7.2 Structural Steel Design

Structural Steel Design per AISC

Structural steel design on the PE Civil exam is based on the American Institute of Steel Construction (AISC) Steel Construction Manual. The exam covers design specifications for tension members, compression members, flexural members, and connections (bolted and welded) using either Load and Resistance Factor Design (LRFD) or Allowable Strength Design (ASD).

1. Design of Tension Members

Tension members are structural elements subjected to axial pulling forces. The design of tension members requires evaluating two primary strength limit states: yield of the gross section, rupture of the net section, and block shear rupture.

Tensile Yielding Limit State

Yielding of the gross section occurs along the member's length away from connections. It is intended to limit excessive elongation:

• Nominal strength: $P_n = F_y A_g$
• LRFD: $\phi = 0.90$, Design Strength = $\phi P_n$
• ASD: $\Omega = 1.67$, Allowable Strength = $P_n / \Omega$

Where $F_y$ is the specified minimum yield stress of the steel, and $A_g$ is the gross cross-sectional area.

Tensile Rupture Limit State

Rupture of the net section occurs at the connection where bolt holes reduce the effective cross-sectional area:

• Nominal strength: $P_n = F_u A_e$
• LRFD: $\phi = 0.75$, Design Strength = $\phi P_n$
• ASD: $\Omega = 2.00$, Allowable Strength = $P_n / \Omega$

Where $F_u$ is the specified minimum tensile strength of the steel, and $A_e$ is the effective net area, calculated as:

$$A_e = U A_n$$

Here, $A_n$ is the net area of the section (gross area minus the projected area of bolt holes, where the nominal bolt hole diameter is typically taken as the bolt diameter plus 1/16 inch for design, plus another 1/16 inch for damage, totaling bolt diameter + 1/8 inch for structural calculations). The factor $U$ is the shear lag factor, which accounts for the non-uniform distribution of tensile stress when some, but not all, elements of the cross-section are connected. It is computed as:

$$U = 1 - \frac{\bar{x}}{L}$$

Where $\bar{x}$ is the connection eccentricity, and $L$ is the length of the connection.

Block Shear Rupture Limit State

Block shear rupture is a tear-out failure mode that occurs at the boundary of a bolt pattern. It involves a combination of shear yielding or shear rupture along the failure path parallel to the force, and tensile yielding or tensile rupture along the path perpendicular to the force. The nominal strength ($R_n$) is defined by AISC as:

$$R_n = 0.60 F_u A_{nv} + U_{bs} F_u A_{nt} \le 0.60 F_y A_{gv} + U_{bs} F_u A_{nt}$$

Where:

• $A_{gv}$ is the gross area subject to shear.
• $A_{nv}$ is the net area subject to shear.
• $A_{nt}$ is the net area subject to tension.
• $U_{bs}$ is the block shear tension distribution factor (1.0 for uniform tension, 0.5 for non-uniform).

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2. Design of Compression Members (Columns)

Compression members (columns) fail through structural buckling rather than material crushing when the member is slender.

Slenderness Ratio and Effective Length

The column design capacity is governed by its slenderness ratio:

$$\text{Slenderness Ratio} = \frac{KL}{r}$$

Where:

• $K$ is the effective length factor, which depends on the column end boundary conditions.
• $L$ is the unbraced length of the column.
• $r$ is the radius of gyration ($r = \sqrt{I/A}$).

The AISC limits the maximum slenderness ratio $KL/r$ to 200 for compression members.

Nominal Compressive Strength

The nominal compressive strength ($P_n$) is calculated as:

$$P_n = F_{cr} A_g$$

Where the critical stress ($F_{cr}$) is determined based on the elastic vs. inelastic buckling regime, divided by the limit:

$$\frac{KL}{r} \le 4.71 \sqrt{\frac{E}{F_y}} \quad \text{or} \quad \frac{F_y}{F_e} \le 2.25$$

If this condition holds (inelastic buckling controls):

$$F_{cr} = \left[ 0.658^{\frac{F_y}{F_e}} \right] F_y$$

If $\frac{KL}{r} > 4.71 \sqrt{\frac{E}{F_y}}$ (elastic buckling controls):

$$F_{cr} = 0.877 F_e$$

Where $F_e$ is the elastic Euler buckling stress:

$$F_e = \frac{\pi^2 E}{\left(\frac{KL}{r}\right)^2}$$

The LRFD resistance factor for compression is $\phi = 0.90$, and the ASD safety factor is $\Omega = 1.67$.

