Bridge Loads and Load Combinations

The original Woodrow Wilson Bridge, carrying I-95/I-495 across the Potomac between Alexandria, Virginia and Prince George's County, Maryland, opened in 1961 for a projected 75,000 vehicles per day. By 2000 it carried more than 200,000 vehicles per day — a live-load environment far in excess of the loading assumptions used in its original 1950s design. A landmark FHWA and NCHRP investigation of the crossing (a driver behind the development of the HL-93 notional load model) documented growing fatigue damage, deck deterioration, and a rising fraction of permit trucks. It was demolished after the replacement twin bascule bridges opened in 2006–2008.

The Woodrow Wilson replacement was designed to AASHTO LRFD and rated for HL-93 plus MDOT SHA / VDOT permit envelopes. That project — plus MDOT SHA's ongoing evaluation of the Chesapeake Bay Bridge and the recovery of the Francis Scott Key Bridge site — illustrates why loads (not resistance) drive most modern bridge decisions in the Mid-Atlantic.

Pre-lecture — Section properties & load computation

Before you compute demands: how a bridge cross-section actually resists load

Every demand you will compute in §3.1 – §3.8 is only meaningful when you also know the section that resists it. In a composite steel–concrete bridge the resisting section is not one section — it is three or four, depending on the age of the concrete, the direction of bending, and whether the deck is cracked. This pre-lecture builds the vocabulary and the arithmetic you need to turn a plate girder + deck sketch into a set of design section properties (A, I, S_top, S_bot) and then apply the correct permanent-load pieces (self-weight, deck, SIP formwork, haunch, barriers, wearing surface, cross-frames) at the correct sequence in construction.

Why the sequence matters

The wet concrete deck is resisted by the steel girder alone. The barrier and future wearing surface, poured months later, are resisted by the long-term composite section. HL-93 truck traffic is resisted by the short-term composite section. Using the wrong section under a given load is the single most common student error in bridge design.

PL.1 — Terminology

Naming every piece of a composite bridge cross-section

Section properties are fundamental to load and stress computations, structural analysis, and resistance/capacity checks. In this course they include cross-sectional area A, centroid location ȳ, moment of inertia I, section modulus S, and slenderness ratios. Before we compute any of them, we need agreement on names.

Term	Definition	Typical value
Total slab thickness	Full deck depth as poured (IWS + structural).	8 in – 9½ in
Integral Wearing Surface (IWS)	Sacrificial top layer of the deck assumed to wear off over the design life.	¼ in – ½ in
Structural slab thickness	Deck depth remaining after IWS is deducted. Used for capacity.	7½ in – 9 in
Haunch	Gap between top of steel web and underside of the deck. Accommodates screed elevation and formwork tolerance.	2 in – 4 in
Effective flange width, b_eff	Portion of the deck that acts compositely with a given girder.	See §4.6.2.6 (AASHTO)
Top flange	Compression flange of a positive-moment section. Sized against local buckling.	b_tf × t_tf
Web	Vertical plate resisting shear. Slenderness limited by D/t_w.	D × t_w
Bottom flange	Tension flange in positive-moment regions; typically 1.5× top flange area.	b_bf × t_bf

IWS split-use rule

For loads use the total slab thickness (all the concrete you actually pour weighs on the girder). For resistance use the structural slab thickness (the IWS is assumed gone at end of life).

PL.2 — Effective flange width

Why only part of the deck acts as a girder flange

AASHTO LRFD §4.6.2.6

Concrete compression stress in a wide deck is highest directly above the girder web and drops off toward mid-span between girders — this is shear lag. Rather than integrate a curved stress block, AASHTO defines an effective flange width b_E over which the peak stress f_c is assumed uniform, delivering the same total compressive force.

AASHTO LRFD Reference

§4.6.2.6.1 — General. Unless specified otherwise, the effective flange width of a concrete deck slab in composite or monolithic construction may be taken as the tributary width perpendicular to the axis of the member for determining cross-section stiffnesses for analysis and for determining flexural resistances.

Tributary width means: for an interior girder, half the spacing to each adjacent girder — i.e. the full girder spacing S. For an exterior girder, half the spacing to the adjacent interior girder plus the deck overhang. That's why the same bridge often has two sets of composite section properties — one interior, one exterior. Always label your section with which one you are computing.

PL.3 — Materials & modular ratio

Transforming concrete into equivalent steel

AASHTO LRFD §5.4.2.4 / §6.4.1

The girder is steel and the deck is concrete — two materials with very different stiffnesses. To perform a single beam-bending analysis, we transform the concrete into an equivalent area of steel using the modular ratio n.

