AASHTO LRFD Prestressed Girder Design Calculations & Methods

Design responsibility: This is a software code reference, not a substitute for the adopted AASHTO LRFD edition, project plans, jurisdictional amendments, approved material properties, or review by a qualified bridge engineer. Confirm applicable limit states, load factors, resistance factors, distribution-factor applicability, constructability, and detailing requirements for each project.

1. Scope and Design Sequence

Normalize girder, bridge, material, strand, deck, and load inputs.
Build gross and transformed composite section properties.
Calculate live-load distribution factors and an HL-93 truck/tandem/lane envelope.
Build uniform dead-load envelopes for the girder, slab, barriers, utilities, and wearing surface.
Calculate elastic shortening, approximate long-term losses, and strand force states.
Check transfer, initial, final-service, live-load-plus-dead-load, and deck-fiber stresses.
Check flexural strength, shear strength, transverse reinforcement, longitudinal reinforcement, anchorage-zone reinforcement, interface shear, deflection, camber, development length, and top reinforcement.

2. Main Inputs and Units

Group	Representative inputs	Units / convention
Bridge	`bridge_span`, `girder_length`, `bridge_width`, `nb`, `girder_spacing`, `theta`	ft, ft, ft, count, ft, degrees.
Girder family	`girderType`, `girderPosition`, standard AASHTO, box, plank, or custom section data	Interior/exterior position controls the distribution-factor branch.
Concrete	`fc_deck`, `fci`, `fc`, `gamma2`, `K1`	psi, psi, psi, density factor, and aggregate correction factor.
Prestressing	`fpu`, `fpy`, `Ep`, `db`, `aps1`, `total_no_strands`, `e`, mask data	ksi, ksi, ksi, in, in2, count, and in.
Mild reinforcement	`fy`, `As`, `As_prime`	ksi or psi as normalized, and in2.
Loads	barrier, utility, future wearing surface, lane, truck, tandem, and impact data	klf, ksf, kip/ft, kip-ft, and kip.

3. Section Properties

3.1 Gross girder properties

Standard AASHTO and box section families use the property tables in the calculation module. Custom sections use the supplied area, centroid, and moment of inertia. The basic section moduli are:

\[ S_b=\frac{I_g}{C_b} \qquad S_t=\frac{I_g}{C_t} \]

3.2 Concrete modulus

The module's concrete modulus expression is:

\[ E_c=120000K_1\gamma_2^2\left(\frac{f'_c}{1000}\right)^{0.33} \]

Separate values are calculated for deck concrete, release-strength girder concrete, and final-strength girder concrete.

3.3 Transformed composite section

The deck is transformed to girder-concrete stiffness using:

\[ n=\frac{E_{deck}}{E_{girder}} \qquad A_{deck,tr}=n b_e t_s \]

\[ A_c=A_g+A_{deck,tr} \qquad y_c=\frac{\sum Ay}{A_c} \]

\[ I_c=I_o-A_cy_c^2 \]

Composite section moduli are calculated at the bottom of the composite section, top of the girder, and top of the transformed slab.

4. Live-Load Distribution Factors

The selected girder family chooses the distribution-factor routine. The module performs applicability checks before using the factors. Exterior corrections are applied where the family-specific routine requires them.

4.1 Plank and adjacent box family

For the plank/box-style branch:

\[ I/J=0.54(d/b)+0.16 \]

\[ k=\max\left(2.5N_b^{-0.2},1.5\right) \]

\[ g_{SI,M}=k\left(\frac{b}{33.3L}\right)^{0.5}\left(\frac{I}{J}\right)^{0.25} \]

\[ g_{MI,M}=k\left(\frac{b}{305}\right)^{0.6}\left(\frac{b}{12L}\right)^{0.2}\left(\frac{I}{J}\right)^{0.06} \]

The interior moment factor is the larger of the single-lane and multiple-lane factors. Exterior corrections include:

\[ e_{SE}=\max(1.125+d_e/30,1.0) \qquad e_{ME}=\max(1.04+d_e/25,1.0) \]

For shear, the module calculates one-lane and multiple-lane factors using the family-specific expressions and uses the larger value. If the section-property applicability checks fail, the moment factor is used for shear.

4.2 Standard AASHTO girder branch

For standard AASHTO sections, the module uses the section spacing, slab thickness, span, girder count, girder/deck stiffness ratio, and centroid separation to form the moment and shear factors. AASHTO-family processing is used only when the selected family is AASHTO.

4.3 Skew and exterior behavior

Skew adjustments and exterior-girder corrections are recorded in the output. The report also records the applicable range checks so that factors are not interpreted outside their intended geometry range.

