AASHTO LRFD Formula & Symbol Library

Chapters 1–2

LRFD basis of design

The single inequality every other calculation in the course serves, plus the load modifier that scales demand for ductility, redundancy and operational importance.

LRFD design inequality

AASHTO LRFD §1.3.2.1

\sum \eta_i\,\gamma_i\,Q_i \;\le\; \phi\,R_n = R_r

(1.3.2.1-1)

$\eta_i$: load modifier = η_D · η_R · η_I
$\gamma_i$: load factor for load component i
$Q_i$: nominal force effect [kip, kip-ft]
$\phi$: resistance factor
$R_n$: nominal resistance
$R_r$: factored (design) resistance

Demand on the left, factored resistance on the right. Every limit state in AASHTO is an instance of this line.

Load modifier

AASHTO LRFD §1.3.2.1

\eta_i = \eta_D\,\eta_R\,\eta_I \;\ge\; 0.95 \quad\text{(max loads)};\qquad \eta_i = \frac{1}{\eta_D\,\eta_R\,\eta_I} \le 1.0 \quad\text{(min loads)}

(1.3.2.1-2/3)

$\eta_D$: ductility factor (0.95–1.05)
$\eta_R$: redundancy factor (0.95–1.05)
$\eta_I$: operational importance (0.95–1.05, Strength only)

Service and fatigue combinations

AASHTO LRFD §3.4.1, Table 3.4.1-1

\begin{aligned} Q_{\text{Service I}} &= 1.00\,DC + 1.00\,DW + 1.00\,(LL+IM) \\ Q_{\text{Service III}} &= 1.00\,DC + 1.00\,DW + 0.80\,(LL+IM) \\ Q_{\text{Fatigue I}} &= 1.75\,(LL+IM)_{\text{fat}} \end{aligned}

$DC$: dead load of structural components
$DW$: dead load of wearing surface and utilities
$LL+IM$: live load plus dynamic allowance

Service III is the tension-in-prestressed-concrete check; Fatigue I is the infinite-life check.

Chapter 2

Loads on bridges

HL-93, dynamic allowance, multiple presence, and the Strength I combination that packages them.

HL-93 live-load envelope (per lane)

AASHTO LRFD §3.6.1.2 / §3.6.2

L_{\text{env}} = m\left[\,(1+IM)\max\!\big(L_{\text{truck}},\,L_{\text{tandem}}\big) + L_{\text{lane}}\right]

$m$: multiple-presence factor (1.20 / 1.00 / 0.85 / 0.65)
$IM$: dynamic load allowance: 33% general, 15% fatigue, 75% deck joints
$L_{\text{lane}}$: 0.64 klf uniform lane load, no IM

Lane-load effects, simple span

AASHTO LRFD §3.6.1.2.4

M_{\text{lane}} = \frac{wL^2}{8},\qquad V_{\text{lane}} = \frac{wL}{2},\qquad w = 0.64\ \text{klf}

$L$: span length [ft]
$w$: design lane load [klf]

HS20 / design-truck maximum moment (simple span)

AASHTO LRFD §3.6.1.2.2

M_{\max} = \frac{P_{\text{tot}}}{L}\left(\frac{L}{2}-\frac{e}{2}\right)^{2} \;-\; \sum P_i\,d_i

$P_{\text{tot}}$: total truck axle load on the span [kip]
$e$: distance from resultant to the reference axle [ft]
$d_i$: distance from the reference axle to axle i [ft]

Place the resultant and the nearest axle equidistant about midspan (Barré's rule); maximum moment occurs under that axle.

Strength I force effect

AASHTO LRFD §3.4.1

Q_{u} = 1.25\,DC + 1.50\,DW + 1.75\,(LL+IM)

$Q_u$: factored force effect (moment, shear, axial)

Use γ_DC = 0.90 and γ_DW = 0.65 when minimising dead load helps the demand (uplift, overturning, sliding).

