Problem: A structural engineer is analyzing a 5-story building and deciding between using a moment-resisting frame (unbraced) or a concentrically braced frame for lateral stability. How does this choice affect gravity column design?

Solution: In a braced frame, lateral translation at the floor levels is prevented by the diagonal braces. The columns can be designed with an effective length factor ($K$) near $1.0$.

In an unbraced moment frame, the frame must sway to resist lateral loads (sidesway uninhibited). The columns in an unbraced frame typically have $K$ values greater than $1.0$ (sometimes well over $2.0$), significantly increasing their effective slenderness ($KL/r$) and reducing their axial compressive capacity. Consequently, the columns in the unbraced frame must be much larger and heavier.

Problem: Tall steel storage racks in a warehouse are failing under heavy pallet loads. The columns are built-up channel sections. The racks are buckling, but not in the direction one might expect. The engineers must diagnose the failure mode.

Solution: An inspection reveals that the columns are buckling globally, but specifically about their weak axis (the y-axis). While the beams connecting the columns provide strong-axis bracing, there is insufficient bracing in the longitudinal direction of the aisle.

Because a column will always buckle about the axis with the largest $KL/r$ ratio, the lack of weak-axis bracing caused $KL_y / r_y$ to exceed the critical limit, leading to premature failure. The retrofit involved adding diagonal cross-bracing along the aisles to reduce the unbraced length $L_y$.

Problem: Calculate the theoretical Euler critical buckling load ($P_{cr}$) for a W10x49 column ($E = 29,000 \text{ ksi}$) that is 15 feet long and pinned at both ends ($K = 1.0$). The moment of inertia for the weak axis ($I_y$) is $93.4 \text{ in}^4$ and for the strong axis ($I_x$) is $272 \text{ in}^4$.

Solution:

Problem: Determine the LRFD design compressive strength ($\phi_c P_n$) of an ASTM A992 ($F_y = 50 \text{ ksi}$) W14x90 column with an unbraced length of 15 feet in both directions. The ends are pinned ($K=1.0$). Properties: $A_g = 26.5 \text{ in}^2$, $r_x = 6.14 \text{ in}$, $r_y = 3.70 \text{ in}$. Assume the section is not slender ($Q=1.0$).

Solution:

Problem: A built-up box column has highly slender flanges such that the unstiffened element reduction factor $Q_s = 0.85$ and the stiffened element reduction factor $Q_a = 0.90$. The steel is $F_y = 50 \text{ ksi}$ and $A_g = 20 \text{ in}^2$. The calculated elastic buckling stress is $F_e = 80 \text{ ksi}$. Determine the critical stress $F_{cr}$ and nominal capacity $P_n$.

Solution:

Problem: A W12x50 column ($F_y = 50 \text{ ksi}$) has a total length of 24 ft. The strong axis (x-axis) is unbraced for the entire length, and pinned at both ends ($K_x = 1.0$). The weak axis (y-axis) is braced at mid-height, with both ends and the mid-height brace acting as pinned supports ($K_y = 1.0$ for each 12 ft segment). Determine the governing slenderness ratio. Properties: $r_x = 5.18 \text{ in}$, $r_y = 1.96 \text{ in}$.

Solution:

Problem: Consider a W8x24 column ($F_y = 50 \text{ ksi}$) with an unbraced length of 25 ft and pinned ends ($K=1.0$). Properties: $A_g = 7.08 \text{ in}^2$, $r_x = 3.42 \text{ in}$, $r_y = 1.61 \text{ in}$. Determine the design compressive strength $\phi_c P_n$.

Solution:

Problem: An engineer proposes using a single angle $L4\times4\times1/4$ as a primary compression member in a truss, with pinned ends and an unbraced length of 8 ft. Why must the engineer be cautious?

Solution: Single angles are unsymmetric shapes. The shear center and the centroid do not coincide. When a compressive load is applied, even if perfectly axial at the ends, the member is highly susceptible to flexural-torsional buckling.

The engineer cannot simply calculate the weak-axis flexural buckling capacity using $r_z$ (the minimum radius of gyration for an angle). They must use the complex AISC equations for flexural-torsional buckling, which simultaneously consider bending stiffness, torsional stiffness (St. Venant torsion), and warping stiffness. This combined mode will result in an elastic buckling stress ($F_e$) that is significantly lower than the pure flexural buckling stress.

Problem: A W10x49 column transfers a factored axial load of $P_u = 400 \text{ kips}$ to a concrete footing. The footing has a compressive strength of $f'_c = 3.0 \text{ ksi}$. The base plate covers the entire area of the concrete pier supporting it (i.e., $A_2 = A_1$). Determine the minimum required area of the base plate ($A_1$).

Solution:

Problem: Select the lightest W12 column of ASTM A992 steel ($F_y = 50 \text{ ksi}$) to support a factored axial load of $P_u = 600 \text{ kips}$. The column has an unbraced length of 14 ft and is pinned at both ends ($K_x = K_y = 1.0$). Available trial shapes and their weak-axis capacities ($\phi_c P_n$) for $KL = 14 \text{ ft}$:

• W12x53: $\phi_c P_n = 518 \text{ kips}$
• W12x58: $\phi_c P_n = 569 \text{ kips}$
• W12x65: $\phi_c P_n = 642 \text{ kips}$
• W12x72: $\phi_c P_n = 717 \text{ kips}$

Solution: