next better category raises the threshold; grinding cover-plate ends, removing backing bars and using full-penetration welds are the usual fixes.

Module 39AASHTO LRFD 6.7.4 / 4.6.2.7Chapters 8 · 11

Cross-frames and diaphragms — force and sizing

Wind and stability bracing forces into the cross-frame, the diagonal and strut member demands, the slenderness limits for bracing members, and the connection force to detail.

Major-axis moment Mkip·ftCross-frame spacingftRadius (0 = straight)ftDistance between outer girdersftCross-frame depthinGirder spacingftWind pressure P_zksfExposed depthftFlange forcekip

Compares the V-load (curvature), wind and 2 % stability bracing demands and picks the governing cross-frame force.

V-load H

4.17 kip

Diagonal force

4.56 kip

Wind force

2.40 kip

Stability force

18.00 kip

Design force

18.00 kip

Suggested member

L4×4×3/8

Derivation — equation, substitution, result

V-load (curved girder)

\begin{aligned}&\;H = M L_b/(D R) \;\text{(V-load method)}\\[4pt]&=\;6000(20.0)/(36.0 \times 800)\\[4pt]&=\;\boxed{H = 4.17 \,\text{kip}}\end{aligned}

Diagonal geometry

\begin{aligned}&\;\alpha = \operatorname{arctan}(h_{cf}/s_g)\\[4pt]&=\;\operatorname{arctan}(48.0/108.0)\\[4pt]&=\;\boxed{\alpha = 24.0^{\circ}}\end{aligned}

Diagonal axial force

\begin{aligned}&\;F_{diag} = H/\operatorname{cos} \alpha \\[4pt]&=\;4.17/\operatorname{cos} 24.0^{\circ}\\[4pt]&=\;\boxed{4.56 \,\text{kip}}\end{aligned}

Wind on the exposed superstructure

\begin{aligned}&\;w = P_z d, F = w L_b/2 \,\text{into} \,\text{the} \,\text{cross}-\,\text{frame} (§4.6.2.7)\\[4pt]&=\;0.0300 \,\text{ksf} \times 8.00 \,\text{ft} \times 20.0 \,\text{ft} / 2\\[4pt]&=\;\boxed{2.40 \,\text{kip}}\end{aligned}

Stability bracing

\begin{aligned}&\;F_{br} \approx 0.02 P_{flange} \;\text{(AISC App. 6 nodal bracing)}\\[4pt]&=\;0.02 \times 900\\[4pt]&=\;\boxed{18.00 \,\text{kip}}\end{aligned}

Governing design force

\begin{aligned}&\;\operatorname{max}\;\text{(V-load, wind, stability)}\\[4pt]&=\;\boxed{18.00 \,\text{kip}}\end{aligned}

Detailing — plan, elevation and section

Section — composite

Plan — framing

Elevation

Constructability & detailing notes

Weld the connection plate to both flanges — a gap at the tension flange is a distortion-induced fatigue crack waiting to happen.
Cross-frames control the girder geometry during erection; specify whether they are detailed for no-load, steel-dead-load or full-dead-load fit.
Provide slotted or oversized holes only where the erection tolerance genuinely requires them, and note them on the plans.

Check	Demand	Capacity / limit	Status
Cross-frame spacing ≤ 25 ft (§6.7.4.2)	20.0 ft	25 ft	PASS
Cross-frame depth ≥ 0.5 girder depth (straight) / 0.75 (curved)Connect near the flanges so the frame engages the full girder depth.	—	—	PASS
Members are primary in curved bridges (§6.7.4.1)Design curved-bridge cross-frames as primary members with fatigue checks on the connection plates.	—	—	PASS

Independent verification	Expected	Computed	Status
Governing design force identified	max(V-load, wind, stability)	18.00	VERIFIED

Assumptions & basis of design

V-load method approximates curvature effects by a self-equilibrating couple between adjacent girders.
Wind is distributed to the deck-level bracing over half the cross-frame spacing each side.
Connection plates must be welded (not bolted) to both flanges to avoid a Category C′ distortion-induced fatigue detail.

On a curved bridge the cross-frames carry real load in every stage — check the erection condition, when they are often most heavily loaded.

Module 40AASHTO LRFD 6.13.2 · AISC Manual Part 7Chapters 8 · 11

Eccentric bolt group — instantaneous centre / elastic

Direct and torsional bolt force components for an eccentrically loaded group, the governing extreme bolt resultant, the elastic and IC-method comparison, and the bolt count needed to keep the resultant below φr_n.

Shear V_ukipEccentricity einRows—Columns—Pitch (vertical)inGage (horizontal)inBolt diameterinBolt F_ubksiShear planes—Connected thicknessinPlate F_uksiSurface factor K_s—Bolt pretension P_tkipFlange splice M_ukip·ftFlange lever arm h_effinConnection typeThreads

Elastic vector analysis of an eccentric bolt group plus the flange-splice bolt count from the flange couple.

Bolt capacity

39.00 kip

Direct shear R_v

18.00 kip

Resultant R_max

31.62 kip

Group utilisation

0.811

Flange force F_fl

869.0 kip

Flange bolts per side

23

Derivation — equation, substitution, result

Bolt shear

\begin{aligned}&\;R_n = 0.45 A_b F_{ub} N_s \;\text{(threads included, §6.13.2.7)}\\[4pt]&=\;A_b = 0.6013 \,\text{in}^{2}, F_{ub} = 120 \,\text{ksi}, N_s = 2\\[4pt]&=\;\boxed{\phi R_n = 51.95 \,\text{kip}}\end{aligned}

Bearing on the connected material

\begin{aligned}&\;R_n = 2.4 d t F_u (§6.13.2.9)\\[4pt]&=\;2.4(0.875)(0.500)(65)\\[4pt]&=\;\boxed{\phi R_n = 54.60 \,\text{kip}}\end{aligned}

Slip resistance

\begin{aligned}&\;R_n = K_h K_s N_s P_t (Eq. 6.13.2.8-1)\\[4pt]&=\;K_s = 0.50, P_t = 39.0 \,\text{kip}\\[4pt]&=\;\boxed{39.00 \,\text{kip}/\,\text{bolt}}\end{aligned}

Governing bolt capacity

\begin{aligned}&\;\operatorname{min} \operatorname{of} \,\text{shear}, \,\text{bearing} \operatorname{and} \;\text{(if slip-critical)} \,\text{slip}\\[4pt]&=\;\boxed{39.00 \,\text{kip}/\,\text{bolt}}\end{aligned}

Group geometry

\begin{aligned}&\;\Sum (x^{2} + y^{2}) \,\text{about} \,\text{the} \,\text{group} \,\text{centroid}\\[4pt]&=\;5 \,\text{rows} @ 3.00 \,\text{in} \times 2 \,\text{cols} @ 3.00 \,\text{in}\\[4pt]&=\;\boxed{\Sum r^{2} = 202.5 \,\text{in}^{2}, n_b = 10}\end{aligned}

Direct shear per bolt

\begin{aligned}&\;R_v = V_u/n_b\\[4pt]&=\;180.0/10\\[4pt]&=\;\boxed{18.00 \,\text{kip}}\end{aligned}

Torsional components

\begin{aligned}&\;R_{mx} = M y_{\operatorname{max}}/\Sum r^{2}, R_{my} = M x_{\operatorname{max}}/\Sum r^{2}\\[4pt]&=\;M = V_u e = 720.0 \,\text{kip}\cdot \,\text{in}\\[4pt]&=\;\boxed{R_{mx} = 21.33, R_{my} = 5.33 \,\text{kip}}\end{aligned}

Resultant bolt force

\begin{aligned}&\;R_{\operatorname{max}} = \sqrt{(R_v + R_{my})^{2} + R_{mx}^{2}} \le \phi R_n\\[4pt]&=\;\sqrt{(18.00 + 5.33)^{2} + 21.33^{2}}\\[4pt]&=\;\boxed{31.62 \,\text{kip}}\end{aligned}

Flange splice force

\begin{aligned}&\;F_{fl} = M_u/h_{eff}, n = F_{fl}/\phi R_n (§6.13.6.1.3)\\[4pt]&=\;4200(12)/58.0\\[4pt]&=\;\boxed{F_{fl} = 869.0 \,\text{kip} \rightarrow 23 \,\text{bolts} \operatorname{per} \,\text{side}}\end{aligned}

Detailing — plan, elevation and section

Elevation — bolt group

Section — splice plates

Plan — weld detail

Constructability & detailing notes

Slip-critical faying surfaces must be blast-cleaned Class B and left unpainted — mask them before shop priming.
Use a minimum of two bolts per line and fill every hole; open holes in a splice are a fatigue and corrosion detail.
Erection bolts and drift pins go in first; final pretensioning proceeds from the stiffest point outward.

Check	Demand	Capacity / limit	Status
Eccentric group R_max ≤ φR_n	31.62 kip	39.00 kip	PASS
Bolt pitch ≥ 3d (§6.13.2.6.1)	3.00 in	2.63 in	PASS
Edge distance ≥ 1.25 in for 7/8 in bolts (Table 6.13.2.6.6-1)Confirm the plate geometry provides the sheared/rolled edge distance.	—	—	PASS
Slip-critical check at Service II	31.62 kip	39.00 kip	PASS

Independent verification	Expected	Computed	Status
Resultant bolt force within capacity	39.00	31.62	VERIFIED

Assumptions & basis of design

Elastic (vector) analysis of the bolt group — conservative relative to the instantaneous-centre method.
ASTM F3125 Grade A325 bolts (F_ub = 120 ksi) unless otherwise entered; φ_s = 0.80 for bolts in shear.
Flange splice bolts sized for the flange couple M_u/h_eff; AASHTO also requires design for the smaller of the yield or 75 % of the flange resistance.

Detail 10 bolts in the web group and 23 per side in each flange splice plate.

Module 41AASHTO LRFD 6.13.5 / 6.13.6 / 6.13.4Chapters 8 · 11

Splice plate tension — yielding, rupture & block shear

Gross yielding, net-section rupture with hole deduction, block shear on the tearout path, and the plate thickness and width that satisfies the governing limit state at a flange or web splice.

Factored tension P_ukipPlate widthinPlate thicknessinHoles across the section—Hole diameterinF_yksiF_uksiConnection x̄inConnection length LinGross shear lengthinGross tension lengthinNet shear lengthinNet tension lengthinU_bs—

Gross yielding, net-section rupture with shear lag, and both block-shear paths for a bolted tension splice plate.

A_n/A_g

0.750

Shear-lag U

1.000

Gross yielding

570.0 kip

Net rupture

468.0 kip

Block shear

328.8 kip

Governing capacity

328.8 kip

block shear

Derivation — equation, substitution, result

Gross and net areas

\begin{aligned}&\;A_g = b t, A_n = A_g − n d_h t (§6.8.3)\\[4pt]&=\;16.00 \times 0.750 − 4(1.000)(0.750)\\[4pt]&=\;\boxed{A_g = 12.000, A_n = 9.000 \,\text{in}^{2}}\end{aligned}

Shear lag

\begin{aligned}&\;U = 1 − x̄/L \le 1.0, A_e = U A_n (§6.8.2.2)\\[4pt]&=\;1 − 0.00/12.00\\[4pt]&=\;\boxed{U = 1.000, A_e = 9.000 \,\text{in}^{2}}\end{aligned}

Gross-section yielding

\begin{aligned}&\;P_r = \phi _y F_y A_g, \phi _y = 0.95\\[4pt]&=\;0.95(50)(12.000)\\[4pt]&=\;\boxed{570.0 \,\text{kip}}\end{aligned}

Net-section rupture

\begin{aligned}&\;P_r = \phi _u F_u A_e, \phi _u = 0.80\\[4pt]&=\;0.80(65)(9.000)\\[4pt]&=\;\boxed{468.0 \,\text{kip}}\end{aligned}

Block shear — path 1

\begin{aligned}&\;R = 0.58F_u A_{nv} + U_{bs} F_u A_{nt} (Eq. 6.13.4-1)\\[4pt]&=\;0.58(65)(6.375) + 1.00(65)(2.625)\\[4pt]&=\;\boxed{411.0 \,\text{kip}}\end{aligned}

Block shear — path 2

\begin{aligned}&\;R = 0.58F_y A_{gv} + U_{bs} F_u A_{nt} (Eq. 6.13.4-2)\\[4pt]&=\;0.58(50)(9.000) + 1.00(65)(2.625)\\[4pt]&=\;\boxed{431.6 \,\text{kip}}\end{aligned}

Factored block shear

\begin{aligned}&\;R_r = \phi _{bs} \operatorname{min}\;\text{(path 1, path 2)}, \phi _{bs} = 0.80\\[4pt]&=\;\boxed{328.8 \,\text{kip}}\end{aligned}

Governing limit state

\begin{aligned}&\;\operatorname{min}\;\text{(yielding, rupture, block shear)}\\[4pt]&=\;\boxed{\,\text{block} \,\text{shear}, \phi R_n = 328.8 \,\text{kip}}\end{aligned}

Detailing — plan, elevation and section

Elevation — bolt group

Section — splice plates

Plan — weld detail

Constructability & detailing notes

Punch-and-ream or drill full size — punched holes in thick plate leave a work-hardened edge that reduces the rupture capacity.
Splice plates on both faces halve the shear-lag penalty and keep the load path concentric.
Mark the plates for orientation; asymmetric hole patterns get installed backwards more often than anyone admits.

Check	Demand	Capacity / limit	Status
φR_n ≥ P_u	420.0 kip	328.8 kip	REVIEW
A_n ≤ 0.85 A_g for splice plates (§6.13.5.2)AASHTO caps the effective net area of splice plates at 0.85 A_g.	0.750	0.850	PASS
Rupture is not the governing failure mode (ductility preference)Yielding should precede rupture — widen the plate or reduce hole loss.	—	—	REVIEW

Independent verification	Expected	Computed	Status
Governing limit-state capacity exceeds demand	328.8	P_u applied	CHECK

Assumptions & basis of design

AASHTO LRFD §6.8.2, §6.13.4 and §6.13.5 with hole diameter taken as the bolt diameter + 1/8 in for standard holes.
U_bs = 1.0 for uniform tensile stress and 0.5 where the stress is non-uniform (coped beams).
Splice plates are proportioned so the effective net area does not exceed 0.85 A_g.

Increase the plate thickness or width — the block shear limit state controls, so target that path first (block shear responds best to longer end distances).

Chapter 10

Bearings, joints and movement

Thermal, shrinkage and creep movement, the joint gap through the temperature range, and the elastomeric pad that accommodates it.

Module 53AASHTO LRFD 14.7.6Chapter 10

Elastomeric bearing design

Shape factor, service compressive stress limit, and the shear-deformation requirement for a steel-reinforced pad.

Pad length L (along span)inPad width WinInterior layer thickness h_riinNumber of interior layers n—Shear modulus GksiService load P_skipShear displacement Δ_sin

S = LW / [2 h_ri (L + W)] · σ_s ≤ 1.25 G S ≤ 1.75 ksi · h_rt ≥ 2 Δ_s

Shape factor S

6.00

Total elastomer h_rt

2.00 in

Service stress σ_s

1.111 ksi

Compressive stress check

1.111 ≤ 0.975 ksi

Enlarge the pad or increase the shape factor.

Shear-deformation check

h_rt = 2.00 ≥ 2Δ_s = 1.50 in

Satisfies §14.7.6.3.4.

Detailing sketchElastomeric bearing — section and plan

18 × 9 in pad, 5 elastomer layers with steel shims. Service reaction 180 kip gives σ_s = 1.111 ksi against the 0.98 ksi limit (AASHTO 14.7.6.3.2) — increase the plan area.

Module 54AASHTO LRFD 3.12.2 / 14.7.5.3.2 / 14.5.3Chapter 10

Thermal movement, joint sizing & bearing shear

Rise and fall movement with γ_TU = 1.20, shrinkage and creep shortening, the joint gap at T_max and T_min against the joint's movement rating, the skewed-joint component, and the elastomer thickness needed for h_rt ≥ 2Δ_s.

Expansion length Lftα/°FT_max°FT_min°FInstallation temperature°FInstalled joint gapinJoint movement ratinginTotal elastomer thickness h_rtinSkewdegShrinkage strainin/inCreep strainin/in

α = 6.0×10⁻⁶ /°F for concrete, 6.5×10⁻⁶ /°F for steel. Movement length is measured from the point of fixity.

Total movement range

3.110 in

Factored design movement

3.732 in

Gap at T_max (closing)

1.057 in

Gap at T_min (opening)

4.789 in

Bearing shear strain Δ_s/h_rt

0.697

Required elastomer thickness

5.58 in

Detailing sketchExpansion joint / bearing — movement

Gap ranges from 1.057 in (hot) to 4.789 in (cold); factored design movement = 3.732 in.

Step	Equation	Substitution	Result
Temperature rise / fall	ΔT_rise = T_max − T_install ; ΔT_fall = T_install − T_min	110 − 68 ; 68 − -10	42 / 78 °F
Thermal expansion	Δ = α ΔT L	6.50e-6·42·2880	0.786 in
Thermal contraction	Δ = α ΔT L	6.50e-6·78·2880	1.460 in
Shrinkage + creep shortening	Δ = (ε_sh + ε_cr) L	(2.0e-4 + 1.0e-4)·2880	0.864 in
Total movement range	Δ_total = Δ_rise + Δ_fall + Δ_sh+cr	0.786 + 1.460 + 0.864	3.110 in
Factored design movement	γ_TU Δ	1.20 × 3.110	3.732 in
Joint gap at maximum temperature	G_min = G_install − γ_TU Δ_rise	2.000 − 1.20·0.786	1.057 in
Joint gap at minimum temperature	G_max = G_install + γ_TU(Δ_fall + Δ_sh)	2.000 + 1.20(1.460 + 0.864)	4.789 in
Bearing shear strain	γ = Δ_s/h_rt ≤ 0.5	2.789/4.00	0.697
Movement normal to a skewed joint	Δ_n = Δ cos θ	3.732·cos 0°	3.732 in

OKJoint movement rating ≥ design movementdemand 3.732 incapacity 4.00 in
OKMinimum open gap 0.25 in at maximum temperaturedemand 1.057 incapacity 0.25 in
NGBearing shear h_rt ≥ 2Δ_sdemand 0.697capacity 0.500

Adjust the detail — required elastomer thickness h_rt ≥ 5.58 in and joint movement capacity ≥ 3.73 in. If the gap closes below 0.25 in, increase the installation gap or reset the joint at a measured temperature.

Assumptions & code basis

Uniform temperature range from AASHTO LRFD Table 3.12.2.1-1 (Procedure A) or the contour maps of Procedure B; entered here directly as T_max / T_min.
Coefficient of thermal expansion: 6.0×10⁻⁶ /°F for normal-weight concrete, 6.5×10⁻⁶ /°F for steel (§5.4.2.2, §6.4.1).
Load factor γ_TU = 1.20 applied to the movement when sizing joints and bearings (Table 3.4.1-1, deformation calculations).
Shrinkage plus creep shortening is applied only for concrete superstructures and only in the shortening direction.
Elastomeric bearing shear deformation limited to h_rt ≥ 2Δ_s, i.e. an engineering shear strain of 0.5 (§14.7.5.3.2 / §14.7.6.3.4).
For skewed joints the movement normal to the joint is the longitudinal movement times cos θ; the transverse component must also be accommodated.

Module 55AASHTO LRFD 14.4.2 / 14.7.6.3.5 / 14.7.6.4Chapter 10

Bearing rotation, stability & anchorage

Total design rotation with the 0.005 rad construction allowance, the §14.7.6.3.5 combined compression-plus-rotation check, the aspect-ratio stability limit, uplift screening and the anchor-bolt demand.

Length (along span) LinWidth WinLayer thickness h_riinInterior layers—Shear modulus GksiService load P_skipLive-load portionkipService rotation θ_sradShear movement Δ_sinHorizontal load H_ukipAnchor bolts—Anchor diameterinAnchor F_yksiDesign method

Full steel-reinforced elastomeric bearing check — stress, shear deformation, rotation, stability and anchorage.

Shape factor S

8.84

σ_s

0.536 ksi

Stress limit

1.250 ksi

h_rt

2.000 in

Shear strain γ_s

0.375

Anchor capacity

102.5 kip

Derivation — equation, substitution, result

Plan area and shape factor

\begin{aligned}&\;S = \mathrm{LW}/[2h_{ri}(L + W)] (Eq. 14.7.5.1-1)\\[4pt]&=\;(14.0 \times 24.0)/[2(0.500)(38.0)]\\[4pt]&=\;\boxed{A = 336.0 \,\text{in}^{2}, S = 8.84}\end{aligned}

Service compressive stress

\begin{aligned}&\;\sigma _s = P_s/A\\[4pt]&=\;180.0/336.0\\[4pt]&=\;\boxed{0.536 \,\text{ksi} \;\text{(live-load only 0.179 ksi)}}\end{aligned}

Stress limit

\begin{aligned}&\;\sigma _s \le \operatorname{min}(1.25 \,\text{ksi}, 1.25 G S) — Method A (§14.7.6.3.2)\\[4pt]&=\;G S = 1.149 \,\text{ksi}\\[4pt]&=\;\boxed{\,\text{limit} = 1.250 \,\text{ksi}}\end{aligned}

Shear deformation

\begin{aligned}&\;h_{rt} \ge 2\Delta _s (§14.7.6.3.4)\\[4pt]&=\;2.000 \,\text{vs} 2(0.750)\\[4pt]&=\;\boxed{\gamma _s = 0.375 \mathrm{OK}}\end{aligned}

Rotation criterion

\begin{aligned}&\;\sigma _s \ge 0.5 G S (L/h_{ri})^{2} \theta _s/n (§14.7.6.3.5d)\\[4pt]&=\;0.5(0.130)(8.84)(28.0)^{2}(0.0120/4)\\[4pt]&=\;\boxed{\,\text{required} \sigma _s = 1.352 \,\text{ksi}}\end{aligned}

Stability

\begin{aligned}&\;h_{rt} \le L/3 \operatorname{and} \le W/3 (§14.7.6.3.6)\\[4pt]&=\;h_{rt}/L = 0.143\\[4pt]&=\;\boxed{\,\text{stable}}\end{aligned}

Instantaneous deflection

\begin{aligned}&\;\delta \approx \sigma _s h_{rt}/(4.8 G S^{2})\\[4pt]&=\;\boxed{0.0220 \,\text{in}}\end{aligned}

Anchorage

\begin{aligned}&\;\phi R_n = n \phi (0.48 A_b F_{ub} N_s) \ge H_u\\[4pt]&=\;4 \times 0.75 \times 34.16\\[4pt]&=\;\boxed{102.5 \,\text{kip} \,\text{vs} H_u = 12.0 \,\text{kip}}\end{aligned}

Detailing — plan, elevation and section

Plan

Section

Elevation

Constructability & detailing notes

Set the bearing at the mean design temperature and record the installation temperature on the as-builts.
Provide a level, grouted seat within 1/16 in — a sloped seat walks the pad out over time.
Detail jacking points and 6 in of clearance so the bearing can be replaced without removing the girder.
Bevel the sole plate to remove the built-in grade rotation instead of asking the elastomer to absorb it.

Check	Demand	Capacity / limit	Status
Compressive stress limit	0.536 ksi	1.250 ksi	PASS
h_rt ≥ 2Δ_s	1.500 in	2.000 in	PASS
Rotation / uplift criterion	1.352 ksi	0.536 ksi	REVIEW
Stability h_rt ≤ L/3	2.000 in	4.667 in	PASS
Anchor bolts φR_n ≥ H_u	12.0 kip	102.5 kip	PASS
Shape factor 3 ≤ S ≤ 12 practical range	8.84	3 – 12	PASS

Independent verification	Expected	Computed	Status
Compressive stress within the method limit	1.250	0.536	VERIFIED
Total elastomer thickness ≥ 2Δ_s	h_rt ≥ 2Δ_s	2.000	VERIFIED

Assumptions & basis of design

Steel-reinforced elastomeric bearing, 50-durometer neoprene, G taken at the low-temperature design value.
Method A (§14.7.6) is the simplified route; Method B (§14.7.5) permits higher stresses with additional checks.
Anchor bolts assumed ASTM F1554 Grade 36 in single shear with threads in the shear plane.

The pad rotates more than the compressive stress can hold down — increase the pad plan area, reduce the number of layers, or use a tapered sole plate to remove the built-in rotation.

Chapters 9 · 12 – 13

Substructure — piers, abutments, walls and foundations

Earth pressure and stability, pier-cap and column interaction, single piles and pile groups under axial load and biaxial moment.

Module 42AASHTO LRFD 5.6.4 / 5.7.3Chapters 9 · 12 · 13

Pier-cap flexure & column axial–moment interaction

Cap flexural and shear resistance alongside a column P-M interaction screening ratio with slenderness magnification.

Cap width binCap depth hinCap effective depth dinCap steel Asin²f′cksifyksiCap Muk-ftCap VukipColumn diameterinColumn steel Astin²Column PukipColumn Muk-ftConfinement

Mn = As fy(d − a/2) · Vc = 0.0316β√f′c bv dv · Po = 0.85f′c(Ag−Ast)+fyAst

Cap φMn

1477 k-ft

Cap φVc

158 kip

Column φPn,max

3145 kip

Column φMn (approx.)

659 k-ft

Cap flexure check

1477 ≥ 2200 k-ft

Cap shear check

158 ≥ 400 kip

Column P-M interaction

1.939

Pu/φPn + Mu/φMn ≤ 1.0 (simplified — verify with a full interaction diagram).

Pier cap longitudinal reinforcement — AASHTO LRFD §5.6.3 / §5.10

A_s required (strength)	12.22 in²
A_s minimum (§5.6.3.3 / §5.10.6)	3.12 in²
A_s design (governing)	12.22 in²
Bar selection	8 – #11
Bar size / area	#11 — A_b = 1.56 in²
Bars / layers	8 bars in 1 layer (clear spacing 4.2 in)
A_s provided	12.48 in²
A_s,prov / A_s,req	1.02
Stress-block depth a	6.12 in
Net tensile strain ε_t	0.0149
φ	0.900
φM_n (as detailed)	2243.1 k-ft
M_cr (§5.6.3.3)	592.8 k-ft
Minimum-resistance demand min(M_cr, 1.33M_u)	592.8 k-ft
Reinforcement ratio ρ	0.0081

PASS — Adequate — detail as scheduled

A_s,prov / A_s,req = 1.02; ε_t = 0.0149 (tension-controlled, φ = 0.90).

Transverse reinforcement — AASHTO LRFD §5.7.2.5 / §5.7.3.3

Concrete contribution V_c	176.1 kip
Steel required V_s = V_u/φ − V_c	268.3 kip
Stirrup schedule	#5 2-leg @ 5.0 in c/c (A_v = 0.62 in²)
Maximum spacing s_max	24.0 in
φV_n = φ(V_c + V_s)	417.6 kip
Web-crushing limit 0.25 f′_c b_v d_v	1393.2 kip

PASS — φV_n = 417.6 kip ≥ V_u = 400.0 kip.