Braking force

AASHTO LRFD §3.6.4

BR = \max\big(0.25\,(P_{\text{truck}}\ \text{or}\ P_{\text{tandem}}),\;0.05\,(P_{\text{truck}}+w L)\big)

$BR$: braking force per lane, applied 6 ft above the deck [kip]

Wind pressure on the structure

AASHTO LRFD §3.8.1.2

P_Z = 2.56\times10^{-6}\,V^2\,K_z\,G\,C_D

$V$: design 3-s gust wind speed at 33 ft [mph]
$K_z$: pressure exposure and elevation coefficient
$G$: gust effect factor
$C_D$: drag coefficient
$P_Z$: design wind pressure [ksf]

Vessel collision — equivalent static force and risk

AASHTO LRFD §3.14.5 / §3.14.4

P_S = 8.15\,V\sqrt{DWT}\ \ (\text{kip});\qquad AF = N\cdot PA\cdot PG\cdot PC

$V$: vessel impact speed [knots]
$DWT$: vessel deadweight tonnage [tonne]
$N$: annual number of vessels of the category
$PA$: probability of aberrancy
$PG$: geometric probability of striking the pier
$PC$: probability of collapse given impact

AF ≤ 0.0001 for critical bridges, ≤ 0.001 for regular bridges.

Chapter 3

Section properties & composite action

Neutral axis, transformed sections, and the effective flange width that makes a girder-and-deck act as one member.

Centroid and moment of inertia of a built-up section

AASHTO LRFD Mechanics (used throughout §6.10)

\bar y = \frac{\sum A_i\,y_i}{\sum A_i},\qquad I = \sum\!\left(I_i + A_i\,(y_i-\bar y)^2\right)

$A_i$: area of component i [in²]
$y_i$: centroid of component i from the datum [in]
$I_i$: own moment of inertia of component i [in⁴]

Modular ratio and transformed deck

AASHTO LRFD §6.10.1.1.1b

n=\frac{E_s}{E_c},\qquad b_{\text{tr}}=\frac{b_{\text{eff}}}{n}\ \ (\text{short term}),\qquad b_{\text{tr},\infty}=\frac{b_{\text{eff}}}{3n}\ \ (\text{long term})

$E_s$: steel modulus, 29,000 [ksi]
$E_c$: concrete modulus [ksi]
$b_{\text{eff}}$: effective flange width [in]

Use n for transient loads and 3n for permanent loads to account for concrete creep.

Concrete modulus of elasticity

AASHTO LRFD §5.4.2.4

E_c = 120{,}000\,K_1\,w_c^{2.0}\,f'^{\,0.33}_c

(5.4.2.4-1)

$K_1$: aggregate source factor (1.0 unless tested)
$w_c$: unit weight of concrete [kcf]
$f'_c$: specified compressive strength [ksi]

Effective flange width

AASHTO LRFD §4.6.2.6.1

b_{\text{eff}} = \min\!\left(\text{tributary width},\; \tfrac{1}{2}S_{\text{left}}+\tfrac{1}{2}S_{\text{right}}\right)

$S$: girder spacing [in]

Since 2008, AASHTO simply uses the tributary width for most girder bridges.

Section modulus and flexural stress

AASHTO LRFD Mechanics

S = \frac{I}{c},\qquad f = \frac{M}{S} = \frac{M\,c}{I}

$c$: distance from the neutral axis to the extreme fibre [in]
$S$: elastic section modulus [in³]

Chapter 4

Live-load distribution

Turning a per-lane force effect into a per-girder force effect — the step that decides how many girders you need.

Longitudinal stiffness parameter

AASHTO LRFD §4.6.2.2.1

K_g = n\left(I + A\,e_g^{\,2}\right)

(4.6.2.2.1-1)

$n$: modular ratio E_beam / E_deck
$I$: non-composite girder moment of inertia [in⁴]
$A$: non-composite girder area [in²]
$e_g$: distance between girder and deck centroids [in]

DF — interior girder, moment, one lane loaded

AASHTO LRFD §4.6.2.2.2b-1

g_{M}^{1} = 0.06 + \left(\frac{S}{14}\right)^{0.4}\left(\frac{S}{L}\right)^{0.3}\left(\frac{K_g}{12.0\,L\,t_s^{3}}\right)^{0.1}

$S$: girder spacing [ft]
$L$: span length [ft]
$t_s$: deck slab thickness [in]
$K_g$: longitudinal stiffness parameter [in⁴]

Range of applicability: 3.5 ≤ S ≤ 16 ft, 4.5 ≤ t_s ≤ 12 in, 20 ≤ L ≤ 240 ft, N_b ≥ 4.

