Formula Sheet: Steel Design

Tension Members

Nominal Tensile Strength

The nominal tensile strength \(P_n\) is the lesser of:

• Yielding in gross section: \[P_n = F_y A_g\]
• Fracture in net section: \[P_n = F_u A_e\]

Variables:

• \(F_y\) = specified minimum yield stress (ksi or MPa)
• \(A_g\) = gross area of member (in² or mm²)
• \(F_u\) = specified minimum tensile strength (ksi or MPa)
• \(A_e\) = effective net area (in² or mm²)

Effective Net Area

\[A_e = A_n U\]

Variables:

• \(A_n\) = net area (in² or mm²)
• \(U\) = shear lag factor (dimensionless)

Net Area

\[A_n = A_g - \sum(d_{hole} \times t)\]

Variables:

• \(d_{hole}\) = diameter of bolt hole (in or mm)
• \(t\) = thickness of material (in or mm)

Bolt hole diameter: Standard hole = bolt diameter + 1/16 in (or +2 mm for SI)

Shear Lag Factor (U)

When all cross-sectional elements are connected: \(U = 1.0\)

When some elements are not connected:

\[U = 1 - \frac{\bar{x}}{L}\]

Variables:

• \(\bar{x}\) = connection eccentricity (distance from centroid of connected elements to plane of load transfer) (in or mm)
• \(L\) = length of connection in direction of loading (in or mm)

Tabulated U values for common cases:

• W, M, S, HP shapes with flange connected, \(b_f \geq \frac{2d}{3}\), minimum 3 fasteners per line: \(U = 0.90\)
• W, M, S, HP shapes with flange connected, \(b_f < \frac{2d}{3}\),="" minimum="" 3="" fasteners="" per="" line:="" \(u="">
• Single and double angles, 4 or more fasteners per line: \(U = 0.80\)
• Single and double angles, 3 fasteners per line: \(U = 0.60\)
• Single and double angles, 2 fasteners per line: \(U = 0.40\)

Design Tensile Strength

LRFD: \(\phi_t = 0.90\) for yielding, \(\phi_t = 0.75\) for fracture

\[\phi_t P_n\]

ASD: \(\Omega_t = 1.67\) for yielding, \(\Omega_t = 2.00\) for fracture

\[\frac{P_n}{\Omega_t}\]

Compression Members

Nominal Compressive Strength

\[P_n = F_{cr} A_g\]

Variables:

• \(F_{cr}\) = critical stress (ksi or MPa)
• \(A_g\) = gross area of member (in² or mm²)

Critical Stress for Flexural Buckling

For elastic buckling when \(\frac{KL}{r} \leq 4.71\sqrt{\frac{E}{F_y}}\) or \(F_e \geq 0.44F_y\):

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

For inelastic buckling when \(\frac{KL}{r} > 4.71\sqrt{\frac{E}{F_y}}\) or \(F_e <>

\[F_{cr} = 0.877 F_e\]

Elastic critical buckling stress:

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

Variables:

• \(K\) = effective length factor (dimensionless)
• \(L\) = laterally unbraced length of member (in or mm)
• \(r\) = radius of gyration about axis of buckling (in or mm)
• \(E\) = modulus of elasticity of steel = 29,000 ksi (200,000 MPa)

Effective Length Factor (K)

Theoretical values:

• Both ends pinned: \(K = 1.0\)
• One end fixed, one end pinned: \(K = 0.7\)
• Both ends fixed: \(K = 0.5\)
• One end fixed, one end free: \(K = 2.0\)
• One end fixed, one end fixed against rotation but free to translate: \(K = 1.0\)

Recommended design values (AISC):

• Use alignment charts for frame structures
• For columns in braced frames: \(1.0 \geq K \geq 0.5\)
• For columns in unbraced frames: \(K > 1.0\)

Design Compressive Strength

LRFD: \(\phi_c = 0.90\)

\[\phi_c P_n\]

