Structural Engineering Exam (SE) Vertical Forces (Gravity/Other) and Incidental Lateral — Breadth Syllabus

What is the relationship among load $w$, shear $V$, and moment $M$ along a beam?

$$\frac{dV}{dx} = -w(x), \qquad \frac{dM}{dx} = V(x)$$ Thus the slope of the shear diagram equals the negative distributed load, and the slope of the moment diagram equals the shear. The change in shear equals the area under the load diagram; the change in moment equals the area under the shear diagram. Maximum moment occurs where $V = 0$.

For a simply supported beam of span $L$ under uniform load $w$, give the maximum shear, maximum moment, and maximum deflection.

$$V_{max} = \frac{wL}{2}, \qquad M_{max} = \frac{wL^2}{8}, \qquad \delta_{max} = \frac{5wL^4}{384EI}$$ Maximum moment is at midspan; maximum shear is at the supports.

For a cantilever beam of length $L$ with a point load $P$ at the free end, give the maximum moment and tip deflection.

$$M_{max} = PL\ \text{(at the fixed support)}, \qquad \delta_{tip} = \frac{PL^3}{3EI}$$ The moment is maximum at the fixed end and zero at the free end.

State the moment-area theorems used for beam deflection.

First theorem: the change in slope between two points equals the area of the $M/EI$ diagram between them. Second theorem: the tangential deviation of one point from the tangent at another equals the moment of the $M/EI$ diagram area (about the first point) between them. Used graphically for slopes and deflections.

Write the unit-load (virtual work) equations for deflection of beams and of trusses.

Beam/frame: $$\delta = \int_0^L \frac{m M}{EI}\,dx$$ Truss: $$\delta = \sum \frac{n N L}{A E}$$ where $M, N$ are real-load internal forces and $m, n$ are internal forces from a unit virtual load applied at the point and direction of the desired displacement.

What is compatibility in indeterminate structural analysis?

Compatibility requires that the deformed structure remain continuous and consistent with support conditions, i.e., displacements and rotations match at connections and supports. Force (flexibility) methods enforce compatibility by setting redundant-induced displacements equal to the geometric constraints; equilibrium alone is insufficient for indeterminate structures.

Describe the force (flexibility) method for analyzing indeterminate structures.

Select redundant reactions/forces and release them to form a determinate primary structure. Compute displacements at the released points due to applied loads ($\delta_0$) and due to unit redundants (flexibility coefficients $\delta_{ij}$). Enforce compatibility: $[\delta]\{X\} = -\{\delta_0\}$, solve for redundants $X$, then superpose to get final forces.

Describe the displacement (stiffness) method and its governing matrix equation.

Unknowns are joint displacements/rotations. Assemble member stiffness matrices into a global stiffness matrix $[K]$, apply the equation $\{P\} = [K]\{\Delta\}$, solve for unknown displacements $\{\Delta\}$ after applying boundary conditions, then recover member forces. It is the basis of matrix/finite element structural analysis.

What is the slope-deflection equation for a prismatic member end moment?

$$M_{AB} = \frac{2EI}{L}\left(2\theta_A + \theta_B - 3\psi\right) + FEM_{AB}$$ where $\theta_A, \theta_B$ are end rotations, $\psi = \Delta/L$ is the chord rotation, and $FEM_{AB}$ is the fixed-end moment due to span loads.

Give the fixed-end moments for a prismatic beam under a uniform load $w$ and a central point load $P$.

Uniform load: $FEM = \dfrac{wL^2}{12}$ at each end. Central point load: $FEM = \dfrac{PL}{8}$ at each end. Signs are opposite at the two ends per the sign convention; these are restraining moments for both ends fixed.

What is the moment distribution method, and what is a distribution factor?

Moment distribution is an iterative stiffness-based method that balances joints by distributing unbalanced moments according to relative member stiffnesses and carrying over moments to far ends. The distribution factor for a member at a joint is $DF = \dfrac{k_i}{\sum k}$, where $k = 4EI/L$ (or $3EI/L$ for a far pinned end).

What is the carry-over factor in moment distribution for a prismatic member with a fixed far end?

The carry-over factor is $+\tfrac{1}{2}$: half of the moment applied (distributed) at the near end is carried over to the fixed far end. For a member pinned at the far end, the carry-over factor is $0$ and its stiffness is $3EI/L$.

What is an influence line, and how does it differ from a shear or moment diagram?

An influence line plots the value of a specific response function (a reaction, shear, or moment at one fixed location) as a unit load moves across the structure. A shear/moment diagram shows the response at all locations for one fixed loading. Influence lines locate the position of moving loads that produces the maximum response.

State the Müller-Breslau principle for influence lines.

The influence line for a reaction or internal force is proportional to the deflected shape produced when the corresponding restraint is removed and a unit displacement (or rotation) is imposed in the direction of that force. For determinate structures the shape is piecewise linear; for indeterminate structures it is curved.

