Structural Engineering Exam (SE) Vertical Forces (Gravity/Other) — Bridges Depth Syllabus

What is the lever rule for live load distribution?

A statics method: assume the deck acts as a simple span hinged at each interior girder; sum moments about the adjacent girder to find the reaction (load fraction) at the girder of interest from the wheel loads. The multiple presence factor must then be applied.

What is the principle behind prestressing concrete girders?

Introducing internal compressive force (via tensioned tendons) precompresses the concrete so that under service load the tension at the bottom fiber is reduced or eliminated, controlling cracking and allowing longer spans/shallower sections than reinforced concrete.

Distinguish pretensioning from post-tensioning.

Pretensioning: strands are tensioned against external abutments BEFORE casting; force transfers by bond after release (typical for precast plant girders). Post-tensioning: tendons in ducts are stressed and anchored AFTER concrete cures, then grouted (typical for cast-in-place/segmental).

List the principal sources of prestress loss.

Immediate (instantaneous): elastic shortening, anchorage seating/set, friction (post-tensioning). Time-dependent: concrete creep, concrete shrinkage, and steel relaxation. Total loss is the sum: $\Delta f_{pT} = \Delta f_{pES} + \Delta f_{pLT}$.

Write the elastic shortening prestress loss expression for pretensioned members.

$$\Delta f_{pES} = \frac{E_p}{E_{ci}} f_{cgp}$$ where $E_p$ = strand modulus, $E_{ci}$ = concrete modulus at transfer, and $f_{cgp}$ = concrete stress at the strand centroid due to prestress (at transfer) plus self-weight.

Give the AASHTO concrete stress limits at transfer for a pretensioned girder (no bonded reinforcement in tension zone).

Compression limit $0.60\,f'_{ci}$; tension limit $0.0948\sqrt{f'_{ci}} \leq 0.2\text{ ksi}$ (ksi units). With bonded reinforcement resisting the tensile force, tension may reach $0.24\sqrt{f'_{ci}}$.

State the AASHTO service compressive stress limits for prestressed concrete at the final (service) stage.

$0.45\,f'_c$ for effective prestress + permanent loads; $0.60\,\phi_w f'_c$ for the load combination of all loads including transient (Service I). The $\phi_w$ is a reduction for slender web/flange elements.

What governs flexural design of a typical RC bridge deck under AASHTO?

The deck is designed as a continuous slab spanning transversely between girders. The empirical/equivalent-strip method gives moments per unit width; design provides top steel over girders (negative moment) and bottom steel at midspan (positive moment), checked for strength and crack control.

What is the AASHTO equivalent strip width concept for deck design?

A width of deck (perpendicular to traffic for transverse strips) is assumed to carry one wheel line; e.g., positive moment strip width $= 26.0 + 6.6S$ (in), negative $= 48.0 + 3.0S$ (in), with $S$ in ft, distributing the wheel load over that strip.

Describe the AASHTO empirical deck design method and its key requirement.

For decks satisfying geometric/material conditions (e.g., composite, $6.0 \leq \frac{\text{effective length}}{\text{depth}} \leq 18.0$, min depth $7.0\text{ in}$), reinforcement is specified by ratio rather than analysis: bottom transverse steel $\geq 0.27\ \text{in}^2/\text{ft}$, top $\geq 0.18\ \text{in}^2/\text{ft}$, relying on internal arching (compressive membrane) action.

What distribution reinforcement is required in the bottom of a deck for primary reinforcement perpendicular to traffic?

A percentage of the primary positive-moment steel: $\frac{220}{\sqrt{S}} \leq 67\%$, where $S$ is the effective span in ft. It is placed in the secondary (longitudinal) direction to distribute concentrated wheel loads.

Write the nominal flexural resistance of a rectangular RC section and the strength reduction.

$$M_n = A_s f_y\left(d - \frac{a}{2}\right),\quad a = \frac{A_s f_y}{0.85 f'_c b}$$ and $M_r = \phi M_n$ with $\phi = 0.90$ for tension-controlled RC flexure.

Write the nominal flexural resistance of a bonded prestressed rectangular section.

$$M_n = A_{ps} f_{ps}\left(d_p - \frac{a}{2}\right)$$ where $f_{ps}$ = stress in prestressing steel at nominal flexure (from strain compatibility or the approximate AASHTO $f_{ps} = f_{pu}\left(1 - k\frac{c}{d_p}\right)$).

State the AASHTO LRFD shear resistance equation (sectional/MCFT model).

$$V_n = \min\big(V_c + V_s + V_p,\ 0.25 f'_c b_v d_v + V_p\big)$$ with $V_c = 0.0316\,\beta\sqrt{f'_c}\,b_v d_v$ and $V_s = \frac{A_v f_y d_v \cot\theta}{s}$ (vertical stirrups), $V_p$ = vertical prestress component.