(PL.3-1)

1. Modular ratio

   $$n \;=\; \frac{E_{s}}{E_{c}}$$

$E_{s}$

Modulus of elasticity of structural steel [ksi]

$E_{c}$

Modulus of elasticity of concrete [ksi]

$n$

Modular ratio, rounded to nearest whole number

For all grades of structural steel AASHTO AASHTO LRFD §6.4.1 sets E_s = 29,000 ksi. For normal-weight concrete AASHTO AASHTO LRFD §C5.4.2.4 gives the practical form:

(PL.3-2)

1. Concrete modulus

   $$E_{c} \;=\; 1{,}820\sqrt{f'_{c}}$$

$f'_{c}$

Specified 28-day compressive strength [ksi]

f′_c (ksi)	E_c (ksi)	n = 29,000 / E_c	Rounded n	3n (long-term)
3.0	3152	9.20	9	27
3.5	3405	8.50	8	24
4.0	3640	7.97	8	24
4.5	3861	7.51	8	24
5.0	4070	7.13	7	21
6.0	4458	6.51	7	21
8.0	5148	5.63	6	18

PL.4 — Positive-bending section family

Noncomposite, short-term composite, long-term composite

AASHTO LRFD §6.10.1.1

The class of section that resists a given load depends on when that load arrives on the bridge and how long it acts.

1. Noncomposite (steel only). The bare steel girder — before the deck has cured — resists every load applied during construction: girder self-weight, wet concrete deck, haunch, SIP formwork, and cross-frames.
2. Short-term composite (steel + b_eff / n). After the deck cures, transient loads (traffic, wind on live load) are applied for seconds to minutes. The concrete is transformed by dividing the effective width by n.
3. Long-term composite (steel + b_eff / 3n). Permanent loads added after the deck cures — barriers, sidewalks, future wearing surface, utility conduits, sign structures — act for decades. Concrete creeps under sustained stress; using 3n instead of n reduces the transformed area so the steel picks up the extra strain the creeping concrete cannot sustain.

Why 3n and not 2n or 4n

The 3× factor traces to the AASHTO calibration of long-term creep coefficient for normal-weight concrete under sustained stress. It is a code-mandated approximation — do not derive it from first principles for design; simply apply it whenever a load is sustained (DC applied to composite, DW, utilities, etc.).

PL.5 — Negative-bending sections

What happens over the pier

Over interior supports of continuous girders the top of the deck is in tension. Concrete has essentially no tensile strength for design purposes, so we assume the deck concrete cracks and take steel + longitudinal deck reinforcement as the resisting section. Two methods are permitted:

• Count every longitudinal deck bar within b_eff and include its area and centroid — tedious but exact.
• Use the conservative shortcut of AASHTO LRFD §6.10.1.1.1c: assume 1% longitudinal reinforcement in the effective flange width (two-thirds in the top mat, one-third in the bottom mat).

Design cover requirements (AASHTO §5.10.1)

Locate top-mat longitudinal reinforcement using cover: 2.5 in for deck surfaces exposed to deicing salts (typical for MDOT SHA), 1.0 in bottom cover for the bottom-mat bars up to No. 11. Cover directly drives the effective depth of the cracked-section analysis.

PL.6 — The arithmetic

Parallel-axis theorem, applied to every section in the family

For each section variant (noncomposite, short-term, long-term, steel + rebar) you follow the same 5-step recipe:

1. Choose a datum (usually bottom of bottom flange).
2. List each component: area A_i and centroid height y_i above the datum.
3. Compute the section centroid ȳ = ΣA_iy_i / ΣA_i.
4. Compute each component's d_i = y_i − ȳ, then sum I = Σ(I_o,i + A_id_i²).
5. Get section moduli: S_top = I / c_top and S_bot = I / c_bot.

1. Parallel-axis theorem

   $$I \;=\; \sum\left(I_{o} + A\,d^{2}\right)$$

   (PL.6-1)

2. Rectangle self-inertia

   $$I_{o} \;=\; \frac{b\,h^{3}}{12}\quad\text{(for a rectangle)}$$

   (PL.6-2)

3. Section modulus and bending stress

   $$S \;=\; \frac{I}{c} \quad\Rightarrow\quad \sigma_{max} \;=\; \frac{M}{S}$$

   (PL.6-3)

PL.6b — Preliminary sizing

How to select a trial section for a plate girder

AASHTO LRFD §6.10

Before you can compute section properties you need a starting cross-section. The following rules come from AASHTO slenderness limits, plate-availability tables from the AASHTO/NSBA Steel Bridge Collaboration, and decades of fabricator practice. They give an economical trial girder — you will iterate once demands are known.

#	Dimension	Code limit	Design rule (backed off)	Trial equation
1	Girder depth D	L/D ≤ 30 (Table 2.5.2.6.3-1)	L/D = 27.5	D = L / 27.5
2	Web thickness tw	D/t_w ≤ 150 (unstiffened)	D/t_w = 120	t_w = D / 120
3	Compression flange width bfc	b_f/D ≤ 1/6	b_fc/D = 1/4 (stockier)	b_fc = D / 4
4	Compression flange thickness tfc	b_fc/(2 t_fc) ≤ 0.38 √(E/F_yc)	Fy = 50 ksi → t_fc = b_fc/18.3 Fy = 70 ksi → t_fc = b_fc/15.5	t_fc ≥ b_fc / (0.76 √(E/F_yc))
5	Tension flange thickness tft	Typically 1.5× top flange	Assume b_ft = b_fc	t_ft = 1.5 t_fc

Why these rules of thumb?

