CELE Steel & Timber Design — Steel Beams: Flexure and ShearStudy Notes

When a steel beam is loaded in bending, it passes through three distinct stress regimes: (1) Elastic, where stress is proportional to strain and the neutral axis remains at the geometric center; (2) Partially plastic, where the outer fibers yield but the core remains elastic; (3) Fully plastic, where the entire section has reached $F_y$ and the neutral axis shifts to divide the section into equal strength (not equal area) zones. The plastic moment $M_p = F_y Z_x$ represents the maximum bending strength a section can develop, achievable only if (a) the section is compact (no local buckling of flanges or web), and (b) the compression flange is continuously braced or the unbraced length $L_b \le L_p$. The plastic modulus $Z_x$ is significantly larger than the elastic section modulus $S_x$ (typically 1.10–1.15 times for rolled I-shapes), reflecting the capacity gained by allowing outer fibers to yield. Per AISC 360 Section F1, the design flexural strength is $\phi_b M_n$ where $\phi_b = 0.90$ (LRFD) and the nominal moment $M_n$ depends on section compactness and lateral bracing.

As the unbraced length of the compression flange increases, the beam becomes susceptible to lateral-torsional buckling (LTB)—a simultaneous lateral deflection and twist of the cross-section—before the section can yield. AISC 360 Section F1.1 defines two critical unbraced lengths: $L_p$ (plastic lateral-torsional buckling limit) and $L_r$ (inelastic-to-elastic buckling transition). For compact I-shaped beams, the nominal moment is:

• If $L_b \le L_p$: $M_n = M_p$ (no LTB reduction)
• If $L_p < L_b \le L_r$: $M_n$ decreases linearly from $M_p$ to $M_{cr}$ (inelastic LTB)
• If $L_b > L_r$: $M_n = F_{cr} S_x$ (elastic LTB), where $F_{cr}$ varies inversely with unbraced length.

The limit $L_p$ is calculated as:

where $r_y$ is the radius of gyration about the minor (weak) axis, $E = 200\,000 \text{ MPa}$ (steel elastic modulus), and $F_y$ is the yield strength. This formula, based on elastic stability theory, represents the maximum unbraced length at which the section can reach its plastic moment in the presence of realistic imperfections. Beyond $L_p$, the beam buckles elastically (laterally) before achieving full yield. The moment-gradient factor $C_b$ (AISC Table B1.1) modifies the nominal moment for non-uniform bending; typical values range from $C_b = 1.0$ (worst case: uniform moment or cantilever) to $C_b \approx 2.5$ (favorable: moment zero at one end). The effective formula becomes $M_n = C_b M_r$, capped at $M_p$.

Before lateral-torsional buckling can occur, the individual elements of the cross-section (flanges and web) must remain stable under compression. AISC 360 Section B4 defines slenderness limits using the width-to-thickness ratio ($\lambda = b/t$) compared to $\lambda_p$ (compact limit) and $\lambda_r$ (non-compact limit). A section is: (1) Compact if flange $\lambda_f \le \lambda_p$ and web $\lambda_w \le \lambda_p$ (can reach $M_p$ without local buckling); (2) Non-compact if $\lambda_p < \lambda \le \lambda_r$ (local buckling occurs after yield; moment capacity is between $M_p$ and elastic); (3) Slender if $\lambda > \lambda_r$ (local buckling occurs before yield; elastic analysis required). For I-shaped sections under bending, the flange compact limit is typically $\lambda_p = 0.38\sqrt{E/F_y} \approx 10.8$ for $F_y = 250 \text{ MPa}$, and the web limit is $\lambda_p = 3.76\sqrt{E/F_y} \approx 106.5$. Most rolled I-shapes in the Philippines (HEB, IPE, and local equivalents) are designed to be compact in typical strength grades. However, fabricated plate girders or heavily loaded sections may be non-compact or slender, requiring reduction factors. The nominal moment for non-compact sections is reduced according to AISC F2.2 (see formula below); slender sections use elastic analysis.

The web of a steel beam is designed to resist transverse (shear) forces. Per AISC 360 Section G2, the nominal shear strength of the web is:

where:

• $A_w = d \cdot t_w$ (gross area of web = overall depth × web thickness; note: NOT the clear web height)
• $C_v$ is the shear coefficient, depending on web slenderness: $C_v = 1.0$ for stocky webs ($h/t_w \le 2.24\sqrt{E/F_y}$), and $C_v < 1.0$ for slender webs (reduced by shear buckling).
• The resistance factor is $\phi_v = 1.0$ for most rolled shapes (some fabricated girders with $\phi_v = 0.9$).

For stocky webs (typical of rolled I-shapes in the Philippines), $C_v = 1.0$ and shear yield governs:

The design shear strength is:

For slender webs, shear buckling reduces capacity; $C_v$ must be calculated per AISC Section G2.1 or simplified formulas. In most practical rolled sections used in Philippine buildings, $h/t_w$ is well below the slender limit, so $C_v = 1.0$ applies. Note: The shear formula uses the gross web area (full depth), not the area minus corner radii or fillets; this is a common design error on exams. Shear rarely governs for typical beam-column frames but becomes critical in:

• Webs with high aspect ratios (deep, thin webs)
• Near supports where shear is large
• Beams with concentrated loads close to reactions