Define $b_v$ and $d_v$ in the AASHTO shear provisions.

$b_v$ = effective web width (minimum width within $d_v$, adjusted for duct diameters in post-tensioned members). $d_v$ = effective shear depth = distance between resultants of tensile and compressive flexural forces, taken $\geq \max(0.9 d_e,\ 0.72 h)$.

In the MCFT-based shear method, what do $\beta$ and $\theta$ represent?

$\beta$ = factor for the tensile stress carried by cracked concrete (aggregate interlock), and $\theta$ = angle of inclination of diagonal compressive stresses. Both depend on the net longitudinal strain $\varepsilon_s$ at mid-depth and are taken from AASHTO tables/equations.

What is the purpose of the longitudinal reinforcement check in AASHTO shear design?

It ensures the longitudinal steel can resist the additional tension demanded by shear (truss action): $A_{ps}f_{ps} + A_s f_y \geq \frac{M_u}{d_v \phi_f} + 0.5\frac{N_u}{\phi_c} + \left(\frac{V_u}{\phi_v} - 0.5 V_s - V_p\right)\cot\theta$.

What is the cause of, and design treatment for, continuity moments in precast-girder bridges made continuous for live load?

Simple-span precast girders are joined by a cast-in-place deck/diaphragm to act continuous for superimposed dead and live loads, creating negative restraint moments at piers. Continuity reinforcement in the deck over the pier resists negative moment; positive restraint moment from creep/shrinkage is handled at the bottom connection.

Explain restraint moments due to time-dependent effects in continuous prestressed structures.

Differential creep, shrinkage, and thermal gradients in a structure made continuous develop secondary (restraint) moments because the supports prevent free deformation. They can be positive at midspan or piers and must be combined with applied-load moments; magnitude grows over time toward a creep-adjusted final value.

What is the modular-ratio aging coefficient method (age-adjusted effective modulus) used for?

It evaluates time-dependent stress redistribution from creep/shrinkage using $E_c^{*} = \frac{E_c}{1 + \chi\,\phi(t,t_0)}$, where $\phi$ is the creep coefficient and $\chi \approx 0.8$ is the aging coefficient, accounting for gradually applied restraint stresses.

What are the main components of a welded steel plate girder?

Two flange plates (top and bottom) and a web plate joined by fillet welds, plus transverse (and sometimes longitudinal) stiffeners and bearing stiffeners. Plate dimensions are tailored along the span, unlike a rolled beam of fixed cross-section.

Why and where are transverse stiffeners and bearing stiffeners used on plate girders?

Transverse (intermediate) stiffeners increase web shear buckling resistance (tension-field action) and are placed at spacing $d_o$. Bearing stiffeners are placed at supports and concentrated loads to transfer reactions and prevent web crippling/local buckling.

State the web slenderness limit distinguishing a compact/noncompact web from a slender web in AASHTO.

A web is slender (subject to bend-buckling) when $\frac{2D_c}{t_w} > 5.7\sqrt{\frac{E}{F_{yc}}}$, where $D_c$ is the depth of web in compression. Below the compactness limit $\frac{2D_{cp}}{t_w} \leq 3.76\sqrt{\frac{E}{F_{yc}}}$ the section can reach $M_p$.

What is tension-field action in a stiffened plate girder web?

After the web buckles in shear, a diagonal tension band forms that, anchored by the flanges and transverse stiffeners (acting like a truss), carries additional shear beyond the elastic buckling load—giving post-buckling shear strength $V_n$ greater than $V_{cr}$.

Define composite action in a steel girder bridge and how it is achieved.

The concrete deck and steel girder act as one unit, with the deck resisting compression and the steel resisting tension, increasing stiffness and capacity. It is achieved via shear connectors (typically welded headed studs) that transfer horizontal shear at the steel-concrete interface.

How are the number of shear studs for strength (fatigue and strength) governed in AASHTO?

Fatigue governs spacing/pitch from the horizontal fatigue shear range $V_{sr}$ and stud fatigue resistance $Z_r$; strength governs the total number between maximum-moment and zero-moment points from $n = \frac{P}{Q_r}$, where $P$ is the smaller of the deck or steel force and $Q_r$ is factored stud strength.

What is the effective flange width of the concrete deck in a composite section per current AASHTO?

Taken as the tributary width—the lesser of one-half the distance to the adjacent girder plus the overhang—i.e., effective width equals the center-to-center girder spacing for interior girders (the older S/4, 12t, spacing rules were simplified to tributary width in LRFD).