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3. Design of Flexural Members (Beams)

The design of flexural members (beams) involves evaluating plastic bending capacity, lateral-torsional buckling, and local buckling.

Compactness and Local Buckling

Steel sections are classified as compact, non-compact, or slender based on the width-to-thickness ratio ($\lambda = b/t$) of their flange and web elements. Compact sections can develop their full plastic moment ($M_p$) before local buckling occurs:

$$M_p = F_y Z_x$$

Where $Z_x$ is the plastic section modulus. Non-compact sections are limited to the elastic yield moment ($M_y = F_y S_x$), where $S_x$ is the elastic section modulus.

The exact flange width-to-thickness ratio limit is $\lambda_p = 0.38\sqrt{E/F_y}$ for unstiffened elements (flanges) and the web slenderness limit is $\lambda_p = 3.76\sqrt{E/F_y}$ for stiffened elements in flexure.

Lateral-Torsional Buckling (LTB)

When a beam is bent about its major axis, the compression flange acts like a column and tries to buckle laterally. The tension flange keeps it from doing so fully, resulting in a lateral displacement and twisting known as lateral-torsional buckling. The LTB strength depends on the unbraced length ($L_b$), which is the distance between points of lateral support.

AISC defines three regions for LTB:

1. Plastic Region ($L_b \le L_p$): No LTB occurs. The nominal moment capacity is the full plastic capacity: $$M_n = M_p = F_y Z_x$$
2. Inelastic LTB Region ($L_p < L_b \le L_r$): Buckling occurs after some yielding. The moment capacity decreases linearly: $$M_n = C_b \left[ M_p - (M_p - 0.7 F_y S_x) \left( \frac{L_b - L_p}{L_r - L_p} \right) \right] \le M_p$$
3. Elastic LTB Region ($L_b > L_r$): Buckling occurs before yielding. The moment capacity is governed by elastic buckling theory: $$M_n = F_{cr} S_x \le M_p$$

Where $C_b$ is the lateral-torsional buckling modification factor for non-uniform moment diagrams:

$$C_b = \frac{12.5 M_{max}}{2.5 M_{max} + 3 M_A + 4 M_B + 3 M_C}$$

$C_b$ can be conservatively taken as 1.0 for uniform moment profiles, cantilever beams, or when design tables are used.

The LRFD resistance factor for bending is $\phi = 0.90$.

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4. Connection Design

Connections transfer forces between members and are designed for shear, tension, or bearing.

Bolted Connections

Bolted connections are divided into bearing-type connections (where bolts bear against the sides of the holes to transfer shear) and slip-critical connections (where high-strength bolts are tensioned to create friction between clamping surfaces).

• Bolt Shear Strength: $$R_n = F_{nv} A_b$$ Where $F_{nv}$ is the nominal shear stress (which is higher if threads are excluded from the shear plane, denoted as 'X', and lower if threads are included, denoted as 'N').
• Bolt Bearing Strength: $$R_n = 1.2 L_c t F_u \le 2.4 d t F_u$$ Where $L_c$ is the clear distance between the edge of the hole and the edge of the adjacent hole or member edge, $t$ is the thickness of the connected part, and $d$ is the nominal bolt diameter.

Welded Connections

The most common weld in structural steel is the fillet weld. Fillet welds are assumed to fail in shear through the effective throat thickness ($t_e$):

$$t_e = 0.707 w$$

Where $w$ is the weld leg size.

The nominal shear strength ($R_n$) per unit length of a fillet weld is:

$$R_n = 0.60 F_{exx} t_e$$

Where $F_{exx}$ is the classification number of the weld electrode (e.g., $F_{exx} = 70 \text{ ksi}$ for E70 electrodes). The design strength (LRFD) is $\phi R_n$ with $\phi = 0.75$.