Rule 1 sets a stiffness that keeps live-load deflection manageable. Rule 2 keeps the web from bend-buckling before we've done any real analysis. Rule 3 keeps the flange stocky enough that we don't spend the whole design budget on lateral-torsional buckling. Rule 4 is the AASHTO local-buckling limit rearranged for t_fc. Rule 5 reflects that in positive-moment regions the bottom flange (tension) usually needs more area than the top (compression) because the compression flange gets help from the composite deck.

Worked trial — span L = 150 ft (1,800 in):

PL.6b

Trial section — span L = 150 ft (1,800 in)

1. Girder depth

   $$D \;=\; \tfrac{L}{27.5} \;=\; \tfrac{1{,}800}{27.5} \;=\; 65.45\ \text{in}\ \Rightarrow\ D \approx 66\ \text{in}$$

2. Web thickness

   $$t_{w} \;=\; \tfrac{D}{120} \;=\; \tfrac{66}{120} \;=\; 0.55\ \text{in}\ \Rightarrow\ t_{w} = \tfrac{5}{8}\ \text{in}$$

3. Flange width

   $$b_{f} \;=\; \tfrac{D}{4} \;=\; \tfrac{66}{4} \;=\; 16.5\ \text{in}\ \Rightarrow\ b_{f} \approx 18\ \text{in}$$

4. Flange thicknesses

   $$t_{fc} \;=\; \tfrac{b_{fc}}{18.3} \approx 1.0\ \text{in},\quad t_{ft} \;=\; 1.5\,t_{fc} \;=\; 1\tfrac{1}{2}\ \text{in}$$

5. Trial girder

   $$18{\times}1\ \text{top flange},\ \ 66{\times}\tfrac{5}{8}\ \text{web},\ \ 18{\times}1\tfrac{1}{2}\ \text{bottom flange}$$

Layout considerations before you draw plates

• Girder spacing. Wider is generally more economical (fewer girders, cross-frames, bearings). For L < 140 ft use S = 10–11 ft; for L > 140 ft use S = 11–14 ft. Wider spacing → thicker deck.
• Minimum plate thicknesses. Stiffeners & connection plates 7/16 in min (½ in preferred). Webs 7/16 in min (½ in preferred), 1/16-in increments. Flanges ¾ in min, 1/8-in increments up to 2½ in then ¼-in increments.
• Never change flange width along the girder — change thickness. Butt-weld thicker flange plates with CJP welds, torch-cut a 60-in min transition length. Constant flange width simplifies fabrication.
• Plate availability. Use AASHTO/NSBA plate-length tables (e.g. ½-in × 72-in plate available up to ~972 in long, 3-in × 96-in only ~734 in) so you don't specify a plate the mill can't roll.

PL.7 — Worked example

Noncomposite section properties of a 42-in plate girder

Compute ȳ, I, and both section moduli for the steel-only cross-section below. This is the section that resists deck-pour loading — before composite action develops.

Component	A (in²)	y (in)	Ay (in³)	I_o (in⁴)	d = y − ȳ (in)	Ad² (in⁴)
Top flange (14 × ¾)	10.500	41.625	437.06	0.492	+24.512	6,308.5
Web (40 × 7/16)	17.500	21.250	371.88	2,333.33	+4.137	299.5
Bottom flange (16 × 1¼)	20.000	0.625	12.50	2.60	−16.488	5,437.0
Σ	48.000	—	821.44	2,336.43	—	12,045.0

PL.7

Steel-only section properties (bare plate girder)

1. Centroid

   $$\bar{y} \;=\; \frac{\sum A_i y_i}{\sum A_i} \;=\; \frac{821.44}{48.000}$$

2. Moment of inertia

   $$I \;=\; \sum I_{o,i} + \sum A_i d_i^{2} \;=\; 2{,}336.43 + 12{,}045.0$$

3. Section modulus — top

   $$S_{top} \;=\; \frac{I}{c_{top}} \;=\; \frac{14{,}381}{42.00-17.11} \;=\; \frac{14{,}381}{24.89}$$

4. Section modulus — bottom

   $$S_{bot} \;=\; \frac{I}{c_{bot}} \;=\; \frac{14{,}381}{17.11}$$

5. Bare-steel properties

   $$\bar{y}=17.11\ \text{in},\ \ I=14{,}381\ \text{in}^{4},\ \ S_{top}=577.8\ \text{in}^{3},\ \ S_{bot}=840.5\ \text{in}^{3}$$

$\bar{y}$

centroid above the datum [in]

$I$

second moment of area about the neutral axis [in^4]

$c$

distance from neutral axis to extreme fibre [in]

Short-term composite — same recipe, one extra row

Add the transformed deck (b_eff / n × t_s) centered at the mid-depth of the slab, above the top flange by (haunch + t_s/2). For b_eff = 120 in, t_s = 8 in, n = 8, the transformed slab area = 120 in². The composite centroid rises to ≈39.2 in above datum and I_ST ≈ 47,700 in⁴ — more than 3× the bare-steel I.