In composite section analysis, why are two modular ratios ($n$ and $3n$) used?

$n = E_s/E_c$ is used for short-term (live) loads; $3n$ accounts for concrete creep under sustained (long-term/superimposed dead) loads, reducing the effective deck stiffness so long-term stresses redistribute toward the steel.

What defines a fatigue detail category in AASHTO, and what is the consequence of a lower category?

Each connection/weld geometry is assigned a category (A, B, B', C, C', D, E, E') reflecting stress-concentration severity. Lower categories (E, E') have lower constant-amplitude fatigue thresholds $(\Delta F)_{TH}$ and detail constants $A$, meaning shorter fatigue life for a given stress range.

Write the AASHTO nominal fatigue resistance equation for finite life (Fatigue II).

$$(\Delta F)_n = \left(\frac{A}{N}\right)^{1/3}$$ where $A$ is the detail-category constant and $N = (365)(75)(n)(\text{ADTT})_{SL}$ is the number of cycles over a 75-year design life ($n$ = cycles per truck passage).

What is the constant-amplitude fatigue threshold (CAFL) and how is it used (Fatigue I / infinite life)?

$(\Delta F)_{TH}$ is the stress range below which essentially infinite fatigue life is assumed. For infinite-life (Fatigue I) design, the factored fatigue stress range must satisfy $\gamma(\Delta f) \leq (\Delta F)_{TH}$ (or $\leq \tfrac{1}{2}(\Delta F)_{TH}$ in load-induced checks per edition).

Distinguish load-induced from distortion-induced fatigue.

Load-induced fatigue arises from primary in-plane stress ranges at details and is checked by category. Distortion-induced fatigue arises from secondary out-of-plane deformations (e.g., web gaps at connection plates not attached to flanges) and is controlled by detailing—rigidly attaching connection plates to both flanges—rather than by stress calculation.

What is fracture toughness and how is it addressed for steel bridges (Charpy/temperature zones)?

Fracture toughness is a material's resistance to crack propagation, measured by Charpy V-notch (CVN) impact testing. AASHTO requires minimum CVN energy at a specified test temperature based on the service Temperature Zone (1, 2, or 3) and whether the member is fracture-critical (FCM/nonredundant).

Define a fracture-critical member (FCM) and the design implication.

An FCM is a steel tension member or tension component whose failure would cause collapse of the bridge (nonredundant load path). It requires more stringent material toughness (CVN), fabrication, and in-service inspection requirements under the AASHTO Fracture Control Plan.

What is the function of diaphragms and cross-frames in girder bridges?

They provide lateral stability to girder compression flanges during erection and under load, distribute live load transversely among girders, resist lateral wind and seismic forces, and control differential girder deflection and twisting—especially critical on curved/skewed bridges.

Compare diaphragms and cross-frames structurally.

A diaphragm is a solid (e.g., channel or rolled-beam) full-depth transverse member between girders; a cross-frame is a truss-type assembly of angles in an X or K configuration. Cross-frames are lighter and typically used between deeper plate girders; diaphragms suit shallower rolled-beam bridges.

How do bracing members participate in lateral load distribution on a girder bridge?

Wind and other lateral loads applied to the windward girder/deck are transferred through the deck and cross-frames/diaphragms to all girders and then to the bearings/substructure, so the lateral system spreads the load rather than overloading a single girder.

What are the principal force effects a bridge pier must be designed for?

Axial load from superstructure dead/live load; longitudinal forces (braking, temperature, friction at bearings); transverse forces (wind, seismic, centrifugal); and resulting biaxial bending and shear in the columns, plus load transfer to the footing/foundation.

Compare a hammerhead pier, a multi-column bent, and a wall pier.

Hammerhead: single column with a cantilevered cap (good for limited footprint/waterways). Multi-column bent: row of columns with a cap beam (economical for wide bridges, redundant). Wall pier: solid wall (streamlined for flowing water/ice, stiff transversely but flexible longitudinally).

What loads and design checks govern a bridge abutment?

Vertical superstructure reaction plus self-weight; lateral earth pressure (active/at-rest) and surcharge from approach fill and live load; and design checks for sliding, overturning, bearing capacity, and global stability, plus structural design of stem, backwall, and footing.

Write the active earth pressure coefficient (Rankine) used in abutment design.

$$K_a = \tan^2\left(45^\circ - \frac{\phi}{2}\right)$$ for a level backfill, where $\phi$ is the soil friction angle. The lateral pressure at depth $z$ is $p = K_a \gamma z$ (plus surcharge $K_a q$).

Compare a spread footing with a deep (pile/drilled shaft) foundation for bridges.