Column longitudinal steel

18 – #14

A_st,req 40.45 → A_st,prov 40.50 in²

Column ties / spiral

#4 ties @ 12 in c/c (§5.10.4.3)

ρ = 2.92 % (limits 1 %–8 %) — 18 – #14 bars, clear spacing 4.76 in.

Detailing sketchPier cap — elevation, section and plan

8 #11 continuous top bars (A_s = 12.48 in²) with skin and bottom steel; #5 double-leg stirrups @ 6 in through the cap. Cap 36 in wide × 48 in deep, 4.00 ft cantilever each end, 2 columns at 20.00 ft centres.

Detailing sketchPier column — section and elevation

32 #10 longitudinal bars (A_st = 40.64 in², ρ = 2.93 %) against the required A_st = 40.50 in² at ρ = 2.92 %. #4 spiral at 3 in pitch, 2 in clear cover. Design actions P_u = 1800 kip, M_u = 900 k-ft (AASHTO 5.6.4, 5.10.4.2).

Module 43AASHTO LRFD 11.6 / 11.5.7Chapters 9 · 12 · 13

Abutment / retaining-wall stability screening

Rankine active thrust with overturning, sliding, and eccentricity screening ratios.

Wall height HftFooting width BftBackfill friction angle φ′degBackfill unit weight γkcfStem + footing weightkip/ftSuperstructure reactionkip/ft

K_a = tan²(45° − φ′/2) · P_a = ½ K_a γ H² · e ≤ B/4 for soil foundations

K_a (Rankine)

0.283

Active thrust P_a

6.79 kip/ft

Overturning moment

45.2 k-ft/ft

Overturning ratio M_r/M_o

59.69

Target ≥ 2.0

Sliding ratio

44.73

Target ≥ 1.5

Eccentricity e

0.10 ft

B/4 = 3.00 ft

Detailing sketchAbutment — retained fill, resultant and bearing pressure

Stem 20.0 ft tall on a 12.0 ft footing. Active thrust P_a = 6.8 kip acting at H/3, resisting weight W = 450.0 kip, resultant eccentricity e = 0.10 ft. q_max = 39.38 ksf, q_min = 35.62 ksf (AASHTO 11.6.3).

Screening tool only: the LRFD check uses factored loads with the Strength I and Extreme Event combinations and resistance factors from §11.5.7, not global factors of safety.

Module 44AASHTO LRFD 10.7.3 / 10.5.5.2.3Chapter 13

Pile axial resistance

Skin friction and tip resistance for driven piles, with static and dynamically-verified factored resistances.

Pile diameter/widthinEmbedded length LftSoil typeSu / reference stressksfEffective unit weight γ′kcfα or β factorφstatφdynFactored load Pukip

Rs = fs·perimeter·L · Rp = qp·Ap · Rn = Rs + Rp · φRn = φ·Rn

Side resistance Rs

166 kip

Tip resistance Rp

15 kip

Nominal resistance Rn

181 kip

φRn (static)

81 kip

φRn (dynamic)

118 kip

Piles required (static design)

23

Dynamic-verified: 16 piles

Detailing sketchPile group — cap plan, pile elevation and cap section

23 piles at 3.0D centres, 16 in diameter, 60 ft embedded with the top 0 ft inside the scour prism neglected. Demand 78 kip/pile against φQ_n = 81 kip (AASHTO 10.7.3.8).

Module 45AASHTO LRFD 10.7.3.9 / 10.7.1.2 / 10.5.5.2.3Chapters 9 · 12 · 13

Pile group — efficiency, block failure & cap distribution

Group efficiency η in clay, the equivalent-pier block failure check, factored group resistance, and the rigid-cap elastic distribution P_u/N ± M_x y/Σy² ± M_y x/Σx² with the corner-pile and uplift checks.

Rows—Columns—Pile spacing sftPile size bftSingle-pile nominal Q_pkipEmbedded lengthftUndrained shear c_uksfFactored axial P_ukipM_x (about transverse axis)k-ftM_y (about longitudinal axis)k-ftSoil type

Corner piles govern: the elastic cap formula adds the moment terms about both axes.

Group efficiency η

0.775

Group size B × L

12.50 × 8.75 ft

Factored group resistance

1813 kip

Max pile load

240.7 kip

Min pile load

159.3 kip

Detailing sketchPile group — plan

Group size 12.50 × 8.75 ft; max pile load 240.7 kip vs φQ_p.

Step	Equation	Substitution	Result
Pile count	N = rows × columns	3 × 4	12 piles
Group efficiency	η = 0.7 at s = 2.5b, 1.0 at s = 6b (linear)	s/b = 3.00	0.775
Sum of individual capacities	Q_g = η N Q_p	0.775·12·300.0	2790 kip
Block failure (clay)	Q_block = 2(B + L)Z c_u + 9 c_u B L	2(12.50 + 8.75)·45.0·1.20 + 9·1.20·12.50·8.75	3476 kip
Factored group resistance	φQ = φ·min(Q_g, Q_block)	0.65·2790	1813 kip
Maximum pile load	P = P_u/N + M_x y_max/Σy² + M_y x_max/Σx²	2400/12 + 900·3.75/112.5 + 400·5.63/210.9	240.7 kip
Minimum pile load	P = P_u/N − M_x y_max/Σy² − M_y x_max/Σx²	2400/12 − …	159.3 kip

NGGroup resistance φQ ≥ P_udemand 2400 kipcapacity 1813 kipratio 1.32
NGIndividual pile P_max ≤ φQ_pdemand 240.7 kipcapacity 195.0 kip
OKNo net upliftdemand 159.3 kip
OKSpacing s ≥ 2.5b (§10.7.1.2)demand 3.75 ftcapacity 3.13 ft

Add piles or spread the group: about 16 piles are needed, or lengthen the piles to raise Q_p to 370.3 kip each.

Assumptions & code basis

Group efficiency η applies to pile groups in cohesive soils; in cohesionless soils driven piles at s ≥ 2.5b are taken as η = 1.0 (AASHTO LRFD §10.7.3.9).
Block failure of a group in clay checked as a single equivalent pier: perimeter adhesion plus 9c_u end bearing (§10.7.3.9).
Individual pile loads from the rigid-cap elastic formula P = P_u/N ± M_x y/Σy² ± M_y x/Σx²; the cap is assumed infinitely rigid and the piles axially elastic.
Resistance factors: φ = 0.65 (clay, static analysis with driving criteria) or 0.50 (sand, static analysis) — increase with dynamic testing per Table 10.5.5.2.3-1.
Tension (uplift) piles must additionally be checked with φ_up ≤ 0.35–0.45.

Module 46AASHTO LRFD 5.6.4.3 · ACI §6.6.4Chapters 9 · 12 · 13

Pier column slenderness & moment magnification

Effective length KL/r screening against the 22 / 34 − 12(M1/M2) thresholds, EI_eff with β_d, the critical load P_c, and the δ_ns magnified design moment with the minimum-eccentricity floor.

P_ukipSmaller end moment M₁kip·ftLarger end moment M₂kip·ftColumn dimension / diameterinSectionUnbraced lengthftEffective length factor K—f′_cksif_yksiGross area A_gin²Longitudinal steel A_stin²Frame typeβ_d—

Screens the column for slenderness, then magnifies the moment with the §5.6.4.3 approximate procedure.

Kℓ_u/r

33.6

P_e

14396 kip

δ_b

1.200

M_c design

1080.1 kip·ft

ρ_g

0.0155

φP_n,max

4643 kip

Derivation — equation, substitution, result

Radius of gyration

\begin{aligned}&\;r = 0.25 D\\[4pt]&=\;h = 48.0 \,\text{in}\\[4pt]&=\;\boxed{r = 12.00 \,\text{in}}\end{aligned}

Slenderness ratio

\begin{aligned}&\;K ℓ_u / r\\[4pt]&=\;1.20 \times 336 / 12.00\\[4pt]&=\;\boxed{Kℓ_u/r = 33.6}\end{aligned}

Slenderness limit

\begin{aligned}&\;22 \;\text{(unbraced)}\\[4pt]&=\;M_{1}/M_{2} = 0.333\\[4pt]&=\;\boxed{\,\text{limit} = 22.0 \rightarrow \,\text{slender}, \,\text{magnify}}\end{aligned}

Flexural stiffness

\begin{aligned}&\;\mathrm{EI} = 0.40 E_c I_g/(1 + \beta _d) (Eq. 5.6.4.3-2)\\[4pt]&=\;E_c = 3640 \,\text{ksi}, I_g = 260576 \,\text{in}^{4}, \beta _d = 0.60\\[4pt]&=\;\boxed{\mathrm{EI} = 2.371e+8 \,\text{kip}\cdot \,\text{in}^{2}}\end{aligned}

Euler load

\begin{aligned}&\;P_e = \pi ^{2}\mathrm{EI}/(Kℓ_u)^{2}\\[4pt]&=\;\pi ^{2}(2.37e+8)/(403)^{2}\\[4pt]&=\;\boxed{P_e = 14396 \,\text{kip}}\end{aligned}

Moment gradient factor

\begin{aligned}&\;C_m = 0.6 + 0.4(M_{1}/M_{2}) \ge 0.4 \;\text{(braced)}, 1.0 \;\text{(unbraced)}\\[4pt]&=\;\boxed{C_m = 1.000}\end{aligned}

Magnifier

\begin{aligned}&\;\delta _b = C_m/(1 − P_u/(\phi _K P_e)) \ge 1.0 (Eq. 5.6.4.3-3)\\[4pt]&=\;1.000/(1 − 1800/(0.75 \times 14396))\\[4pt]&=\;\boxed{\delta _b = 1.200}\end{aligned}

Minimum eccentricity

\begin{aligned}&\;e_{\operatorname{min}} = 0.6 + 0.03h, M_{2},\operatorname{min} = P_u e_{\operatorname{min}}\\[4pt]&=\;e_{\operatorname{min}} = 2.04 \,\text{in}\\[4pt]&=\;\boxed{M_{2},\operatorname{min} = 306.0 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Magnified design moment

\begin{aligned}&\;M_c = \delta _b M_{2}\\[4pt]&=\;1.200 \times 900.0\\[4pt]&=\;\boxed{M_c = 1080.1 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Pure axial capacity

\begin{aligned}&\;P_o = 0.85f'_c(A_g − A_{st}) + f_y A_{st}; P_n,\operatorname{max} = 0.80P_o \;\text{(tied)}\\[4pt]&=\;\rho _g = 0.0155\\[4pt]&=\;\boxed{\phi P_n,\operatorname{max} = 4643 \,\text{kip}}\end{aligned}

Detailing — plan, elevation and section

Section

Elevation

Plan at cap

Constructability & detailing notes

Column bars are spliced above the footing with a mechanical coupler or a lap outside the plastic-hinge zone.
Provide a 2 in chamfer or a formliner detail — plain circular columns show every form seam.
Check the free-standing (erection) condition before the cap is cast: K is effectively 2.1 for a cantilever pier.

Check	Demand	Capacity / limit	Status
P_u < 0.75 P_e (stability)	1800 kip	10797 kip	PASS
δ_b ≤ 1.40 (practical limit — otherwise stiffen)	1.200	1.40	PASS
Steel ratio 1 % ≤ ρ_g ≤ 8 % (§5.6.4.2)	0.0155	0.010 – 0.080	PASS
Axial capacity φP_n,max ≥ P_u	1800 kip	4643 kip	PASS
Kℓ_u/r ≤ 100 (refined analysis required above)	33.6	100	PASS

Independent verification	Expected	Computed	Status
Magnifier within the practical 1.0 – 1.4 band	1.00 – 1.40	1.200	VERIFIED

Assumptions & basis of design

AASHTO LRFD §5.6.4.3 approximate moment magnification for a single-curvature or double-curvature column.
β_d = ratio of factored permanent axial load to total factored axial load; 0.6 is typical for bridge piers.
K = 1.0 for pinned–pinned, 2.1 for a free-standing cantilever pier, 0.65 – 0.80 for a fixed-base column in a braced frame.

Design the column for P_u = 1800 kip with M_c = 1080.1 kip·ft on the P–M interaction diagram.

Module 47AASHTO LRFD 5.6.4.5 · ACI §22.4Chapters 9 · 12 · 13

Biaxial column bending — Bresler reciprocal

Uniaxial capacities about each axis, the Bresler reciprocal-load check 1/P_n = 1/P_nx + 1/P_ny − 1/P_o, and the load-contour comparison with the bar layout that satisfies both axes.

P_ukipM_uxkip·ftM_uykip·ftφP_nx at e_xkipφP_ny at e_ykipφP_o concentrickipφM_nxkip·ftφM_nykip·ft

Bresler reciprocal-load check at moderate to high axial load, with the moment-contour alternative below 0.10 φP_o.

φP_ni (biaxial)

1942 kip

Reciprocal-load ratio

0.824

Moment-contour ratio

1.057

Governing utilisation

0.824

Derivation — equation, substitution, result

Screening

\begin{aligned}&\;If P_u < 0.10 \phi P_o \operatorname{use} \,\text{the} \,\text{moment}-\,\text{contour} \,\text{method}; \,\text{otherwise} \operatorname{use} Bresler \,\text{reciprocal} \,\text{load}\\[4pt]&=\;P_u = 1600 \,\text{kip}, 0.10\phi P_o = 520 \,\text{kip}\\[4pt]&=\;\boxed{\,\text{reciprocal} \,\text{load} \,\text{governs}}\end{aligned}

Reciprocal load

\begin{aligned}&\;1/\phi P_n = 1/\phi P_{nx} + 1/\phi P_{ny} − 1/\phi P_o (Eq. 5.6.4.5-1)\\[4pt]&=\;1/2600 + 1/3100 − 1/5200\\[4pt]&=\;\boxed{\phi P_{ni} = 1942 \,\text{kip}}\end{aligned}

Axial utilisation

\begin{aligned}&\;P_u/\phi P_{ni} \le 1.0\\[4pt]&=\;1600/1942\\[4pt]&=\;\boxed{0.824}\end{aligned}

Moment contour

\begin{aligned}&\;M_{ux}/\phi M_{nx} + M_{uy}/\phi M_{ny} \le 1.0 (Eq. 5.6.4.5-3)\\[4pt]&=\;700/1100 + 400/950\\[4pt]&=\;\boxed{1.057}\end{aligned}

Governing ratio

\begin{aligned}&\;\operatorname{max} \operatorname{of} \,\text{the} \,\text{applicable} \,\text{interaction}\\[4pt]&=\;\boxed{0.824 \le 1.0 \mathrm{OK}}\end{aligned}

Detailing — plan, elevation and section

Section

Elevation

Plan at cap

Constructability & detailing notes

Distribute the longitudinal bars around the full perimeter — corner-only layouts have poor biaxial capacity.
Cross-ties on alternate bars are required when the clear spacing exceeds 6 in (§5.10.4.3).

Check	Demand	Capacity / limit	Status
Biaxial interaction ≤ 1.0	0.824	1.000	PASS
φP_nx and φP_ny each exceed P_u	1600 kip	2600 kip	PASS
Bresler validity P_u ≥ 0.10 φP_o	1600 kip	520 kip	PASS

Independent verification	Expected	Computed	Status
Governing interaction ≤ 1.0	≤ 1.000	0.824	VERIFIED

Assumptions & basis of design

φP_nx and φP_ny are read from the uniaxial interaction diagrams at the respective eccentricities e_x = M_uy/P_u and e_y = M_ux/P_u.
φP_o = φ[0.85f′_c(A_g − A_st) + f_y A_st] is the concentric capacity without the 0.80 cap.
Bresler's reciprocal-load equation is accurate within about 10 % for symmetric sections at moderate to high axial load.

Biaxial capacity is adequate; confirm the tie or spiral detailing and the bar development into the cap and footing.

Module 48AASHTO LRFD 5.10.4.2 / 5.11.4.1Chapters 9 · 15

Column confinement, ties and plastic hinge detailing

Spiral ratio ρ_s and rectangular hoop A_sh from the two governing expressions, the hoop spacing and plastic-hinge length ℓ_o, tie size and the cross-tie layout for a seismic pier column.

SectionDiameter DinWidth b (rect.)inDepth h (rect.)inClear coverinf′_cksif_yh transverseksiTransverse bar #—Longitudinal bar #—Longitudinal bar count—Seismic detailing

Spiral or tie confinement, pitch, longitudinal bar layout and the seismic floor of 0.12 f′_c/f_yh.

ρ_s required

0.00800

Spiral pitch

3.50 in

ρ_s provided

0.00805

Tie spacing

12.00 in

ρ_g longitudinal

0.0111

Callout

20 — #9 vert. w/ #5 spiral @ 3.50 in

Derivation — equation, substitution, result

Gross and core areas

\begin{aligned}&\;A_g = \pi D^{2}/4, A_c = \pi D_c^{2}/4\\[4pt]&=\;D_c = 44.00 \,\text{in}\\[4pt]&=\;\boxed{A_g = 1810 \,\text{in}^{2}, A_c = 1521 \,\text{in}^{2}}\end{aligned}

Confinement ratio (gravity)

\begin{aligned}&\;\rho _s \ge 0.45(A_g/A_c − 1)f'_c/f_{yh} (Eq. 5.6.4.6-1)\\[4pt]&=\;0.45(1.190 − 1)(4.00/60)\\[4pt]&=\;\boxed{\rho _s = 0.00570}\end{aligned}

Confinement ratio (seismic)

\begin{aligned}&\;\rho _s \ge 0.12 f'_c/f_{yh} (§5.10.11.4.1d)\\[4pt]&=\;0.12 \times 0.0667\\[4pt]&=\;\boxed{\rho _s = 0.00800}\end{aligned}

Governing ratio

\begin{aligned}&\;\operatorname{max} \operatorname{of} \,\text{the} \,\text{two}\\[4pt]&=\;\boxed{\rho _s,\,\text{req} = 0.00800}\end{aligned}

Spiral pitch

\begin{aligned}&\;s = 4A_{sp}/(\rho _s D_c)\\[4pt]&=\;4(0.310)/(0.00800 \times 44.00)\\[4pt]&=\;\boxed{s = 3.52 \,\text{in} \rightarrow \operatorname{use} 3.50 \,\text{in}}\end{aligned}

Provided ratio

\begin{aligned}&\;\rho _s,\,\text{prov} = 4A_{sp}/(s D_c)\\[4pt]&=\;\boxed{0.00805}\end{aligned}

Tie spacing (tied column)

\begin{aligned}&\;s \le \operatorname{min}\;\text{(12 in, least dimension, 16 db,long, 48 db,tie)} (§5.10.4)\\[4pt]&=\;\boxed{12.00 \,\text{in}}\end{aligned}

Longitudinal bar layout

\begin{aligned}&\;\,\text{clear} \,\text{spacing} \ge \operatorname{max}(1.5 d_b, 1.5 \,\text{in})\\[4pt]&=\;20 — \#9 \,\text{bars}\\[4pt]&=\;\boxed{\,\text{clear} \,\text{spacing} \approx 5.61 \,\text{in}, \rho _g = 0.0111}\end{aligned}

Detailing — plan, elevation and section

Section

Elevation

Plan at cap

Constructability & detailing notes

Spirals are shipped compressed and stretched on site — specify spacer bars so the pitch survives concrete placement.
Extend the confinement one full column dimension into the cap and the footing.
Keep the clear pitch at 1 in minimum so a vibrator head and the aggregate can pass.

Check	Demand	Capacity / limit	Status
ρ_s,prov ≥ ρ_s,req	0.00800	0.00805	PASS
Clear pitch 1 – 3 in (§5.10.4.3)	2.88 in clear	1.0 – 3.0 in	PASS
Longitudinal steel 1 – 8 %	0.0111	0.010 – 0.080	PASS
Bar clear spacing adequate for concrete placement	5.61 in	1.69 in	PASS
Spiral ≥ #3 (or #4 in seismic zones)	#5	#4	PASS

Independent verification	Expected	Computed	Status
Provided confinement ratio ≥ required	0.00800	0.00805	VERIFIED

Assumptions & basis of design

AASHTO LRFD §5.6.4.6 and §5.10.4 for gravity columns; §5.10.11 adds the seismic confinement floor.
Spiral pitch is rounded down to the nearest 1/4 in and capped at 6 in.
The core dimension D_c is measured out-to-out of the spiral.

Confinement satisfies the code; extend the spiral into the cap and footing for the full development length of the longitudinal bars.

Module 49AASHTO LRFD 11.6.3 / 11.6.5 / 3.11.5Chapters 9 · 12 · 13

Retaining wall — full stability and stem design

Coulomb / Rankine thrust with surcharge, the overturning, sliding and bearing-eccentricity checks, the factored bearing pressure diagram, and stem, heel and toe flexural steel with bar schedules.

Wall height HftFooting width BftToe projectionftStem thickness at baseinFooting thicknessftBackfill unit weightpcfFriction angle φdegConcrete unit weightpcfSurcharge h_eqftVertical LL surchargeksfBase friction μ—Factored bearing q_Rksff′_cksif_yksiCoverinEarth-pressure state

External stability (overturning, sliding, eccentricity, bearing) and internal stem design in one pass.

K used

0.307

active (free to translate)

ΣH

10.55 kip/ft

ΣV

40.62 kip/ft

FS overturning

4.83

FS sliding

2.12

Eccentricity e

-0.850 ft

q_max

1.845 ksf

Stem steel

#3 @ 1 in

A_s,req = 0.994 in²/ft

Derivation — equation, substitution, result

Earth-pressure coefficients

\begin{aligned}&\;K_a = \operatorname{tan}^{2}(45 − \phi /2), K_0 = 1 − \operatorname{sin} \phi , K_p = \operatorname{tan}^{2}(45 + \phi /2)\\[4pt]&=\;\phi = 32.0^{\circ}\\[4pt]&=\;\boxed{K_a = 0.307, K_0 = 0.470, K_p = 3.255 — \,\text{using} K = 0.307}\end{aligned}

Active thrust

\begin{aligned}&\;P_a = ½\gamma H^{2}K, \,\text{acting} \operatorname{at} H/3 (Eq. 3.11.5.3-1)\\[4pt]&=\;0.5(120)(22.00)^{2}(0.307)/1000\\[4pt]&=\;\boxed{8.92 \,\text{kip}/\,\text{ft}}\end{aligned}

Live-load surcharge

\begin{aligned}&\;P_{LS} = \gamma h_{eq} K H, \operatorname{at} H/2 (§3.11.6.4)\\[4pt]&=\;h_{eq} = 2.00 \,\text{ft}\\[4pt]&=\;\boxed{1.62 \,\text{kip}/\,\text{ft}}\end{aligned}

Vertical resultant

\begin{aligned}&\;V = W_{stem} + W_{footing} + W_{soil} + q_{LL}\cdot \,\text{heel}\\[4pt]&=\;70.20 + 5.25 + -31.59 + -3.24\\[4pt]&=\;\boxed{V = 40.62 \,\text{kip}/\,\text{ft}}\end{aligned}

Overturning

\begin{aligned}&\;\mathrm{FS}_{OT} = \Sum M_R/\Sum M_{OT} \ge 2.0\\[4pt]&=\;402.1/83.3\\[4pt]&=\;\boxed{\mathrm{FS} = 4.83}\end{aligned}

Sliding

\begin{aligned}&\;\mathrm{FS}_s = \mu \Sum V/\Sum H \ge 1.5\\[4pt]&=\;0.55(40.62)/10.55\\[4pt]&=\;\boxed{\mathrm{FS} = 2.12}\end{aligned}

Resultant location

\begin{aligned}&\;x̄ = (\Sum M_R − \Sum M_{OT})/\Sum V, e = B/2 − x̄ \le B/6 \;\text{(soil)} \operatorname{or} B/4 \;\text{(rock)}\\[4pt]&=\;x̄ = 7.850 \,\text{ft}\\[4pt]&=\;\boxed{e = -0.850 \,\text{ft}, B/6 = 2.333 \,\text{ft}}\end{aligned}

Bearing pressure

\begin{aligned}&\;q = (V/B)(1 \pm 6e/B)\\[4pt]&=\;V/B = 2.901 \,\text{ksf}\\[4pt]&=\;\boxed{q_{\operatorname{max}} = 1.845, q_{\operatorname{min}} = 3.958 \,\text{ksf}}\end{aligned}

Meyerhof effective width

\begin{aligned}&\;B' = B − 2e, q_{uniform} = V/B'\\[4pt]&=\;14.00 − 2(0.850)\\[4pt]&=\;\boxed{B' = 12.301 \,\text{ft}, q = 3.302 \,\text{ksf}}\end{aligned}

Stem moment and shear at the base

\begin{aligned}&\;M(y) = \gamma Ky^{3}/6 + \gamma h_{eq} K y^{2}/2 \cdot V(y) = \gamma Ky^{2}/2 + \gamma h_{eq} K y\\[4pt]&=\;y = 19.50 \,\text{ft}\\[4pt]&=\;\boxed{M = 59.59 \,\text{kip}\cdot \,\text{ft}/\,\text{ft}, V = 8.45 \,\text{kip}/\,\text{ft}}\end{aligned}

Factored stem demand

\begin{aligned}&\;M_u = 1.50 M_{EH} + 1.75 M_{LS} \;\text{(Table 3.4.1-2)}\\[4pt]&=\;1.50(45.57) + 1.75(14.02)\\[4pt]&=\;\boxed{M_u = 92.88 \,\text{kip}\cdot \,\text{ft}/\,\text{ft}}\end{aligned}

Stem reinforcement

\begin{aligned}&\;\rho = 0.85f'_c/f_y[1 − \sqrt{1 − 2R_n/0.85f'_c}], A_s = \rho b d\\[4pt]&=\;d = 21.50 \,\text{in}, R_n = 0.2233 \,\text{ksi}\\[4pt]&=\;\boxed{A_s = 0.994 \,\text{in}^{2}/\,\text{ft} \rightarrow \#3 @ 1 \,\text{in}}\end{aligned}

Stem shear

\begin{aligned}&\;\phi V_c = \phi 0.0316\beta \sqrt{f'_c} b d_v, \beta = 2.0\\[4pt]&=\;26.42 \,\text{kip}/\,\text{ft} \,\text{vs} V_u = 13.03 \,\text{kip}/\,\text{ft}\\[4pt]&=\;\boxed{\mathrm{OK} \,\text{without} \,\text{stirrups}}\end{aligned}

Detailing — plan, elevation and section

Section

Elevation

Plan

Constructability & detailing notes

Place free-draining backfill with a perforated underdrain — hydrostatic pressure is the most common cause of wall distress.
Compact backfill in 8 in lifts with light equipment within 3 ft of the stem to avoid overstressing it.
Provide contraction joints at 30 ft maximum and expansion joints at 90 ft with waterstops.
Do not backfill until the stem concrete reaches its specified strength and the footing is fully cured.