How do you find the maximum response from a moving distributed load and from a set of moving concentrated loads using an influence line?

For a uniform moving load $w$, the response equals $w$ times the net area of the influence line over the loaded region (place load over positive area for maximum). For concentrated loads, response $= \sum P_i y_i$, where $y_i$ is the influence-line ordinate under each load; position the load group to maximize $\sum P_i y_i$.

What is the absolute maximum moment criterion for a series of moving loads on a simple beam?

The absolute maximum moment occurs under one of the loads when the beam centerline bisects the distance between that load and the resultant of all loads on the span. Check each candidate load; the moment under it (when positioned per this rule) is computed and the largest is the absolute maximum.

Explain idealization of support boundary conditions: pin, roller, fixed, and free.

Roller: resists one translation (1 reaction), allows rotation and one translation. Pin: resists two translations (2 reactions), allows rotation. Fixed: resists two translations and rotation (3 reactions). Free: no restraint. Idealization choice strongly affects computed forces and deflections; real connections are usually semi-rigid between pinned and fixed.

Why does modeling a connection as pinned versus fixed matter, and how does rotational stiffness affect moment distribution?

A pinned connection transmits no moment (forces redistribute to other members and deflections increase), while a fixed connection attracts moment and reduces deflection. Real semi-rigid connections carry partial moment based on their rotational stiffness; over-idealizing as fully fixed can unconservatively underestimate deflections and member demands elsewhere.

Write the general bearing capacity equation for a shallow foundation (Terzaghi/general form).

$$q_{ult} = c N_c + q N_q + \tfrac{1}{2}\gamma B N_\gamma$$ where $c$ is cohesion, $q = \gamma D_f$ is surcharge at footing base, $\gamma$ is soil unit weight, $B$ is footing width, and $N_c, N_q, N_\gamma$ are bearing capacity factors that depend on $\phi$. Shape, depth, and inclination factors modify each term.

How is the allowable bearing capacity obtained from ultimate bearing capacity, and what is net bearing pressure?

Apply a factor of safety (typically $FS = 3$): $q_{all} = \dfrac{q_{ult}}{FS}$. The net allowable bearing capacity subtracts the overburden replaced by the footing: $q_{net} = q_{ult} - \gamma D_f$, then divided by FS. Net pressure governs settlement-controlled design.

For a concentrically loaded rectangular footing, how is soil bearing pressure computed, and when does uplift (no contact) occur under eccentric load?

Concentric: $q = \dfrac{P}{A} = \dfrac{P}{BL}$. Eccentric (one-way): $q = \dfrac{P}{A} \pm \dfrac{Mc}{I} = \dfrac{P}{BL}\left(1 \pm \dfrac{6e}{L}\right)$. If eccentricity $e > L/6$ (outside the kern), part of the footing lifts off and the pressure distribution becomes triangular over partial contact.

What are the two main failure checks for a shallow footing besides bearing, and what controls them?

One-way (beam) shear, checked at distance $d$ from the face of the column, and two-way (punching) shear, checked on a perimeter at $d/2$ around the column. Footing thickness is usually governed by shear (concrete shear strength without stirrups); flexure governs reinforcement.

Distinguish end-bearing piles from friction piles, and give the pile capacity equation.

End-bearing piles transfer load mainly through the tip to a firm stratum; friction (floating) piles transfer load through side (skin) friction along the shaft. Total capacity: $$Q_{ult} = Q_p + Q_s = q_p A_p + \sum f_s A_s$$ where $Q_p$ is tip resistance and $Q_s$ is shaft resistance.

What is the difference between efficiency of a pile group and block failure, and why can group capacity be less than the sum of individual piles?

Overlapping stress zones of closely spaced piles reduce the group's capacity below the sum of single-pile capacities, expressed by a group efficiency factor $\eta < 1$. The group may also fail as a single block (perimeter shear plus block end bearing); design capacity is the lesser of $\eta \times \sum Q_{single}$ and the block capacity.

What are the four stability checks for a gravity/cantilever retaining wall?

1) Overturning about the toe ($FS \geq 1.5\text{–}2.0$); 2) Sliding along the base ($FS \geq 1.5$); 3) Bearing capacity of foundation soil ($FS \geq 3$, resultant within the middle third); 4) Overall (global) slope stability. Structural design of stem, toe, and heel follows after stability is satisfied.

For a cantilever retaining wall, how are the factors of safety against overturning and sliding defined?

$$FS_{OT} = \frac{\sum M_{resisting}}{\sum M_{overturning}}, \qquad FS_{sliding} = \frac{\sum F_{resisting}}{\sum F_{driving}}$$ Resisting moment comes from wall and soil weight about the toe; sliding resistance comes from base friction ($\mu \sum V$) plus passive pressure and any key. Driving force/moment comes from active earth pressure.

Define key geotechnical parameters: $\phi$, $c$, $\gamma$, and OCR.

