Determine the design flexural strength ($\phi_b M_n$) of a continuously braced W16x31 beam of A992 steel ($F_y = 50$ ksi). The plastic section modulus $Z_x$ is $54.0 \text{ in}^3$.

A 40-foot span steel beam is designed to support a composite concrete floor slab.

Scenario: During the concrete pouring phase, before the concrete cures and bonds to the shear studs, the steel beam must support the entire wet weight of the concrete by itself.

Solution: Once the concrete cures, it provides continuous lateral bracing to the top flange of the beam ($L_b = 0$). However, during construction, the unbraced length ($L_b$) is the full 40-foot span.

Because $40 \text{ ft} > L_r$, the bare steel beam would fail via elastic lateral-torsional buckling (LTB) under the wet concrete load. To solve this, the contractor must install temporary lateral bracing (such as bridging or strut members) at 10-foot intervals. This reduces $L_b$ to 10 feet, keeping it within Zone 2 (inelastic LTB) and ensuring the beam has sufficient temporary strength until the concrete hardens.

An existing older building is being evaluated for a change in use that will significantly increase floor loads.

Scenario: The existing floor beams are older rolled shapes where the flanges are classified as "non-compact" according to modern AISC specifications.

Solution: Because the flanges are non-compact, they are susceptible to local buckling before the entire cross-section can reach its full plastic moment capacity ($M_p$). The nominal moment capacity ($M_n$) is governed by Flange Local Buckling (FLB) rather than yielding.

To increase the flexural capacity without replacing the beams, the engineers design a retrofit. They specify welding continuous steel cover plates to the top and bottom flanges. This not only increases the section modulus but also effectively thickens the flanges, changing their classification from non-compact to compact and allowing the modified section to reach its full plastic capacity.

Determine the nominal flexural strength ($M_n$) of a W18x50 beam ($F_y = 50 \text{ ksi}$) if it is braced at its ends and at midspan. The total span is 30 ft. Given properties: $L_p = 5.83 \text{ ft}$, $L_r = 16.9 \text{ ft}$, $M_p = 379 \text{ kip-ft}$, and for $C_b = 1.0$, the inelastic buckling capacity at $L_b = 15 \text{ ft}$ is calculated as $M_{LTB} = 285 \text{ kip-ft}$. The actual moment gradient modifier $C_b$ for this segment is $1.30$.

A built-up I-shaped beam has a non-compact flange but a compact web. The yield stress is $F_y = 50 \text{ ksi}$. The plastic moment capacity is $M_p = 600 \text{ kip-ft}$. The flange width-to-thickness ratio is $\lambda = 10.5$. The limits are $\lambda_p = 9.15$ and $\lambda_r = 24.1$. The yield moment accounting for residual stresses ($0.7 F_y S_x$) is $410 \text{ kip-ft}$. Determine the nominal moment capacity $M_n$ assuming continuous bracing ($L_b = 0$).

A W14x30 beam is subjected to bending with an unbraced length $L_b = 20 \text{ ft}$. The steel is A992 ($F_y = 50 \text{ ksi}$). Assume $C_b = 1.0$. The limiting lengths are $L_p = 5.26 \text{ ft}$ and $L_r = 15.0 \text{ ft}$. The critical buckling stress $F_{cr}$ has been calculated as $38.5 \text{ ksi}$ for this unbraced length. The section modulus $S_x = 42.0 \text{ in}^3$ and $Z_x = 47.3 \text{ in}^3$. Determine the nominal moment capacity $M_n$.

An unbraced beam segment of length $L_b = 20 \text{ ft}$ has the following moment distribution under factored loads: maximum moment $M_{max} = 150 \text{ kip-ft}$ at midspan. The moment at the quarter point ($M_A$) is $75 \text{ kip-ft}$, the moment at the midpoint ($M_B$) is $150 \text{ kip-ft}$, and the moment at the three-quarter point ($M_C$) is $75 \text{ kip-ft}$. Determine the moment gradient factor $C_b$.

A W10x49 column in a frame is subjected to bending about both principal axes. The factored required moments are $M_{ux} = 120 \text{ kip-ft}$ and $M_{uy} = 30 \text{ kip-ft}$. The design flexural capacities have been determined as $\phi M_{nx} = 227 \text{ kip-ft}$ and $\phi M_{ny} = 105 \text{ kip-ft}$. Verify if the section is adequate using the simplified AISC linear interaction equation.

A W18x50 beam has a tension flange spliced using 1-inch diameter bolts. There are two bolts in the cross-section. The gross area of the flange is $A_{fg} = b_f \times t_f = 7.50 \text{ in} \times 0.570 \text{ in} = 4.275 \text{ in}^2$. The net area $A_{fn}$ is $A_{fg} - 2 \times (1.0 \text{ in} + 0.125 \text{ in}) \times 0.570 \text{ in} = 2.99 \text{ in}^2$. The steel is A992 ($F_y = 50 \text{ ksi}$, $F_u = 65 \text{ ksi}$). Determine if a reduction in nominal moment capacity is required.

A propped cantilever beam (fixed at one end, simply supported at the other) supports a uniform load $w$. Elastic analysis shows the maximum moment occurs at the fixed support, $M = wL^2 / 8$, and the positive moment is $9wL^2 / 128$. When the moment at the fixed support reaches the plastic moment $M_p$, what happens if additional load is applied?

A simply supported beam with a span of $24 \text{ ft}$ must carry a factored uniform load of $w_u = 3.0 \text{ kip/ft}$ (including beam self-weight). The beam is continuously braced ($L_b = 0$). Select the lightest W-shape of A992 steel ($F_y = 50 \text{ ksi}$) from the AISC manual that can safely support the load.

A W21x44 beam ($F_y = 50 \text{ ksi}$) has an unbraced length $L_b = 10 \text{ ft}$. The plastic moment is $M_p = 399 \text{ kip-ft}$. The properties are: $L_p = 5.25 \text{ ft}$, $L_r = 14.8 \text{ ft}$, and $0.7 F_y S_x = 241 \text{ kip-ft}$. Assume $C_b = 1.0$. Determine the nominal moment capacity $M_n$.