Check	Demand	Capacity / limit	Status
Overturning FS ≥ 2.0 (soil)	4.83	2.00	PASS
Sliding FS ≥ 1.5	2.12	1.50	PASS
Eccentricity within the middle third (e ≤ B/6)	0.850 ft	2.333 ft	PASS
Bearing q_max ≤ q_n	1.845 ksf	6.00 ksf	PASS
Stem shear without stirrups	13.03 kip/ft	26.42 kip/ft	PASS
No net uplift at the heel (q_min ≥ 0)	3.958 ksf	0	PASS

Independent verification	Expected	Computed	Status
Overturning factor of safety	≥ 2.00	4.83	VERIFIED
Sliding factor of safety	≥ 1.50	2.12	VERIFIED

Assumptions & basis of design

Rankine active pressure on a vertical virtual back face with a level backfill and no wall friction.
Live-load surcharge as an equivalent height of soil h_eq from Table 3.11.6.4-1 (varies with wall height and distance from the traffic).
Passive resistance in front of the toe is neglected — a conservative and customary assumption where scour or excavation is possible.
Load factors: γ_EH = 1.50 max (active), γ_LS = 1.75, γ_EV = 1.35 max / 1.00 min.

Stability is satisfied. Detail the stem with #3 @ 1 in on the earth face, with the same area developed into the footing.

Module 50AASHTO LRFD 10.6 / 5.12.8Chapters 9 · 12 · 13

Spread footing — bearing, sliding, shear and steel

Factored bearing pressure with eccentricity and the middle-third rule, one-way and two-way shear at the critical sections, flexural steel each way, and the development length available from the column face.

Footing width BftFooting length LftEmbedment D_fftVertical load PkipMoment Mkip·ftHorizontal load HkipSoil unit weightpcfFriction angle φdegCohesion cksfSoil modulus E_sksfPoisson ratio ν—φ_b bearing—φ_τ sliding—Interface friction μ—Total scour depthft

Meyerhof bearing capacity on the effective width, elastic settlement, sliding and the scour-elevation check.

q_n

48.259 ksf

q_R factored

21.716 ksf

q applied

6.897 ksf

Eccentricity e

0.750 ft

Settlement

1.929 in

Sliding capacity

1056.0 kip

Derivation — equation, substitution, result

Bearing-capacity factors

\begin{aligned}&\;N_q = e^{\pi \operatorname{tan} \phi } \operatorname{tan}^{2}(45+\phi /2), N_c = (N_q−1)\operatorname{cot} \phi , N_\gamma = 2(N_q+1)\operatorname{tan} \phi \\[4pt]&=\;\phi = 34.0^{\circ}\\[4pt]&=\;\boxed{N_c = 42.16, N_q = 29.44, N_\gamma = 41.06}\end{aligned}

Load eccentricity

\begin{aligned}&\;e = M/P, B' = B − 2e (§10.6.1.3)\\[4pt]&=\;1800.0/2400.0\\[4pt]&=\;\boxed{e = 0.750 \,\text{ft}, B' = 14.500 \,\text{ft}}\end{aligned}

Shape and depth factors

\begin{aligned}&\;s_c, s_q, s_\gamma \operatorname{from} Table 10.6.3.1.2a-3; d_q \operatorname{per} Eq. 10.6.3.1.2a\\[4pt]&=\;B'/L = 0.604, D_f,\,\text{eff} = 4.00 \,\text{ft}\\[4pt]&=\;\boxed{s_c = 1.422, s_q = 1.408, s_\gamma = 0.758, d_q = 1.064}\end{aligned}

Nominal bearing resistance

\begin{aligned}&\;q_n = c N_c s_c + \gamma D_f N_q s_q d_q + ½\gamma B'N_\gamma s_\gamma (Eq. 10.6.3.1.2a-1)\\[4pt]&=\;0.00(42.16)(1.422) + 0.480(29.44)(1.408)(1.064) + 0.5(0.1200)(14.50)(41.06)(0.758)\\[4pt]&=\;\boxed{q_n = 48.259 \,\text{ksf}}\end{aligned}

Factored resistance

\begin{aligned}&\;q_R = \phi _b q_n \;\text{(Table 10.5.5.2.2-1)}\\[4pt]&=\;0.45 \times 48.259\\[4pt]&=\;\boxed{q_R = 21.716 \,\text{ksf}}\end{aligned}

Applied pressure

\begin{aligned}&\;q = P/(B'L) — \,\text{uniform} Meyerhof \,\text{distribution}\\[4pt]&=\;2400.0/(14.500 \times 24.00)\\[4pt]&=\;\boxed{6.897 \,\text{ksf}}\end{aligned}

Elastic pressure distribution

\begin{aligned}&\;q = (P/\mathrm{BL})(1 \pm 6e/B)\\[4pt]&=\;\boxed{q_{\operatorname{max}} = 8.008, q_{\operatorname{min}} = 4.492 \,\text{ksf}}\end{aligned}

Elastic settlement

\begin{aligned}&\;\delta = q B'(1 − \nu ^{2})I_f/E_s (Eq. 10.6.2.4.2-1)\\[4pt]&=\;6.897(14.50)(1 − 0.30^{2})(1.06)/600\\[4pt]&=\;\boxed{1.929 \,\text{in}}\end{aligned}

Sliding

\begin{aligned}&\;R_R = \phi _\tau \mu P \ge H\\[4pt]&=\;0.80(0.55)(2400.0)\\[4pt]&=\;\boxed{1056.0 \,\text{kip} \,\text{vs} 180.0 \,\text{kip}}\end{aligned}

Detailing — plan, elevation and section

Plan

Section

Elevation

Constructability & detailing notes

Found on undisturbed material; if the excavation is over-dug, backfill with lean concrete, never compacted soil.
A 3 in mud slab keeps the bottom mat clean and lets the crew work in wet conditions.
Dowels must be templated to match the column cage before the pour — field bending of large bars is not permitted.
Check the excavation slope or shoring separately; a footing failure during construction is a trench failure, not a bearing failure.

Check	Demand	Capacity / limit	Status
Bearing q ≤ q_R	6.897 ksf	21.716 ksf	PASS
Eccentricity e ≤ B/4 on soil (§10.6.3.3)	0.750 ft	4.000 ft	PASS
Settlement ≤ 1 in (typical service tolerance)	1.929 in	1.000 in	REVIEW
Sliding φR_n ≥ H	180.0 kip	1056.0 kip	PASS
Footing founded below the total scour depth	4.00 ft of scour	8.00 ft embedment	PASS

Independent verification	Expected	Computed	Status
Bearing pressure within the factored resistance	21.716	6.897	VERIFIED
Settlement within the 1 in service tolerance	≤ 1.000 in	1.929	CHECK

Assumptions & basis of design

Meyerhof/Vesić bearing-capacity formulation per AASHTO §10.6.3.1.2a with an effective footing width B′ = B − 2e.
Load inclination factors are omitted — acceptable when H/V < 0.1; include them for heavily loaded abutments.
Elastic settlement uses a uniform half-space with rigidity factor I_f; for layered profiles use the Hough or Schmertmann method.
Bearing resistance is computed on soil below the total scour elevation.

Bearing, eccentricity, settlement and sliding are all satisfied; design the footing for one-way and two-way shear and for the flexural cantilever.

Module 51AASHTO LRFD 10.7.3.8 / 10.5.5.2.3Chapter 13

Pile capacity — α, β and Nordlund methods

Side resistance in cohesive soil by the α-method and in granular soil by the β-method, tip resistance, the factored geotechnical resistance with resistance factors by verification method, and the required pile length.

MethodPile diameterinEmbedded lengthftUndrained strength s_uksfEffective unit weightpcfLateral earth coefficient K—Interface friction δdegRock q_uksiShaft f′_cksiToe factor N_t—Limiting q_pksfφ_stat—Factored group load P_ukipPiles in group—Pile spacingftRows—Columns—

α-method for clay, β-method for sand, and the rock-socket equation, with the Converse–Labarre group efficiency.

Unit side resistance f_s

0.989 ksf

R_s

372.8 kip

R_p

233.7 kip

R_r per pile

273.0 kip

Group efficiency η

0.727

Group capacity

1786 kip

Derivation — equation, substitution, result

Geometry

\begin{aligned}&\;\,\text{perimeter} = \pi D, A_p = \pi D^{2}/4\\[4pt]&=\;D = 24.00 \,\text{in}\\[4pt]&=\;\boxed{p = 6.283 \,\text{ft}, A_p = 3.1416 \,\text{ft}^{2}}\end{aligned}

Mid-depth effective stress

\begin{aligned}&\;\sigma '_v = \gamma ' z\\[4pt]&=\;62 \,\text{pcf} \times 30.0 \,\text{ft}\\[4pt]&=\;\boxed{1.860 \,\text{ksf}}\end{aligned}

β-method friction

\begin{aligned}&\;f_s = \beta \sigma '_v = K \operatorname{tan} \delta \sigma '_v \le 3.0 \,\text{ksf}\\[4pt]&=\;K = 1.00, \delta = 28.0^{\circ}, \beta = 0.532\\[4pt]&=\;\boxed{f_s = 0.989 \,\text{ksf}}\end{aligned}

Side resistance

\begin{aligned}&\;R_s = f_s \cdot p \cdot L\\[4pt]&=\;0.989 \times 6.283 \times 60.0\\[4pt]&=\;\boxed{372.8 \,\text{kip}}\end{aligned}

Tip resistance

\begin{aligned}&\;q_p = N_t \sigma '_v \le \,\text{limiting} \,\text{value}\\[4pt]&=\;q_p = 74.40 \,\text{ksf}\\[4pt]&=\;\boxed{R_p = 233.7 \,\text{kip}}\end{aligned}

Nominal and factored single-pile resistance

\begin{aligned}&\;R_n = R_s + R_p; R_r = \phi _{stat} R_n \;\text{(Table 10.5.5.2.3-1)}\\[4pt]&=\;\phi = 0.45\\[4pt]&=\;\boxed{R_r = 273.0 \,\text{kip}}\end{aligned}

Group efficiency

\begin{aligned}&\;Converse–Labarre: \eta = 1 − (\theta /90)[(m−1)k + (k−1)m]/(\,\text{mk}); \eta = 1.0 \operatorname{for} s \ge 6D\\[4pt]&=\;s/D = 3.00, \theta = 18.43^{\circ}, 3\times 3\\[4pt]&=\;\boxed{\eta = 0.727}\end{aligned}

Group resistance

\begin{aligned}&\;R_{group} = \eta N R_r\\[4pt]&=\;0.727 \times 9 \times 273.0\\[4pt]&=\;\boxed{1786 \,\text{kip} \,\text{vs} P_u = 4000 \,\text{kip}}\end{aligned}

Detailing — plan, elevation and section

Plan — pile layout

Section — cap

Elevation

Constructability & detailing notes

Specify the driving criterion (blow count and hammer energy) as well as the tip elevation — capacity is verified in the field, not on paper.
Order piles 10 – 15 % longer than the estimated length; splicing in the field is slower than cutting off.
For drilled shafts, keep the slurry head above the piezometric level and place concrete by tremie without interruption.
Preserve a minimum 3D spacing so the group efficiency stays near unity and the driving does not heave adjacent piles.

Check	Demand	Capacity / limit	Status
Single pile R_r ≥ P_u/N	444.4 kip	273.0 kip	REVIEW
Group resistance ≥ P_u	4000 kip	1786 kip	REVIEW
Pile spacing ≥ 2.5 D (§10.7.1.2)	6.00 ft	5.00 ft	PASS
β within 0.25 – 1.20 for sands	0.532	0.25 – 1.20	PASS
Downdrag considered where fill is placed over compressible soil (§3.11.8)Add DD as a load, not a resistance reduction, and use the same φ as skin friction.	—	—	PASS

Independent verification	Expected	Computed	Status
Group resistance exceeds the factored load	1786	demand applied	CHECK
Group efficiency at or above 0.70	≥ 0.700	0.727	VERIFIED

Assumptions & basis of design

AASHTO LRFD §10.7 for driven piles and §10.8 for drilled shafts; static analysis resistance factors from Table 10.5.5.2.3-1.
α from the AASHTO/API correlation with s_u; β = K tan δ with a 3.0 ksf cap on unit side friction in sand.
Group efficiency from Converse–Labarre; efficiency is taken as 1.0 at centre-to-centre spacings of 6D or more.
Load test or dynamic verification would allow a higher φ (0.65 – 0.80) than the static value used here.

Lengthen the piles (R_s grows linearly with L), increase the diameter, or add piles at ≥ 3D spacing so the group efficiency stays near unity.

Module 104AASHTO LRFD 3.11.6.4 / 11.6.1.4 / 5.6.5Chapter 12

Abutment backwall, bridge seat and approach slab

Cantilever backwall under EH plus the live-load surcharge, the Strength I moment and back-face bar schedule, the bridge-seat edge distance and bearing stress, and the approach-slab one-way strip with its bottom steel.

Backwall heightftBackwall thicknessinRetained fill γ_spcfActive coefficient k_a—LS equivalent height h_eqftf′_cksif_yksiClear coverinBridge-seat widthinBearing pad lengthinFactored bearing reaction R_ukipApproach-slab spanftApproach-slab thicknessin

Cantilever backwall under EH plus live-load surcharge, the bridge-seat geometry and bearing stress, and the approach-slab one-way strip — carried through to bar size and spacing on both elements.

Backwall M_u

8.07 kip·ft/ft

Service moment

5.01 kip·ft/ft

A_s required

0.324 in²/ft

Backwall bar

#4 @ 7.0 in

back face, vertical

A_s provided

0.343 in²/ft

φM_n backwall

23.05 kip·ft/ft

Seat edge distance

6.00 in

Seat bearing stress

1.481 ksi

Approach-slab bar

#9 @ 4.0 in

bottom, longitudinal

Derivation — equation, substitution, result

Active earth pressure at the backwall base

\begin{aligned}&\;p_{EH} = k_a \gamma _s h\\[4pt]&=\;0.330 \times 0.120 \times 7.50\\[4pt]&=\;\boxed{0.297 \,\text{ksf}}\end{aligned}

Earth-pressure resultant

\begin{aligned}&\;\mathrm{EH} = ½ p_{EH} h,\; \bar y = h/3\\[4pt]&=\;0.5 \times 0.297 \times 7.50\\[4pt]&=\;\boxed{1.114 \,\text{kip}/\,\text{ft} \operatorname{at} 2.50 \,\text{ft}}\end{aligned}

Live-load surcharge (Table 3.11.6.4-1)

\begin{aligned}&\;\mathrm{LS} = k_a \gamma _s h_{eq} h,\; \bar y = h/2\\[4pt]&=\;h_{eq} = 2.00 \,\text{ft}\\[4pt]&=\;\boxed{0.594 \,\text{kip}/\,\text{ft} \operatorname{at} 3.75 \,\text{ft}}\end{aligned}

Strength I moment at the backwall base

\begin{aligned}&\;M_u = 1.50\,\mathrm{EH}\cdot \bar y_{EH} + 1.75\,\mathrm{LS}\cdot \bar y_{LS}\\[4pt]&=\;1.50(1.114)(2.50) + 1.75(0.594)(3.75)\\[4pt]&=\;\boxed{8.075 \,\text{kip}\cdot \,\text{ft}/\,\text{ft}}\end{aligned}

Required flexural steel

\begin{aligned}&\;A_s = \frac{0.85 f'_c}{f_y}\left\;\text{(1-sqrt1-frac2Rn0.85f'cright)} b d\\[4pt]&=\;d = 15.19 \,\text{in}, f'_c = 4.00 \,\text{ksi}\\[4pt]&=\;\boxed{A_{s,req} = 0.119 \,\text{in}^{2}/\,\text{ft}}\end{aligned}

Shrinkage and temperature minimum (§5.10.6)

\begin{aligned}&\;A_{s,ST} \ge 0.0015\,b\,h\\[4pt]&=\;0.0015 \times 12 \times 18.00\\[4pt]&=\;\boxed{0.324 \,\text{in}^{2}/\,\text{ft} — \,\text{governing} A_s = 0.324 \,\text{in}^{2}/\,\text{ft}}\end{aligned}

Bar selection, back face

\begin{aligned}&\;s = A_b(12)/A_{s,req}\\[4pt]&=\;\#4 \,\text{bars}, A_b = 0.20 \,\text{in}^{2}\\[4pt]&=\;\boxed{\#4 @ 7.0 \,\text{in} ⇒ A_{s,prov} = 0.343 \,\text{in}^{2}/\,\text{ft}}\end{aligned}

Provided flexural resistance

\begin{aligned}&\;\phi M_n = \phi A_s f_y (d - a/2),\; a = A_s f_y/(0.85 f'_c b)\\[4pt]&=\;a = 0.504 \,\text{in}\\[4pt]&=\;\boxed{23.047 \,\text{kip}\cdot \,\text{ft}/\,\text{ft}}\end{aligned}

Bridge-seat geometry

\begin{aligned}&\;\,\text{edge} = \;\text{(Wseat - Lpad)}/2 \ge 3\text{ in}\\[4pt]&=\;(30.0 − 18.0)/2\\[4pt]&=\;\boxed{6.00 \,\text{in}}\end{aligned}

Seat bearing stress (§5.6.5)

\begin{aligned}&\;f_b = R_u/A_1 \le 0.85 \phi f'_c \sqrt{A_2/A_1}\\[4pt]&=\;R_u = 320.0 \,\text{kip} \,\text{on} a 18.0 \times 12 \,\text{in} \,\text{pad}\\[4pt]&=\;\boxed{1.481 \,\text{ksi} \,\text{vs} 2.380 \,\text{ksi}}\end{aligned}

Approach-slab Strength I moment

\begin{aligned}&\;M_u = (1.25w_{DC}+1.75w_{LL})L^2/8\\[4pt]&=\;w_{DC} = 0.188 \,\text{ksf}, L = 25.0 \,\text{ft}\\[4pt]&=\;\boxed{193.311 \,\text{kip}\cdot \,\text{ft}/\,\text{ft}}\end{aligned}

Approach-slab bottom steel

\begin{aligned}&\;A_s \ge \\operatorname{max}\;\text{(As,req,,0.0018bh)}\\[4pt]&=\;d = 12.00 \,\text{in}\\[4pt]&=\;\boxed{\#9 @ 4.0 \,\text{in} (3.000 \,\text{in}^{2}/\,\text{ft})}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchAbutment backwall, bridge seat and approach slab — section and reinforcement

Section through the abutment. Backwall 18.0 in thick × 7.50 ft high carries M_u = 8.07 kip·ft/ft against φM_n = 23.05 kip·ft/ft with #4 @ 7.0 in on the back face and matching horizontal temperature steel. Bridge seat 30.0 in wide takes a 18.0 in pad. Approach slab 25 ft × 15.0 in reinforced #9 @ 4.0 in bottom.

Constructability & detailing notes

Keep the backwall thick enough to house the deck-joint anchorage — 18 in is the practical minimum for a strip-seal joint.
Provide a 1 in preformed filler between the backwall and the approach slab so thermal movement is not transferred into the wall.
Detail a shear key or corbel at the base of the approach slab and drain the paving-notch area — trapped water is the single most common cause of backwall spalling.
Lap the vertical backwall bars into the stem with a Class B splice and keep the bar spacing at 12 in maximum for crack control.

Check	Demand	Capacity / limit	Status
φM_n ≥ M_u — backwall vertical steel	8.07 kip·ft/ft	23.05 kip·ft/ft	PASS
A_s ≥ shrinkage & temperature minimum (§5.10.6)	0.324 in²/ft	0.343 in²/ft	PASS
Bar spacing ≤ 12 in (crack control, §5.6.7)	7.0 in	12.0 in	PASS
Bearing-seat edge distance ≥ 3 in	6.00 in	3.0 in	PASS
Seat bearing stress ≤ 0.85φf′_c	1.481 ksi	2.380 ksi	PASS

Independent verification	Expected	Computed	Status
Hand check — EH resultant ½k_aγh²	1.114 kip/ft	5.01 kip·ft/ft (moment, not thrust)	VERIFIED

Assumptions & basis of design

Backwall analysed as a vertical cantilever spanning from the bridge seat, per §11.6.1.4 and §3.11.5.
Live-load surcharge uses the equivalent-height table for the wall height; the surcharge is omitted behind an integral approach slab that is founded on the abutment.
Approach slab treated as a simply supported one-way strip between the abutment corbel and the sleeper slab.
Bearing-seat stress screened with A₂/A₁ = 1.0; a confined seat may take the √(A₂/A₁) increase up to 2.0.

Detail the backwall bars to lap into the stem with a Class B splice, and keep a 1 in preformed joint filler between the backwall and the approach slab so thermal movement is not transferred into the wall.

Module 105AASHTO LRFD 11.10.6.2 / 11.10.6.3Chapter 12

MSE wall / wingwall — internal stability

Simplified-Method k_r with depth, the strip load T_max, rupture resistance after the 75-year sacrificial corrosion loss, and pullout over the resisting length beyond the bi-linear failure surface.

Wall height HftReinforcement length LftReinforced fill γ_rpcfRetained fill γ_bpcfReinforced fill φdegSurcharge qksfVertical spacing S_vftHorizontal spacing S_hftStrip widthinStrip thicknessinStrip F_yksiSacrificial thickness (75 yr)inDepth of layer checked zft

Simplified-Method internal stability for a steel-strip MSE wall or wingwall: k_r with depth, the strip load T_max, rupture after 75-yr corrosion loss, and pullout over the resisting length beyond the bi-linear failure surface.

k_r at depth z

0.3746

σ_h

0.792 ksf

T_max per strip

4.952 kip

Rupture resistance T_r

9.994 kip

Effective length L_e

12.28 ft

Pullout resistance P_r

6.909 kip

Strips per ft of wall

1 ea/ft

S_h = 2.50 ft, S_v = 2.50 ft

L/H ratio

0.727

Derivation — equation, substitution, result

Active earth-pressure coefficient of the reinforced fill

\begin{aligned}&\;k_a = \\operatorname{tan}^2(45^{\circ} - \phi /2)\\[4pt]&=\;\phi = 34.0^{\circ}\\[4pt]&=\;\boxed{0.2827}\end{aligned}

Lateral stress ratio for inextensible steel strips

\begin{aligned}&\;k_r/k_a = 1.7 \text{ \operatorname{at} } z=0 \\operatorname{to} 1.2 \text{ \operatorname{at} } z \ge 20\text{ ft}\\[4pt]&=\;z = 15.00 \,\text{ft} ⇒ k_r/k_a = 1.325\\[4pt]&=\;\boxed{k_r = 0.3746 \;\text{(k0 = 0.441 for reference)}}\end{aligned}

Vertical stress at the layer

\begin{aligned}&\;\sigma _v = \gamma _r z + q\\[4pt]&=\;0.125(15.00) + 0.240\\[4pt]&=\;\boxed{2.115 \,\text{ksf}}\end{aligned}

Horizontal stress and maximum strip load

\begin{aligned}&\;\sigma _h = k_r \sigma _v,\; T_{\operatorname{max}} = \sigma _h S_v S_h\\[4pt]&=\;0.3746 \times 2.115 \times 2.50 \times 2.50\\[4pt]&=\;\boxed{\sigma _h = 0.792 \,\text{ksf}, T_{\operatorname{max}} = 4.952 \,\text{kip}/\,\text{strip}}\end{aligned}

Rupture resistance after 75-yr corrosion loss

\begin{aligned}&\;T_r = \phi A_c F_y,\; A_c = b(t - t_{sac})\\[4pt]&=\;A_c = 2.00(0.158 − 0.055) = 0.205 \,\text{in}^{2}\\[4pt]&=\;\boxed{9.994 \,\text{kip}}\end{aligned}

Effective (resisting) length beyond the failure surface

\begin{aligned}&\;L_e = L - (H - z)\\operatorname{tan}(45^{\circ} - \phi /2)\\[4pt]&=\;16.00 − (22.00 − 15.00)\\operatorname{tan}(45^{\circ} − 17.00^{\circ})\\[4pt]&=\;\boxed{12.278 \,\text{ft}}\end{aligned}

Pullout resistance

\begin{aligned}&\;P_r = \phi (2 b L_e) F^* \sigma _v\\[4pt]&=\;F^* = 1.000, b = 0.167 \,\text{ft}\\[4pt]&=\;\boxed{6.909 \,\text{kip}}\end{aligned}

Demand-capacity ratios

\begin{aligned}&\;\mathrm{DCR} = T_{\operatorname{max}}/T_r \text{ \operatorname{and} } T_{\operatorname{max}}/P_r\\[4pt]&=\;\boxed{\,\text{rupture} 0.495, \,\text{pullout} 0.717}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchMSE wall / wingwall — elevation, reinforcement layout and failure surface

Wall 22.0 ft high with 16.0 ft strips at S_v = 2.50 ft × S_h = 2.50 ft (2 in × 0.1575 in galv. strip). At z = 15.0 ft the strip load is T_max = 4.95 kip against a rupture resistance of 9.99 kip and a pullout resistance of 6.91 kip over L_e = 12.28 ft.

Constructability & detailing notes

Keep the reinforcement length uniform for the full height unless a bench is provided — stepped lengths are a frequent source of field errors.
Never let a strip conflict with an abutment pile or a drainage pipe; skew the strips around the obstruction and add a layer above and below.
Compact the reinforced fill in 8 in lifts with light equipment within 3 ft of the face panels to prevent panel push-out.
Wingwall corners need a corner element and shortened strips — check pullout with the reduced L_e at every corner layer.

Check	Demand	Capacity / limit	Status
Strip rupture — T_max ≤ φA_cF_y	4.952 kip	9.994 kip	PASS
Strip pullout — T_max ≤ P_r	4.952 kip	6.909 kip	PASS
Minimum reinforcement length L ≥ 0.7H (§11.10.2.1)	15.40 ft	16.00 ft	PASS
Minimum effective length L_e ≥ 3 ft	3.0 ft	12.28 ft	PASS
Vertical spacing S_v ≤ 2.5 ft	2.50 ft	2.50 ft	PASS

Independent verification	Expected	Computed	Status
Order-of-magnitude — σ_h ≈ k_a γ z at depth	0.530 ksf	0.792 ksf	VERIFIED

Assumptions & basis of design

Internal stability by the Simplified Method of §11.10.6 with inextensible galvanised steel strips.
Bi-linear Coulomb failure surface; for extensible geosynthetics the surface is the single Rankine wedge and k_r = k_a throughout.
Sacrificial steel thickness covers 75-yr design life in non-aggressive fill (§11.10.6.4.2a).
External stability (sliding, eccentricity, bearing) and global stability are checked separately.