DF — interior girder, moment, two or more lanes

AASHTO LRFD §4.6.2.2.2b-1

g_{M}^{2+} = 0.075 + \left(\frac{S}{9.5}\right)^{0.6}\left(\frac{S}{L}\right)^{0.2}\left(\frac{K_g}{12.0\,L\,t_s^{3}}\right)^{0.1}

$g_M$: moment distribution factor, lanes per girder

DF — interior girder, shear

AASHTO LRFD §4.6.2.2.3a-1

g_{V}^{1} = 0.36 + \frac{S}{25.0},\qquad g_{V}^{2+} = 0.2 + \frac{S}{12} - \left(\frac{S}{35}\right)^{2.0}

$g_V$: shear distribution factor, lanes per girder

DF — exterior girder

AASHTO LRFD §4.6.2.2.2d / §4.6.2.2.3b

g_{\text{ext}} = e\,g_{\text{int}},\qquad e = 0.77 + \frac{d_e}{9.1}\ (\text{moment}),\quad e = 0.6 + \frac{d_e}{10}\ (\text{shear})

$d_e$: distance from exterior web to interior edge of barrier [ft]

Also check the lever rule and, for cross-framed bridges, the rigid-cross-section (special analysis) equation.

Rigid cross-section (special analysis) for exterior girders

AASHTO LRFD §C4.6.2.2.2d-1

R = \frac{N_L}{N_b} + \frac{X_{\text{ext}}\sum_{}^{N_L} e}{\sum_{}^{N_b} x^{2}}

$N_L$: number of loaded lanes
$N_b$: number of girders
$X_{\text{ext}}$: distance from CG of girders to exterior girder [ft]
$e$: eccentricity of a lane from the CG of girders [ft]

Skew correction factors

AASHTO LRFD §4.6.2.2.2e / §4.6.2.2.3c

\text{Moment: } 1 - c_1\left(\tan\theta\right)^{1.5};\qquad \text{Shear (obtuse corner): } 1.0 + 0.20\left(\frac{12.0\,L\,t_s^{3}}{K_g}\right)^{0.3}\tan\theta

$\theta$: skew angle [deg]

Chapter 5

Deck design

Equivalent strips, overhang design and the empirical design method.

Equivalent strip widths

AASHTO LRFD §4.6.2.1.3, Table 4.6.2.1.3-1

\begin{aligned} \text{Overhang:}&\quad w = 45.0 + 10.0\,X \\ \text{Positive moment:}&\quad w = 26.0 + 6.6\,S \\ \text{Negative moment:}&\quad w = 48.0 + 3.0\,S \end{aligned}

$X$: distance from load to centreline of support [ft]
$S$: girder spacing [ft]
$w$: equivalent strip width [in]

Deck strip design moment

AASHTO LRFD §4.6.2.1

M_{\text{strip}} = \frac{(1+IM)\,m\,P}{w/12}\times(\text{influence coefficient})

$P$: wheel load, 16 kip design truck [kip]
$m$: multiple-presence factor

Minimum flexural reinforcement

AASHTO LRFD §5.6.3.3

M_r \ge \min\!\left(1.33\,M_u,\; \gamma_3\,\gamma_1 f_r S_c\right)

$f_r$: modulus of rupture = 0.24 λ √f'_c [ksi]
$S_c$: section modulus for the extreme tension fibre [in³]
$\gamma_1,\gamma_3$: flexural cracking variability and reinforcement ratio factors

Crack control — reinforcement spacing

AASHTO LRFD §5.6.7

s \le \frac{700\,\gamma_e}{\beta_s f_{ss}} - 2 d_c,\qquad \beta_s = 1+\frac{d_c}{0.7\,(h-d_c)}

