Steel Beams and Girders

Steel Beams and Girders

Keep the formats paired: Compare factored demand M_u,V_u with LRFD design resistance φM_n,φV_n, or compare service/ASD demand M_a,V_a with allowable resistance M_n/Ω,V_n/Ω. Never compare factored demand with an ASD allowable value or apply both φ and Ω.

From Load Path to Member Demand

Establish tributary loads, combinations, supports, releases, and bracing before selecting a shape. Structural analysis provides moment, shear, reactions, and service deflection. Member design then finds the controlling resistance among applicable limit states. A girder supporting beams may also receive concentrated reactions that a uniform-load sketch hides.

Check the actual segment under consideration. Positive bending normally puts the top flange in compression; negative bending can put the bottom flange in compression. The brace must restrain the relevant flange and provide the stiffness/strength required by AISC 15th; physical contact alone is not automatically an effective brace.

Compactness and Local Buckling

AISC classifies compression elements from width-to-thickness ratios and element boundary conditions. A compact section can develop the plastic stress distribution assumed by the applicable flexural expression before local buckling. A noncompact element can reach yield but local buckling reduces available inelastic rotation or strength. A slender element buckles elastically and requires the specified reduction.

Classify flange and web separately using the correct material yield stress and table limits. The least favorable applicable element can control the section behavior. “Wide-flange” is a shape family, not proof of compactness, and a compact cross-section can still undergo lateral-torsional buckling.

Lateral-Torsional Buckling

Lateral-torsional buckling, or LTB, couples lateral movement of the compression flange with twist. The key length L_b is the distance between points that adequately brace the compression flange against lateral displacement and twist—not necessarily the beam span. AISC flexural strength regions use limiting lengths and the applicable moment-gradient modifier. Do not choose a favorable C_b without calculating or obtaining it under the permitted expression.

For a 30-ft simple beam braced at both supports and at an effective midpoint brace, the two unbraced segments are 15 ft. Use L_b = 15 ft for a positive-moment compression flange if all three points provide the required restraint. If that midpoint connection restrains only the tension flange or lacks torsional restraint, it does not automatically halve L_b. Re-evaluate bracing where the moment sign changes.

Worked Nominal-to-Design Strength Screen

Analysis gives M_u = 255 kip-ft and V_u = 150 kips. After checking compactness, LTB with the actual L_b, and all applicable section limit states, suppose the controlling nominal strengths are M_n = 300 kip-ft and V_n = 180 kips. The problem supplies LRFD factors φ_b = 0.90 and φ_v = 1.00 for these limit states.

Flexural design strength is

φ_bM_n = 0.90(300) = 270 kip-ft.

The utilization is

M_u/(φ_bM_n) = 255/270 = 0.944.

Shear design strength and utilization are

φ_vV_n = 1.00(180) = 180 kips

and

V_u/(φ_vV_n) = 150/180 = 0.833.

Both screens pass. Comparing M_u directly with M_n would report 0.850 and overstate available design resistance. For ASD, restart with compatible service-load demand and the specified M_n/Ω; do not divide the already factored LRFD result by Ω.

Shear and Plate-Girder Behavior

Web shear resistance depends on web slenderness, material, stiffening, and the applicable buckling/postbuckling provisions. A slender web may buckle before shear yielding; a stiffened girder may use additional behavior only when AISC requirements are satisfied. Openings interrupt shear flow and require an explicit check. High moment and high shear occurring together may trigger an interaction provision rather than two unrelated unity checks.

Concentrated Forces and Connections

At supports, column reactions, hangers, and point loads, investigate the applicable web local yielding, web crippling, sidesway buckling, and compression-buckling limit states. Bearing length, distance to the end, flange restraint, load application to one or both flanges, and stiffeners matter. A stiffener needs its own bearing, buckling, weld, and load-path checks; drawing one does not create unlimited capacity.

Connections must transfer the beam force into the support. Check bolt/weld demand, copes, block shear or net-section effects where applicable, and erection stability. A member adequate between supports can still fail at its end connection.

Serviceability and Final Workflow

Use service-level loads and the correct elastic stiffness for deflection and vibration. Include continuity and composite action only when they are actually present. Camber changes initial geometry but is not added structural stiffness.

1. Analyze compatible strength and service combinations.
2. Classify flange and web compactness.
3. Map every braced segment and compression flange; calculate applicable LTB strength.
4. Find controlling M_n and V_n, then apply exactly one design format.
5. Check interaction, concentrated forces, openings, stiffeners, and connections.
6. Check deflection, vibration, ponding susceptibility, and construction stage.
7. Trace reactions into the supporting system.

Use only the supplied AISC 15th edition for a 2026 exam; later AISC editions can change equations, tables, and organization.