PL.8 — Permanent loads

What weighs on the bridge, and which section catches it

AASHTO LRFD §3.5.1 / §3.3.2

Permanent loads are grouped by which section resists them, because that drives the modular ratio used in analysis. AASHTO breaks the components into two design codes — DC (structural components and non-structural attachments) and DW (wearing surfaces and utilities) — because they carry different load factors (γ_p = 1.25/0.90 for DC vs. 1.50/0.65 for DW).

Permanent load	Code	Applied to	Typical value	Formula per girder
Steel plate girder self-weight	DC1	Noncomposite (steel only)	ρ_s = 490 pcf ≈ 0.490 kcf	w_g = A_s × ρ_s
Wet concrete deck (total thickness)	DC1	Noncomposite (steel only)	ρ_c = 150 pcf; add 5 pcf for reinforcement → 155 pcf	w_deck = t_total × S × ρ_c
Concrete haunch	DC1	Noncomposite (steel only)	0.150 kcf × haunch height × top-flange width	w_h = t_haunch × b_tf × ρ_c
SIP metal deck formwork + ponded concrete in flutes	DC1	Noncomposite (steel only)	≈15 psf of deck area (form) + concrete fill (add ¼ in equivalent)	w_SIP = 15 psf × (S − b_tf)
Cross-frames / diaphragms	DC1	Noncomposite (steel only)	Point loads at brace lines; ~200 – 500 lb each end reaction	Distribute to nearest node
Stiffeners, splice plates, bolts	DC1	Noncomposite (steel only)	5 – 10% uplift on girder self-weight	w_misc = 0.08 × w_g
Concrete parapets / Jersey barriers	DC2	Long-term composite (3n)	MDOT SHA F-shape ≈ 470 plf per barrier	w_b = (Σ barriers) / N_g (or distribute to exterior only)
Sidewalks / raised medians (if cast after deck)	DC2	Long-term composite (3n)	0.150 kcf × sidewalk cross-section	w_sw = A_sw × ρ_c / N_g
Bituminous wearing surface / future overlay	DW	Long-term composite (3n)	MDOT SHA: 25 – 30 psf (2-in asphalt)	w_ws = t_ws × ρ_asp × (roadway width) / N_g
Utilities (waterline, gas, conduit)	DW	Long-term composite (3n)	Project-specific; get value from utility drawings	w_u = (weight per foot) / N_g

Distribution to girders

Deck weight distributes to each girder by tributary width. Barriers are typically divided equally among all girders or assigned entirely to the exterior girder (MDOT SHA allows either — the exterior-girder-only method governs for exterior-girder design). Future wearing surface is distributed by tributary width of the roadway carried.

PL.8b — Worked example

Computing every permanent load on an interior girder

Given the bridge cross-section of Figure PL.4 (out-to-out width 36 ft, four girders at S = 10 ft 6 in, deck total t_total = 8½ in with ½-in IWS, haunch = 2 in, top flange b_tf = 14 in, two F-shape barriers) and the trial girder from §PL.6b (A_s = 48.875 in²), compute the DC1, DC2, and DW loads carried by one interior girder. Use ρ_c = 0.150 kcf and ρ_s = 0.490 kcf.

Step 1 — DC1: steel girder self-weight

1. Formula

   $$w_{g} \;=\; A_{s}\cdot \rho_{s}$$

2. Substitute

   $$w_{g} \;=\; \frac{48.875\ \text{in}^{2}}{144}\cdot 0.490\ \text{kcf}$$

3. Result

   $$w_{g} = 0.166\ \text{klf}$$

Add ~8% for stiffeners, splices, connection plates:

1. Formula

   $$w_{g,misc} \;=\; 1.08 \cdot 0.166$$

2. Result

   $$w_{g,misc} = 0.180\ \text{klf}$$

Step 2 — DC1: wet concrete deck

1. Formula

   $$w_{deck} \;=\; t_{total}\cdot S \cdot \rho_{c}$$

2. Substitute

   $$w_{deck} \;=\; \tfrac{8.5}{12}\cdot 10.5 \cdot 0.150$$

3. Result

   $$w_{deck} = 1.116\ \text{klf}$$

Step 3 — DC1: haunch

1. Formula

   $$w_{h} \;=\; t_{haunch}\cdot b_{tf} \cdot \rho_{c}$$

2. Substitute

   $$w_{h} \;=\; \tfrac{2}{12}\cdot \tfrac{14}{12}\cdot 0.150$$

3. Result

   $$w_{h} = 0.029\ \text{klf}$$

Step 4 — DC1: SIP formwork + concrete in flutes

Take 15 psf of the tributary form area (girder spacing minus top flange width):

1. Formula

   $$w_{SIP} \;=\; 0.015\ \text{ksf}\cdot (S - b_{tf})$$

2. Substitute

   $$w_{SIP} \;=\; 0.015 \cdot \left(10.5 - \tfrac{14}{12}\right)$$

3. Result

   $$w_{SIP} = 0.140\ \text{klf}$$

Step 5 — DC1: cross-frames (as an equivalent UDL)

Assume K-frames at 25-ft spacing, each weighing ~350 lb; end reaction on one girder ~175 lb. Smeared over 25 ft:

1. Substitute

   $$w_{xf} \;\approx\; \frac{0.175\ \text{kip}}{25\ \text{ft}}$$

2. Result

   $$w_{xf} = 0.007\ \text{klf}$$

DC1 component	w (klf)
Steel girder + details	0.180
Wet concrete deck	1.116
Haunch	0.029
SIP formwork + fill	0.140
Cross-frames (smeared)	0.007
Σ wDC1	1.472 klf

Applied to the noncomposite steel section.