(5.6.7-1)

$\gamma_e$: exposure factor: 1.00 Class 1, 0.75 Class 2
$d_c$: cover to the centre of the nearest bar [in]
$f_ss$: tensile stress in reinforcement at Service I [ksi]

Barrier collision — yield-line resistance

AASHTO LRFD §A13.3.1

R_w = \left(\frac{2}{2L_c-L_t}\right)\left(8M_b+8M_w+\frac{M_c L_c^{2}}{H}\right)

$M_b$: flexural resistance of the beam at the top of the wall [kip-ft]
$M_w$: flexural resistance about the vertical axis [kip-ft/ft]
$M_c$: flexural resistance about the horizontal axis [kip-ft/ft]
$L_c$: critical length of the yield-line pattern [ft]
$H$: wall height [ft]

Interior-region form. Compare to F_t = 54 kip (TL-4) applied over L_t = 3.5 ft.

Chapters 6–7

Reinforced concrete — flexure, shear and torsion

Whitney stress block, strain-based φ, MCFT shear, and the torsion design chain.

Stress-block factor

AASHTO LRFD §5.6.2.2

\beta_1 = 0.85 - 0.05\,\frac{f'_c-4.0}{1.0},\qquad 0.65 \le \beta_1 \le 0.85

$f'_c$: concrete compressive strength [ksi]

Nominal flexural resistance — rectangular behaviour

AASHTO LRFD §5.6.3.2.3

a = \frac{A_s f_y}{0.85 f'_c b},\qquad M_n = A_s f_y\left(d_s - \frac{a}{2}\right),\qquad M_r = \phi M_n

$A_s$: area of tension reinforcement [in²]
$f_y$: reinforcement yield strength [ksi]
$d_s$: effective depth to tension steel [in]
$b$: compression-face width [in]
$\phi$: 0.90 tension-controlled RC flexure

Strain-based resistance factor

AASHTO LRFD §5.5.4.2

\varepsilon_t = 0.003\,\frac{d_t-c}{c};\qquad \phi = 0.75 + 0.15\,\frac{\varepsilon_t-\varepsilon_{cl}}{\varepsilon_{tl}-\varepsilon_{cl}},\quad 0.75\le\phi\le0.90

$\varepsilon_t$: net tensile strain in the extreme tension steel
$c$: neutral-axis depth [in]
$d_t$: depth to extreme tension steel [in]

Shear resistance — sectional (MCFT) model

AASHTO LRFD §5.7.3.3

V_n = \min\!\big(V_c+V_s+V_p,\; 0.25 f'_c b_v d_v + V_p\big)

$b_v$: effective web width [in]
$d_v$: effective shear depth = max(0.9d_e, 0.72h) [in]
$V_p$: vertical component of prestress [kip]

Concrete and steel shear contributions

AASHTO LRFD §5.7.3.3-3/4

V_c = 0.0316\,\beta\,\lambda\sqrt{f'_c}\,b_v d_v,\qquad V_s = \frac{A_v f_y d_v\left(\cot\theta+\cot\alpha\right)\sin\alpha}{s}

$\beta$: tensile stress transmission factor
$\theta$: angle of diagonal compression [deg]
$A_v$: area of shear reinforcement within s [in²]
$s$: stirrup spacing [in]

Longitudinal strain for β and θ

AASHTO LRFD §5.7.3.4.2

\varepsilon_s = \frac{\dfrac{|M_u|}{d_v}+0.5N_u+|V_u-V_p|-A_{ps}f_{po}}{E_sA_s+E_pA_{ps}}

$N_u$: factored axial force, tension positive [kip]
$f_po$: ≈ 0.7 f_pu, locked-in prestress [ksi]

Then β = 4.8/(1+750ε_s) and θ = 29 + 3500ε_s for sections with minimum transverse reinforcement.