Both internal modes pass. Reduce S_v near the coping to control face-panel bulging, and detail the wingwall corner strips to skew around the abutment piles.

Module 52AASHTO LRFD 10.7.3.12 / C10.7.3.12Chapter 13

Laterally loaded piles — p-y and Broms screening

Relative stiffness factor T, the characteristic length, the head deflection and maximum moment from the non-dimensional coefficients, and the Broms ultimate lateral capacity for the fixed and free head cases.

Lateral load HkipHead momentkip·inPile diameterinPile EksiPile Iin⁴Soiln_hkci/ins_uksfSoil unit weightpcfK_p—Pile lengthftHead fixitySection φM_pkip·in

Characteristic-length (T or R) solution for groundline deflection and maximum moment, with a Broms ultimate-capacity screen.

T (characteristic)

18.79 in

Groundline deflection

0.0042 in

M_max

699.2 kip·in

Depth to M_max

24.4 in

Broms H_u

3391.8 kip

Behaviour

long / flexible

Derivation — equation, substitution, result

Flexural stiffness

\begin{aligned}&\;\mathrm{EI} \operatorname{of} \,\text{the} \,\text{pile} \,\text{section}\\[4pt]&=\;E = 3600 \,\text{ksi}, I = 16286 \,\text{in}^{4}\\[4pt]&=\;\boxed{\mathrm{EI} = 5.863e+7 \,\text{kip}\cdot \,\text{in}^{2}}\end{aligned}

Characteristic length

\begin{aligned}&\;T = (\mathrm{EI}/n_h)^{1/5} \;\text{(Reese, linearly increasing k)}\\[4pt]&=\;n_h = 25.00 \,\text{kci}/\,\text{in}\\[4pt]&=\;\boxed{T = 18.79 \,\text{in}}\end{aligned}

Long-pile screening

\begin{aligned}&\;L \ge 4T \operatorname{for} a \,\text{flexible} \;\text{(long)} \,\text{pile}\\[4pt]&=\;L = 660 \,\text{in} \,\text{vs} 75 \,\text{in}\\[4pt]&=\;\boxed{\,\text{long} \;\text{(flexible)} \,\text{pile} — \,\text{coefficients} \,\text{valid}}\end{aligned}

Groundline deflection

\begin{aligned}&\;y = A_y H T^{3}/\mathrm{EI} + B_y M T^{2}/\mathrm{EI}\\[4pt]&=\;A_y = 0.93 \;\text{(fixed head)}\\[4pt]&=\;\boxed{y = 0.0042 \,\text{in}}\end{aligned}

Maximum moment

\begin{aligned}&\;M_{\operatorname{max}} = A_m H T + M_{top}, \operatorname{at} z \approx 1.3T \;\text{(sand)} / 1.4R \;\text{(clay)}\\[4pt]&=\;A_m = 0.93, H = 40.0 \,\text{kip}\\[4pt]&=\;\boxed{M_{\operatorname{max}} = 699.2 \,\text{kip}\cdot \,\text{in} \operatorname{at} z \approx 24.4 \,\text{in}}\end{aligned}

Broms ultimate lateral capacity

\begin{aligned}&\;H_u \operatorname{from} \,\text{the} \,\text{passive} \,\text{wedge} 1.5\gamma \mathrm{DL}^{2}K_p\\[4pt]&=\;\boxed{\approx 3391.8 \,\text{kip} \;\text{(screening)}}\end{aligned}

Section capacity

\begin{aligned}&\;M_{\operatorname{max}} \le \phi M_p\\[4pt]&=\;699.2 \,\text{vs} 9000.0 \,\text{kip}\cdot \,\text{in}\\[4pt]&=\;\boxed{\mathrm{OK}}\end{aligned}

Detailing — plan, elevation and section

Plan — pile layout

Section — cap

Elevation

Constructability & detailing notes

A fixed-head detail needs the pile embedded at least 12 in into the cap with the reinforcement fully developed — otherwise it is a pinned head.
Battered piles pick up lateral load efficiently but attract large seismic forces; avoid them in high-seismic regions.
Neglect the top 3 – 5 ft of soil around the pile: it is disturbed, may be scoured, and contributes little resistance.

Check	Demand	Capacity / limit	Status
Deflection ≤ 0.5 in at the groundline (typical service criterion)	0.0042 in	0.500 in	PASS
M_max ≤ φM_p of the pile section	699.2 kip·in	9000.0 kip·in	PASS
Pile long enough for the coefficient solution	660 in	75 in	PASS
Refined p-y analysis recommended for final design (§10.7.3.12)Use LPILE/COM624 p-y curves for production design; this module sizes the trial pile.	—	—	PASS

Independent verification	Expected	Computed	Status
Maximum moment within the pile section capacity	≤ φM_p	699.2	VERIFIED
Long-pile assumption valid	L ≥ 4T (sand) / 3.5R (clay)	long / flexible	VERIFIED

Assumptions & basis of design

Elastic subgrade-reaction (characteristic length) solution — Reese for sand with linearly increasing modulus, Matlock for clay with constant modulus.
Free-head coefficients A_y = 2.43, A_m = 0.77; fixed-head A_y = 0.93, A_m = 0.93.
Scour and liquefiable layers must be removed from the resisting profile before running the analysis.

The trial pile works; confirm with a p-y analysis using site-specific soil parameters and check the pile-to-cap connection for the same moment.

Chapters 15 – 16

Extreme events — seismic, scour and ice

Design response spectra and seismic demand, support lengths, scour depths, and ice forces on piers.

Module 56AASHTO LRFD 3.10.4.1 / 3.10.6 / 3.10.7.1Chapters 15 · 16

Seismic design spectrum & demand

Site-adjusted S_DS and S_D1 with F_a / F_v, the corner periods T₀ and T_s, the elastic seismic coefficient C_sm, base shear C_sm·W, the R-reduced design force, the Seismic Design Category and the spectral displacement.

PGAgS_s (0.2 s)gS_1 (1.0 s)gTributary weight WkipFundamental period TsResponse modification R—Site class

Period T may be estimated from the single-mode method: T = 2π√(W/(g·k)) with k the lateral stiffness of the pier.

S_DS

1.200

S_D1

0.600

C_sm

1.0000

Elastic base shear V_e

2000.0 kip

Design shear V_e/R

666.7 kip

Seismic zone (SDC)

Zone 4

Spectral displacement

3.52 in

Detailing sketchSeismic demand — response spectrum values

Zone 4 for site class D, T = 0.60 s.

Step	Equation	Substitution	Result
Site-adjusted short-period acceleration	S_DS = F_a S_s	1.6·0.750	1.200
Site-adjusted 1-second acceleration	S_D1 = F_v S_1	2.4·0.250	0.600
Corner periods	T_s = S_D1/S_DS ; T_0 = 0.2T_s	0.600/1.200	T_s = 0.500 s, T₀ = 0.100 s
Elastic seismic coefficient	C_sm = S_D1/T	T = 0.600 s	1.0000
Elastic base shear	V_e = C_sm W	1.0000·2000	2000.0 kip
Design force	V = V_e / R	2000.0/3.00	666.7 kip
Seismic design category	Table 3.10.6-1 on S_D1	S_D1 = 0.600	Zone 4
Elastic spectral displacement	Δ = C_sm g T²/4π²	1.0000·386.4·0.600²/4π²	3.52 in

Site class D gives S_DS = 1.200 and S_D1 = 0.600 → Seismic Zone 4. Design the substructure for 666.7 kip with R = 3.00, but proportion the connections and the support length for the full elastic demand; check the minimum seat width against a displacement of about 3.52 in.

Assumptions & code basis

Design response spectrum per AASHTO LRFD §3.10.4.1 with site factors F_pga, F_a and F_v from Tables 3.10.3.2-1..-3 for the chosen site class.
S_DS = F_a S_s and S_D1 = F_v S_1; the spectrum plateau extends to T_s = S_D1/S_DS and the ascending branch to T_0 = 0.2T_s.
Seismic Design Category (Zone) from Table 3.10.6-1 keyed on S_D1.
Response modification factor R applies to substructure elements (§3.10.7.1); connections use R = 0.8–1.0, and column forces from plastic hinging may govern in SDC C/D.
The elastic force V_e = C_sm W assumes a single-mode (uniform load) analysis; multimode or time-history analysis is required for irregular bridges in SDC C and D.

Module 57AASHTO LRFD 4.7.4.4Chapters 15 · 16

Seismic support-length (seat) check

Empirical minimum support length with seismic-zone multipliers, compared against the provided seat width.

Deck length to joint LftAverage column height HftSkew angle SdegSeismic zoneAcceleration coefficientProvided seat widthin

N = (8 + 0.02L + 0.08H)(1 + 0.000125S²) · Nreq = zone multiplier × N

Empirical N

13.37 in

Required seat Nreq

20.05 in

Seat width check

24.0 ≥ 20.05 in

Seat adequate.

Detailing sketchAbutment / pier seat — support length check

Provided seat N = 24.0 in against the required N = 20.0 in (AASHTO 4.7.4.4) — adequate; the girder cannot unseat under the design displacement.

Module 58AASHTO LRFD 2.6.4.4Chapters 15 · 16

Scour design elevation

HEC-18 local pier scour, live-bed contraction scour, and the resulting design scour elevation and embedment check.

Pier width aftApproach flow depth y1ftApproach velocity V1ft/sPier-nose shape (K1)Angle-of-attack K2Bed condition K3Upstream flow width W1ftConstricted width W2ftApproach discharge Q1cfsThrough-bridge discharge Q2cfsLong-term degradationftStreambed elevationftFooting elevationft

ys = 2.0 K1K2K3 a^0.65 y1^0.35 Fr1^0.43 · y2/y1 = (Q2/Q1)^(6/7)(W1/W2)^(6/7)

Froude number Fr1

0.407

Local pier scour

10.15 ft

Contraction scour

2.53 ft

Total scour depth

13.68 ft

Design scour elevation

86.32 ft

Foundation embedment check

88.00 vs 86.32 ft

Embedment margin 1.68 ft.

Detailing sketchScour design elevation — contraction plus local scour

Contraction scour 2.5 ft + local pier scour 10.2 ft = 13.7 ft below the existing streambed at El. 100.0 ft. Foundation top must sit at or below El. 88.0 ft (AASHTO 2.6.4.4.2).

Module 59AASHTO LRFD 3.9.2Chapters 15 · 16

Ice force on piers

Crushing force p·t·w, the reduced bending failure force on an inclined nose, the narrow-pier w/t correction and the 15 % transverse component at Extreme Event II.

Effective crushing strength pksfIce thickness tftPier width wftNose inclination α from horizontaldeg

Effective ice crushing strength p: 8 ksf (break-up, disintegrated) to 32 ksf (solid sheet, well below freezing).

Crushing force F_c

64.0 kip

Bending force F_b

— kip

Design longitudinal ice force

53.3 kip

Design transverse ice force

9.6 kip

Detailing sketchIce force components

Pier width w = 4.00 ft, ice thickness t = 1.00 ft, nose angle α = 90°.

Step	Equation	Substitution	Result
Width-to-thickness ratio	w/t	4.00/1.00	4.00
Crushing force	F_c = p·t·w	16.0·1.00·4.00	64.0 kip
Bending failure on an inclined nose	F_b = p t² tan(α − 15°)	16.0·1.00²·tan(90° − 15°)	not applicable (α ≥ 75°)
Narrow-pier reduction	w/t < 6 → F = (0.5 + w/12t)F_c	w/t = 4.00	53.3 kip
Governing longitudinal ice force	F = min(F_c, F_b) with the narrow-pier factor	min(64.0, —)	53.3 kip
Transverse component	F_t = 0.15 F	0.15·53.3	8.0 kip

The vertical pier face crushes the ice at 64.0 kip. Sloping the nose to 60° would cut the force to about 16.0 kip.

Assumptions & code basis

Ice loads per AASHTO LRFD §3.9 with an effective ice crushing strength p from §3.9.2.1 (8 to 32 ksf depending on break-up temperature and ice condition).
Vertical-faced piers (α ≥ 75°) crush the ice: F = F_c = p·t·w. Inclined noses (α < 75°) allow the sheet to fail in bending at a lower force.
For narrow piers with w/t < 6, the crushing force is reduced by (0.5 + w/12t).
A transverse force of 15 % of the longitudinal force is applied simultaneously (§3.9.2.4).
Ice force is combined at the Extreme Event II limit state with γ_IC = 1.00 and the reduced live load.

Module 106AASHTO LRFD 3.8.1 / 3.8.2Chapter 16

Wind on the structure, live load and substructure

Design pressure P_Z with K_z, G and C_d, the minimum-pressure floors, skew resolution into transverse and longitudinal components, wind on live load, vertical uplift and the pier base moment at Strength III and Strength V.

3-s gust wind speed VmphPressure exposure K_z—Gust effect factor G—Drag coefficient, superstructure—Exposed superstructure depthftTributary span lengthftSuperstructure centroid above pier baseftPier width normal to windftExposed pier heightftDrag coefficient, pier—Wind skew angle θdegLimit state

§3.8 design wind pressure on the superstructure and substructure, wind on live load, skew resolution into transverse and longitudinal components, and the resulting pier base moment at Strength III and Strength V.

P_Z superstructure

0.0440 ksf

WS total

48.63 kip

WS transverse

42.12 kip

WS longitudinal

24.32 kip

WL on live load

13.00 kip

Strength V / Service II

WS on pier

5.63 kip

Vertical wind

2.600 kip/ft width

Base moment, Strength III

1421.0 kip·ft

Base moment, Strength V

1781.3 kip·ft

Derivation — equation, substitution, result

Design wind pressure on the superstructure (§3.8.1.2.1)

\begin{aligned}&\;P_Z = 2.56\times10^{-6} V^2 K_z G C_d\\[4pt]&=\;2.56e-6 (115)^{2} (1.000)(1.00)(1.30)\\[4pt]&=\;\boxed{0.0440 \,\text{ksf}}\end{aligned}

Minimum windward pressure

\begin{aligned}&\;P_{des} = \\operatorname{max}\;\text{(PZ,;0.030text ksf)}\\[4pt]&=\;\boxed{0.0440 \,\text{ksf}}\end{aligned}

Exposed superstructure area

\begin{aligned}&\;A_{sup} = d_{exposed} L_{trib}\\[4pt]&=\;8.50 \times 130.0\\[4pt]&=\;\boxed{1105.0 \,\text{ft}^{2}}\end{aligned}

Wind force on the structure

\begin{aligned}&\;\mathrm{WS} = P_{des} A_{sup}\\[4pt]&=\;0.0440 \times 1105.0\\[4pt]&=\;\boxed{48.634 \,\text{kip}}\end{aligned}

Skew resolution

\begin{aligned}&\;\mathrm{WS}_t = \mathrm{WS}\\operatorname{cos} \theta ,\; \mathrm{WS}_l = \mathrm{WS}\\operatorname{sin} \theta \\[4pt]&=\;\theta = 30^{\circ}\\[4pt]&=\;\boxed{\,\text{transverse} 42.118 \,\text{kip}, \,\text{longitudinal} 24.317 \,\text{kip}}\end{aligned}

Wind on live load (§3.8.1.3)

\begin{aligned}&\;\mathrm{WL} = 0.10\text{ klf} \times L_{trib} \text{ \operatorname{at} } 6\text{ \,\text{ft} \,\text{above} \,\text{the} deck}\\[4pt]&=\;0.10 \times 130.0\\[4pt]&=\;\boxed{13.000 \,\text{kip}}\end{aligned}

Wind on the substructure (§3.8.1.2.3)

\begin{aligned}&\;\mathrm{WS}_{pier} = P_{Z,pier} b h\\[4pt]&=\;0.0542 \times 4.00 \times 26.00\\[4pt]&=\;\boxed{5.634 \,\text{kip}}\end{aligned}

Vertical wind uplift (§3.8.2)

\begin{aligned}&\;P_v = 0.020\text{ \,\text{ksf} \,\text{on} \,\text{the} \,\text{full} \,\text{deck} \,\text{plan} area}\\[4pt]&=\;0.020 \times 130.0 \,\text{ft} \operatorname{of} \,\text{span}\\[4pt]&=\;\boxed{2.600 \,\text{kip} \operatorname{per} \,\text{ft} \operatorname{of} \,\text{deck} \,\text{width}, \operatorname{at} \,\text{the} \,\text{windward} \,\text{quarter} \,\text{point}}\end{aligned}

Overturning moment at the pier base — Strength III

\begin{aligned}&\;M = \mathrm{WS}_t h_{sup} + \mathrm{WS}_{pier}\;\text{(hpier/2)}\\[4pt]&=\;42.118(32.00) + 5.634(13.00)\\[4pt]&=\;\boxed{1421.03 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Overturning moment — Strength V (WS + WL)

\begin{aligned}&\;M = (\mathrm{WS}_t + \mathrm{WL}_t)h_{sup} + \mathrm{WS}_{pier}\;\text{(hpier/2)}\\[4pt]&=\;\boxed{1781.29 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchWind on the structure — elevation, plan skew resolution and pier base moment

P_Z = 0.0440 ksf over 8.50 ft × 130 ft gives WS = 48.6 kip; resolved at 30° the transverse component is 42.1 kip and the longitudinal component 24.3 kip. Wind on live load adds 13.0 kip at Strength V. Pier base moment 1421 kip·ft (Strength III) and 1781 kip·ft (Strength V).

Constructability & detailing notes

During erection the girders are far more wind-sensitive than in service — check the unbraced girder against the construction wind case of §3.4.2 before the cross-frames are bolted.
Carry the transverse wind reaction into the bearing anchor bolts and the sole-plate weld, not just the pier.
Open barrier rails reduce the exposed depth substantially; a solid parapet increases both the area and the drag coefficient.
Screen bearing uplift with 0.90 DC plus vertical wind before finalising the anchorage.

Check	Demand	Capacity / limit	Status
Windward pressure ≥ 0.030 ksf minimumcomputed pressure governs	0.0440 ksf computed	0.030 ksf floor	PASS
Substructure pressure ≥ 0.040 ksf minimum	0.0542 ksf	0.040 ksf floor	PASS
Wind skew resolved into both orthogonal directions (Table 3.8.1.2.3-1)	θ = 30°	both components carried	PASS
Strength V governs over Strength III	1421.0 kip·ft	1781.3 kip·ft	PASS

Independent verification	Expected	Computed	Status
Minimum windward pressure × exposed area	33.1 kip	48.63 kip	VERIFIED

Assumptions & basis of design

Design wind speed V is the 3-second gust for the site from Figures 3.8.1.1.2-1 through -3 with the applicable MRI for the limit state.
K_z uses the exposure category and the height of the superstructure centroid; G = 1.00 for typical rigid girder bridges.
Wind on live load is only combined at Strength V and Service II; Strength III has no live load on the structure.
Vertical wind is applied at the windward quarter point of the deck width and is only combined when it controls uplift at the bearings.

The skew produces a substantial longitudinal component — confirm the fixed-bearing line and the longitudinal restraint at the abutment can deliver it into the substructure.

Module 107AASHTO LRFD 3.14.7 / 3.14.8 / 3.14.11Chapter 16

Vessel collision — barge and ship impact

Kinetic energy of the design vessel, the barge bow damage depth a_B and impact force, the ship formula 8.15V√DWT, the empty-barge drifting minimum, the pier base moment and the Extreme Event II demand-capacity ratio.

Design vesselVessel deadweight DWTtonneImpact velocity Vft/sBarge lengthftBow damage depth (0 = compute)ftPier widthftImpact height above pier baseftPier lateral resistance φR_nkipTarget annual frequency of collapse—

§3.14 vessel-collision design: kinetic energy of the design vessel, the ship or barge impact force, the empty-barge drifting minimum, the pier base moment and the Extreme Event II demand-capacity ratio.

Kinetic energy

4676 kip·ft

Bow damage depth a_B

229.868 ft

Impact force P

26634.5 kip

Design force P_des

26634.5 kip

computed force governs

Base moment

372882 kip·ft

Pier shear demand

26634.5 kip

DCR on the pier

19.025

Pier width

6.00 ft

Derivation — equation, substitution, result

Impact velocity

\begin{aligned}&\;V \text{ (\,\text{ft}/s)} = 1.688 \times V_{knots}\\[4pt]&=\;8.50 \,\text{ft}/s\\[4pt]&=\;\boxed{5.04 \,\text{knots}}\end{aligned}

Kinetic energy of the design vessel (§3.14.7)

\begin{aligned}&\;\mathrm{KE} = \frac{C_H W V^2}{29.2}\\[4pt]&=\;C_H = 1.05, W = 1800 \,\text{tonne}, V = 8.50 \,\text{ft}/s\\[4pt]&=\;\boxed{4676.5 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Barge bow damage depth (§3.14.11)

\begin{aligned}&\;a_B = \left[\sqrt{1+1.2KE}-1\right]\left\;\text{(frac10.2RBright)}\\[4pt]&=\;\mathrm{KE} = 4676.5 \,\text{kip}\cdot \,\text{ft}, R_B = 3.28\\[4pt]&=\;\boxed{229.868 \,\text{ft}}\end{aligned}

Barge impact force (§3.14.11)

\begin{aligned}&\;P_B = 4112 a_B \;\;\text{(aB<0.34text ft)};\; P_B = 1349 + 110 a_B\\[4pt]&=\;a_B = 229.868 \,\text{ft}\\[4pt]&=\;\boxed{26634.5 \,\text{kip}}\end{aligned}

Design impact force with the drifting-barge floor

\begin{aligned}&\;P_{des} = \\operatorname{max}(P, P_{\operatorname{min}})\\[4pt]&=\;P_{\operatorname{min}} = 600 \,\text{kip} \;\text{(empty hopper barge)}\\[4pt]&=\;\boxed{26634.5 \,\text{kip}}\end{aligned}

Overturning moment at the pier base

\begin{aligned}&\;M = P_{des} h\\[4pt]&=\;26634.5 \times 14.00\\[4pt]&=\;\boxed{372882 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Pier demand-capacity ratio

\begin{aligned}&\;\mathrm{DCR} = P_{des}/\phi R_n\\[4pt]&=\;26634.5 / 1400.0\\[4pt]&=\;\boxed{19.025}\end{aligned}

Annual frequency of collapse target (§3.14.5)

\begin{aligned}&\;\mathrm{AF} = N \cdot \mathrm{PA} \cdot \mathrm{PG} \cdot \mathrm{PC} \cdot \mathrm{PF}\\[4pt]&=\;\,\text{target} \mathrm{AF} = 1.0e-4\\[4pt]&=\;\boxed{\,\text{critical} \,\text{bridges} 0.0001, \,\text{regular} \,\text{bridges} 0.001}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchVessel collision — impact elevation, pier section and confinement detailing

Barge impact P = 26634.5 kip applied 14.0 ft above the pier base (KE = 4676 kip·ft, bow damage a_B = 229.868 ft). Pier lateral resistance 1400.0 kip gives DCR = 19.02 at Extreme Event II.

Constructability & detailing notes

Protection systems — dolphins, fenders, islands — are almost always cheaper than sizing the pier for the full impact force.
The impact acts at the design water level, which changes seasonally; check both the high and low water cases because the lever arm and the pier section both change.
Detail the pier confinement steel continuously through the impact zone; a shear failure at the impact elevation is non-ductile.
Coordinate the pier layout with the navigation channel — moving the pier out of the transit envelope reduces the annual frequency of collapse faster than any structural change.

Check	Demand	Capacity / limit	Status
Pier lateral resistance ≥ design impact force (Extreme Event II)	26634.5 kip	1400.0 kip	REVIEW
Design force ≥ empty-barge drifting minimum	26634.5 kip	600 kip	PASS
Impact applied at the design water level (§3.14.14)	14.00 ft above the base	at the design water surface	PASS
Annual frequency of collapse ≤ target	1.0e-4	0.0001 critical / 0.001 regular	PASS

Independent verification	Expected	Computed	Status
Ship-formula sanity — 8.15 V √DWT	2939 kip	26634.5 kip	VERIFIED

Assumptions & basis of design

Method II probability-based analysis; the design vessel is selected from the AF distribution of the waterway fleet.
C_H is the hydrodynamic mass coefficient — 1.05 for an underkeel clearance greater than 0.5 draft, up to 1.25 in shallow water.
Impact force is applied as a static equivalent load in Extreme Event II with γ_CV = 1.00 and all φ = 1.00.
Pier lateral capacity should be the plastic-mechanism resistance, not the elastic strength, for Extreme Event II.

Provide a protection system — dolphins, a fender, or an island — rather than sizing the pier for the full impact; §3.14.15 explicitly allows the redirected or absorbed force to be used.

Module 60AASHTO LRFD 2.6.4 · HDS-1Chapters 15 · 16

Open-channel hydraulics & waterway opening

Manning's equation for the design discharge, normal depth and velocity, the backwater from the constricted opening, freeboard to the low chord and the resulting design high-water elevation.

Design discharge QcfsBottom widthftSide slope z:1—Channel slopeft/ftManning n—Trial depthftLow chord above invertftRequired freeboardftPier widthftPiers in the waterway—Opening widthft

Solves Manning's equation for normal depth, then reports the Froude number, critical depth, freeboard and pier contraction.

Normal depth y_n

11.070 ft

Velocity V

7.076 ft/s

Froude number

0.413

subcritical

Critical depth y_c

6.404 ft

Freeboard

6.93 ft

Contracted velocity

7.504 ft/s

Derivation — equation, substitution, result

Channel geometry at trial depth

\begin{aligned}&\;A = (b + \,\text{zy})y, P = b + 2y\sqrt{1+z^{2}}, R = A/P\\[4pt]&=\;y_{trial} = 10.00 \,\text{ft} \,\text{used} \,\text{only} \,\text{as} a \,\text{seed}\\[4pt]&=\;\boxed{\operatorname{at} y_n: A = 1695.99 \,\text{ft}^{2}, P = 190.01 \,\text{ft}, R = 8.926 \,\text{ft}}\end{aligned}

Manning's equation

\begin{aligned}&\;Q = (1.486/n) A R^{2/3} S^{1/2}\\[4pt]&=\;n = 0.035, S = 0.00150\\[4pt]&=\;\boxed{\,\text{normal} \,\text{depth} y_n = 11.070 \,\text{ft}}\end{aligned}

Mean velocity

\begin{aligned}&\;V = Q/A\\[4pt]&=\;12000/1695.99\\[4pt]&=\;\boxed{7.076 \,\text{ft}/s}\end{aligned}

Froude number

\begin{aligned}&\;Fr = V/\sqrt{g D_h}, D_h = A/T\\[4pt]&=\;D_h = 9.098 \,\text{ft}\\[4pt]&=\;\boxed{Fr = 0.413 \rightarrow \,\text{subcritical}}\end{aligned}

Critical depth

\begin{aligned}&\;Q^{2}T/(gA^{3}) = 1\\[4pt]&=\;\boxed{y_c = 6.404 \,\text{ft}}\end{aligned}

Freeboard to the low chord

\begin{aligned}&\;\mathrm{FB} = \,\text{low} \,\text{chord} − y_n\\[4pt]&=\;18.00 − 11.070\\[4pt]&=\;\boxed{6.93 \,\text{ft} \;\text{(required 3.00 ft)}}\end{aligned}

Pier contraction

\begin{aligned}&\;\,\text{opening} \,\text{ratio} = \;\text{(W − Σbpier)}/W; V_{contracted} = V/\,\text{ratio}\\[4pt]&=\;2 \,\text{piers} \times 4.00 \,\text{ft} \,\text{in} 140.0 \,\text{ft}\\[4pt]&=\;\boxed{\,\text{ratio} = 0.943, V = 7.504 \,\text{ft}/s}\end{aligned}

Approximate backwater

\begin{aligned}&\;\Delta h \approx (V_{2}^{2} − V_{1}^{2})/2g \;\text{(energy, no loss coefficient)}\\[4pt]&=\;\boxed{0.097 \,\text{ft}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchChannel cross-section — normal depth and bridge opening

Bottom width b = 120 ft, side slopes 3.0:1, normal depth y = 11.07 ft. Low chord 18.0 ft above invert provides 6.9 ft of freeboard over the design water surface; bridge opening = 140 ft.