Step 6 — DC2: barriers

Two F-shape barriers at 0.470 klf each, distributed equally to the four girders:

1. Substitute

   $$w_{b} \;=\; \frac{2 \cdot 0.470}{4}$$

2. Result

   $$w_{b} = 0.235\ \text{klf}$$

Step 7 — DW: future wearing surface

Assume 25 psf over the 34-ft clear roadway, distributed to all four girders:

1. Substitute

   $$w_{ws} \;=\; \frac{0.025\ \text{ksf}\cdot 34\ \text{ft}}{4}$$

2. Result

   $$w_{ws} = 0.213\ \text{klf}$$

Applied later	Code	w (klf)	Resisting section
Barriers	DC2	0.235	Long-term composite (3n)
Future wearing surface	DW	0.213	Long-term composite (3n)

Sanity check the numbers

For an interior girder on a typical 4-girder composite highway bridge, DC1 usually lands between 1.2 and 1.7 klf — dominated by the wet deck. DC2 (barriers) is typically 0.15 – 0.30 klf per girder, and DW (2-in future overlay) is 0.15 – 0.25 klf. If your totals are far outside these ranges, check your tributary width, unit weights, and whether you accidentally distributed barrier weight only to the exterior girder.

PL.9 — Live loads, at a glance

What §3.1 – §3.4 will compute for you

AASHTO LRFD §3.6

The three live-load streams you will meet in §3.2 are all resisted by the short-term composite section:

• HL-93 design truck (8 + 32 + 32 kip axles) — see §3.2.
• HL-93 design tandem (25 + 25 kip axles, 4 ft apart) — governs short spans.
• Design lane load (0.64 klf uniform) — the "traffic queue" component, always superimposed on truck or tandem.

Each is amplified by the dynamic load allowance IM (33% for strength, 15% for fatigue) and factored for how many lanes are simultaneously loaded via the multiple-presence factor m. Finally, an approximate live-load distribution factor DF (AASHTO Table 4.6.2.2.2b-1) assigns a fraction of the axle line to each girder — because you almost never do a full 3D analysis for routine composite steel bridges.

(PL.9-1)

1. Live-load moment

   $$M_{LL+IM} \;=\; DF \cdot (1 + IM) \cdot m \cdot M_{lane\text{-}line}$$

With M_LL+IM in hand and S_ST from §PL.7 you can already compute the live-load bending stress at any fiber:

1. Bending stress

   $$\sigma_{LL+IM,\ bottom} \;=\; \frac{M_{LL+IM}}{S_{ST,\ bot}}$$

…which you'll then combine with the DC1 stress (on noncomposite S) and DC2 + DW stresses (on long-term S) in the Strength I equation from §3.4.

PL.10 — Design workflow

Putting section properties + loads together

1. Sketch the cross-section with dimensions. Note interior vs exterior girder, spacing, overhang, deck thickness, IWS, haunch.
2. Compute the noncomposite section properties (§PL.7 recipe).
3. Compute b_eff (§PL.2), then the short-term composite (÷n) and long-term composite (÷3n) properties.
4. For continuous spans, also compute the steel + longitudinal reinforcement properties for negative-moment regions (§PL.5).
5. List every permanent load (§PL.8) and tag it DC1 / DC2 / DW. Compute its unfactored moment/shear per girder using structural analysis (Chapter 4 tools).
6. Compute the live-load force effects (§3.2 – §3.3) and apply the distribution factor.
7. Combine with the AASHTO load factors from Table 3.4.1-1 (§3.4). Check stresses on the correct section for each load type.

Common mistakes to avoid

• Using short-term S for the barrier or the future wearing surface.
• Forgetting the haunch weight (it can be 20 – 40 plf per girder).
• Forgetting SIP formwork (~15 psf) — often 5 – 10% of the deck DC1.
• Using total slab thickness for capacity or structural slab thickness for weight.
• Using interior b_eff for both interior and exterior girders.
• Dividing barrier weight by the number of girders when only the exterior girder should carry it.

With this pre-lecture complete you're ready for §3.1 Load Categories (below), where we formalize the AASHTO notation for every load type you just met, and then move into the HL-93 live-load model, dynamic allowance, and Strength/Service combinations.

Every highway bridge is subject to two broad classes of loads: permanent (present throughout the design life, essentially constant) and transient (variable in space, time, or occurrence). AASHTO LRFD subdivides these into notation each engineer must memorize, because every subsequent chapter references it.

Estimating dead load (DC, DW). Dead load of a structural component is its volume times its unit weight. AASHTO LRFD Table 3.5.1-1 tabulates the unit weights the designer must use so that every engineer computing DC and DW on the same girder arrives at the same number. The most-used values are reproduced below (kip/ft³):

Adapted from AASHTO LRFD §3.5.1 (Table 3.5.1-1), 10th Edition (2024). Verify current values against the specification for each new design; older editions used slightly different concrete-strength thresholds.