When torsion must be considered

AASHTO LRFD §5.7.2.1

T_u > 0.25\,\phi\,T_{cr},\qquad T_{cr} = 0.126\,\lambda\sqrt{f'_c}\,\frac{A_{cp}^{2}}{p_c}\sqrt{1+\frac{f_{pc}}{0.126\lambda\sqrt{f'_c}}}

$A_cp$: total area enclosed by the outside perimeter [in²]
$p_c$: outside perimeter of the concrete section [in]
$f_pc$: compressive stress at the centroid after losses [ksi]

Nominal torsional resistance

AASHTO LRFD §5.7.3.6.2

T_n = \frac{2\,A_o\,A_t\,f_y\,\cot\theta}{s}

(5.7.3.6.2-1)

$A_o$: area enclosed by the shear flow path ≈ 0.85 A_oh [in²]
$A_t$: area of one leg of closed transverse torsion reinforcement [in²]
$\theta$: angle of the compression diagonals [deg]
$s$: spacing of the closed stirrups [in]

Longitudinal torsion reinforcement

AASHTO LRFD §5.7.3.6.3

A_\ell \ge \frac{T_u\,p_h}{2\phi A_o f_y \tan\theta}

$p_h$: perimeter of the centreline of the closed stirrups [in]

Interface (cold-joint) shear transfer

AASHTO LRFD §5.7.4.3

V_{ni} = c\,A_{cv} + \mu\left(A_{vf}f_y + P_c\right) \le \min\!\left(K_1 f'_c A_{cv},\; K_2 A_{cv}\right)

$c$: cohesion factor [ksi]
$\mu$: friction factor
$A_cv$: interface area engaged in shear transfer [in²]
$A_vf$: interface shear reinforcement crossing the plane [in²]
$P_c$: permanent net compressive force normal to the plane [kip]

The provision the FIU pedestrian bridge design failed.

Chapter 8

Prestressed concrete

Stress limits, loss estimation, and the Service III tension check that usually governs a girder.

Stress at a fibre, any stage

AASHTO LRFD §5.9.2 (sign convention: compression negative)

f = -\frac{P}{A} \mp \frac{P e\,c}{I} \pm \frac{M c}{I}

$P$: effective prestress force [kip]
$e$: strand eccentricity from the section centroid [in]

Concrete stress limits

AASHTO LRFD §5.9.2.3

\begin{aligned} \text{At transfer:}&\quad f_c \le 0.65 f'_{ci},\qquad f_t \le 0.24\lambda\sqrt{f'_{ci}} \\ \text{At service:}&\quad f_c \le 0.45 f'_c\ (\text{perm.}),\qquad f_t \le 0.19\lambda\sqrt{f'_c}\ (\text{Service III}) \end{aligned}

$f'_ci$: concrete strength at transfer [ksi]
$\lambda$: concrete density modification factor

Approximate time-dependent losses

AASHTO LRFD §5.9.3.3

\Delta f_{pLT} = 10.0\,\frac{f_{pi}A_{ps}}{A_g}\gamma_h\gamma_{st} + 12.0\,\gamma_h\gamma_{st} + \Delta f_{pR}

(5.9.3.3-1)

$\gamma_h$: 1.7 − 0.01H, humidity factor
$\gamma_{st}$: 5/(1+f'_ci), strength factor
$\Delta f_{pR}$: relaxation loss, 2.4 ksi for low-relaxation strand [ksi]

Elastic shortening loss

AASHTO LRFD §5.9.3.2.3

\Delta f_{pES} = \frac{E_p}{E_{ci}}f_{cgp}

$f_cgp$: concrete stress at the strand centroid at transfer [ksi]
$E_ci$: concrete modulus at transfer [ksi]

Flexural resistance with prestressing steel

AASHTO LRFD §5.6.3.2.2

M_n = A_{ps}f_{ps}\left(d_p-\frac{a}{2}\right) + A_sf_y\left(d_s-\frac{a}{2}\right) - A'_sf'_y\left(d'_s-\frac{a}{2}\right)

$f_ps$: average stress in prestressing steel at nominal resistance [ksi]
$d_p$: depth to the strand centroid [in]

Strand stress at nominal resistance (bonded)

AASHTO LRFD §5.6.3.1.1

f_{ps} = f_{pu}\left(1-k\frac{c}{d_p}\right),\qquad k = 2\left(1.04-\frac{f_{py}}{f_{pu}}\right)

$f_pu$: ultimate strand strength, 270 ksi typical [ksi]
$f_py$: yield strength of the strand [ksi]

Chapters 9–10

Steel girders — flexure, shear and stability

Compactness, plastic and yield moments, web shear, and the fatigue check.