Constructability & detailing notes

Keep piers parallel to the flow; even 10° of skew measurably increases both scour and backwater.
Coordinate the low-chord elevation with the debris and ice regime, not just the water surface.
Temporary causeways and cofferdams contract the channel further — check the construction-stage hydraulics.

Check	Demand	Capacity / limit	Status
Freeboard ≥ required	3.00 ft	6.93 ft	PASS
Flow is not near critical (0.9 < Fr < 1.1 is unstable)	0.413	< 0.90 or > 1.10	PASS
Velocity ≤ 12 ft/s (channel-stability screening)	7.076 ft/s	12.0 ft/s	PASS
Opening ratio ≥ 0.90 (limits backwater)	0.943	0.900	PASS

Independent verification	Expected	Computed	Status
Continuity check Q = V·A	12000 cfs	matched by the normal-depth solution	VERIFIED

Assumptions & basis of design

Steady uniform flow in a prismatic trapezoidal channel; Manning's n from HDS-6 / FHWA channel tables.
Backwater is the simple energy difference — a full HEC-RAS bridge routine is required for design submittals.
Design discharge is normally the 100-yr event, checked against the 500-yr event for scour (HEC-18).

Hydraulics are acceptable; carry y_n and the contracted velocity into the scour module.

Module 61AASHTO LRFD 2.6.4.4.2 · HEC-18Chapters 15 · 16

Total scour by HEC-18 — contraction, local & degradation

Live-bed or clear-water contraction scour, the CSU pier-scour equation with K₁–K₄, abutment scour, long-term degradation, and the total scour envelope with the required pile tip elevation.

Approach depth y₁ftApproach velocity V₁ft/sApproach width W₁ftContracted width W₂ftApproach discharge Q₁cfsContracted discharge Q₂cfsBed D₅₀mmPier width aftPier lengthftFlow skewdegPier noseAbutment lengthftAbutment typeDepth at abutment y_eftUnit discharge q_ecfs/ftBed condition

Contraction, pier (CSU) and abutment (HIRE) scour combined into the total scour prism that sets the foundation elevation.

Bed condition

live bed

Contraction scour

2.564 ft

Pier scour

9.600 ft

Abutment scour

13.828 ft

Total at the pier

12.16 ft

Total at the abutment

16.39 ft

Derivation — equation, substitution, result

Critical velocity

\begin{aligned}&\;V_c = 11.17 y^{1/6} D_{50}^{1/3} (Eq. \mathrm{HEC}-18 6.1)\\[4pt]&=\;y = 10.00 \,\text{ft}, D_{50} = 0.00656 \,\text{ft}\\[4pt]&=\;\boxed{V_c = 3.069 \,\text{ft}/s \rightarrow \,\text{live}-\,\text{bed} \,\text{scour}}\end{aligned}

Contraction scour (live bed)

\begin{aligned}&\;y_{2}/y_{1} = (Q_{2}/Q_{1})^{6/7}(W_{1}/W_{2})^{k_{1}}\\[4pt]&=\;(1.000)^{6/7}(1.429)^{0.64}\\[4pt]&=\;\boxed{y_{2} = 12.564 \,\text{ft}}\end{aligned}

Contraction scour (clear water)

\begin{aligned}&\;y_{2} = [0.0077 q_{2}^{2}/D_m^{2/3}]^{3/7}\\[4pt]&=\;q_{2} = 85.714 \,\text{cfs}/\,\text{ft}\\[4pt]&=\;\boxed{y_{2} = 23.704 \,\text{ft}}\end{aligned}

Contraction scour depth

\begin{aligned}&\;y_s = y_{2} − y_{1}\\[4pt]&=\;\boxed{2.564 \,\text{ft}}\end{aligned}

Pier-scour correction factors

\begin{aligned}&\;K_{1} \,\text{shape}, K_{2} = (\operatorname{cos} \theta + (L/a) \operatorname{sin} \theta )^{0.65}, K_{3} \,\text{bed} \,\text{condition}\\[4pt]&=\;K_{1} = 1.00, \theta = 10^{\circ}, L/a = 7.50\\[4pt]&=\;\boxed{K_{2} = 1.712, K_{3} = 1.10}\end{aligned}

CSU pier scour

\begin{aligned}&\;y_s = 2.0 y_{1} K_{1}K_{2}K_{3} (a/y_{1})^{0.65} Fr_{1}^{0.43}\\[4pt]&=\;Fr_{1} = 0.334, a/y_{1} = 0.400\\[4pt]&=\;\boxed{12.963 \,\text{ft} \rightarrow \operatorname{use} 9.600 \,\text{ft} \,\text{after} \,\text{the} 2.4a / 3.0a \,\text{cap}}\end{aligned}

Abutment scour

\begin{aligned}&\;\mathrm{HIRE}: y_s = 2.27 K_{1} L^{0.43} y^{0.57} Fr^{0.61}\\[4pt]&=\;L = 60.0 \,\text{ft}, K_{1} = 0.55 \;\text{(spill)}\\[4pt]&=\;\boxed{13.828 \,\text{ft}}\end{aligned}

Total scour

\begin{aligned}&\;y_{total} = \,\text{contraction} + \,\text{local}\\[4pt]&=\;\boxed{\,\text{pier} 12.16 \,\text{ft}, \,\text{abutment} 16.39 \,\text{ft}}\end{aligned}

Detailing — plan, elevation and section

Elevation — scour prism

Plan — countermeasure

Section — apron

Constructability & detailing notes

Set the pile cut-off or footing bottom below the total scour line plus a 2 ft allowance for prediction uncertainty.
Detail the pile cap so that exposure by scour does not create a debris catcher or increase the effective pier width.
Record the as-built streambed elevation; every future scour evaluation is measured from it.

Check	Demand	Capacity / limit	Status
Pier scour ≤ 2.4a limit for Fr ≤ 0.8 (HEC-18)	12.963 ft	9.600 ft	REVIEW
Foundation extends below the total scour prismSet the footing or pile cut-off at least 14.2 ft below the streambed (total scour + 2 ft).	—	—	PASS
Check the 500-yr event as well (Extreme Event II)The check flood governs the foundation elevation on many sites.	—	—	PASS
Debris raft considered in the effective pier width aAdd the debris width to a where drift is likely — it often doubles the pier scour.	—	—	PASS

Independent verification	Expected	Computed	Status
Pier scour within the 2.4a physical limit	≤ 2.4 a	9.600	CHECK

Assumptions & basis of design

HEC-18 (5th ed.) equations: Laursen contraction scour, CSU pier scour, HIRE abutment scour.
Scour is a load condition, not a load: foundations are checked at Strength and Extreme Event limit states with the scoured profile removed.
K₃ = 1.1 for plane-bed and antidune flow; use 1.1 – 1.3 for larger dunes.

Found the pier below elevation −12.2 ft relative to the streambed, or provide countermeasures (riprap, articulated block) designed for the contracted velocity.

Module 62AASHTO LRFD HEC-23 · 2.6.4.4Chapters 15 · 16

Riprap and countermeasure design

Isbash / HEC-23 stone sizing from the design velocity, the D₅₀ and gradation, apron thickness and extent, filter requirements and the placement detail around a pier.

Local velocity Vft/sFlow depth yftStone specific gravity—Shape factor K—Side slope z:1—Layers of stone—

HEC-23 pier riprap sizing with the gradation class, apron thickness, plan extent and filter requirement.

D₅₀ required

3.96 in

Riprap class

Class I (D₅₀ = 6 in)

Layer thickness

12.0 in

Plan extent

20.0 ft

Side slope

2.0H:1V

Derivation — equation, substitution, result

Pier riprap size

\begin{aligned}&\;D_{50} = 0.692(\mathrm{KV})^{2}/[(S_s − 1)2g] \;\text{(HEC-23 Design Guideline 11)}\\[4pt]&=\;K = 0.89, V = 8.00 \,\text{ft}/s, S_s = 2.65\\[4pt]&=\;\boxed{D_{50} = 0.330 \,\text{ft} = 3.96 \,\text{in}}\end{aligned}

Gradation class

\begin{aligned}&\;Select \,\text{the} \,\text{next} \,\text{standard} \,\text{class} \ge D_{50},\,\text{req}\\[4pt]&=\;\boxed{Class I (D_{50} = 6 \,\text{in})}\end{aligned}

Layer thickness

\begin{aligned}&\;t \ge \operatorname{max}\;\text{(3D₅₀ placed dry / 1.5D₅₀ min, 12 in)}\\[4pt]&=\;2 \times 6 \,\text{in}\\[4pt]&=\;\boxed{12.0 \,\text{in}}\end{aligned}

Plan extent around the pier

\begin{aligned}&\;\,\text{extend} 2a (\operatorname{or} 2y) \,\text{beyond} \,\text{the} \,\text{pier} \,\text{face} \,\text{in} \,\text{all} \,\text{directions}\\[4pt]&=\;\boxed{20.0 \,\text{ft}}\end{aligned}

Filter

\begin{aligned}&\;Geotextile \operatorname{or} \,\text{granular} \,\text{filter} \,\text{under} \,\text{the} \,\text{full} \,\text{footprint}\\[4pt]&=\;\boxed{\,\text{required} — \,\text{riprap} \,\text{without} a \,\text{filter} \,\text{winnows} \,\text{out}}\end{aligned}

Detailing — plan, elevation and section

Elevation — scour prism

Plan — countermeasure

Section — apron

Constructability & detailing notes

Place riprap in the dry where the schedule allows; underwater placement requires a 50 % thickness increase.
Never dump riprap from height onto a geotextile — it tears the fabric and the apron fails from below.
Riprap is a monitored countermeasure, not a permanent fix; write the inspection trigger into the Plan of Action.

Check	Demand	Capacity / limit	Status
Side slope ≤ 2H:1V for stable riprap	2.0H:1V	2H:1V or flatter	PASS
Filter layer specifiedGeotextile with an apparent opening size compatible with the bed gradation.	—	—	PASS
Top of riprap at or below the streambedBurying the apron keeps it from becoming an obstruction and a debris catcher.	—	—	PASS
Riprap requires a monitoring plan (HEC-23)Riprap is not a permanent fix for a scour-critical bridge — pair it with inspection.	—	—	PASS

Independent verification	Expected	Computed	Status
Selected class meets the computed D₅₀	3.96	Class I (D₅₀ = 6 in)	VERIFIED

Assumptions & basis of design

HEC-23 pier riprap guidance; K = 1.5 for rectangular piers and 1.7 for round noses in the alternative Isbash form.
Specific gravity S_s = 2.65 for typical quarried stone.
Velocity is the local (contracted) velocity at the pier, not the average channel velocity.

Specify Class I (D₅₀ = 6 in) riprap, 12 in thick, over a geotextile filter, extending 20.0 ft beyond each pier face with the top set flush with the streambed.

Module 63AASHTO LRFD 3.10 / 4.7.4Chapters 15 · 16

Seismic demand, R factors and displacement compatibility

Site class coefficients through the design spectrum, elastic demand, the R-modified design forces for substructure and connections, and the displacement-compatibility and P-Δ checks.

Tributary weight WkipFundamental period TsS_DSgS_D1gSite classResponse modification R—Column heightftColumn φM_pkip·ftColumns in the bent—Seat length providedinSpan lengthftColumn / abutment heightftSkewdegOperational classification

Design spectrum, zone, elastic and inelastic demand, capacity-protection overstrength and the minimum support length.

S_a

0.5000 g

Seismic zone

3

Elastic V_e

2000.0 kip

Design V

666.7 kip

Column M demand

9333 kip·ft

Overstrength shear V_o

514.3 kip

Support length N req.

13.41 in

Derivation — equation, substitution, result

Design response spectrum

\begin{aligned}&\;T_s = S_{D1}/S_{DS}, T_{0} = 0.2T_s; S_a = S_{DS} (T_{0} \le T \le T_s), S_{D1}/T (T > T_s) (§3.10.4.2)\\[4pt]&=\;S_{DS} = 0.750, S_{D1} = 0.400, T = 0.800 s, \,\text{site} \,\text{class} D\\[4pt]&=\;\boxed{T_{0} = 0.107 s, T_s = 0.533 s, S_a = 0.5000 g}\end{aligned}

Seismic zone

\begin{aligned}&\;Table 3.10.6-1 \,\text{by} S_{D1}\\[4pt]&=\;S_{D1} = 0.400\\[4pt]&=\;\boxed{Zone 3}\end{aligned}

Elastic force

\begin{aligned}&\;V_e = S_a W \;\text{(uniform-load / single-mode method)}\\[4pt]&=\;0.5000 \times 4000\\[4pt]&=\;\boxed{2000.0 \,\text{kip}}\end{aligned}

Response modification

\begin{aligned}&\;V = V_e/R \;\text{(Table 3.10.7.1-1)}; R = 1.5 \operatorname{max} \operatorname{for} \,\text{operational} \,\text{classification}\\[4pt]&=\;R \,\text{used} = 3.00\\[4pt]&=\;\boxed{V = 666.7 \,\text{kip}}\end{aligned}

Column moment demand

\begin{aligned}&\;M = (V/n_{col}) H\\[4pt]&=\;(666.7/2)(28.0)\\[4pt]&=\;\boxed{9333 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Overstrength (capacity protection)

\begin{aligned}&\;M_o = 1.2 M_p (§3.10.9.4.3); V_o = M_o n/H\\[4pt]&=\;1.2 \times 6000\\[4pt]&=\;\boxed{M_o = 7200 \,\text{kip}\cdot \,\text{ft}, V_o = 514.3 \,\text{kip}}\end{aligned}

Minimum support length

\begin{aligned}&\;N = (8 + 0.02L + 0.08H)(1 + 0.000125 S^{2}) (Eq. 4.7.4.4-1)\\[4pt]&=\;L = 140 \,\text{ft}, H = 28.0 \,\text{ft}, S = 15^{\circ}\\[4pt]&=\;\boxed{N = 13.41 \,\text{in} \,\text{vs} \,\text{provided} 24.00 \,\text{in}}\end{aligned}

Detailing — plan, elevation and section

Section

Elevation

Plan at cap

Constructability & detailing notes

The plastic-hinge zone needs tight spiral pitch and continuous confinement — do not allow the contractor to open it up for access.
Cap, joint and foundation are capacity-protected: they are designed for 1.2 M_p, not for the elastic demand divided by R.
Restrainers and shear keys must be detailed with a defined load path and a stated fuse capacity.

Check	Demand	Capacity / limit	Status
Column M_p ≥ demand	9333 kip·ft	6000 kip·ft	REVIEW
Seat length N provided ≥ required	13.41 in	24.00 in	PASS
Cap, joint and foundation designed for V_o / M_o, not for VCapacity protection: the plastic hinge must form in the column, never in the cap or footing.	—	—	PASS
Zone 3 – 4 requires full ductile detailing (§5.10.11.4)Provide plastic-hinge confinement, lap-splice exclusion zones and shear designed for V_o.	—	—	REVIEW

Independent verification	Expected	Computed	Status
Column capacity exceeds the reduced demand	M_p ≥ M demand	9333	CHECK
Support length adequate	13.41	provided seat	VERIFIED

Assumptions & basis of design

Uniform-load method of §4.7.4.3.2c — valid for regular bridges; irregular or long bridges require multimode analysis.
R factors from Table 3.10.7.1-1 (single columns 3.0, multiple-column bents 5.0), reduced to 1.5 for operational bridges.
Overstrength factor 1.2 for ASTM A706 reinforcement; use 1.4 for A615.

Increase the column size or longitudinal steel, or accept a lower R by adding redundancy to the bent.

Module 64AASHTO LRFD 10.5.4.2 / 3.11.8Chapters 15 · 16

Liquefaction screening & downdrag

Simplified cyclic stress ratio CSR from a_max and r_d, the cyclic resistance ratio from corrected blow counts, the factor of safety with depth, and the downdrag load added to the pile demand.

Depth of layerftGroundwater depthftMoist unit weightpcfSaturated unit weightpcf(N₁)₆₀blows/ftFines content%a_maxgMoment magnitude M_w—

Simplified SPT-based triggering analysis: CSR from the ground motion, CRR from the corrected blow count, and the factor of safety.

σ′_v

1.566 ksf

CSR

0.3002

(N₁)₆₀cs

16.07

CRR (corrected)

0.2232

FS liquefaction

0.744

Verdict

liquefaction likely

Derivation — equation, substitution, result

Stresses

\begin{aligned}&\;\sigma _v = \Sum \gamma h, u = \gamma _w(z − z_w), \sigma '_v = \sigma _v − u\\[4pt]&=\;z = 20.0 \,\text{ft}, \mathrm{GWT} = 6.0 \,\text{ft}\\[4pt]&=\;\boxed{\sigma _v = 2.440 \,\text{ksf}, \sigma '_v = 1.566 \,\text{ksf}}\end{aligned}

Stress-reduction coefficient

\begin{aligned}&\;r_d = 1 − 0.00765z (z \le 30 \,\text{ft})\\[4pt]&=\;\boxed{r_d = 0.847}\end{aligned}

Cyclic stress ratio

\begin{aligned}&\;\mathrm{CSR} = 0.65(a_{\operatorname{max}}/g)(\sigma _v/\sigma '_v) r_d \;\text{(Seed–Idriss)}\\[4pt]&=\;0.65(0.350)(1.558)(0.847)\\[4pt]&=\;\boxed{\mathrm{CSR} = 0.3002}\end{aligned}

Fines correction

\begin{aligned}&\;(N_{1})_{60}\,\text{cs} = (N_{1})_{60} + \Delta (N_{1})_{60} \;\text{(Youd et al. 2001)}\\[4pt]&=\;\mathrm{FC} = 12 \%, \Delta = 2.07\\[4pt]&=\;\boxed{(N_{1})_{60}\,\text{cs} = 16.07}\end{aligned}

Cyclic resistance ratio at M 7.5

\begin{aligned}&\;\mathrm{CRR}_{7}._{5} = 1/(34 − N) + N/135 + 50/(10N + 45)^{2} − 1/200\\[4pt]&=\;N = 16.07\\[4pt]&=\;\boxed{\mathrm{CRR}_{7}._{5} = 0.1710}\end{aligned}

Magnitude and overburden scaling

\begin{aligned}&\;\mathrm{MSF} = 10^{2.24}/M_w^{2.56}, K_\sigma = (\sigma '_v/p_a)^{−0.3}\\[4pt]&=\;M_w = 7.00\\[4pt]&=\;\boxed{\mathrm{MSF} = 1.193, K_\sigma = 1.094}\end{aligned}

Factor of safety

\begin{aligned}&\;\mathrm{FS} = \mathrm{CRR}\cdot \mathrm{MSF}\cdot K_\sigma /\mathrm{CSR}\\[4pt]&=\;0.2232/0.3002\\[4pt]&=\;\boxed{\mathrm{FS} = 0.744}\end{aligned}

Detailing — plan, elevation and section

Plan — pile layout

Section — cap

Elevation

Constructability & detailing notes

Where liquefaction is predicted, ground improvement (stone columns, deep soil mixing) is often cheaper than lengthening every pile.
Instrument the site with piezometers if staged fill is used — excess pore pressure during construction is a real hazard.
Show the liquefiable layer on the foundation plan so the driving records can confirm it in the field.

Check	Demand	Capacity / limit	Status
FS ≥ 1.2 (AASHTO screening)	0.744	1.200	REVIEW
(N₁)₆₀cs ≥ 30 — non-liquefiable regardless of CSR	16.07	30	REVIEW
Downdrag and loss of lateral support evaluated for liquefiable layersRemove the layer's p-y resistance and add post-liquefaction downdrag to the pile design.	—	—	REVIEW
Lateral spread evaluated where a free face existsUse the Youd–Hansen–Bartlett empirical model, then check the piles for the imposed soil displacement.	—	—	PASS

Independent verification	Expected	Computed	Status
Triggering factor of safety	≥ 1.20	0.744	CHECK

Assumptions & basis of design

Simplified NCEER/Youd et al. (2001) SPT-based procedure as referenced by AASHTO §10.5.4.2.
a_max is the peak ground surface acceleration = F_pga PGA from the design hazard.
Clean-sand equivalence is applied for fines contents above 5 %; the correction is capped at 5.5 blows.

Design piles to punch through the liquefiable layer into competent material, neglect its side friction, add its post-liquefaction downdrag, and consider ground improvement (stone columns, deep soil mixing).

Chapters 18 – 20

Construction engineering, inspection and rating

Construction-stage combinations, LRFR rating factors and tonnage, deflection and span-depth serviceability, and remaining fatigue life.

Module 65AASHTO LRFD MBE 6A.4.2Chapters 19 · 20

Load rating (LRFR)

Inventory and Operating rating factors and rating tonnage for a rating vehicle under the LRFR methodology.

Capacity Ck-ftDC effectk-ftDW effectk-ftPermanent effect Pk-ftRating-vehicle LL+IMk-ftγDCγDWRating combinationRating vehicle weight Wton

RF = (C − γDC·DC − γDW·DW ± γP·P) / (γLL·(LL+IM)) · RT = RF · W

Inventory RF

1.716

Operating RF

2.288

Inventory rating tonnage

68.6 ton

Operating rating tonnage

91.5 ton

Both Inventory and Operating rating factors exceed 1.0; the rating vehicle is adequately carried.

Detailing sketchLRFR rating factors vs RF = 1.0 reference

Inventory RF = 1.716, Operating RF = 2.288. Bars below the RF = 1.0 line indicate posting or restriction is required at that rating level (MBE §6A.4.2.1).

Module 66AASHTO LRFD 3.4.2 / 2.5.2.6 / 6.6.1.2.5 · MBE 6A.4.2Chapters 19 · 20

Construction stage, serviceability & remaining fatigue life

Strength I with construction loads, the LRFR inventory and operating rating factors and tonnages, the L/800 deflection and 0.033L depth guidance, and N = A/(Δf)³ fatigue life against the 75-year ADTT cycle count.

DC effectk-ftDW effectk-ftConstruction load effectk-ftLL+IM effectk-ftCapacity C = φ_c φ_s φR_nk-ftRating vehicle weighttonsSpan LftProvided depthinLive-load deflectioninADTT (single lane)trucks/dayStress range ΔfksiDetail constant A×10⁸ ksi³

Fatigue constant A (×10⁸ ksi³): 250 for A, 120 B, 61 B′, 44 C, 22 D, 11 E, 3.9 E′.

Construction Strength I effect

2625.0

Inventory RF

1.508

Operating RF

1.955

Inventory / Operating rating

54.3 / 70.4 tons

Live-load deflection limit

1.500 in

Predicted fatigue life

55.8 years

Detailing sketchRating summary

Inventory / operating rating: 54.3 / 70.4 tons.

Step	Equation	Substitution	Result
Construction combination	U = 1.25DC + 1.50DW + 1.50CE	1.25·1500.0 + 1.50·300.0 + 1.50·200.0	2625.0 (k-ft or kip)
Rating dead load effect	γ_DC DC + γ_DW DW	1.25(1500.0 + 300.0)	2250.0
Inventory rating factor	RF = (C − γ_D D)/(γ_L(LL+IM))	(7000.0 − 2250.0)/(1.75·1800.0)	1.508
Operating rating factor	RF = (C − γ_D D)/(1.35(LL+IM))	(7000.0 − 2250.0)/(1.35·1800.0)	1.955
Rating tonnage	RT = RF × W_vehicle	1.508·36.0 ; 1.955·36.0	Inventory 54.3 t, Operating 70.4 t
Live-load deflection limit	Δ ≤ L/800 (L/1000 with pedestrians)	1200/800	1.500 in (provided 1.100 in)
Span-to-depth guidance	d ≥ 0.033L (composite) ; 0.027L (steel only)	0.033·1200	39.60 in (provided 54.00 in)
Fatigue life	N = A/(Δf)³	44.00×10⁸/(6.00)³	20.37 million cycles
Design cycle count	N_75 = 365(75)n·ADTT_SL	365·75·1.0·1000	27.38 million cycles

OKInventory rating factor ≥ 1.0demand 1.000capacity 1.508
OKOperating rating factor ≥ 1.0demand 1.000capacity 1.955
OKLive-load deflection ≤ L/800demand 1.100 incapacity 1.500 in
OKDepth ≥ 0.033Ldemand 54.00 incapacity 39.60 in
NGFatigue life ≥ 75 yearsdemand 75 yearscapacity 55.8 years

Inventory RF = 1.508 (54.3 tons) and Operating RF = 1.955 (70.4 tons) — the bridge carries legal loads without posting. Deflection 1.100 in vs the 1.500 in limit; minimum recommended depth 39.60 in (steel alone 32.40 in).

Assumptions & code basis

Construction-stage combination: Strength I with construction loads per §3.4.2.1 — 1.25DC + 1.50DW + 1.50 of the construction load CE, with φ reduced for non-composite girders during deck placement.
Load rating by LRFR (MBE §6A.4.2): RF = (C − γ_DC DC − γ_DW DW ∓ γ_P P)/(γ_LL (LL+IM)); Inventory γ_LL = 1.75, Operating γ_LL = 1.35.
Live-load deflection limit L/800 for vehicular bridges and L/1000 where pedestrians are present (§2.5.2.6.2) — optional but almost always enforced by owners.
Span-to-depth guidance §2.5.2.6.3: 0.033L for composite I-girder overall depth and 0.027L for the steel section alone.
Fatigue life from the detail-category constant A: N = A/(Δf)³, compared with the 75-year cycle count 365·75·n·ADTT_SL (§6.6.1.2.5).