Spread footing: bears directly on competent soil/rock at shallow depth; economical where bearing strata are near grade and scour is limited. Deep foundations (driven piles, drilled shafts): transfer load to deeper strata by skin friction and end bearing, used for weak surface soils, high loads, scour-prone or marine sites.

How is the axial capacity of a single pile expressed?

$$Q_{ult} = Q_s + Q_p = f_s A_s + q_p A_p$$ where $Q_s$ = skin-friction resistance ($f_s$ unit friction over shaft area $A_s$) and $Q_p$ = end-bearing resistance ($q_p$ unit tip resistance over tip area $A_p$); factored for LRFD by resistance factors.

What is scour and why is it critical to bridge foundation design?

Scour is the erosion of streambed/bank material around piers and abutments by flowing water (general, contraction, and local scour). It is a leading cause of bridge failure; foundations must be embedded below the predicted scour depth, and scour is treated under the extreme-event/strength limit states.

What is the function of bridge bearings?

Bearings transmit superstructure loads (vertical and horizontal) to the substructure while accommodating movements—translation (thermal expansion/contraction, creep, shrinkage) and rotation (live-load and dead-load end rotation)—at the support.

Compare fixed and expansion bearings.

A fixed bearing permits rotation but restrains translation (transfers longitudinal/horizontal forces to that support). An expansion bearing permits both rotation and longitudinal translation, releasing the superstructure to move thermally so it does not induce large axial forces in the substructure.

Describe a steel-reinforced elastomeric bearing pad and how it accommodates movement.

Alternating layers of elastomer (neoprene/natural rubber) bonded to thin steel shim plates. The steel shims confine the elastomer against bulging under vertical load (raising compressive stiffness), while the elastomer shears to accommodate horizontal movement and deforms to allow rotation.

What is the role of expansion joints, and name common types.

Expansion joints provide a gap that lets the deck expand/contract while giving a smooth riding surface and sealing out water/debris. Types include strip seals, compression seals, finger joints, modular joints (for large movements), and poured/asphaltic-plug joints (for small movements).

How are superstructure loads transmitted through the substructure to the foundation?

Vertical and horizontal reactions pass from the girders through bearings to the pier cap, then down the columns/wall (as axial + bending + shear) to the footing, which spreads them to the soil (spread footing) or to piles/shafts that carry them to deeper strata.

How are longitudinal forces (braking, thermal) distributed among bridge substructure units?

They distribute to the supporting piers/abutments in proportion to the relative stiffness of each substructure unit and its bearing fixity—fixed bearings and stiffer (shorter/larger) columns attract more force, while expansion bearings shed longitudinal force to other units.

What is the AASHTO braking force (BR) and how is it applied?

The greater of $25\%$ of the design truck/tandem axle weights, or $5\%$ of the design truck/tandem plus lane load. It is applied horizontally at $6\text{ ft}$ above the deck in either longitudinal direction, in all loaded lanes carrying traffic in the same direction (with multiple presence factors).

Why must bridge bearings and substructure design account for thermal movement, and how is the movement estimated?

Temperature change causes the superstructure to expand/contract; if restrained, it induces large forces in bearings and substructure. The movement is $\Delta L = \alpha L \Delta T$, where $\alpha$ is the coefficient of thermal expansion, $L$ the expansion length, and $\Delta T$ the design temperature range—bearings/joints are sized to accommodate it.

What is the difference between the strength and service checks for a prestressed bridge girder design sequence?

Service checks (Service I and III) verify concrete compressive and tensile stresses at transfer and at final stage to control cracking and overstress under elastic behavior. Strength checks (Strength I) verify factored moment/shear capacity ($\phi M_n$, $\phi V_n$) against ultimate demand. A girder must satisfy both; prestressed designs are frequently controlled by service tension (Service III).

What is the minimum reinforcement (cracking moment) requirement for flexural members in AASHTO?

The factored flexural resistance must satisfy $M_r \geq \min(1.33 M_u,\ M_{cr})$, where the cracking moment $M_{cr} = \gamma_3\left(\gamma_1 f_r + \gamma_2 f_{cpe}\right) S_c - M_{dnc}\left(\frac{S_c}{S_{nc}} - 1\right)$ ensures the section does not fail immediately upon cracking.

What controls lateral-torsional buckling resistance of a steel I-girder, and what are the key unbraced-length limits?

The compression-flange unbraced length $L_b$ relative to limits $L_p = 1.0 r_t\sqrt{\frac{E}{F_{yc}}}$ (yielding/plastic) and $L_r = \pi r_t\sqrt{\frac{E}{F_{yr}}}$ (onset of elastic LTB). For $L_b \leq L_p$ full capacity is available; between $L_p$ and $L_r$ inelastic LTB governs; beyond $L_r$ elastic buckling governs.