Web compactness / slenderness

AASHTO LRFD §6.10.6.2.2 / §6.10.1.9

\frac{2D_{cp}}{t_w}\le 3.76\sqrt{\frac{E}{F_{yc}}}\quad(\text{compact});\qquad \frac{2D_c}{t_w} \le 5.7\sqrt{\frac{E}{F_{yc}}}\quad(\text{non-slender})

$D_cp$: web depth in compression at plastic moment [in]
$t_w$: web thickness [in]
$F_yc$: compression-flange yield strength [ksi]

Compression-flange slenderness

AASHTO LRFD §6.10.8.2.2

\lambda_f=\frac{b_{fc}}{2t_{fc}},\qquad \lambda_{pf}=0.38\sqrt{\frac{E}{F_{yc}}},\qquad \lambda_{rf}=0.56\sqrt{\frac{E}{F_{yr}}}

$b_fc$: compression flange width [in]
$t_fc$: compression flange thickness [in]

Lateral-torsional buckling resistance

AASHTO LRFD §6.10.8.2.3

F_{nc} = C_b\left[1-\left(1-\frac{F_{yr}}{R_hF_{yc}}\right)\left(\frac{L_b-L_p}{L_r-L_p}\right)\right]R_bR_hF_{yc}\;\le\;R_bR_hF_{yc}

$L_b$: unbraced length [in]
$L_p$: 1.0 r_t √(E/F_yc) [in]
$L_r$: π r_t √(E/F_yr) [in]
$R_b, R_h$: web load-shedding and hybrid factors
$C_b$: moment gradient modifier

Plastic and yield moments (composite)

AASHTO LRFD §D6.1 / §D6.2

M_p = \sum P_i\,\bar d_i,\qquad M_y = M_{D1}+M_{D2}+M_{AD}

$P_i$: plastic force in element i [kip]
$\bar d_i$: distance from the PNA to the element centroid [in]
$M_AD$: additional moment causing first yield [kip-in]

Web shear resistance

AASHTO LRFD §6.10.9.2 / §6.10.9.3.2

V_n = C\,V_p,\qquad V_p = 0.58\,F_{yw}D\,t_w

$C$: ratio of shear-buckling to shear-yield stress
$D$: web depth [in]
$d_o$: transverse stiffener spacing [in]

For stiffened interior panels, add the tension-field term: V_n = V_p[C + 0.87(1−C)/√(1+(d_o/D)²)].

Fatigue resistance

AASHTO LRFD §6.6.1.2.5

\left(\Delta F\right)_n = \left(\frac{A}{N}\right)^{1/3} \ge \tfrac{1}{2}\left(\Delta F\right)_{TH};\qquad N = (365)(75)\,n\,(ADTT)_{SL}

$A$: detail-category constant [ksi³]
$(\Delta F)_{TH}$: constant-amplitude fatigue threshold [ksi]
$n$: stress cycles per truck passage
$(ADTT)_SL$: single-lane average daily truck traffic

Bolted connection resistance

AASHTO LRFD §6.13.2.7 / §6.13.2.8

R_n = 0.38\,A_b F_{ub} N_s\ (\text{shear, threads incl.});\qquad R_n = K_h K_s N_s P_t\ (\text{slip})

$A_b$: bolt area [in²]
$N_s$: number of shear planes / slip planes
$P_t$: minimum required bolt tension [kip]
$K_h, K_s$: hole-size and surface-condition factors

Chapters 12–13

Substructure — piers, abutments and walls

Earth pressure, stability screening, and column interaction.

