Structural Engineering
Structural engineering is the largest slice of the PrepPass PE Civil bank because every civil discipline eventually relies on members that carry load safely to the ground. This chapter builds from statics of beams and columns through stress analysis to reinforced-concrete and steel design, so you can both compute closed-form answers and recognize the code checks NCEES expects. Master this material and roughly one question in five falls into place.
Beam Reactions, Shear, and Moment
Every beam problem starts with equilibrium: sum of vertical forces = 0, sum of moments = 0. For a simply supported beam under a full-span uniform load w, symmetry splits the load, so each reaction equals wL/2 and the maximum moment at midspan is wL^2/8. A central point load P gives reactions of P/2 and a peak moment of PL/4. For an off-center point load at distance a from the left support (span L), the far reaction is Pa/L. Cantilevers behave differently: a tip load P gives a fixed-end moment of PL, and a full uniform load gives wL^2/2. Commit these four results to memory. Two diagram rules tie shear and moment together: the slope of the moment diagram equals the shear (dM/dx = V), and the slope of the shear diagram equals the negative of the distributed load (dV/dx = -w). It follows that the maximum moment occurs exactly where shear passes through zero, and a point of contraflexure (inflection point) is where the moment, not the shear, equals zero.
Section Properties: I, S, r, and Z
Bending resistance is geometry. For a rectangle of width b and depth h: moment of inertia I = bh^3/12, elastic section modulus S = bh^2/6, and radius of gyration r = sqrt(I/A) = h/sqrt(12) ≈ 0.289h about the strong axis. Because depth is cubed in I, doubling the depth multiplies stiffness by eight — that is why beams stand tall and why the strong axis of a wide-flange (W-shape) is the one with the web vertical. The plastic section modulus Z = bh^2/4 for a rectangle; the ratio Z/S is the shape factor, equal to 1.5 for a solid rectangle and about 1.1–1.2 for a typical W-shape. Bending stress in the elastic range is σ = M/S (or the general form σ = Mc/I), and this is the workhorse equation for sizing a beam: compute the demand moment, divide by the allowable stress, and pick a section whose S exceeds M/σ_allow.
Axial Stress, Strain, Thermal, and Combined Loading
Axial members obey σ = P/A and Hooke's law σ = Eε, so elongation δ = PL/(AE). Steel's modulus is E ≈ 29,000 ksi (200 GPa) — a value worth memorizing because it appears in deflection, buckling, and modular-ratio questions. Poisson's ratio for steel is about 0.3, so a longitudinal strain produces a lateral strain of 0.3 times its magnitude. Thermal effects split into two cases the exam loves to contrast: a free bar expands by δ = αL(ΔT) and develops zero stress, whereas a fully restrained bar cannot move and develops a thermal stress σ = EαΔT independent of length. When axial and bending act together on a short member, superpose them: σ = P/A ± M/S, giving a larger compressive fiber on one face and a smaller (or tensile) stress on the other.
Column Buckling and Effective Length
Slender columns fail by buckling, not crushing. The Euler critical load is P_cr = π^2EI/(KL)^2, and the corresponding critical stress is π^2E/(KL/r)^2, where KL/r is the slenderness ratio. The effective-length factor K captures end conditions: K = 1.0 pinned-pinned, K = 0.5 fixed-fixed, K = 0.7 fixed-pinned, and K = 2.0 for a fixed-free (flagpole) column. A classic trap: because P_cr varies with 1/(KL)^2, doubling the unbraced length cuts the buckling load to one quarter, not one half. Always buckle about the axis with the larger slenderness — usually the weak axis unless bracing changes the story.
Reinforced-Concrete Flexure (ACI Strength Design)
ACI 318 uses ultimate strength design. For a singly reinforced rectangular beam, force equilibrium sets the depth of the equivalent Whitney stress block: a = As·fy / (0.85·f'c·b). The nominal moment is then M_n = As·fy·(d - a/2), and the design strength is φM_n with φ = 0.90 for a tension-controlled section. ACI deliberately requires beams to be under-reinforced (tension-controlled) so the steel yields and the member deflects with warning before the concrete crushes — a brittle over-reinforced failure is prohibited. The balanced condition is the exact state where the steel reaches yield strain just as the concrete reaches its 0.003 crushing strain. Key constants to memorize: usable concrete strain = 0.003, φ = 0.90 flexure, φ = 0.75 shear, and concrete shear strength V_c = 2·sqrt(f'c)·b·d (psi units).
Steel Design: LRFD, ASD, and Load Combinations
Steel is designed by AISC using either LRFD or ASD. LRFD factors loads up and multiplies capacity by φ; the basic gravity combination is 1.2D + 1.6L. When wind or seismic can uplift and relieve dead load, the governing case flips to 0.9D + 1.0W, which checks whether the structure can hold down against uplift. For tension yielding on the gross section, LRFD uses φ = 0.90 (design strength φP_n = 0.90·Fy·Ag), while ASD divides the same nominal strength by Ω = 1.67. Know the common yield strengths: A36 = 36 ksi, A992 (W-shapes) = 50 ksi, A615 Grade 60 rebar = 60 ksi. Two connection facts round out the topic: a fillet weld's effective throat is 0.707 times the leg size, and bolt shear stress is simply the force divided by the bolt area. Watch for lateral-torsional buckling, which governs a long, unbraced compression flange in bending.
Deflection, Serviceability, and Loads
Strength is not the only limit state — a beam that is strong but too flexible fails serviceability. The midspan deflection of a simply supported beam under uniform load is 5wL^4/(384EI); under a central point load it is PL^3/(48EI). Note the L^4 and L^3 terms: deflection is far more span-sensitive than moment, so long spans are usually deflection-governed. A common code limit for live-load deflection on a floor beam is L/360. On the load side, remember dead load includes self-weight — normal-weight reinforced concrete is taken as 150 pcf (about 23.6 kN/m^3), so a 12 in × 24 in beam weighs 150 × (1 × 2) = 300 lb/ft. Getting self-weight into the load path is a frequent exam step that candidates forget.
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