Beam and Slab Flexural Load Effects

Beam and Slab Flexural Load Effects

Model before formula: wL²/8 is a simply supported result, not a universal beam moment. A correct load on the wrong support model still produces the wrong answer.

Read the Structural Idealization

Start by marking each support as a pin, roller, fixed end, elastic restraint, or internal release. Then identify member continuity and relative stiffness. A simple span has zero end moment at ideal pins. A cantilever develops its largest bending moment at the fixed end. A continuous beam transfers moment across supports; equilibrium alone cannot determine all reactions and moments because deformation compatibility is also required.

Use one sign convention throughout. This section calls sagging moment positive, normally placing the bottom face in tension, and hogging moment negative, normally placing the top face in tension. The physical tension face matters for reinforcement, bracing, connection force, and detailing even when a multiple-choice answer lists only a magnitude.

Real connections may be semirigid, and adjacent members may provide finite rotational restraint. Use the idealization stated by the problem or justified by the structural system; do not choose the smallest tabulated moment.

Worked Support-Condition Comparison

A 24-ft member carries w = 2.0 kip/ft over its full span. Compare three idealizations. First calculate wL² = 2(24²) = 1,152 kip-ft.

Simple span:

M_mid = 1,152/8 = +144 kip-ft.

Cantilever:

M_fixed = -1,152/2 = -576 kip-ft.

Fixed–fixed span:

M_A = M_B = -1,152/12 = -96 kip-ft

and

M_mid = +1,152/24 = +48 kip-ft.

The load and length are identical, yet the governing moment magnitude ranges from 96 to 576 kip-ft and occurs at different locations. Full fixity reduces positive midspan moment by developing negative end moments; it does not make bending disappear. If an end releases, settlements occur, or EI varies, these fixed-end results cannot be copied unchanged.

Continuity and Indeterminate Behavior

For continuous beams and frames, use an applicable compatibility method: slope-deflection, moment distribution, stiffness analysis, or a supplied coefficient method whose limitations are satisfied. A disciplined sequence is:

1. Establish nodes, spans, releases, support translations, and member EI.
2. Apply fixed-end effects with the chosen sign convention.
3. Enforce joint equilibrium and deformation compatibility.
4. Recover reactions, shears, and span moments.
5. Check that the final diagram satisfies free-end, hinge, and support conditions.

Live load may be movable or patternable. Loading every span does not necessarily maximize every effect. Alternate-span loading can increase positive moment in one span, while adjacent loaded spans can increase negative support moment. Construct envelopes for positive span moment, negative support moment, and reactions using arrangements allowed by the governing load provisions. Dead load is normally present where the structure exists; do not pattern it as though it could be removed. Redistribution of elastic moment is allowed only when the applicable material standard and its limits permit it.

Beam Load Width and Units

Convert area load to line load through a defensible tributary width before applying beam formulas. For a one-foot slab strip under q = 120 psf, the strip line load is

w = q(1 ft) = 120 plf,

not 120 psf inserted directly into wL². Include self-weight once, and distinguish service load from factored design load. Keep feet with kip/ft, or convert all geometry consistently before combining with section properties in inches.

One-Way and Two-Way Slab Action

A one-way slab primarily spans between two opposite support lines. A unit-width strip can be modeled as a beam with the slab's actual continuity, and primary flexural reinforcement follows the spanning direction. A two-way slab distributes load and moment in two directions to multiple support lines or columns. Its column-strip, middle-strip, support, and span regions cannot be replaced automatically by one independent beam strip.

Classify action from support geometry and the current ACI 318-14 provisions, not from appearance alone. Check whether the chosen direct-design, equivalent-frame, coefficient, or analysis method is permitted by its regularity, span, loading, and support assumptions. At a continuous support, negative moment creates top tension; at midspan, positive moment generally creates bottom tension. Discontinuous edges and corner restraint change the distribution and detailing.

2026 Reference Discipline

Use the PE Civil handbook active for the test date for mechanics relationships, ACI 318-14 for concrete and slab provisions, and the AISC Steel Construction Manual, 15th edition, for steel design context. Do not use a coefficient remembered from a newer office edition. Separate load-effect analysis from member resistance: first obtain the correct M_u or service moment diagram, then apply the appropriate material strength or serviceability check.

Exam Audit

Before selecting an answer, write: support model, span, load width, load level, sign, target location. Sketch the qualitative moment diagram first. It should be zero at an ideal internal hinge, negative over a continuous restraining support under gravity load, and positive in a typical span region. A numerical answer that violates those boundary conditions is evidence of a model error, even if the algebra is flawless.