Rankine earth-pressure coefficients

AASHTO LRFD §3.11.5

K_a = \tan^2\!\left(45^\circ-\frac{\phi'}{2}\right),\qquad K_p = \tan^2\!\left(45^\circ+\frac{\phi'}{2}\right),\qquad K_0 = 1-\sin\phi'

$\phi'$: effective angle of internal friction [deg]

Lateral earth force and live-load surcharge

AASHTO LRFD §3.11.5.1 / §3.11.6.4

P_a = \tfrac{1}{2}\gamma_s K_a H^2,\qquad \Delta p = k\,\gamma_s\,h_{eq}

$\gamma_s$: unit weight of soil [kcf]
$H$: wall height [ft]
$h_eq$: equivalent height of soil surcharge [ft]

Wall stability — eccentricity, sliding and bearing

AASHTO LRFD §11.6.3

e = \frac{B}{2}-\frac{\sum M_R-\sum M_O}{\sum V},\quad e\le\frac{B}{4}\ (\text{soil});\qquad \frac{\phi_\tau R_\tau}{H} \ge 1.0;\qquad q_{\max}=\frac{\sum V}{B-2e}\le q_R

$B$: footing width [ft]
$e$: eccentricity of the resultant [ft]
$q_R$: factored bearing resistance [ksf]

Biaxial column interaction (reciprocal load method)

AASHTO LRFD §5.6.4.5

\frac{1}{P_{rxy}} = \frac{1}{P_{rx}}+\frac{1}{P_{ry}}-\frac{1}{\phi P_o},\qquad P_o = 0.85f'_c(A_g-A_{st})+A_{st}f_y

$P_rx, P_ry$: factored axial resistance with eccentricity in one direction only [kip]
$A_st$: total longitudinal reinforcement [in²]

Column slenderness and moment magnification

AASHTO LRFD §5.6.4.3

\frac{K\ell_u}{r} \le 22\ (\text{unbraced, neglect});\qquad \delta_b=\frac{C_m}{1-\dfrac{P_u}{\phi_K P_e}}\ge1.0,\quad P_e=\frac{\pi^2EI}{(K\ell_u)^2}

$K$: effective length factor
$\ell_u$: unbraced length [in]
$r$: radius of gyration [in]

Pile group axial capacity

AASHTO LRFD §10.7.3.8

R_r = \phi_{qp}R_p + \phi_{qs}R_s = \phi_{qp}q_pA_p + \phi_{qs}\sum q_{s,i}A_{s,i}

$q_p$: unit end-bearing resistance [ksf]
$q_s$: unit side (skin) resistance [ksf]
$\phi_{qp},\phi_{qs}$: resistance factors by method of determination

Chapter 15

Bearings, joints and movement

Thermal movement, shape factor, and the shear/rotation checks for elastomeric bearings.

Thermal movement range

AASHTO LRFD §3.12.2 / §14.7.5.3.2

\Delta_T = \alpha\,L\,\left(T_{\text{MaxDesign}}-T_{\text{MinDesign}}\right)\times\gamma_{TU}

$\alpha$: coefficient of thermal expansion (6.0×10⁻⁶ /°F steel, 6.0×10⁻⁶ concrete)
$L$: expansion length [in]
$\gamma_{TU}$: load factor 1.2 for movement design

Elastomeric bearing shape factor

AASHTO LRFD §14.7.5.1

S_i = \frac{L\,W}{2\,h_{ri}\,(L+W)}

$L, W$: plan dimensions of the bearing [in]
$h_ri$: thickness of the i-th elastomer layer [in]

Bearing compressive stress and shear deformation

AASHTO LRFD §14.7.6.3.2 / §14.7.5.3.2

\sigma_s = \frac{P_{\text{total}}}{LW} \le 1.25\ \text{ksi (Method A, steel-reinforced)};\qquad h_{rt} \ge 2\,\Delta_s

$\Delta_s$: maximum total shear deformation [in]
$h_rt$: total elastomer thickness [in]

Chapters 16–17

Extreme events — seismic, scour, ice

Response spectrum, R-factors, and the scour depth that has to be assumed gone.