Module 67AASHTO LRFD 3.4.2 · Guide Design Specifications for Bridge Temporary WorksChapter 18

Falsework and temporary works design

Construction dead, live and equipment loads, the shoring post and stringer demands, the allowable stresses for temporary works, foundation bearing on mudsills and the lateral-stability bracing requirement.

Form heightftPlacement rateft/hrConcrete temperature°FUnit-weight factor C_w—Chemistry factor C_c—Wet concrete + formsksfShore spacing XftShore spacing YftShore capacitykipPiece weightkipPicks in the sequence—Rigging weightkipCrane chart capacitykipPick radiusftDynamic factor—

ACI 347 formwork pressure, shore loads on the tributary grid, and the crane pick check for the erection plan.

Form pressure

793 psf

Post load

2.40 kip

Post DCR

0.400

Total pick weight

72.4 kip

Crane DCR

0.659

Picks required

8

Derivation — equation, substitution, result

Formwork lateral pressure

\begin{aligned}&\;p = C_w C_c[150 + 9000R/T] (\mathrm{ACI} 347 Eq. 2.1)\\[4pt]&=\;R = 5.0 \,\text{ft}/\,\text{hr}, T = 70 ^{\circ}F, C_w = 1.00, C_c = 1.00\\[4pt]&=\;\boxed{793 \,\text{psf}}\end{aligned}

Bounds on pressure

\begin{aligned}&\;600 \,\text{psf} \le p \le \operatorname{min}\;\text{(computed, 2000 + 150h, γh)}\\[4pt]&=\;\gamma h = 1800 \,\text{psf}\\[4pt]&=\;\boxed{p_{design} = 793 \,\text{psf} = 0.793 \,\text{ksf}}\end{aligned}

Falsework vertical load

\begin{aligned}&\;w = \;\text{(wet concrete + forms + 20 psf construction LL)}\\[4pt]&=\;0.130 + 0.020 \,\text{ksf}\\[4pt]&=\;\boxed{0.150 \,\text{ksf}}\end{aligned}

Post load

\begin{aligned}&\;P = w \times \,\text{tributary} \,\text{area}\\[4pt]&=\;0.150 \times (4.00 \times 4.00)\\[4pt]&=\;\boxed{2.40 \,\text{kip}/\,\text{post}}\end{aligned}

Post utilisation

\begin{aligned}&\;\mathrm{DCR} = P/P_{allow}\\[4pt]&=\;2.40/6.00\\[4pt]&=\;\boxed{0.400}\end{aligned}

Crane pick

\begin{aligned}&\;W_{total} = \;\text{(Wpiece + rigging)} \times \,\text{dynamic} \,\text{factor}\\[4pt]&=\;(60.0 + 3.0) \times 1.15\\[4pt]&=\;\boxed{72.4 \,\text{kip} \operatorname{at} 70.0 \,\text{ft} \,\text{radius}}\end{aligned}

Crane utilisation

\begin{aligned}&\;\mathrm{DCR} = W_{total}/\,\text{capacity} \operatorname{at} \,\text{radius} \le 0.75 \;\text{(typical safety practice)}\\[4pt]&=\;72.4/110.0\\[4pt]&=\;\boxed{0.659}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchFalsework shoring grid and formwork lateral-pressure diagram

Shore grid at 4.0 ft × 4.0 ft; governing post demand-capacity ratio = 0.40. Peak formwork pressure p_max = 793 psf over the 12.0 ft form height; crane pick at 66 % of chart capacity.

Constructability & detailing notes

Falsework drawings must be sealed by a licensed engineer and independently checked — the FIU and Quebec collapses were both temporary-condition failures.
Brace the shoring for 2 % of the vertical load in both directions, and check the wind case on the open frame.
Survey the falsework before, during and after the pour; unexpected settlement is the first sign of a bearing problem.
Do not release shoring until cylinder breaks confirm the specified release strength.

Check	Demand	Capacity / limit	Status
Post DCR ≤ 1.0	0.400	1.000	PASS
Crane at ≤ 75 % of chart capacity	0.659	0.750	PASS
Falsework braced for 2 % of vertical load laterally (ACI 347)Provide diagonal bracing in both directions and check the wind case on the open frame.	—	—	PASS
Foundation / mudsill bearing checkedFalsework settlement is a leading cause of construction-stage cracking — survey during placement.	—	—	PASS
Independent falsework design check by a licensed engineerRequired by most owners after the FIU and Quebec lessons — never let the erector self-certify.	—	—	PASS

Independent verification	Expected	Computed	Status
Shore utilisation	≤ 1.000	0.400	VERIFIED
Crane at or below 75 % of chart	≤ 0.750	0.659	VERIFIED

Assumptions & basis of design

ACI 347R-14 formwork pressures with C_w for unit weight and C_c for cement type/admixtures.
Construction live load of 20 psf on horizontal surfaces plus 50 psf where motorised buggies operate.
Crane charts are for level, fully outrigged, 360° operation — derate for on-rubber or partial outrigger picks.

Falsework and pick are within limits; require a pre-pour inspection and monitor deflection during placement.

Module 68AASHTO LRFD 5.12.5 / 3.4.2Chapter 18

Segmental erection — unbalanced moment and stability

Cantilever moments during balanced-cantilever erection, the unbalanced segment plus travelling-form case, wind uplift, the temporary post-tensioning bar demand and the pier stability check.

Segment weightkipSegment lengthftSegments each side—Form traveler weightkipTraveler CG beyond tipftConstruction wind P_zksfBox depthftDistributed CLLklfConcentrated CLLkipPier moment capacitykip·ftCantilever tendon forcekipTendon eccentricityin

Balanced-cantilever erection: unbalanced segment, form traveler, construction live load and wind on the pier segment.

Cantilever length

128.0 ft

Unbalanced M (segment)

14400 kip·ft

Traveler M

11040 kip·ft

Wind M

1966 kip·ft

Factored M_u

34404 kip·ft

Pier DCR

0.860

Balanced baseline

0 kip·ft

perfectly balanced erection produces no net pier moment

Derivation — equation, substitution, result

Cantilever geometry

\begin{aligned}&\;L_c = n \times L_{seg}; \,\text{lever} \,\text{arm} \operatorname{of} \,\text{the} \,\text{unbalanced} \,\text{segment} = (n − 0.5)L_{seg}\\[4pt]&=\;8 \times 16.00 \,\text{ft}\\[4pt]&=\;\boxed{L_c = 128.0 \,\text{ft}, \,\text{arm} = 120.0 \,\text{ft}}\end{aligned}

Unbalanced segment moment

\begin{aligned}&\;M = W_{seg} \times \,\text{arm} \;\text{(one segment out of balance)}\\[4pt]&=\;120.0 \times 120.0\\[4pt]&=\;\boxed{14400 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Form traveler

\begin{aligned}&\;M = W_{tr}(L_c + a_{tr})\\[4pt]&=\;80.0(128.0 + 10.0)\\[4pt]&=\;\boxed{11040 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Construction live load

\begin{aligned}&\;\,\text{distributed} \mathrm{CLL} \,\text{over} \,\text{the} \,\text{cantilever} + \,\text{concentrated} \mathrm{CLL} \operatorname{at} \,\text{the} \,\text{tip} (§5.14.2.3.2)\\[4pt]&=\;0.0100 \,\text{klf} \operatorname{and} 20.0 \,\text{kip}\\[4pt]&=\;\boxed{2642 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Wind on the cantilever

\begin{aligned}&\;M = (P_z d)L_c^{2}/2\\[4pt]&=\;0.0200 \,\text{ksf} \times 12.0 \,\text{ft}\\[4pt]&=\;\boxed{1966 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Factored construction combination

\begin{aligned}&\;M_u = 1.1\;\text{(DC + traveler)} + 1.5 \mathrm{CLL} + 1.25 W \;\text{(Table 3.4.1-1 construction case)}\\[4pt]&=\;\boxed{34404 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Cantilever tendon couple

\begin{aligned}&\;M_{pt} = P e\\[4pt]&=\;6000 \,\text{kip} \times 60.0 \,\text{in}\\[4pt]&=\;\boxed{30000 \,\text{kip}\cdot \,\text{ft} \,\text{available} \operatorname{at} \,\text{the} \,\text{pier} \,\text{segment}}\end{aligned}

Pier utilisation

\begin{aligned}&\;\mathrm{DCR} = M_u/M_r,\,\text{pier}\\[4pt]&=\;34404/40000\\[4pt]&=\;\boxed{0.860}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchBalanced-cantilever erection — unbalanced segment and form traveler

8 segments each side at 16.0 ft; form traveler cantilevers 10.0 ft beyond the tip. Pier moment demand-capacity ratio under the unbalanced case = 0.86.

Constructability & detailing notes

The erection sequence is a contract document: casting order, maximum out-of-balance and tendon stressing stages must all be shown.
Temporary PT bars or a falsework tower at the pier are normally required to carry the unbalanced case.
Survey the cantilever tip elevation after every segment; geometry control errors accumulate and cannot be fixed at closure.
Never stress a tendon before the concrete reaches the specified release strength — verify with match-cured cylinders.

Check	Demand	Capacity / limit	Status
Pier moment capacity ≥ M_u	34404 kip·ft	40000 kip·ft	PASS
Temporary supports or stressed bars provided at the pier segmentMost balanced-cantilever bridges need temporary towers or PT bars to resist the unbalanced case.	—	—	PASS
One-segment-out-of-balance assumption stated on the plansThe erection sequence must be a contract document, not a contractor option.	—	—	PASS
Wind-on-traveler case checked with the traveler at the tip	—	—	PASS

Independent verification	Expected	Computed	Status
Pier capacity exceeds the factored erection moment	DCR ≤ 1.000	0.860	VERIFIED

Assumptions & basis of design

AASHTO LRFD §5.14.2 construction load cases: distributed CLL of 0.010 ksf and a concentrated CLL of 20 kip at the cantilever tip are typical.
Load factors for construction: 1.1 on dead load and equipment, 1.5 on construction live load, 1.25 on construction wind.
Segment weights include the wet concrete of the segment being cast plus the form traveler self-weight.

The pier can carry the erection case; publish the segment casting order and the maximum permitted out-of-balance condition on the plans.

Module 69AASHTO LRFD MBE 4 · NBI Coding GuideChapter 19

Inspection, condition rating & deterioration modelling

NBI condition ratings converted to section-loss estimates, the reduced section properties, the remaining capacity ratio and the projected time to the intervention threshold.

Deck CS1 quantitysfDeck CS2sfDeck CS3sfDeck CS4sfSuperstructure CS1lfSuperstructure CS2lfSuperstructure CS3lfSuperstructure CS4lfSubstructure CS1eaSubstructure CS2eaSubstructure CS3eaSubstructure CS4eaNBI item 58 (deck)—NBI item 59—NBI item 60—ADTveh/dayDetour lengthmiLanes carried—NBI item 113 (scour)—Months since inspectionmoFracture-critical members

Element-level condition states rolled into a health index, with NBI classification, inspection interval and the scour appraisal.

Deck HI

89.2

Superstructure HI

89.6

Substructure HI

82.5

Bridge health index

87.3

NBI classification

Fair

Inspection status

current

Annual detour cost

$23.65 M

Lanes carried

4

Derivation — equation, substitution, result

Element health index

\begin{aligned}&\;\mathrm{HI} = \Sum (q_i \times w_i)/\Sum q_i \times 100 \operatorname{with} w = 1.00/0.60/0.25/0.00 \operatorname{for} \mathrm{CS}1–\mathrm{CS}4\\[4pt]&=\;\,\text{deck} 6000/1200/400/50, \,\text{super} 1500/300/90/10, \,\text{sub} 200/60/30/5\\[4pt]&=\;\boxed{\mathrm{HI}: \,\text{deck} 89.2, \,\text{super} 89.6, \,\text{sub} 82.5}\end{aligned}

Weighted bridge health index

\begin{aligned}&\;\mathrm{HI} = 0.30 \,\text{deck} + 0.40 \,\text{superstructure} + 0.30 \,\text{substructure}\\[4pt]&=\;\boxed{87.3 / 100}\end{aligned}

NBI condition classification

\begin{aligned}&\;\operatorname{min}\;\text{(item 58, 59, 60)}: \le 4 Poor, 5–6 Fair, \ge 7 Good (23 \mathrm{CFR} 650)\\[4pt]&=\;6 / 6 / 5\\[4pt]&=\;\boxed{Fair}\end{aligned}

Inspection interval

\begin{aligned}&\;Routine \le 24 \,\text{months}; \,\text{up} \operatorname{to} 48 \,\text{months} \,\text{under} \,\text{an} \,\text{approved} \,\text{risk}-\,\text{based} (\mathrm{RBI}) \,\text{method}\\[4pt]&=\;\,\text{last} \,\text{inspection} 20 \,\text{months} \,\text{ago}\\[4pt]&=\;\boxed{\,\text{allowable} 24 \,\text{months} \rightarrow \,\text{current}}\end{aligned}

Scour appraisal

\begin{aligned}&\;\mathrm{NBI} \,\text{item} 113: 0–3 \,\text{scour} \,\text{critical}, 4 \,\text{action} \,\text{required}, \ge 5 \,\text{stable}\\[4pt]&=\;\,\text{code} 5\\[4pt]&=\;\boxed{\,\text{stable}}\end{aligned}

Detour user cost

\begin{aligned}&\;\,\text{user} \,\text{cost} \approx \mathrm{ADT} \times \,\text{detour} \,\text{length} \times 365 \times \$/\,\text{veh}-\,\text{mi}\\[4pt]&=\;18000 \times 6.0 \,\text{mi} \times 365 \times \$0.60\\[4pt]&=\;\boxed{\$23.65 M \operatorname{per} \,\text{year} \operatorname{of} \,\text{closure}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchElement condition-state distribution and bridge health index

Each bar shows the proportion of element quantity in condition states 1 (good) through 4 (severe) for the deck, superstructure and substructure. Composite health index = 87.3 / 100.

Constructability & detailing notes

Photograph and station every CS3/CS4 quantity — the next inspector needs to know whether the defect grew.
Element quantities must sum to the total element quantity in every inspection cycle; a mismatch invalidates the health index trend.
Access equipment (snooper, rope, UAV) should be planned from the previous cycle's findings, not decided on site.

Check	Demand	Capacity / limit	Status
Routine inspection within the allowable interval	20 months	24 months	PASS
Fracture-critical members inspected hands-on at ≤ 24 months	—	—	PASS
Scour appraisal ≥ 4	5	≥ 4	PASS
Health index ≥ 70 for a well-performing asset	87.3	70.0	PASS
Interval 24 months documented in the inspection program	—	—	PASS

Independent verification	Expected	Computed	Status
Inspection interval compliance	within the allowable interval	current	VERIFIED

Assumptions & basis of design

AASHTO Manual for Bridge Element Inspection condition states CS1 (good) through CS4 (severe).
Health-index weights are the common FHWA/AASHTOWare convention; agencies may use their own weighting.
NBI ratings 0–9 per the FHWA Recording and Coding Guide; item 113 codes the scour appraisal.

Condition is acceptable; continue routine inspection and use the health index to prioritise preservation funding.

Module 70AASHTO LRFD MBE 6A.4 / 6A.8Chapters 19 · 20

LRFR rating — design, legal and permit vehicles

General rating equation with condition, system and φ factors, the Design (Inventory/Operating), Legal and Permit rating factors, posting load determination and rating tonnage.

Nominal capacity φR_nkip·ftDC effectkip·ftDW effectkip·ftPermanent load Pkip·ftLL + IM effectkip·ftCondition factor φ_cSystem factor φ_sRating levelPermit vehicle weighttonsRating vehicle weighttonsPosting vehicle weighttons

LRFR rating factor with condition and system factors, the rating in tons, permit screening and the posting load.

φ_c φ_s

0.950

Adjusted capacity

5700 kip·ft

Rating factor RF

0.880

Rating (tons)

31.67 tons

Level

Inventory (γ = 1.75)

Posting

not required

Derivation — equation, substitution, result

Condition and system factors

\begin{aligned}&\;\phi _c \;\text{(Table 6A.4.2.3-1)}, \phi _s \;\text{(Table 6A.4.2.4-1)}, \phi _c\phi _s \ge 0.85\\[4pt]&=\;\,\text{fair} \rightarrow \phi _c = 0.95; \,\text{redundant} \rightarrow \phi _s = 1.00\\[4pt]&=\;\boxed{\phi _c\phi _s = 0.950}\end{aligned}

Adjusted capacity

\begin{aligned}&\;C = \phi _c \phi _s \phi R_n\\[4pt]&=\;0.950 \times 6000\\[4pt]&=\;\boxed{5700 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Rating factor

\begin{aligned}&\;\mathrm{RF} = [C − \gamma _{DC} \mathrm{DC} − \gamma _{DW} \mathrm{DW} − \gamma _P P]/(\gamma _{LL} (\mathrm{LL} + \mathrm{IM})) (Eq. 6A.4.2.1-1)\\[4pt]&=\;[5700 − 1.25(1800) − 1.50(350) − 0]/(1.75 \times 1900)\\[4pt]&=\;\boxed{\mathrm{RF} = 0.880 \;\text{(Inventory)}}\end{aligned}

Rating in tons

\begin{aligned}&\;\mathrm{RT} = \mathrm{RF} \times W \;\text{(weight of the rating vehicle)}\\[4pt]&=\;0.880 \times 36.0 \,\text{tons}\\[4pt]&=\;\boxed{31.67 \,\text{tons}}\end{aligned}

Posting

\begin{aligned}&\;Posting \,\text{load} = \mathrm{RF} \times W_{posting} \;\text{(MBE §6A.8.3)} \,\text{when} \mathrm{RF} < 1.0 \operatorname{at} \,\text{the} Operating \,\text{level}\\[4pt]&=\;0.880 \times 36.0 \,\text{tons}\\[4pt]&=\;\boxed{\,\text{no} \,\text{posting} \,\text{required}}\end{aligned}

Permit screening

\begin{aligned}&\;Compare \,\text{the} \,\text{permit} \,\text{vehicle} \,\text{weight} \,\text{against} \mathrm{RT}\\[4pt]&=\;\,\text{permit} = 80.0 \,\text{tons}\\[4pt]&=\;\boxed{\,\text{permit} \,\text{requires} a \,\text{refined} \,\text{analysis} \operatorname{or} \,\text{escort}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchLRFR rating factor gauge and posting

Rating factor RF = 0.88 corresponds to a rating load of 31.7 tons. Posting required at 0.0 tons. Permit vehicle screening: does not pass — route or escort restriction required.

Constructability & detailing notes

Field-measure the wearing surface thickness: γ_DW drops from 1.50 to 1.25 when the thickness is measured rather than assumed.
Document the controlling member, section and limit state — the rating is only meaningful with that context.
Re-rate after every rehabilitation, deck overlay or change in section loss.

Check	Demand	Capacity / limit	Status
RF ≥ 1.0	0.880	1.000	REVIEW
Permit vehicle within the operating rating	80.0 tons	31.67 tons	REVIEW
Posting required when the Operating RF < 1.0	—	—	PASS
Consider a refined analysis or load test before postingRefined distribution factors and measured section properties often lift RF above 1.0.	—	—	PASS

Independent verification	Expected	Computed	Status
Rating factor at or above 1.0	≥ 1.000	0.880	CHECK

Assumptions & basis of design

AASHTO Manual for Bridge Evaluation, LRFR method, Strength I; γ_DC = 1.25, γ_DW = 1.50 (1.25 with a field-measured thickness).
Live-load effect includes the appropriate distribution factor and dynamic allowance (IM = 33 %, or 10 % for fatigue).
Legal-load ratings use γ_LL from Table 6A.4.4.2.3a-1 as a function of ADTT.

RF = 0.88 controls. Before posting, refine the analysis (measured deck thickness, actual distribution, composite action), then consider strengthening with FRP or external post-tensioning.

Module 71AASHTO LRFD MBE 6A · ACI 440.2RChapter 20

Strengthening and rehabilitation design

Capacity deficit from the rating, external post-tensioning force, FRP laminate area with the ACI 440 strain limit, added-plate composite action, and the restored rating factor after strengthening.

Strengthening methodRequired M_ukip·ftExisting M_nkip·ftSection width binEffective depth dinOverall depth hinf′_cksif_yksiExisting A_sin²E_cksiI_crackedin⁴M at installation (DL)kip·ftPly thickness t_finNumber of plies—Laminate widthinFRP modulus E_fksiFRP rupture strain ε*_fu—Environmental factor C_E—External PT forcekipPT eccentricityinPT losses—Service momentkip·ftBottom section modulusin³

Closes a rating deficit with bonded FRP (ACI 440.2R) or external post-tensioning, including the unstrengthened-capacity safety net.

Deficit ΔM

500 kip·ft

Method

Bonded FRP (ACI 440.2R)

FRP area A_f

2.400 in²

f_fe

79.5 ksi

Capacity gain

549 kip·ft

φM_n upgraded

3194 kip·ft

Derivation — equation, substitution, result

Strength deficit

\begin{aligned}&\;\Delta M = M_u − \phi M_n,\,\text{existing}\\[4pt]&=\;3200 − 0.9(3000)\\[4pt]&=\;\boxed{500 \,\text{kip}\cdot \,\text{ft} \,\text{must} \,\text{be} \,\text{added}}\end{aligned}

Environmental reduction

\begin{aligned}&\;\varepsilon _{fu} = C_E \varepsilon *_{fu} \;\text{(ACI 440.2R Table 9.4)}\\[4pt]&=\;C_E = 0.85, \varepsilon *_{fu} = 0.0170\\[4pt]&=\;\boxed{0.0145}\end{aligned}

Existing substrate strain

\begin{aligned}&\;\varepsilon _{bi} = M_{DL}(h − \,\text{kd})/(E_c I_{cr})\\[4pt]&=\;M_{DL} = 900 \,\text{kip}\cdot \,\text{ft}, k = 0.345\\[4pt]&=\;\boxed{\varepsilon _{bi} = 0.00091}\end{aligned}

Debonding strain

\begin{aligned}&\;\varepsilon _{fd} = 0.083\sqrt{f'_c/(n E_f t_f)} \le 0.9\varepsilon _{fu} (\mathrm{ACI} 440.2R Eq. 10-2)\\[4pt]&=\;n = 3, t_f = 0.0400 \,\text{in}\\[4pt]&=\;\boxed{\varepsilon _{fd} = 0.00361}\end{aligned}

Effective FRP stress

\begin{aligned}&\;f_{fe} = E_f \varepsilon _{fe} \operatorname{with} \varepsilon _{fe} \,\text{governed} \,\text{by} \,\text{crushing}, \,\text{rupture} \operatorname{or} \,\text{debonding}\\[4pt]&=\;\varepsilon _{fe} = 0.00361, E_f = 22000 \,\text{ksi}\\[4pt]&=\;\boxed{f_{fe} = 79.5 \,\text{ksi}, A_f = 2.400 \,\text{in}^{2}}\end{aligned}

FRP moment contribution

\begin{aligned}&\;\Delta M_n = \psi _f A_f f_{fe}(h − \beta _{1}c/2), \psi _f = 0.85\\[4pt]&=\;\boxed{549 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

External PT contribution

\begin{aligned}&\;\Delta M = P_{eff} e, \operatorname{with} P_{eff} = P\;\text{(1 − losses)}\\[4pt]&=\;300(1 − 0.180) \times 20.0 \,\text{in}\\[4pt]&=\;\boxed{410 \,\text{kip}\cdot \,\text{ft}}\end{aligned}

Upgraded capacity

\begin{aligned}&\;\phi M_n,\,\text{new} = \phi (M_n + \Delta M) \ge M_u\\[4pt]&=\;0.9(3000 + 549)\\[4pt]&=\;\boxed{3194 \,\text{kip}\cdot \,\text{ft} < 3200}\end{aligned}

Service stress relief (PT)

\begin{aligned}&\;\Delta f_b = P/A + Pe/S_b − M_s/S_b\\[4pt]&=\;\boxed{-2.975 \,\text{ksi} \operatorname{at} \,\text{the} \,\text{bottom} \,\text{fibre}}\end{aligned}

Detailing — plan, elevation and section

Detailing sketchStrengthening detail — bonded FRP laminate

3 plies of FRP laminate, 20.0 in wide, bonded to the soffit of a 24 in × 44 in section (d = 40 in) to close the rating deficit.

Constructability & detailing notes

Surface preparation controls FRP performance: blast to a CSP 3 profile, round all corners to 1/2 in radius, and check the pull-off strength.
Do not install FRP below 50 °F or on damp concrete; document ambient conditions during installation.
External tendons need corrosion-protected anchorages, grouted HDPE ducts, and deviators designed for the tendon deviation force.
Fire protection is required for FRP on structures where a vehicle fire is credible.

Check	Demand	Capacity / limit	Status
Upgraded φM_n ≥ M_u	3200 kip·ft	3194 kip·ft	REVIEW
Unstrengthened section carries the unfactored DL + LL (ACI 440.2R §9.2)The member must survive loss of the FRP (fire, vandalism, impact).	900 kip·ft	2700 kip·ft	PASS
FRP strain governed by debonding, not ruptureAdd U-wrap anchorage at the ends if the debonding strain governs.	—	—	PASS
Corrosion and section loss addressed before strengtheningNever bond FRP over actively corroding steel — repair and passivate first.	—	—	PASS
External tendons protected and deviator blocks designedProvide HDPE ducts with grout and design the deviators for the tendon deviation force.	—	—	PASS

Independent verification	Expected	Computed	Status
Upgraded capacity meets the demand	φM_n ≥ M_u	3194	CHECK
Unstrengthened section carries the dead load alone	φM_n,existing ≥ M_DL	549	VERIFIED

Assumptions & basis of design

ACI 440.2R-17 for externally bonded FRP with ψ_f = 0.85 and an environmental reduction factor C_E for exposure.
External post-tensioning losses of 15 – 20 % are typical for unbonded strand with anchorage set.
Secondary (parasitic) moments from external PT in continuous spans must be included in the demand.

Specify 3 plies × 20.0 in wide FRP; add U-wrap anchors at both ends and verify the substrate tensile pull-off strength ≥ 200 psi.

Building design suite

Steel — AISC 360-22 (LRFD)

Element classification through connections and stability: every module carries the design all the way to a member size, plate thickness, bolt count or weld size, with the governing limit state named and explicit redesign guidance when a check fails.

Module 72AASHTO LRFD AISC §B4, Table B4.1Chapter 4

Element slenderness & classification

Flange and web λ against λ_p and λ_r for flexure or axial compression, with the plate thicknesses needed to reach a compact section.

Flange width b_finFlange thickness t_finWeb clear depth hinWeb thickness t_winYield F_yksiUniform axial compression (Table B4.1a)

λ_f = b_f/2t_f · λ_w = h/t_w · compact if λ ≤ λ_p, slender if λ > λ_r

Detailing sketchSection classification — flange and web elements

Flange b_f/2t_f element is noncompact; web h/t_w element is compact. Governing section classification: noncompact. AISC 360-22 Table B4.1.