$\phi$ is the effective angle of internal friction (shear strength's frictional component). $c$ is cohesion (shear strength at zero normal stress). $\gamma$ is soil unit weight (total or effective $\gamma' = \gamma_{sat} - \gamma_w$ below water). OCR (overconsolidation ratio) is the ratio of maximum past effective stress to current effective stress; OCR $=1$ is normally consolidated.

State the Mohr-Coulomb shear strength equation and the effective stress principle.

Shear strength: $$\tau_f = c' + \sigma' \tan\phi'$$ in effective stress terms. Effective stress: $$\sigma' = \sigma - u$$ where $\sigma$ is total stress and $u$ is pore water pressure. Soil strength and volume change are governed by effective, not total, stress.

What is soil-structure interaction, and why does foundation flexibility matter?

Soil-structure interaction is the mutual influence between a structure's deformation and the supporting soil's response. A rigid footing imposes uniform settlement but non-uniform pressure; a flexible footing gives more uniform pressure but differential settlement. Modeling soil as springs (subgrade modulus $k_s$) captures this; differential settlement induces additional moments in indeterminate structures.

Define load path and explain its importance for gravity systems.

The load path is the continuous route a load follows from its point of application through slab, beam, girder, column, and foundation into the ground. Each element and connection must have adequate strength and stiffness along the path; a discontinuity (missing connection or member) creates a weak link that can cause progressive or local collapse. A complete, redundant load path is essential.

Compare the elastic moduli and key behavior of structural steel versus normal-weight concrete.

Steel: $E_s \approx 29{,}000\ \text{ksi}$, isotropic, ductile, equal strength in tension and compression, yields at $F_y$. Concrete: $E_c = 57{,}000\sqrt{f'_c}$ (psi), strong in compression but weak in tension ($\approx 10\%$ of compressive), brittle, requires reinforcement for tension. Concrete also exhibits creep and shrinkage; steel does not significantly.

Why must wood and masonry be treated as direction-dependent materials in design?

Wood is orthotropic: strength parallel to grain greatly exceeds strength perpendicular to grain, and design values are adjusted by factors (load duration, moisture, size, etc.). Masonry is strong in compression but weak in tension; unreinforced masonry relies on gravity for stability, so flexural/shear capacity depends on net mortared area and the direction of the bed joints.

What is the modular ratio $n$ in composite/reinforced section analysis, and how is it used?

$$n = \frac{E_s}{E_c}$$ It converts steel area to an equivalent concrete area (transformed section) so the composite section can be analyzed as a single homogeneous material for stresses and the location of the neutral axis. Larger $n$ (lower $f'_c$) gives steel a greater relative stiffness contribution.

Distinguish simple (shear/pinned) connections from moment (rigid) connections in force transfer.

Simple connections transfer primarily shear (and small axial) with negligible moment, allowing rotation; they idealize as pins (e.g., single-plate shear tabs, clip angles). Moment connections transfer shear plus full or partial bending moment, maintaining angle continuity; they require flanges/welds developing the beam's moment (e.g., welded flange or end-plate connections).

In gravity connection design, what are the principal limit states for a bolted shear connection?

Bolt shear, bolt bearing/tearout on connected plies, block shear rupture, plate gross-section yielding, plate net-section rupture, and weld strength (if welded). The connection capacity is the minimum (governing) of these limit states; ductile limit states (yielding, bolt bearing) are generally preferred over brittle ones (rupture).

What is constructability, and name detailing considerations that improve it for gravity systems.

Constructability is the ease and economy with which a design can be built safely. Considerations: erection sequence and temporary stability/bracing, member repetition and standard sizes, adequate access and clearance for bolting/welding, shoring and reshoring of concrete, simple repeatable connections, tolerances for fit-up, and avoiding congested reinforcement. Good detailing reduces field errors and cost.

What is the difference between nominal, factored (design), and service loads in structural design?

Nominal loads are the code-specified reference magnitudes (e.g., $L_0$, $p_g$). Factored (design) loads are nominal loads multiplied by LRFD load factors for strength design, compared against $\phi R_n$. Service loads are unfactored (ASD or serviceability) combinations used to check deflection, vibration, crack control, and stresses under everyday conditions.

How does ASCE 7 treat snow drift and unbalanced snow loads on roofs?

In addition to balanced snow, ASCE 7 requires drift loads where snow accumulates at roof steps, parapets, and against projections, modeled as a triangular surcharge with height $h_d$ (a function of upwind fetch and ground snow). Unbalanced loads on gable/curved roofs and sliding snow from higher roofs onto lower roofs must also be considered.

What is ponding instability and how is it prevented in flat roof design?

Ponding instability occurs when roof deflection under accumulated water allows more water to collect, increasing load and deflection in a potentially divergent cycle. It is prevented by providing adequate roof slope/drainage (positive drainage, secondary drains/scuppers) and by ensuring sufficient roof stiffness so the structure is stable against the incremental ponding load.