The Load and Resistance Factor Design (LRFD) and Allowable Stress Design (ASD) methods are both permitted by NSCP 2015 and AISC 360. LRFD factors loads (multiplies by $\gamma$) and checks against factored resistance ($\phi M_n$ or $\phi V_n$); ASD divides nominal strength by a safety factor ($\Omega$) to obtain allowable strength. For flexure of compact, braced sections: LRFD uses $\phi_b = 0.90$ and $M_n = M_p = F_y Z_x$, so $\text{Design strength} = 0.90 M_p$. ASD uses $\Omega_b = 1.67$ (approximately), giving $\text{Allowable strength} = M_p / 1.67 \approx 0.60 M_p$. The ASD allowable bending stress is often stated as 0.66$F_y$. For shear, LRFD uses $\phi_v = 1.0$ (rolled shapes) and $V_n = 0.6 F_y A_w C_v$; ASD uses $\Omega_v = 1.50$, giving allowable shear as $V_n / 1.50 = 0.4 F_y A_w C_v$. Both methods require the same checks (compactness, bracing, stability); they differ only in the safety margin (factor). The AISC 360 code provides formulas for both, and the engineer chooses one approach for a project. In practice, LRFD is more common in modern design due to its explicit treatment of load combinations and more refined safety factors. For the PRC exam, both approaches must be understood, and the student should be able to convert between them or apply either method as specified in a problem.

In real structures, lateral bracing of the compression flange is essential to limit unbraced length and ensure that beams achieve high moment capacity. Common lateral bracing methods include: (1) Composite slabs (reinforced concrete or steel deck + topping), which provide continuous lateral support to the top flange and effectively set $L_b = 0$ for the top flange; (2) Discrete braces (channels, angles, rods) connecting the compression flange to a vertical element (wall, column, or bracing system) at regular intervals; (3) Continuous channels or edge beams running parallel to the main beam. In Philippine building practice, composite construction is common in office and residential floors; the floor slab is typically connected to the top flange via shear studs, providing full lateral support. For exposed steel frames (parking structures, industrial buildings), lateral braces must be explicitly designed. The engineer calculates $L_p$ for the beam and specifies brace locations such that $L_b \le L_p$. If $L_b > L_p$, the moment capacity drops significantly, and either (a) closer braces are added, or (b) a larger section is chosen. Practical exam problems often ask: (1) Calculate $L_p$ for a given section and yield strength; (2) Determine if a beam with specified unbraced length can develop $M_p$ or what moment capacity results from LTB; (3) Design the required spacing of lateral braces. For typical grade 250 or 355 steel, $L_p$ ranges from 1.5 to 3.5 m for rolled I-shapes, making it feasible to achieve full plastic capacity in most floor spans with intermediate bracing.

While flexure and shear are analyzed separately, they interact in the cross-section. When a beam is subjected to both bending moment $M$ and shear force $V$, the combined stress state must be checked. Per AISC 360 Section H1, when both flexure and shear are significant, the interaction formula is: $$\frac{M_u}{\phi_b M_n} + \frac{V_u}{\phi_v V_n} \le 1.0 \quad (\text{LRFD})$$ where $M_u$ and $V_u$ are the required (factored) moment and shear, and $\phi_b M_n$ and $\phi_v V_n$ are the design strengths. This is a linear interaction and is conservative for most sections. In practice, shear rarely governs because the web carries shear while the flanges carry moment. Shear becomes critical only when: (1) the beam has a large concentrated load near a support (high shear, low moment), (2) the web is very slender (low $V_n$), or (3) unusual geometry (short, deep beams). For typical rolled I-shapes in building frames, shear capacity is 2–5 times the required shear, and flexure controls the design. The interaction formula is primarily a checkpoint, not the controlling limit. Some advanced texts address shear-flexure interaction via reduced moment capacity when shear is high, but AISC 360 Chapter H (Stability and Other Limit States) provides the framework.

The following errors are frequently encountered in PRC exam problems and student solutions: (1) Using $S_x$ instead of $Z_x$ for plastic moment: The plastic section modulus $Z_x$ is ~1.12 times larger than the elastic $S_x$. Students often mistakenly use $M_p = F_y S_x$, which significantly underestimates capacity. Always verify which modulus applies—$Z_x$ for plastic/LRFD, $S_x$ for elastic/ASD analysis. (2) Ignoring unbraced length and LTB: A common trap is calculating $M_p = F_y Z_x$ without checking whether the compression flange is braced. If $L_b > L_p$, the actual $M_n$ is much less than $M_p$. Always calculate $L_p$ and compare to $L_b$. (3) Confusing web area in shear: The shear formula uses $A_w = d \times t_w$ (overall depth times web thickness), not the clear web height or reduced area. Using the wrong area leads to a 10–30% error in shear capacity. (4) Mixing LRFD and ASD without consistency: The $\phi$ factors, load factors, and allowable stresses differ. If a problem states LRFD, use $M_u / \phi_b M_n$ format; if ASD, use $M_a / F_b S_x$ format. Mixing them yields nonsensical results. (5) Forgetting compactness checks: Some problems expect verification that a section is compact before using $M_p$. Non-compact sections have reduced capacity. (6) Not accounting for $C_b$ in LTB calculations: For non-uniform moment, the moment-gradient factor $C_b > 1.0$ increases capacity. Ignoring it is conservative but may waste material or unfairly reject a design. (7) Misidentifying limit states: The three primary limits are: (a) yield (compact, braced sections), (b) LTB (unbraced length), (c) local buckling (slender flanges or web). A complete analysis must identify which limit governs.