5. HL-93 Live-Load Envelope

The live-load routine evaluates a simple-span envelope at stations including the bearing, (60d_b), (0.1L) through (0.9L), and midspan. It moves the HL-93 truck and tandem using the configured search increment and includes a lane load.

\[ M_{lane}=w_{lane}\frac{x(L-x)}{2} \]

\[ M_{HL93}(x)=\max(M_{truck}+M_{lane},M_{tandem}+M_{lane}) \]

\[ V^+(x)=\max(V_{truck}+V_{lane},V_{tandem}+V_{lane}) \]

The governing positive moment, positive shear, negative shear, station, truck/tandem placement, and live-load calculation trail are returned in the report.

6. Dead-Load Envelopes and Strength Demands

Uniform-load envelopes are created for:

As-fabricated girder self-weight.
Installed girder self-weight.
Top slab dead load.
Barrier wall dead load.
Utility dead load.
Future wearing surface dead load.

For the reported Strength I demand at a section, the implementation uses:

\[ M_u=1.25(M_{dc1}+M_{dc2}+M_{ub})+1.50M_{fw}+1.75(DFM M_{LL}) \]

\[ V_u=1.25(V_{dc1}+V_{dc2}+V_{ub})+1.50V_{fw}+1.75(DFV V_{LL}) \]

Strength II uses the same dead-load and wearing-surface factors with the live-load factor changed to 1.35. The controlling flexural and shear combinations are selected independently.

7. Prestress Losses

7.1 Stress immediately before transfer

\[ f_{pbt}=0.75f_{pu} \]

7.2 Elastic shortening

The implementation uses the installed-girder dead-load midspan moment $M_g$, converted to kip-in:

\[ C=I_g+e_m^2A_g \]

\[ \Delta f_{pES}=\frac{A_{ps}f_{pbt}C-e_mM_gA_g}{A_{ps}C+A_gI_gE_{ci}/E_p} \]

\[ P_i=(f_{pbt}-\Delta f_{pES})A_{ps} \]

7.3 Approximate time-dependent losses

The current implementation selects the approximate method for creep, shrinkage, and relaxation:

\[ \gamma_h=\max(0,1.7-0.01H) \qquad \gamma_{st}=\frac{5}{1+f'_{ci}} \]

\[ \Delta f_{pCR}=10\left(\frac{f_{pi}A_{ps}}{A_g}\right)\gamma_h\gamma_{st} \]

\[ \Delta f_{pSH}=12\gamma_h\gamma_{st} \qquad \Delta f_{pR}=2.4\text{ ksi} \]

\[ \Delta f_{pLT}=\Delta f_{pCR}+\Delta f_{pSH}+\Delta f_{pR} \]

Release time is captured and reported, but it is not used in the current approximate-loss equation. The code assumes normal-weight concrete, moist/steam curing, low-relaxation strand, and average exposure conditions for this method.

8. Strand Force States

Masked strands use the reduced force block inside the mask length. The two transfer forces and final effective force are:

\[ f_{base}=f_{pbt}-(\Delta f_{pES}+\Delta f_{pR1}) \]

\[ P_{t1}=f_{base}A_{ps} \qquad P_{t2}=f_{base}(A_{ps}-A_{ps,mask}) \]

\[ P_e=(f_{pbt}-\Delta f_{pT})A_{ps} \qquad \Delta f_{pT}=\Delta f_{pES}+\Delta f_{pLT} \]

Sections within the mask length use $P_{t2}$; sections outside the mask use $P_{t1}$.

9. Section Stress Checks

Stress checks are evaluated at $60d_b$, $0.1L$, $0.2L$, $0.3L$, $0.4L$, and $0.5L$.

9.1 Initial and transfer stress

\[ f_{it}=\frac{P}{A}-\frac{Pe}{S_t}+\frac{M_{dp,fab}}{S_t} \]

\[ f_{ib}=\frac{P}{A}+\frac{Pe}{S_b}-\frac{M_{dp,fab}}{S_b} \]

The same section equations are used at transfer with the installed-girder dead-load moment.

9.2 Final service stress

\[ f_{ft}=\frac{P_e}{A}-\frac{P_ee}{S_t}+\frac{M_{dp,inst}+M_{ds}}{S_t}+\frac{M_{ub}+M_{fw}+DFM M_{LL}}{S_{tc}} \]

\[ f_{fb}=\frac{P_e}{A}+\frac{P_ee}{S_b}-\frac{M_{dp,inst}+M_{ds}}{S_b}-\frac{M_{ub}+M_{fw}+0.8(DFM M_{LL})}{S_{bc}} \]

\[ f_{dt}=\frac{M_{ub}+M_{fw}+DFM M_{LL}}{S_{t,slab}} \]

The report compares actual stresses with the limits generated from $f'_{ci}$, girder $f'_c$, and deck $f'_c$, and preserves signed stress values for audit.

