Beams: Shear and Serviceability

While flexure (bending moment) often governs the size of steel beams, they must also be explicitly checked for shear capacity and serviceability requirements. Shear failure can occur near supports or under heavy concentrated loads. Serviceability involves ensuring the beam performs well under everyday use without excessive deflection, vibration, or localized damage.

Shear Yielding: Occurs in compact, thick webs where the entire web cross-section reaches its shear yield stress ($\tau_{yield} \approx 0.60 F_y$). The web deforms plastically in shear. This is the dominant failure mode for almost all standard rolled W-shapes.

Shear Buckling: Occurs in slender webs (like plate girders) before the material yields. The web buckles diagonally out-of-plane due to the principal compressive stresses induced by the shear force. Once buckled, the web's capacity to carry shear drops significantly unless tension field action (TFA) is permitted.

The value of $C_v$ heavily depends on the web's slenderness ratio ($h/t_w$, where $h$ is the clear distance between flanges). The web slenderness determines which buckling regime controls the shear capacity:

• Yielding ($C_v = 1.0$): Compact webs where the entire web yields before any out-of-plane buckling occurs.
• Inelastic Buckling ($C_v < 1.0$): Moderately slender webs where some yielding occurs, but buckling initiates before full plastic shear capacity is reached.
• Elastic Buckling ($C_v \ll 1.0$): Slender webs (like deep plate girders) that buckle completely elastically without reaching the shear yield stress.

For most standard rolled W-shapes, the webs are thick enough (compact) that $C_v = 1.0$.

For standard W-shapes with $h/t_w \le 2.24\sqrt{E/F_y}$ (where $C_v = 1.0$), the LRFD resistance factor is $\phi_v = 1.00$.

For all other webs (built-up sections, slender webs, channels, etc.), the LRFD resistance factor is $\phi_v = 0.90$.

When a beam is subjected to heavy concentrated loads (like a column bearing on the top flange) or at reaction points, the highly localized compressive force can crush or buckle the thin web beneath it. If the web fails these checks, bearing stiffeners must be added.

Occurs when the compression flange is restrained against rotation but the tension flange is not. If the beam is subjected to a heavy concentrated load, the entire web can buckle sideways.

The nominal capacity $R_n$ depends on the ratio of $h/t_w$ and the ratio of $L_b/b_f$.

If the web fails this check, it must be braced laterally at the load point, or bearing stiffeners must be installed.

Serviceability limits ensure the structure remains functional, aesthetically pleasing, and comfortable for occupants during normal everyday use. Excessive deflection can crack plaster ceilings, misalign doors, or cause ponding.

Standard industry deflection limits depend on the application and the attached non-structural elements:

• Live Load Only: $L/360$ (typical for floors supporting plaster ceilings to prevent cracking).
• Total Load (Dead + Live): $L/240$ for typical floors.
• Roof Members (Total Load): $L/180$ or $L/240$ depending on whether the roofing material is susceptible to damage.

Flat roofs are uniquely susceptible to ponding, a progressive instability.

Prevention: The primary defense is providing adequate roof slope (minimum 1/4 inch per foot) and proper overflow drainage.

Design Check (AISC Appendix 2): The entire roof system must be explicitly checked for stiffness ($C_p$ and $C_s$) to ensure the framing is stiff enough to resist the cascading weight of accumulated water.

Long-span, lightweight steel floors can experience unacceptable vibrations caused by human activity.

Natural Frequency ($f_n$): A natural frequency greater than 3 to 5 Hz is often recommended to avoid resonance with the walking excitation frequency ($1.5 - 2.5$ Hz).

Solution: Increasing the stiffness (moment of inertia, $I$) of the floor beams and girders is the primary way to increase the natural frequency and reduce peak acceleration.

Standard W-shapes perform exceptionally poorly when subjected to torsion because they are "open" sections.

Pure (St. Venant) Torsion: Resisted by shear stresses. Closed shapes (HSS) have massive torsional constants ($J$) and are highly efficient. Open shapes (W-shapes) have tiny $J$ values.

Warping Torsion: When an open section twists, its cross-section distorts.

Design Rule: For significant torsion, HSS (Hollow Structural Sections) or box girders are strongly preferred over open I-shapes.