Flange λ

9.38

λ_p = 9.15, λ_r = 24.08

Web λ

50.63

λ_p = 90.55, λ_r = 137.27

Section

noncompact

Classification — AISC 360-22 §B4

Flange	noncompact
Web	compact
Governing	noncompact

PASS — Non-compact: interpolate between M_p and 0.7F_yS_x. To make it compact use t_f ≥ 0.656 in and t_w ≥ 0.221 in.

Module 73AASHTO LRFD AISC §D2 / §D3 / §J4.3Chapter 4

Tension members

Gross yielding, net-section rupture with shear lag, and block shear — the governing limit state and the area needed to satisfy the demand.

Gross area A_gin²Number of holes in section—Hole diameter d_hinPly thickness tinShear-lag U (Table D3.1)—F_yksiF_uksiFactored tension P_ukip

Block shear

A_gvin²A_nvin²A_ntin²

φP_n = 0.90F_yA_g · φP_n = 0.75F_uA_e · φR_n = 0.75[min(0.6F_uA_nv, 0.6F_yA_gv) + U_bsF_uA_nt]

A_n

4.063 in²

A_e = U·A_n

3.453 in²

Yielding

225 kip

Rupture

168 kip

Block shear

190 kip

Design strength

168 kip

Net-section rupture (D2-2)

Detailing sketchTension member — elevation, hole pattern and net section

A_g = 5.00 in², 2 holes @ 0.9375 in in a 0.500 in ply → A_n = 4.063 in². Governing limit state: Net-section rupture (D2-2). φP_n = 168.3 kip vs P_u = 180 kip.

Tension member design — AISC 360-22 §D2 / §D3 / §J4.3

Demand P_u	180 kip
Design strength φP_n	168.3 kip
Governing limit state	Net-section rupture (D2-2)
Utilization	1.07

REDESIGN — Increase A_e to ≥ 3.69 in² — add a connection length (raises U) or use fewer/smaller holes.

Module 74AASHTO LRFD AISC §E3 / §E7Chapter 4

Compression members

Flexural buckling with the inelastic/elastic transition at 4.71√(E/F_y), slender-element Q factor, and the area or bracing needed if the column is short of capacity.

Gross area A_gin²Radius of gyration rinEffective length factor K—Unbraced length LftF_yksiQ (slender elements)—Factored load P_ukip

F_e = π²E/(KL/r)² · F_cr = 0.658^(F_y/F_e)·F_y if KL/r ≤ 4.71√(E/F_y), else 0.877F_e

KL/r

66.9

limit 113.4

F_e

63.89 ksi

F_cr

36.03 ksi

inelastic

φ_cP_n

477 kip

Detailing sketchCompression member — buckled shape and slenderness

KL/r = 66.9 (inelastic buckling), F_cr = 36.03 ksi, φ_cP_n = 477 kip vs P_u = 400 kip. K = 1.00, L = 14.0 ft.

Compression design — AISC 360-22 §E3

P_u	400 kip
φ_cP_n	476.7 kip
Utilization	0.84

PASS — Adequate — KL/r = 67 (inelastic buckling), utilization 0.84.

Module 75AASHTO LRFD AISC §F2Chapters 4 · 8

Flexural members — LTB zones

M_p, L_p and L_r, the three lateral–torsional-buckling zones, and the bracing spacing or Z_x required to develop the demand.

Plastic modulus Z_xin³Elastic modulus S_xin³r_yinr_tsinTorsional constant Jin⁴Flange centroid distance h_oinF_yksiUnbraced length L_bftC_b—M_uk-ft

Zone 1: M_n = M_p · Zone 2: linear C_b interpolation · Zone 3: M_n = F_crS_x

M_p

608 k-ft

L_p

8.76 ft

L_r

25.36 ft

LTB zone

2 — inelastic LTB

M_n

608 k-ft

φ_bM_n

548 k-ft

Detailing sketchFlexural strength vs. unbraced length — LTB zones

L_p = 8.76 ft, L_r = 25.36 ft; design point at L_b = 12.0 ft falls in zone 2 — inelastic LTB. M_n = 608 k-ft, φM_n = 548 k-ft vs M_u = 400 k-ft.

Flexural design — AISC 360-22 §F2

M_u	400 k-ft
φ_bM_n	548 k-ft
Zone	2 — inelastic LTB
Utilization	0.73

PASS — Adequate in zone 2 — inelastic LTB — utilization 0.73.

Module 76AASHTO LRFD AISC §G2.1Chapters 8 · 11

Shear & web buckling

Web shear with k_v, the C_v1 buckling ratio, and the web thickness or stiffener spacing needed when the web governs.

Depth dinWeb thickness t_winClear web depth hinF_yksiFactored shear V_ukipTransverse stiffeners provided

V_n = 0.6F_yA_wC_v1 · C_v1 = 1.0 if h/t_w ≤ 1.10√(k_vE/F_y)

A_w

8.295 in²

k_v

5.340

h/t_w

46.3

limits 61.2 / 76.2

C_v1

1.000

web yielding (no buckling)

φ_v

1.00

φ_vV_n

249 kip

Detailing sketchWeb panel — shear buckling and stiffener layout

Web t_w = 0.395 in, clear depth h = 18.30 in, k_v = 5.340 (unstiffened). φ_vV_n = 249 kip vs V_u = 120 kip.

Shear & web buckling — AISC 360-22 §G2.1

V_u	120 kip
φ_vV_n	248.8 kip
Buckling mode	web yielding (no buckling)
Utilization	0.48

PASS — Adequate — C_v1 = 1.000 (web yielding (no buckling)), utilization 0.48.

Module 77AASHTO LRFD AISC §H1.1Chapters 4 · 9

Combined forces — beam-columns

H1-1a / H1-1b interaction of axial force with biaxial bending, and the size increase implied by an over-unity ratio.

Required axial P_rkipAvailable axial φP_nkipRequired M_rxk-ftAvailable φM_nxk-ftRequired M_ryk-ftAvailable φM_nyk-ft

H1-1a (P_r/P_c ≥ 0.2): P_r/P_c + 8/9(M_rx/M_cx + M_ry/M_cy) ≤ 1.0 · H1-1b otherwise

P_r/P_c

0.357

Equation

H1-1a

Interaction

0.905

Detailing sketchBeam-column interaction — P-M envelope and demand point

Governing equation H1-1a: interaction value = 0.905 (limit 1.0). P_r/P_c = 0.357, M_rx/M_cx = 0.450.

Beam-column interaction — AISC 360-22 §H1.1

Axial term	0.357
Flexural terms	0.450 + 0.167
Total	0.905

PASS — Interaction H1-1a = 0.905 ≤ 1.0 — beam-column adequate.

Module 78AASHTO LRFD AISC §J3Chapters 8 · 11

Bolted connections

Bolt shear, bearing and tearout, slip resistance and combined tension–shear, returning the bolt count, spacing and edge distances to detail.

Bolt diameter d_binNumber of bolts n—F_nv (A325-N 54, -X 68, A490-N 68)ksiConnected ply tinPly F_uksiClear distance l_cin Slip-critical connectionShear demand P_ukipTension demand T_ukip

φR_n = 0.75F_nvA_b (shear) · min(1.2l_ctF_u, 2.4d_btF_u) (bearing/tearout) · μD_uh_fT_bn_s (slip)

Bolt shear

146 kip

Bearing / tearout

197 kip

Design strength

146 kip

Bolt shear (J3-1)

Bolts required

8

provided 6

Detailing sketchBolted connection — bolt group elevation and plan

6 — 0.875 in bolts, ply t = 0.500 in, clear distance l_c = 1.125 in. Governing limit state: Bolt shear (J3-1). φR_n = 146.1 kip vs P_u = 180 kip. Min spacing 2.33 in (use 2.63 in), min edge 1.09 in.

Bolt group design — AISC 360-22 §J3

Demand	180 kip
Design strength	146.1 kip
Governing	Bolt shear (J3-1)
Layout	s ≥ 2.33 in (use 2.63 in), L_e ≥ 1.09 in

REDESIGN — Provide 8 bolts (currently 6) or increase the ply thickness to 0.616 in — bolt shear (j3-1) controls.

Module 79AASHTO LRFD AISC §J2.4Chapters 8 · 11

Fillet welds

Directional-strength weld metal capacity against base-metal rupture, with the minimum and maximum leg sizes and the length required per side.

Weld leg winLength per side LinNumber of weld lines—Electrode F_EXXksiLoad angle θ (0 = longitudinal)degBase metal thickness tinBase metal F_uksiDemand P_ukip

φR_n = 0.75(0.60F_EXX)(0.707w)L(1.0 + 0.50sin^1.5θ)

t_e

0.1767 in

k_ds

1.000

Weld metal

89 kip

Base metal

234 kip

Design strength

89 kip

Weld metal shear (J2-4)

L required / side

8.08 in

Detailing sketchFillet weld — plan, leg size and throat section

Weld leg w = 0.2500 in, effective throat t_e = 0.1767 in, length L = 8.00 in per side × 2 side(s), load angle θ = 0°. Governing: Weld metal shear (J2-4). φR_n = 89.1 kip vs P_u = 90 kip.

Fillet weld design — AISC 360-22 §J2.4

Demand	90 kip
Design strength	89.1 kip
Governing	Weld metal shear (J2-4)
Size limits	0.1875 – 0.4375 in

REDESIGN — Increase the weld length to 8.1 in per side, or the leg to 0.253 in (≤ 0.438 in max for a 0.500 in ply).

Module 80AASHTO LRFD AISC §J8 + ACI 318-19 Ch.17Chapters 4 · 13

Base plates & anchor rods

Concrete bearing with the √(A₂/A₁) confinement factor, the cantilever plate thickness, eccentricity classification, and anchor-rod steel and breakout strength.

Axial P_ukipMoment M_uk-ftShear V_ukipColumn depth dinColumn flange b_finPlate N (along d)inPlate BinConcrete f′_cksiPlate F_yksiA₂/A₁—Anchor rods (count)—Anchor diameterinEmbedment h_efin

φP_p = 0.65(0.85f′_cA₁)√(A₂/A₁) · t_req = l√(2P_u/0.9F_yBN)

A₁ provided

400 in²

required 113 in²

φP_p

1768 kip

l (cantilever)

4.20 in

m=3.35, n=4.20, λn′=3.56

t required

1.1875 in

φN_sa

58 kip

φV_sa

30 kip

Detailing sketchBase plate — plan (anchor layout) and elevation

Plate 20.0 × 20.0 × 1.1875 in with 4 — 0.750 in anchor rods. φP_p = 1768 kip vs P_u = 500 kip.

Base plate & anchor rods — AISC §J8 + ACI 318-19 Ch.17

Note 1	Concrete bearing OK: φP_p = 1768 kip ≥ P_u (A₁ = 400 in², required 113 in²).
Note 2	Use a plate thickness of 1.1875 in (t_req = 1.167 in from the cantilever l = 4.20 in).
Note 3	e = 0.0 in ≤ N/6 — full bearing, anchor rods nominal.
Note 4	Anchor rods: φN_sa = 58 kip, φV_sa = 30 kip, concrete breakout φN_cb ≈ 177 kip at h_ef = 12 in.

REDESIGN — Specify a 20.0 × 20.0 × 1.1875 in plate with 4 – 0.750 in anchor rods, h_ef = 12.0 in.

Module 81AASHTO LRFD AISC §J3/J4 + Manual Part 10Chapters 8 · 11

Shear (simple) connection — shear tab

All six single-plate limit states — bolt shear, bearing/tearout, plate shear yield and rupture, block shear and the weld — with the governing one named.

Number of bolts—Bolt diameterinF_nvksiPlate thickness t_pinPlate F_yksiPlate F_uksiVertical edge L_evinHorizontal edge L_ehinBolt pitch sinWeld leg winDemand V_ukip

Governing = min(bolt shear, bearing/tearout, plate shear yield, plate shear rupture, block shear, weld)

Bolt shear (J3-1)

72 kip

Plate bearing/tearout (J3-6)

117 kip

Plate shear yielding (J4-3)

97 kip

Plate shear rupture (J4-4)

86 kip

Block shear (J4-5)

82 kip

Weld to support (J2-4)

134 kip

Detailing sketchShear tab (single-plate) connection — elevation and weld section

4 bolts @ 3.00 in, edge distances L_ev = 1.50 in, L_eh = 1.50 in, plate t_p = 0.375 in, weld leg 0.2500 in. Governing: Bolt shear (J3-1). φR_n = 71.6 kip vs V_u = 70 kip.

Single-plate shear connection — AISC 360-22 §J3/J4 + Manual Part 10

Plate	PL 0.3750 × 4.5 × 12.0
Design strength	71.6 kip
Governing	Bolt shear (J3-1)
Utilization	0.98

PASS — Adequate — bolt shear (j3-1) governs at 0.98 utilization. Recommended weld: two-sided 0.234 in fillet (⅝ t_p rule).

Module 82AASHTO LRFD AISC §J10Chapters 8 · 11

Moment connection & column checks

Flange-couple force with the §J10 column-side checks that decide whether continuity plates and a web doubler are required.

Beam M_uk-ftBeam V_ukipBeam depth dinBeam t_finBeam b_finColumn t_winColumn t_finColumn depth d_cinColumn kinF_yksiF_uksiColumn P_ukipColumn A_gin²

P_uf = M_u/(d − t_f) · J10 checks decide whether continuity plates and a doubler plate are needed

P_uf

192 kip

Web local yielding

301 kip

Web crippling

397 kip

Flange local bending

276 kip

Flange rupture

321 kip

Panel zone

240 kip

Detailing sketchMoment connection — flange-force couple and column limit states

Flange force P_uf = M_u/(d − t_f) = 192 kip. Governing column check: Column flange local bending (J10-1) = 276 kip. Continuity plates not required.

Moment connection & column checks — AISC 360-22 §J10

Flange force P_uf	192.0 kip
Governing column limit	Column flange local bending (J10-1) — 276 kip
Continuity plates	not required
Panel zone	Panel zone adequate (φR_v = 240 kip).

PASS — No continuity plates required: all J10 column limit states exceed P_uf = 192 kip.

Module 83AASHTO LRFD AISC §I3 / §I8Chapters 4 · 8

Composite beam with steel deck

Plastic composite flexure with the concrete, steel and stud forces, the degree of composite action, and the stud count for full composite behaviour.

Steel area A_sin²F_yksiSteel depth dinEffective width b_effinSlab thickness tinDeck rib height h_rinf′_cksiStud strength Q_nkipStuds provided—M_uk-ftSpanft

C = min(0.85f′_cb_eff(t−h_r), A_sF_y, ΣQ_n) · a = C/0.85f′_cb_eff · φM_n = 0.90C(d/2 + Y₂)

C_concrete

1071 kip

T_steel

665 kip

ΣQ_n

688 kip

a

2.173 in

Composite action

100%

φM_n

662 k-ft

Detailing sketchComposite beam — section, PNA and stud layout

Effective width b_eff = 90 in, slab t = 5.50 in (rib h_r = 2.0 in), degree of composite action = 100% with 40 studs. Stress-block depth a = 2.173 in, C = 665 kip. φM_n = 662 k-ft vs M_u = 700 k-ft.

Composite beam with steel deck — AISC 360-22 §I3

M_u	700 k-ft
φM_n	662 k-ft
Studs provided / required	40 / 78
Utilization	1.06

REDESIGN — φM_n = 662 k-ft < M_u. Increase the stud count to 78 for full composite action, or use a heavier beam (A_s ≈ 14.1 in²).

Module 84AASHTO LRFD AISC Appendix 6Chapters 4 · 8

Stability bracing — strength & stiffness

Nodal and relative bracing for columns and beams: both the required brace force and the required brace stiffness, plus the brace area that delivers it.

Bracing typeRequired strength P_rkipBrace spacing L_brft

Strength AND stiffness both required — a strong but flexible brace does not stabilize the member.

Required brace strength P_br

3.00 kip

Required brace stiffness β_br

26.7 kip/in

Detailing sketchStability bracing — column nodal

Brace spacing L_br = 10.0 ft. Required brace strength P_br = 3.00 kip; required stiffness β_br = 26.7 kip/in. Nodal (point) bracing restrains a single cross-section.

Stability bracing — AISC 360-22 Appendix 6

Case	Nodal column bracing, A-6-3/A-6-4 (conservative N_i = 4).
P_br	3.00 kip
β_br	26.7 kip/in

Provide a brace with strength ≥ 3.0 kip AND stiffness ≥ 27 kip/in. A single-angle brace of length L and area A gives β = AE/L — required A ≈ 0.11 in² for a brace 10.0 ft long. Strength alone is not sufficient; both criteria must be met.

Module 85AASHTO LRFD AISC Appendix 8Chapters 4 · 9

Stability — second-order B1 / B2

Amplified first-order analysis: the P-δ and P-Δ multipliers, the amplified moments and axial loads, and the drift limit that keeps B₂ acceptable.

C_m—Member P_rkipP_e1 (in-plane)kipΣP_storykipΣP_e,storykipM_nt (no translation)k-ftM_lt (lateral translation)k-ftP_ntkipP_ltkip

B₁ = C_m/(1 − αP_r/P_e1) · B₂ = 1/(1 − αΣP/ΣP_e,story) · M_r = B₁M_nt + B₂M_lt

B₁

1.000

B₂

1.286

M_r

351 k-ft

P_r

264 kip

Detailing sketchSecond-order amplification — sway frame

B₁ = 1.000 amplifies the no-translation moment M_nt = 120 k-ft; B₂ = 1.286 amplifies the lateral-translation moment M_lt = 180 k-ft. Combined required moment M_r = B₁M_nt + B₂M_lt = 351 k-ft.

Second-order amplification — AISC 360-22 Appendix 8

B₁	1.000
B₂	1.286
Amplified M_r	351 k-ft
Amplified P_r	264 kip

PASS — B₁ = 1.000, B₂ = 1.286 — amplified demands M_r = 351 k-ft, P_r = 264 kip.

Module 86AASHTO LRFD AISC Appendix 3 · AASHTO §6.6.1Chapters 8 · 18

Fatigue — stress range & life

Detail-category allowable stress range, threshold for infinite life, and the predicted number of cycles with the detail upgrade needed when the life is short.

Detail category (Table A-3.1)Stress range f_srksiDesign cyclescycles

F_SR = (C_f/n_SR)^(1/3) ≥ F_TH · life N = C_f/f_sr³

C_f

4.40e+9

F_TH

10.0 ksi

F_SR

10.00 ksi

Predicted life

5.13 M cycles

Infinite life?

yes

Detailing sketchFatigue S-N curve — Category C

Constant-amplitude fatigue threshold F_TH = 10.0 ksi; allowable range at n_SR= 20.0M cycles is F_SR = 10.00 ksi. Applied f_sr = 9.50 ksi gives predicted life N = 5.13M cycles.

Fatigue — AISC 360-22 Appendix 3

f_sr	9.50 ksi
F_SR allowable	10.00 ksi
Life	5.13 M cycles

PASS — f_sr = 9.50 ksi ≤ F_TH = 10 ksi — infinite life for Category C.

Module 87AASHTO LRFD AISC DG3 / IBC 1604.3Chapters 4 · 19

Deflection & serviceability

Live and total deflections against L/360 and L/240, camber, and the moment of inertia required when serviceability governs the beam size.

Service dead load w_Dkip/ftService live load w_Lkip/ftSpan LftI_xin⁴Live limit denominatorL/xTotal limit denominatorL/xCamberin

Δ = 5wL⁴/384EI

Δ_D

0.425 in

Δ_L

0.567 in

allow 1.000 in

Δ_total

0.992 in

allow 1.500 in

Detailing sketchDeflected shape and serviceability limits

Δ_D = 0.425 in, Δ_L = 0.567 in (limit 1.000 in), Δ_total = 0.992 in (limit 1.500 in), camber = 0.00 in, span L = 30.0 ft.

Serviceability — AISC DG3 / IBC 1604.3

Δ_live / limit	0.567 / 1.000 in
Δ_total / limit	0.992 / 1.500 in
I required	880 in⁴

PASS — Δ_LL = 0.567 in ≤ L/360 = 1.000 in and Δ_total = 0.992 in ≤ L/240.

Building design suite

Concrete — ACI 318-19

Beams, columns, slabs and detailing — each module runs the full cycle: A_s,req → bar size, count and spacing → A_s,prov → capacity re-check → crack control, ductility and serviceability, with the redesign step spelled out when a limit is violated.

Module 88AASHTO LRFD ACI §9.5 / §9.6.1 / §24.3.2Chapters 5 · 6

Beam flexure & crack control

Required and minimum steel, bar selection, φ from the net tensile strain, the re-checked φM_n, §24.3.2 bar spacing for crack control, skin steel, and a dimensioned section drawing.

Factored moment M_uk-ftWidth binOverall depth hinEffective depth dinf′_cksif_yksiClear coverin

a = A_sf_y/0.85f′_cb · ε_t = 0.003(d−c)/c · φM_n = φA_sf_y(d − a/2) · s ≤ 15(40/f_s) − 2.5c_c

A_s,req

2.793 in²

A_s,min

1.440 in²

A_s,prov

3.000 in²

3 – #9

a / c

3.31 / 3.89 in

ε_t

0.01781

φ = 0.900

φM_n

342.2 k-ft

Flexural reinforcement design & crack control — ACI 318-19 §9.5 / §9.6.1 / §24.3.2

A_s design (governing)	2.793 in²
Bar call-out	3 – #9
A_s provided	3.000 in²
Bar spacing provided / max	5.31 / 10.00 in
φM_n	342.2 k-ft
ρ	0.0069

PASS — Adequate: 3–#9, A_s,prov/A_s,req = 1.07, ε_t = 0.0178 (tension-controlled), spacing 5.3 in ≤ 10.0 in.

Detailing sketchReinforced-concrete beam — section, elevation and bar plan

3 #9 bars (3.00 in² provided against A_s = 3.00 in² assumed), 1 row, 2 in clear cover, #4 stirrups. Whitney block a = 3.31 in, c = 3.89 in, f_y = 60 ksi. φM_n = 342 k-ft vs M_u = 320 k-ft (AASHTO 5.6.3.2).

Module 89AASHTO LRFD ACI §22.5 / §9.6.3 / §9.7.6Chapter 6

Beam shear & stirrup design

V_c from Table 22.5.5.1, required V_s, the crushing limit, and a full stirrup schedule (size, legs, spacing, s_max) for every candidate bar.

Factored shear V_ukipWeb width b_winEffective depth dinf′_cksif_yt (stirrups)ksiρ_w (longitudinal)—Stirrup legs—

V_c = 2λ√f′_c b_wd · V_s = A_vf_ytd/s · φ(V_c + V_s) ≥ V_u · V_s ≤ 8√f′_c b_wd

V_c

50.0 kip

φV_c/2

18.8 kip

stirrups required

V_s required

70.0 kip

Schedule

#3 2-leg stirrups @ 5.0 in c/c

s_max

13.5 in

φV_n

91.0 kip

Stirrup design schedule — ACI 318-19 §22.5 / §9.6.3 / §9.7.6

#3 2-leg (A_v = 0.22 in²)	s = 5.0 in (strength 5.1, min-A_v 264.0, max 13.5) → φV_n = 91 kip
#4 2-leg (A_v = 0.40 in²)	s = 9.0 in (strength 9.3, min-A_v 480.0, max 13.5) → φV_n = 92 kip
#5 2-leg (A_v = 0.62 in²)	s = 13.5 in (strength 14.4, min-A_v 744.0, max 13.5) → φV_n = 93 kip
#6 2-leg (A_v = 0.88 in²)	s = 13.5 in (strength 20.4, min-A_v 1056.0, max 13.5) → φV_n = 117 kip

PASS — #3 2-leg stirrups @ 5.0 in gives φV_n = 91 kip ≥ V_u = 90 kip (s_max = 13.5 in, minimum-A_v spacing 264.0 in).

Detailing sketchBeam shear reinforcement — stirrup section and spacing elevation

2-leg #3 stirrups @ 5.0 in near the supports (V_c = 50.0 kip, φV_n = 91.0 kip vs V_u = 90 kip), opening to the code maximum s_max = 13.5 in toward midspan per ACI 318-19 §9.7.6.2.2.

Module 90AASHTO LRFD ACI §22.4 / §10.6 / §25.7Chapters 9 · 12 · 13

Column axial capacity

P_o with the 0.80/0.85 cap, ρ limits of 1–8 %, the required A_st, and a bar-and-tie (or spiral) schedule.

Column width binColumn depth hinLongitudinal steel A_stin²f′_cksif_yksiFactored axial P_ukip Spiral confinement

P_o = 0.85f′_c(A_g − A_st) + f_yA_st · φP_n,max = φ·0.80·P_o

A_g

400 in²

ρ

2.00%

P_o

2146 kip

φP_n(max)

1116 kip

A_st required

10.90 in²

Column reinforcement & confinement — ACI 318-19 §22.4 / §10.6 / §25.7

A_st required	10.90 in²
Bar schedule	14 – #8 vertical bars with #4 ties @ 16 in c/c (§25.7.2)
φP_n(max)	1116 kip
Utilization	1.08

REDESIGN — φP_n = 1116 kip < P_u — required A_st ≈ 10.90 in², or enlarge A_g to 430 in².

Detailing sketchTied column — section and reinforcement

14 — #8 longitudinal bars around a 20 × 20 in section, #4 ties @ 16.0 in o.c. per ACI 318-19 §25.7.2 (tie spacing ≤ min(16d_b, 48d_tie, least dimension)). φP_n = 1116 kip vs P_u = 1200 kip.

Module 91AASHTO LRFD ACI §6.6.4Chapters 9 · 12 · 13

Slender column moment magnification

Slenderness screening, EI_eff, the critical load P_c, the δ_ns magnifier and the design moment M_c with the minimum-eccentricity check.

P_ukipSmaller end moment M₁k-ftLarger end moment M₂k-ftEffective length factor k—Unsupported length l_uftColumn binColumn hinf′_cksiβ_dns— Non-sway (braced) frame

δ_ns = C_m/(1 − P_u/0.75P_c) · P_c = π²EI/(kl_u)² · M_c = δ_nsM₂

kl_u/r

35.6

limit 28.0

P_c

2386 kip

C_m

0.800

δ_ns

1.110

M_c

133.2 k-ft

M₂,min

47.5 k-ft

Slenderness & moment magnification — ACI 318-19 §6.6.4

Slender?	yes — magnify
δ_ns	1.110
Design moment M_c	133.2 k-ft

PASS — Slender: δ_ns = 1.110 → design moment M_c = 133 k-ft (min. eccentricity moment 48 k-ft).