Design response spectrum

AASHTO LRFD §3.10.4.2

C_{sm} = \begin{cases} A_S + (S_{DS}-A_S)\dfrac{T_m}{T_0}, & T_m < T_0\\[4pt] S_{DS}, & T_0 \le T_m \le T_S\\[4pt] \dfrac{S_{D1}}{T_m}, & T_m > T_S \end{cases}

$S_DS$: design spectral acceleration at 0.2 s = F_a S_S
$S_D1$: design spectral acceleration at 1.0 s = F_v S_1
$T_m$: period of vibration of mode m [s]
$T_S$: S_D1 / S_DS [s]

Seismic design force with response modification

AASHTO LRFD §3.10.9 / Table 3.4.1-1

EQ = \frac{C_{sm}\,W}{R},\qquad \text{Extreme Event I: } 1.25DC+1.50DW+\gamma_{EQ}(LL+IM)+EQ

$R$: response modification factor by element and operational class
$W$: tributary seismic weight [kip]
$\gamma_{EQ}$: live-load factor for Extreme Event I (0 to 0.5)

Local pier scour (HEC-18)

AASHTO LRFD §2.6.4.4 / HEC-18

\frac{y_s}{y_1} = 2.0\,K_1K_2K_3\left(\frac{a}{y_1}\right)^{0.65}\!\mathrm{Fr}_1^{0.43}

$y_s$: local scour depth [ft]
$y_1$: flow depth upstream of the pier [ft]
$a$: pier width [ft]
$\mathrm{Fr}_1$: upstream Froude number
$K_1,K_2,K_3$: nose shape, attack angle and bed condition factors

Ice force on a pier

AASHTO LRFD §3.9.2.2

F = C_a\,p\,t\,w,\qquad C_a = \left(5\frac{t}{w}+1\right)^{0.5}

$p$: effective ice crushing strength [ksf]
$t$: ice thickness [ft]
$w$: pier width at the ice level [ft]

Chapters 18–20

Construction engineering, inspection and rating

Erection-stage stability, load rating, and remaining fatigue life.

Strength combinations during construction

AASHTO LRFD §3.4.2 / §6.10.3.2

Q = 1.25\,DC + 1.50\,C + 1.25\,WC,\qquad f_{bu}+f_\ell \le \phi_f R_h F_{yc}

$C$: construction loads (equipment, formwork, personnel)
$f_bu$: flange major-axis stress without lateral bending [ksi]
$f_\ell$: flange lateral bending stress [ksi]

The construction stage is a limit state, not a contractor problem.

Load and resistance factor rating

AASHTO LRFD MBE §6A.4.2

RF = \frac{C-\gamma_{DC}DC-\gamma_{DW}DW\mp\gamma_P P}{\gamma_{LL}\,(LL+IM)},\qquad RT = RF\times W

$C$: capacity = φ_c φ_s φ R_n [kip, kip-ft]
$RF$: rating factor; RF ≥ 1.0 is adequate
$W$: weight of the rating vehicle [ton]

Remaining fatigue life

AASHTO LRFD MBE §7.2

Y = \frac{R_R A}{n\,(365)\,(ADTT)_{SL}\left(\Delta f\right)^{3}} - a

$Y$: remaining life [yr]
$R_R$: resistance factor for evaluation (0.81 mean, 1.0 minimum)
$a$: present age of the detail [yr]
$\Delta f$: effective stress range [ksi]

Optional live-load deflection limits

AASHTO LRFD §2.5.2.6.2

\Delta \le \frac{L}{800}\ \ (\text{vehicular});\qquad \Delta \le \frac{L}{1000}\ \ (\text{vehicular + pedestrian})

$\Delta$: live-load deflection under HL-93 with all lanes loaded [in]
$L$: span length [in]

Traditional minimum depth

AASHTO LRFD Table 2.5.2.6.3-1

h_{\min} = 0.040L\ (\text{RC T-beam}),\quad 0.045L\ (\text{PC I-girder}),\quad 0.033L\ (\text{composite steel, overall})

$L$: span length [ft]

Every equation in the course