The table above lists the notation; this lecture is what the notation actually means in a design calculation. For each load, you need to know (a) the physical mechanism, (b) the AASHTO-prescribed equation or table used to quantify it, and (c) the units and typical magnitude on a Mid-Atlantic bridge. Every worked example in this course draws from the equations below.

DC — Dead load of structural components AASHTO LRFD §3.5.1

The self-weight of everything that forms part of the structure: girders, deck slab, diaphragms, cross-frames, stiffeners, bearings, barriers, and end blocks. Computed by multiplying volume by unit weight from AASHTO Table 3.5.1-1 (γ_c = 150 pcf for normal-weight reinforced concrete, γ_s = 490 pcf for structural steel).

Example: a 9-in cast-in-place deck at 10-ft girder spacing gives w_DC,deck = 0.150 · 0.75 · 10 = 1.125 klf on each interior girder.

DW — Dead load of wearing surface and utilities AASHTO LRFD §3.5.1

Everything not part of the structural cross-section: asphalt overlay, waterproofing membrane, utilities (waterlines, conduits, gas lines). DW is separated from DC because it has a higher load factor (γ_DW = 1.50 vs. γ_DC = 1.25) — future overlays and utilities are far less certain than the structural steel weight.

MDOT SHA standard future-wearing-surface allowance: q_FWS = 0.025 ksf (equivalent to a 2-in bituminous overlay at 150 pcf).

LL — Vehicular live load AASHTO LRFD §3.6.1

HL-93 as introduced in §3.2. The force effect from LL is obtained by positioning the design truck (or tandem) and design lane load on the influence line for the response of interest. A closed-form simple-span expression that every graduate should have memorized:

BR — Vehicular braking force AASHTO LRFD §3.6.4

Horizontal force at 6 ft above the deck, applied longitudinally to the superstructure to represent decelerating trucks. AASHTO requires the larger of:

WA — Water load and stream pressure AASHTO LRFD §3.7

Longitudinal pressure on a pier submerged in flowing water:

WS — Wind on structure AASHTO LRFD §3.8

The 10th Edition adopts the same wind-hazard framework used by ASCE 7 (basic wind speed V, exposure category, gust factor, pressure coefficients). Design pressure on the exposed area of the girder web and barriers:

TU — Uniform thermal expansion AASHTO LRFD §3.12.2

Change in bridge length due to uniform temperature change; drives bearing displacement and pier flexure. For Maryland (moderate climate zone, cold-climate steel bridges): ΔT = 150 °F design range for steel and ΔT = 80 °F for concrete.

EQ — Earthquake AASHTO LRFD §3.10

Elastic seismic response coefficient C_sm as a function of site class, S_DS, S_D1, and structural period T:

Maryland is a low-to-moderate seismic region; most MDOT SHA bridges fall in Seismic Zone 1 (S_D1 ≤ 0.15) — minimum detailing requirements apply but explicit seismic analysis rarely governs member sizes.

CV / CT — Vessel or vehicle collision AASHTO LRFD §3.14

Extreme-event load on piers exposed to navigable waterways or highway traffic. Vessel impact force is a function of vessel dead-weight tonnage, transit speed, and pier geometry — the 2024 Francis Scott Key Bridge collapse is the reference case for the Mid-Atlantic and has driven MDOT SHA's revised vessel-collision risk assessments on the Bay and Patapsco crossings.

CR / SH — Creep and shrinkage AASHTO LRFD §5.4.2.3

Time-dependent volume changes in concrete. Critical for prestressed girders because they cause loss of prestress and long-term camber growth. Handled in detail in Chapter 9.

HL-93 is a notional loading — its combination of design truck and design lane load was calibrated by NCHRP Project 12-33 to envelope the force effects of a fleet of real heavy vehicles observed on U.S. bridges. The design engineer selects, for each response of interest:

1. (design truck + design lane load), or
2. (design tandem + design lane load),

whichever produces the larger force effect.

The governing document

All rules, factors, and combinations in this chapter derive from the AASHTO LRFD Bridge Design Specifications, 10th Edition (2024). Every equation cited with an AASHTO reference is verified against this edition. Older editions used different HL-93 load factors and IM values — always check the edition printed on the cover before using published examples.

The static HL-93 forces are amplified for dynamic vehicle–bridge interaction (bouncing, pothole strike, joint impact) using the dynamic load allowance IM, and reduced for the low likelihood that all traffic lanes are simultaneously loaded to maximum using the multiple-presence factor m.

AASHTO LRFD requires the engineer to check every relevant limit state — a boundary between acceptable and unacceptable performance. Each limit state is expressed as a linear combination of factored loads and compared to the corresponding factored resistance:

The following table lists the most commonly used highway load factors from the AASHTO LRFD 10th Edition. Values shown are maximum load factors; minimum values (used when a load is favorable to the response) are noted where required.

Values are max factors; Service I TU shows force-effect / deformation split. Consult the current AASHTO LRFD table for the complete matrix including wave, ice, temperature-gradient, and vehicle-collision entries.