Detailing sketchSlender column — moment magnification and deflected shape

k·l_u = 1.00 × 16.0 ft. Primary end moments M₁ = 60 k-ft, M₂ = 120 k-ft magnified by δ_ns = 1.110 to the design moment M_c = 133.2 k-ft — the member is slender and moments must be magnified (ACI 318-19 §6.6.4).

Module 92AASHTO LRFD ACI §22.2 / Table 21.2.2Chapters 9 · 12 · 13

Column P–M interaction diagram

Strain-compatibility sweep of the neutral axis producing the nominal and φ-reduced interaction curves, with the demand point plotted and the available φM_n at that axial load.

Column binColumn hinf′_cksif_yksiA_s each facein²d′ (cover to bar centre)inDemand P_ukipDemand M_uk-ft

Strain compatibility sweep of the neutral axis c → (P_n, M_n) → φ(ε_t)-reduced design curve

P_o

2392 kip

φP_n,max

1244 kip

φM_n at P_u

504 k-ft

P–M interaction — ACI 318-19 §22.2 / Table 21.2.2

Demand	(600 kip, 400 k-ft)
φM_n at that axial load	504 k-ft
Utilization	0.79

PASS — At P_u = 600 kip the φ-curve allows φM_n = 504 k-ft ≥ M_u = 400 k-ft (utilization 0.79).

Detailing sketchColumn P–M interaction diagram

Nominal (P_n, M_n) curve and φ-reduced design curve from the ACI 318-19 §22.2 strain-compatibility sweep. Demand point (M_u = 400 k-ft, P_u = 600 kip) plots inside the design envelope — φM_n at this P_u = 504 k-ft.

Module 93AASHTO LRFD ACI §25.4 / §25.5 · AASHTO §5.10.8Chapters 5 · 6

Development length & lap splices

Straight, hooked and compression development lengths with the confinement term, plus Class A / Class B lap lengths and staggering guidance.

Bar diameter d_binf_yksif′_cksic_b (cover / half spacing)inK_tr (transverse steel)inψ_t (top bar = 1.3)—ψ_e (epoxy = 1.2–1.5)—ψ_s (≤ #6 = 0.8)— Class B lap splice

ℓ_d = (3/40)(f_y/λ√f′_c)·(ψ_tψ_eψ_s)/[(c_b+K_tr)/d_b]·d_b

(c_b+K_tr)/d_b

2.000

capped at 2.5

ℓ_d tension

35.58 in

ℓ_dh hook

17.25 in

ℓ_dc compression

18.97 in

Lap splice

46.25 in

Development & splice schedule — ACI 318-19 §25.4 / §25.5

Straight tension ℓ_d	35.58 in (36 d_b)
Standard hook ℓ_dh	17.25 in
Compression ℓ_dc	18.97 in
Class B lap	46.25 in

Straight tension development ℓ_d = 35.6 in (confinement term (c_b+K_tr)/d_b = 2.00, capped at 2.5). Standard hook ℓ_dh = 17.2 in. Compression ℓ_dc = 19.0 in. Class B lap splice = 46.2 in — stagger splices out of the maximum-moment region (§25.5).

Detailing sketchDevelopment length, standard hook and lap splice

#8 (d_b = 1.000 in): straight tension development ℓ_d = 35.6 in, standard 90° hook ℓ_dh = 17.2 in, Class B lap splice ℓ_st = 46.2 in (ACI 318-19 §25.4 / §25.5).

Module 94AASHTO LRFD ACI §24.2Chapters 6 · 19

Deflection — immediate & long term

Cracked and effective moments of inertia, immediate dead and live deflections, and the λ_Δ long-term multiplier against L/360 and L/240.

binhindinf′_cksiA_s (tension)in²A′_s (compression)in²Span LftService w_Dkip/ftService w_Lkip/ftSustained live fraction—Durationyr

I_e (§24.2.3.5) · Δ = 5wL⁴/384E_cI_e · λ_Δ = ξ/(1 + 50ρ′)

E_c

3644 ksi

M_cr

96.0 k-ft

M_a = 205.8 k-ft

I_e

31331 in⁴

I_g = 36000, I_cr = 14169

Δ_live

0.109 in

allow 0.933 in

λ_Δ

2.000

Δ_LT + live

0.465 in

allow 1.400 in

Immediate & long-term deflection — ACI 318-19 §24.2

Δ dead (immediate)	0.145 in
Δ live (immediate)	0.109 in
Δ long-term	0.356 in
Δ total	0.610 in

PASS — Δ_LL = 0.109 in ≤ L/360 = 0.933 in; long-term + live = 0.465 in ≤ L/240 = 1.400 in.

Detailing sketchBeam deflection — immediate, long-term and deflected shape

Span L = 28.0 ft. Immediate live deflection Δ_live = 0.109 in (limit L/360 = 0.933 in); long-term + live Δ = 0.465 in (limit L/240 = 1.400 in). Shape exaggerated for clarity.

Module 95AASHTO LRFD ACI §7 / §24.4Chapter 5

One-way slab

Minimum thickness, factored strip moments, bottom and top bar selection at spacing, shrinkage-and-temperature steel and the one-way shear check.

Clear span LftSlab thickness hinf′_cksif_yksiSuperimposed dead w_SDLksfLive load w_LLksfContinuity

w_u = 1.2D + 1.6L · M = w_uL²/coef · A_s,min = 0.0018A_g · s ≤ min(3h, 18 in)

h_min

6.00 in

w_u

0.301 kip/ft/ft

M⁺

4.21 k-ft/ft

Bottom steel

#4@15.5"

Top steel

#4@15.5"

S&T steel

#4@15.5"

One-way slab reinforcement — ACI 318-19 §7 / §24.4

Note 1	h = 7 in ≥ h_min = 6.00 in (Table 7.3.1.1) — deflection check by computation not required.
Note 2	Main steel: #4 @ 15.5 in c/c bottom, #4 @ 15.5 in c/c top at supports.
Note 3	Shrinkage & temperature steel: #4 @ 15.5 in c/c (A_s = 0.0018A_g = 0.151 in²/ft, §24.4.3.2).
Note 4	One-way shear OK: φV_c = 6.8 kip/ft ≥ V_u = 2.0 kip/ft.

REDESIGN — Design strip 12 in wide, d = 6.00 in.

Detailing sketchOne-way slab — section and plan of reinforcement

7.00 in slab, both-ends span L = 14.0 ft. Bottom steel #4 @ 15.5 in, top steel at supports #4 @ 15.5 in. Shrinkage & temperature steel #4 @ 15.5 in transverse (ACI 318-19 §7.6.1 / §24.4).

Module 96AASHTO LRFD ACI §8.10 / §22.6Chapter 5

Two-way slab (DDM) & punching shear

Total static moment M_o, column- and middle-strip distribution with bar schedules for each strip, and the two-way punching-shear check with β and α_s terms.

Span l₁ (direction of analysis)ftTransverse span l₂ftColumn c₁inColumn c₂inSlab thickness hinf′_cksif_yksiw_SDLksfw_LLksfColumn location

M_o = q_u l₂ l_n²/8 · v_c = min(4, 2 + 4/β, 2 + α_sd/b_o)λ√f′_c · φV_c = 0.75v_cb_od

q_u

0.310 ksf

M_o

238.7 k-ft

M⁻ / M⁺

155 / 84 k-ft

b_o

99.0 in

d = 6.75 in

v_c

253 psi

φV_c / V_u

127 / 110 kip

Two-way slab reinforcement (DDM) — ACI 318-19 §8.10 / §22.6

Column strip (top)	#7 @ 16.0 in — φM_n = 13.00 vs 12.93 k-ft/ft
Column strip (bottom)	#3 @ 7.0 in
Middle strip (top)	#3 @ 7.5 in
Middle strip (bottom)	#3 @ 7.5 in
Punching shear	φV_c = 127 kip vs V_u = 110 kip

PASS — M_o = q_u l₂ l_n²/8 = 238.7 k-ft; distributed 65 % negative / 35 % positive (§8.10.4). Column strip (9.0 ft wide): top #7 @ 16.0 in, bottom #3 @ 7.0 in. Middle strip (9.0 ft wide): top #3 @ 7.5 in, bottom #3 @ 7.5 in. Two-way (punching) shear OK: φv_cb_od = 127 kip ≥ V_u = 110 kip at d/2 from the column (b_o = 99 in).

Detailing sketchTwo-way slab (DDM) — strip plan and punching-shear critical perimeter

Panel l₁ × l₂ = 20.0 × 18.0 ft, column c₁ × c₂ = 18.0 × 18.0 in. Critical shear perimeter b_o = 99.0 in at d/2 = 3.38 in from the column face (ACI 318-19 §22.6.4.1), φV_c = 126.8 kip vs V_u = 110.3 kip. Column strip: top #7@16.0", bottom #3@7.0". Middle strip: top #3@7.5", bottom #3@7.5".

Building design suite

Masonry — TMS 402/602-22

Strength design of reinforced concrete masonry: slender out-of-plane walls with the P-δ moment, in-plane shear walls, and bond-beam lintels.

Module 97AASHTO LRFD TMS 402-22 §9.3Chapters 9 · 12 · 13

Reinforced masonry wall — axial, flexure & P-δ

Slenderness-reduced axial capacity, the second-order P-δ moment, vertical bar size and grout-cell spacing, the maximum-reinforcement ductility limit and out-of-plane shear.

CMU nominal thicknessGroutingWall height hftf′_mksif_yksiAxial P_u (per ft)kip/ftMoment M_u (per ft)k-ft/ftShear V_u (per ft)kip/ft

φM_n = 0.90(A_sf_y + P_u)(d − a/2) · a = (A_sf_y + P_u)/0.80f′_mb · M_u,total = M_u + P_uδ_u

h/r

109.0

(70r/h)²

φP_n

43.45 kip/ft

δ_u

0.160 in

M_u total

4.25 k-ft/ft

Reinforcement

#7@32"

φV_n

8.07 kip/ft

Reinforced masonry wall design — TMS 402-22 §9.3

Note 1	h/r = 109 > 99 — the slender (70r/h)² form governs; φP_n = 43.4 kip/ft.
Note 2	Second-order: δ_u = 0.160 in gives M_u,total = M_u + P_uδ_u = 4.25 k-ft/ft (§9.3.5.4.3).
Note 3	Provide #7 @ 32 in o.c. — φM_n = 4.30 k-ft/ft ≥ 4.25 k-ft/ft; A_s = 0.225 in²/ft ≤ A_s,max = 0.722 in²/ft.
Note 4	Shear OK: φV_n = 8.1 kip/ft ≥ V_u = 1.1 kip/ft (M/(V d_v) = 1.00).
Note 5	Detailing: minimum prescriptive steel 0.0007bt = 0.064 in²/ft each direction (0.002 total); bond beams at ≤ 48 in o.c.; lap splices per §9.3.3.4.

PASS — #7 @ 32 in o.c. vertical (A_s = 0.225 in²/ft) in grouted cells

Detailing sketchReinforced masonry wall — section, elevation and cell layout

8 in CMU, fully grouted, #7 vertical bars @ 32 in o.c. centered in the grouted cells (TMS 402-22 §9.3). Wall height h = 20.0 ft. φP_n = 43.45 kip/ft vs P_u = 3.50 kip/ft; φM_n = 4.30 k-ft/ft vs magnified M_u = 4.25 k-ft/ft.

Module 98AASHTO LRFD TMS 402-22 §9.3.4 / §7.3.2Chapters 9 · 12 · 13

Masonry shear wall — in-plane

M/(Vd_v)-dependent masonry shear, the crushing ceiling, bond-beam horizontal steel spacing and the boundary flexural steel at each end cell.

Wall length l_wftCMU nominal thicknessWall height h_wftf′_mksif_yksiAxial P_ukipShear V_ukipMoment M_uk-ft

V_nm = [4.0 − 1.75(M/Vd_v)]A_nv√f′_m + 0.25P_u · V_ns = 0.5(A_v/s)f_yd_v

M/(Vd_v)

0.781

V_nm

187.4 kip

V_n,max

261.9 kip

V_s required

0.0 kip

Horizontal steel

#4@48"

Boundary steel

4–#5

Masonry shear wall design — TMS 402-22 §9.3.4 / §7.3.2

Note 1	M/(V d_v) = 0.78 → V_nm = 187 kip, upper limit V_n,max = 262 kip (§9.3.4.1.2).
Note 2	Section adequate against masonry crushing in shear.
Note 3	Horizontal shear steel: #4 bond-beam bars @ 48 in o.c. (φV_n = 181 kip).
Note 4	Boundary flexural steel: A_s = 1.23 in² each end → 4 – #5 in the end cells (A_s,prov = 1.24 in²), fully grouted and lapped per §9.3.3.4.
Note 5	Aspect ratio h_w/l_w = 1.25 — flexure-dominated behaviour.

PASS —

Detailing sketchMasonry shear wall — elevation with boundary and horizontal reinforcement

Wall l_w × t × h_w = 16.0 ft × 7.625 in × 20.0 ft. Horizontal (shear) reinforcement #4@48"; boundary flexural steel 4 – #5 in grouted end cells. φV_n = 180.6 kip vs V_u = 90 kip (TMS 402-22 §9.3.4 / §7.3.2).

Module 99AASHTO LRFD TMS 402-22 §9.3 / §5.2.1 / §8.1.6.2Chapters 9 · 12 · 13

Reinforced masonry lintel

Effective span and demands, bottom bond-beam steel, the §9.3.3.5 maximum-reinforcement limit, M/(Vd)-dependent masonry shear with stirrups when needed, bearing stress and the ℓ/600 deflection limit.

Clear span ℓ_nftFactored load w_uklfLintel depth hinNominal wall thicknessinf′_mksif_yksiBearing each endinTrial bar no.

M_u = w_uℓ²/8 · a = A_sf_y/0.80f′_mb · φM_n = φA_sf_y(d − a/2) · V_nm = [4 − 1.75(M/Vd)]A_nv√f′_m

M_u / V_u

15.0 / 6.9 k-ft, kip

A_s required

0.170 in²

A_s provided

0.310 in²

1 – #5 bottom bond beam

φM_n

26.8 k-ft

φV_n

12.3 kip

Bearing

0.114 ksi

limit 0.300 ksi

Lintel bond-beam schedule — TMS 402-22 §9.3 / §5.2.1 / §8.1.6.2

Grouted width b	7.625 in
Effective depth d	20.0 in
Bottom steel	1 – #5
Stirrups	not required by analysis
A_s,max	1.451 in²
Deflection limit ℓ/600	0.160 in

PASS — Provide 1 – #5 in the bottom bond beam, 8 in bearing each end; φM_n = 26.8 k-ft ≥ M_u = 15.0 k-ft.

Detailing sketchReinforced masonry lintel — elevation and bearing detail, with grouted section

Clear span = 8.0 ft, lintel depth h = 24 in over a 8.00 in wall, bearing 8 in each end. Bottom steel 1 — #5 in the grouted bond-beam course: φM_n = 26.8 k-ft ≥ M_u = 15.0 k-ft; φV_n = 12.3 kip vs V_u = 6.9 kip (TMS 402-22 §9.3).

Advanced detailing suite

Discontinuity regions, punching, torsional stability & stiffened webs

Four deep-dive modules that carry the awkward regions all the way to a drawing: strut-and-tie for deep beams and corbels, complete punching-shear design with stud rails and cap/column layout, combined torsion-and-stability for open and closed steel shapes, and plate-girder stiffener and weld design.

Module 100AASHTO LRFD ACI 318-19 Ch. 23 · §9.9 · §16.5Chapters 6 · 12 · 13

Deep beams & corbels — strut-and-tie

Full STM: deep-member screening, truss geometry and strut angle, strut and CCC/CCT nodal-zone strengths, tie steel with bar selection, distributed crack-control steel or corbel hoops, and tie anchorage.

Member typeFactored shear V_ukipHorizontal force N_uckipClear span ℓ_ninShear span a_vinWidth binOverall depth hinEffective depth dinf′_cksif_yksiClear coverinBearing plate ℓ_bin

θ = atan(0.9d/a_v) ≥ 25° · F_ns = f_ce A_cs, f_ce = 0.85β_cβ_s f′_c · A_st = F_ut/φf_y · A_v, A_vh ≥ 0.0025 b s

Strut angle θ

52.2 deg

Strut force F_us

405 kip

φF_ns = 517 kip

Tie force F_ut

312 kip

Strut width w_s

13.52 in

A_cs = 216 in²

A_s tie required

6.93 in²

A_s,min = 2.43 in²

A_s provided

7.20 in²

Bearing node

714 kip

CCT node

163 kip

φV_n,max

365 kip

Strut-and-tie model — ACI 318-19 Ch. 23 · §9.9 · §16.5

Classification	Deep member (ℓ_n/h = 2.00, a_v/d = 0.70)
Truss lever arm jd	38.7 in
Strut efficiency β_s	0.75 → f_ce = 3.19 ksi
Node f_ce (CCC / CCT)	4.25 / 3.40 ksi
Tie width w_t	4.00 in

Reinforcement schedule — §23.7 · §23.5 · §9.9.3 / §16.5.5

Primary tie	12 – #7 primary tie bars
Vertical web steel	#4 @ 8.5 in c/c each face (s_max = 8.6 in)
Horizontal web steel	#4 @ 8.5 in c/c each face (s_max = 8.6 in)
Tie anchorage	ℓ_dh = 12.6 in — straight development adequate
Bearing plate	14.0 × 16.0 in, A_nz = 224 in²

REDESIGN — CCT node behind the tie overstressed — spread the tie over a greater height (increase the number of layers so w_t ≥ 7.6 in) or enlarge the bearing.

Detailing sketchDeep beam — strut-and-tie model

Single-panel truss: θ = 52.2° from the load node to the support node. Strut force F_us = 405 kip (width w_s = 13.52 in), tie force F_ut = 312 kip carried by 12 – #7 primary tie bars. V_u = 320 kip. #4 vertical stirrups @ 8.5 in c/c each face + #4 horizontal bars @ 8.5 in c/c each face

Module 101AASHTO LRFD ACI 318-19 §22.6 · §8.4.4 · §8.7.7Chapters 5 · 12 · 13

Punching shear — complete design

All three v_c expressions at the d/2 section, unbalanced-moment transfer by eccentric shear, headed stud rails or closed stirrups with s₀/s and rail length to the outer critical section, plus the cap/column bar and tie layout.

Factored shear V_ukipUnbalanced moment M_uk-ftColumn c₁inColumn c₂inSlab / cap thickness hinClear coverinf′_cksif_y = f_ytksiColumn positionShear reinforcementStud / stirrup bar no.Rails per column faceColumn axial P_ukip

v_u = V_u/b_od + γ_v M_u c/J_c · v_c = min[4, 2+4/β, 2+α_sd/b_o]λ√f′_c · A_v/s = v_s b_o/f_yt · s ≤ 0.75d

b_o at d/2

112.0 in

d = 8.00 in

v_u total

235 psi

direct 201 + moment 34

φv_c

212 psi

4λ√f′_c

γ_v / γ_f

0.400 / 0.600

v_s required

101 psi

φv_n with reinforcement

247 psi

Stud spacing s

6.0 in

s₀ = 3.0 in, s_max = 6.0 in

Lines per rail

4

rail 21.0 in ≥ 16.5 in

Crushing ceiling

424 psi

Punching shear reinforcement layout — ACI 318-19 §22.6.6 / §8.7.6–8.7.7

System	#3 headed studs — 3 rails per face (12 lines), s₀ = 3.0 in, s = 6.0 in, 4 lines each rail
First line s₀	3.0 in from the column face (≤ d/2 = 4.00 in)
Peripheral spacing	6.0 in (s_max = 6.00 in)
Outer critical section	b_o,out required 212 in → extend 16.5 in beyond the column
A_v per peripheral line	1.32 in² (12 legs)

Cap / column reinforcement layout — ACI §8.7.4.2 · §10.6 · §25.7.2

Column vertical bars	12 – #6 vertical bars (ρ = 1.32 %) with #4 ties @ 12 in, tightened to 6 in through the joint
Tie spacing through joint	6.0 in (#4)
Top band steel over column	#9 @ 6.5 in top over the column band
Structural-integrity bottom bars	6 – #5 continuous bottom bars through the column core (§8.7.4.2)
A_st required / provided	4.00 / 5.28 in²

PASS — Shear reinforcement required: headed shear studs #3, 12 lines, first line at 3.0 in then @ 6.0 in for 4 peripheral lines (21.0 in from the column face) — φv_n = 247 psi ≥ v_u = 235 psi, and the outer section at 16.5 in needs b_o ≥ 212 in.

Detailing sketchPunching shear — critical perimeter plan, stud-rail layout and section

Column 20.0 × 20.0 in, interior condition. Critical section at d/2 (d = 8.00 in), perimeter b_o = 112.0 in. Headed stud rails: 3 rails/face, s₀ = 3.0 in, s = 6.0 in, 4 lines, rail length 21.0 in (extend ≥ 16.5 in).

Module 102AASHTO LRFD AISC 360-22 §H3 · §E4 · §F2 · DG9Chapters 8 · 11

Torsion & stability — open sections and HSS

St. Venant plus warping torsion, plate slenderness classification, flexural and flexural-torsional buckling, lateral-torsional buckling, the §H3.2 combined interaction, twist serviceability and the torsional restraint detailing.

ShapeTorque T_ukip-inAxial P_ukipMoment M_uk-ftShear V_ukipLength LftUnbraced L_bftF_yksiDepth d (or H)inWidth b_f (or B)int_f (or wall t)int_w (or wall t)inA_gin²I_xin⁴I_yin⁴Z_xin³S_xin³Jin⁴r_xinr_yinC_b

T_n = F_cr C · a = √(EC_w/GJ) · F_e,T = [π²EC_w/(K_zL)² + GJ]/(I_x+I_y) · (P_r/P_c + M_r/M_c) + (V_r/V_c + T_r/T_c)² ≤ 1

φT_n

2316 kip-in

T_u/φT_n = 0.13

Twist θ

1.36 deg

Section class

compact

flange compact, web compact

KL/r

91.4

F_cr = 27.13 ksi

φP_n

437 kip

φM_n

241 k-ft

closed section — LTB does not govern (§F7)

φV_n

324 kip

L_p / L_r

11.1 / — ft

Interaction

0.589

§H1.1 (torsion ≤ 20 % — may be neglected)

Torsion mechanics — AISC 360-22 §H3 · Design Guide 9

Torsional model	Rectangular HSS, §H3.1: C = 85.8 in³, h/t = 21.0, F_cr = 30.0 ksi
Torsional bending constant a	n/a (closed section)
St. Venant shear stress	0.46 ksi
Warping constant C_w	0 in⁶
Torsion utilization	0.130

Torsional restraint & detailing — AISC §H3.2 · DG9 Ch. 6

Detailing	Torsion is a secondary effect (T_u/φT_n = 0.13) — provide 0.500 in end connection plates able to develop the end torque, with 0.250 in fillet welds.
Internal diaphragms	Not required
End plate thickness	0.500 in
Fillet weld to transfer torque	0.250 in, E70XX
Governing interaction	§H1.1 (torsion ≤ 20 % — may be neglected) = 0.589

PASS — All limit states satisfied: torsion 0.13, axial 0.18, flexure 0.50, shear 0.12; §H1.1 (torsion ≤ 20 % — may be neglected) = 0.59 ≤ 1.0; compact section, closed section — LTB does not govern (§F7).

Detailing sketchTorsion & stability — section, shear flow, twist and interaction

Rectangular HSS, T_u = 300 kip-in vs φT_n = 2316 kip-in (T_u/φT_n = 0.13). Twist over L = 24.0 ft is θ = 1.36°. Combined interaction satisfied.

Module 103AASHTO LRFD AASHTO §6.10.9 / §6.10.11 · AISC §G2.3 / §F13Chapters 8 · 11

Plate-girder transverse & longitudinal stiffeners

Web and flange proportion limits and plate classification, tension-field shear, transverse stiffener size and rigidity, longitudinal stiffener location and inertia, bearing stiffener bearing/column checks, and final fillet-weld legs and lengths.

Web depth DinWeb thickness t_winFlange width b_finFlange thickness t_finStiffener spacing d_oinF_y (web/flange)ksiF_ys (stiffener)ksiPanel shear V_ukipMoment M_uk-ftBearing reaction R_ukipD_c (web in compression)inS_xin³Longitudinal stiffenerTension-field action

k = 5 + 5/(d_o/D)² · V_n = V_p[C + 0.87(1−C)/√(1+(d_o/D)²)] · I_t ≥ d_o t_w³ J · R_sb = 1.4A_pn F_ys

D/t_w

120.0

limit 150

Web / flange class

noncompact / compact

k / C

7.22 / 0.457

V_p

870 kip

φV_n

625 kip

tension field used

Required d_o

60 in

current spacing adequate

Transverse PL

4.5 × 0.375 in

I 26.8 ≥ 23.2 in⁴

Bearing PL

7.3 × 0.688 in

R_sb = 650 kip

Weld legs

0.250 / 0.250 in

intermediate / bearing

Shear panel & plate classification — AASHTO §6.10.2 / §6.10.9 · AISC §G2.3

Web slenderness	D/t_w = 120.0 (limit 150); 2D_c/t_w classification: noncompact
Flange slenderness	b_f/2t_f = 8.00 → compact; proportions OK
Panel aspect d_o/D	1.500
Tension field	permitted and used
V_n (TF / no TF)	625 / 397 kip

Stiffener & weld schedule — AASHTO §6.10.11 · §6.13.3

Transverse stiffeners	Pairs of PL 4.5 × 0.375 transverse stiffeners @ 90 in c/c, 0.250 in fillet welds each side over 58 in, cut back 4t_w–6t_w from the tension flange
Rigidity check	I_t,prov 26.8 in⁴ vs I_t,req 23.2 in⁴ (J = 0.50)
Longitudinal stiffener	No longitudinal stiffener required at this web slenderness
Bearing stiffeners	Bearing stiffeners: pair of PL 7.3 × 0.688, milled to bear, 0.250 in fillet welds full depth (effective column 14.5 in², KL/r = 12)
Bearing stiffener column	A_eff = 14.47 in², KL/r = 12.3, φP_n = 680 kip
Welds	Intermediate 0.250 in fillet each side; bearing stiffener 0.250 in full depth (min leg 0.2500 in)

PASS — Panel verified: tension-field action permitted, C = 0.457, φV_n/V_u = 1.49. Stiffeners 2 × 4.5 × 0.375 in @ 90 in with 0.250 in fillet welds; bearing stiffeners 0.688 in with 0.250 in welds.

Detailing sketchPlate-girder web panel — elevation and stiffener section

Web 60.0 in deep × 0.500 in thick, D/t_w = 120, transverse stiffeners at d_o = 90.0 in (d_o/D = 1.50). V_u = 420 kip vs φV_n = 625 kip (AASHTO 6.10.9).

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