10. Flexural Strength

The positive-moment section uses prestressing steel, bottom mild steel, and optional top mild steel:

\[ d_p=H/2+t_s+e \]

\[ \beta_1=\max\left(0.65,0.85-0.05(f'_c-4)\right) \]

\[ k=2\left(1.04-\frac{f_{py}}{f_{pu}}\right) \]

\[ c=\frac{A_{ps}f_{pu}+A_sf_s-A'_sf'_s}{0.85f'_c\beta_1b+kA_{ps}f_{pu}/d_p} \]

\[ f_{ps}=f_{pu}\left(1-k\frac{c}{d_p}\right) \qquad a=\beta_1c \]

\[ M_n=A_{ps}f_{ps}(d_p-a/2)+A_sf_s(d_s-a/2)-A'_sf'_s(a/2-d'_s) \]

The report calculates $M_r=\phi M_n$, checks the strain classification, and compares strength demand with resistance.

10.1 Minimum flexural reinforcement

\[ f_r=0.7\sqrt{f'_c} \qquad M_{cr}=f_rS+P_e\left(e+\frac{S}{A}\right) \]

\[ M_{r,required}=\min(1.33M_u,M_{cr}) \]

The module reports the minimum-flexure check separately from the main flexural strength check.

11. Shear Design

11.1 Critical section and factored demand

\[ d_e=H+t_s-y_{bts} \qquad d_v=d_e-a/2 \]

\[ d_v\ge\max(0.9d_e,0.72(H+t_s)) \]

Strength I and Strength II shear and moment demands are evaluated at the critical shear location and the controlling combination is selected.

11.2 MCFT shear parameters

The shear routine iterates the longitudinal strain and uses the AASHTO table lookup for $\theta$ and $\beta$:

\[ \varepsilon_x=\frac{|M_u|/d_v+0.5N_u+(V_u-V_p)\cot\theta-N_uA_{ps}f_{po}}{2(E_sA_s+E_pA_{ps})} \]

\[ v_u=\frac{|V_u-\phi V_p|}{\phi b_vd_v} \]

\[ V_c=0.0316\beta\lambda\sqrt{f'_c}b_vd_v \]

The code selects the lookup row from $v_u/f'_c$ and the lookup column from $1000\varepsilon_x$, then iterates until the selected parameters stabilize.

11.3 Stirrups and limits

\[ V_s=\frac{A_v}{s}f_y d_v(\cot\theta+\cot\alpha)\sin\alpha \]

\[ \phi V_n=\phi(V_c+V_s+V_p) \qquad \phi V_n\ge V_u \]

Transverse reinforcement is required when:

\[ V_u>0.5\phi(V_c+V_p) \]

Additional checks include spacing, minimum transverse steel, maximum nominal shear resistance, and longitudinal reinforcement for shear.

12. Anchorage, Interface, and Development

12.1 Pretensioned anchorage zone

\[ P_i=0.75f_{pu}A_{ps} \qquad P_r=0.04P_i \qquad A_{s,req}=P_r/f_s \]

Required bars are placed within $h/4$ from the girder end and are reported by bar size, number of faces, and selected quantity.

12.2 Interface shear transfer

\[ V_h=\frac{V_u}{d_e} \qquad V_n=V_h/\phi \]

The implementation compares the interface demand with three limits:

\[ V_{n1}=K_1f'_c(deck)A_{cv} \qquad V_{n2}=K_2A_{cv} \]

\[ V_{n3}=cA_{cv}+\mu(A_{vf}f_y+P_c) \]

It also checks required and provided $A_{vf}/s$.

12.3 Development length

\[ f_{pe}=0.75f_{pu}-\Delta f_{pT} \]

\[ l_{d,bonded}=1.6\left(f_{ps}-\frac{2}{3}f_{pe}\right)d_b \]

\[ l_{d,partial}=2.0\left(f_{ps}-\frac{2}{3}f_{pe}\right)d_b \]

The reference also reports the (60d_b) transfer length used in the stress and longitudinal reinforcement checks.

13. Deflection and Camber

Transfer and final deflection are assembled from prestress camber and uniform-load components:

\[ d_1=-\frac{P_{t1}L'^2e}{8E_{ci}I_g} \]

\[ d_2=\frac{5(W_{girder}/12)L'^4}{384E_{ci}I_g} \]

\[ d_3=\frac{5(W_{deck}/12)L^4}{384E_{cp}I_g} \]

\[ d_4=\frac{5(W_{utility}+W_{fw}+W_{barrier})L^4}{384E_{cp}I_c} \]

\[ d_5=\frac{\Delta f_{pT}-(\Delta f_{pES}+\Delta f_{pR1})}{f_{ptr}}|d_1| \]

PCI-style multipliers are applied in the current report for release and final deflection summaries. Live-load deflection uses truck and tandem vehicle searches, impact, lane load, the moment distribution factor, and:

\[ \Delta_{LL,allow}=L/800 \]

Engineering responsibility: This software reference is not a substitute for the adopted AASHTO LRFD edition, project plans, jurisdictional amendments, approved material properties, or review by a qualified bridge engineer. Confirm applicable limit states, load factors, resistance factors, distribution-factor applicability, constructability, and detailing requirements for each project.