Manipulate the truck below to find the maximum moment and shear at any section on a 120-ft simply supported span. This is the same core operation the software performs when computing envelopes, and it should be routine before you trust any commercial output.

Use the interactive calculator to explore how span length, IM, number of loaded lanes, and multiple-presence factor combine into per-lane envelopes. The report at the bottom is exportable for your notes.

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

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

The calculator returns per-lane envelopes; you still need distribution factors (Chapter 4) to obtain girder-line demands.

Section 3.7 works two examples end-to-end. Example 1 is the classical AASHTO reinforced-concrete T-beam bridge — a short simple span where every step (section properties, HL-93 truck / tandem / lane loads, distribution factors, dead loads, Strength I combinations) can be done by hand. Example 2 scales the same workflow up to a three-span continuous steel plate-girder bridge used in Mid-Atlantic practice.

Problem statement. A bridge is to be designed with a span length of L = 50 ft. The superstructure consists of five T-beams spaced at S = 10 ft with a cast-in-place reinforced-concrete deck slab of t_s = 9 in. The overall width is 48 ft and the clear (roadway) width is 44 ft 6 in. Design the superstructure of the T-beam bridge using the specifications below. The three limit states considered are Strength I, Fatigue II, and Service I.

2

Choose deck thickness and stem width

What we are doing: pick minimum trial dimensions that (a) satisfy AASHTO minimums for a T-beam serving as its own deck and (b) leave room for two layers of #11 bars in the stem.

• Minimum deck thickness for a T-beam deck slab ≥ 7 in AASHTO LRFD §5.14.1.5.1a, §9.7.1.1. Try t_s = 9 in.
• Minimum web thickness = 8 in AASHTO LRFD Com. §5.14.1.5.1c.
• Concrete cover for main epoxy-coated bars ≥ 1 in; use 1.5 in for main + stirrups AASHTO LRFD Tbl. 5.12.3-1.

Minimum stem width for four #11 bars in a single row with #4 stirrups and 2.0 in clear cover. Write the formula first, then substitute:

1. Formula

   $$b_{min} = 2 \cdot (\text{cover} + d_{stirrup}) + n \cdot d_{b} + (n - 1) \cdot (1.5 \cdot d_{b})$$

2. Substitute

   $$b_{min} = 2 \cdot (2.0 + 0.5) + 4 \cdot (1.41) + 3 \cdot (1.5 \cdot 1.41) = 5.0 + 5.64 + 6.345 \approx 17\ \text{in}$$

   cover = 2.0 in, d_stirrup (#4) = 0.5 in, n = 4, d_b (#11) = 1.41 in

3. Result

   $$b_{min} \approx 17\ \text{in}$$

Round up: try b_w = 18 in.

5

Interior T-beam section properties

What we are doing: the moment demand from Strength I acts about the composite T-beam's centroid; we need ȳ, I_g, and S to size reinforcement in later chapters.

1. Area

   $$A = (9)(120) + (35)(18) = 1{,}710\ \text{in}^2$$

2. Centroid from bottom

   $$\bar{y}_{t} = \frac{(9 \cdot 120)(35 + 4.5) + (35 \cdot 18)(17.5)}{1{,}710} = 31.4\ \text{in}$$

   Parallel-axis theorem numerator ÷ total area

3. Moment of inertia about the centroid

   $$I_{g} = \frac{120 \cdot 9^{3}}{12} + (120 \cdot 9)(12.5 - 4.5)^{2} + \frac{18 \cdot 35^{3}}{12} + (18 \cdot 35)(31.4 - 17.5)^{2} = 264{,}183.6\ \text{in}^4$$

4. Section modulus

   $$S = I_{g} / \bar{y}_{t} = 264{,}183.6 / 31.4 = 8{,}413.5\ \text{in}^3$$

7

Truck and tandem live-load effects per lane

What we are doing: maximum simple-span moment under HS-20 occurs with the center 32-kip axle at midspan (Barré's theorem gives the same result within rounding for the 14-14 spacing). We take moments about the right support to get the reaction, then compute midspan moment. Simple-beam influence-line ordinates at the load positions give shears.

Formula (midspan moment from statics). Sum axle contributions using the influence-line ordinate at midspan, η_i = min(x_i, L − x_i)·½:

1. Formula

   $$M_{C} = \sum P_{i} \cdot \eta_{i}(x_{i}) \ \text{where} \ \eta_{i} = a_{i} \cdot (L - a_{i}) / L$$

2. Substitute HS-20 truck

   $$M_{tr} = 32 \cdot (12.5 + 5.5) + 8 \cdot (5.5) = 576 + 44 = 620\ \text{k-ft}$$

   Center 32-k axle at midspan; arms a₁ = 5.5 ft (8-k), a₂ = 19.5 ft (center 32-k), a₃ = 33.5 ft (rear 32-k)

3. Substitute HS-20 tandem

   $$M_{tandem} = 25 \cdot (12.5 + 10.5) = 575\ \text{k-ft}$$

   Two 25-k axles at 4 ft, centered at midspan

Formula (shear at left support). Using the shear influence line η_V(x) = (L − x)/L for loads to the right of the support:

1. Formula

   $$V_{A} = \sum P_{i} \cdot \eta_{V}(x_{i}) = \sum P_{i} \cdot (L - x_{i}) / L$$

2. Substitute truck at support

   $$V_{tr} = 32 \cdot (1.00) + 32 \cdot (0.72) + 8 \cdot (0.44) = 32 + 23.04 + 3.52 \approx 58.6\ \text{kips}$$

   Lead 32-k axle at A, so η values are 1.00, 0.72, 0.44

3. Substitute tandem straddling the support

   $$V_{tandem} = 25 \cdot (1.00 + 0.92) = 48\ \text{kips}$$

   η = 1.00, 0.92

9

Distribution factors for moment (DFM)

Why: the truck sits on one lane, but its load spreads to several girders. AASHTO Table 4.6.2.2.2b-1 gives closed-form DFMs for cast-in-place T-beam decks (deck type "e"). Applicability checks: 3.5 ≤ S = 10 ≤ 16 ft, 4.5 ≤ t_s = 9 ≤ 12 in, 20 ≤ L = 50 ≤ 240 ft, N_b = 5 ≥ 4. ✓

Longitudinal stiffness parameter (n = 1 for monolithic T-beam):

(4.6.2.2.1-1)

1. Longitudinal stiffness parameter

   $$I_{stem} = (18 \cdot 35^{3})/12 = 64{,}312.5\ \text{in}^4, \ e_{g} = 17.5 + 4.5 = 22\ \text{in}$$
2. $$K_{g} = n \cdot [I + A \cdot e_{g}^{2}] = 1 \cdot [64{,}312.5 + 1{,}710 \cdot 22^{2}] = 891{,}953\ \text{in}^4$$
3. $$[K_{g} / (12 \cdot L \cdot t_{s}^{3})]^{0.1} = 1.05$$

4. Interior beam — one lane loaded

   $$DFM_{si} = 0.06 + (S/14)^{0.4} \cdot (S/L)^{0.3} \cdot [K_{g}/(12Lt_{s}^{3})]^{0.1} = 0.06 + (10/14)^{0.4} \cdot (10/50)^{0.3} \cdot (1.05) = 0.629\ \text{lane/girder}$$

5. Interior beam — two or more lanes loaded

   $$DFM_{mi} = 0.075 + (S/9.5)^{0.6} \cdot (S/L)^{0.2} \cdot [K_{g}/(12Lt_{s}^{3})]^{0.1} = 0.075 + (10/9.5)^{0.6} \cdot (10/50)^{0.2} \cdot (1.05) = 0.859\ \text{lane/girder}$$

   Governs interior

Exterior beam — one lane loaded (lever rule). Treat the deck strip as simply supported between the exterior girder and the first interior girder. Multiple-presence m = 1.20 applies with the lever rule.

Formula. Sum moments about the interior girder support (ΣM = 0), then apply m:

1. Formula

   $$\sum M_{int} = 0: \ -R \cdot S + (P/2) \cdot a_{1} + (P/2) \cdot a_{2} = 0 \ \Rightarrow \ R = (P/2) \cdot (a_{1} + a_{2}) / S$$
2. $$DFM_{se} = m \cdot (R / P)$$

3. Substitute

   $$R = (P/2) \cdot (10.25 + 4.25) / 10 = 0.725 \cdot P$$

   S = 10 ft, a₁ = 10.25 ft, a₂ = 4.25 ft, m = 1.20

4. $$DFM_{se} = 1.20 \cdot 0.725 = 0.87\ \text{lane/girder}$$

   Governs exterior

Exterior beam — two or more lanes (e-factor on interior DF):

1. $$d_{e} = 2.25\ \text{ft} \ (\text{within} -1.0 \le d_{e} \le 5.5)$$
2. $$e = 0.77 + d_{e}/9.1 = 0.77 + 2.25/9.1 = 1.017 \ \rightarrow \ \text{use}\ e = 1.0$$
3. $$DFM_{me} = m \cdot e \cdot DFM_{mi} = 0.85 \cdot 1.0 \cdot 0.859 = 0.730\ \text{lane/girder}$$

This example uses a representative Mid-Atlantic bridge configured to reflect typical MDOT SHA practice. It follows the same 13-step calculation format used in Example 1.

2

Draw the structural system

Apply everything from Chapter 3 (and the pre-lecture on section properties) to the simply-supported reinforced-concrete T-beam bridge shown below. Produce a complete hand-calculation package and upload it at the bottom of this section.

Twenty questions covering load categories, HL-93, IM, m, combinations, influence-line placement, and the worked example. Your attempt is recorded to your student progress.

• HL-93 = (design truck OR tandem) + design lane load. IM applies only to the truck/tandem.
• Multiple-presence m = 1.20, 1.00, 0.85, 0.65 for 1, 2, 3, ≥4 loaded lanes.
• Strength I governs most normal traffic; Strength II governs permit vehicles (MDOT SHA).
• Negative moment at interior supports uses the two-truck × 0.90 rule.
• Always identify governing combination BEFORE selecting member sizes.