ASD: \(\Omega_c = 1.67\)

\[\frac{P_n}{\Omega_c}\]

Local Buckling - Width-to-Thickness Ratios

Slenderness parameter for unstiffened elements:

\[\lambda = \frac{b}{t}\]

Slenderness parameter for stiffened elements:

\[\lambda = \frac{h}{t} \text{ or } \frac{b}{t}\]

Limiting width-to-thickness ratios:

• Flanges of rolled I-shaped sections: \(\lambda_r = 0.56\sqrt{\frac{E}{F_y}}\)
• Flanges of built-up I-shaped sections: \(\lambda_r = 0.64\sqrt{\frac{kE}{F_y}}\) where \(k = \frac{4}{\sqrt{h/t_w}}\), \(0.35 \leq k \leq 0.76\)
• Webs of I-shaped sections: \(\lambda_r = 1.49\sqrt{\frac{E}{F_y}}\)
• Legs of angles: \(\lambda_r = 0.45\sqrt{\frac{E}{F_y}}\)
• Stems of tees: \(\lambda_r = 0.75\sqrt{\frac{E}{F_y}}\)

Flexural Members (Beams)

Nominal Flexural Strength

\[M_n = \text{lesser of lateral-torsional buckling, local buckling, or plastic moment capacity}\]

Plastic Moment Capacity

\[M_p = F_y Z_x\]

Variables:

• \(Z_x\) = plastic section modulus about major axis (in³ or mm³)

Yielding (Limit State)

For compact sections with adequate lateral bracing:

\[M_n = M_p = F_y Z_x\]

Lateral-Torsional Buckling

When \(L_b \leq L_p\) (fully braced):

\[M_n = M_p = F_y Z_x\]

When \(L_p < l_b="" \leq="" l_r\)="" (inelastic="">

\[M_n = C_b\left[M_p - (M_p - 0.7F_y S_x)\left(\frac{L_b - L_p}{L_r - L_p}\right)\right] \leq M_p\]

When \(L_b > L_r\) (elastic LTB):

\[M_n = F_{cr} S_x \leq M_p\] where: \[F_{cr} = \frac{C_b \pi^2 E}{\left(\frac{L_b}{r_{ts}}\right)^2}\sqrt{1 + 0.078\frac{Jc}{S_x h_o}\left(\frac{L_b}{r_{ts}}\right)^2}\]

Variables:

• \(L_b\) = length between points of lateral bracing (in or mm)
• \(L_p\) = limiting laterally unbraced length for full plastic moment capacity (in or mm)
• \(L_r\) = limiting laterally unbraced length for inelastic lateral-torsional buckling (in or mm)
• \(C_b\) = lateral-torsional buckling modification factor (dimensionless)
• \(S_x\) = elastic section modulus about major axis (in³ or mm³)
• \(r_{ts}\) = effective radius of gyration for lateral-torsional buckling (in or mm)
• \(J\) = torsional constant (in⁴ or mm⁴)
• \(c\) = distance from neutral axis to extreme fiber (in or mm)
• \(h_o\) = distance between flange centroids (in or mm)

Limiting Unbraced Lengths

For doubly symmetric I-shaped members and channels:

\[L_p = 1.76 r_y \sqrt{\frac{E}{F_y}}\] \[L_r = 1.95 r_{ts} \frac{E}{0.7F_y} \sqrt{\frac{Jc}{S_x h_o} + \sqrt{\left(\frac{Jc}{S_x h_o}\right)^2 + 6.76\left(\frac{0.7F_y}{E}\right)^2}}\]

Variables:

• \(r_y\) = radius of gyration about weak axis (in or mm)

Lateral-Torsional Buckling Modification Factor

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

Variables:

• \(M_{max}\) = absolute value of maximum moment in unbraced segment
• \(M_A\) = absolute value of moment at quarter point of unbraced segment
• \(M_B\) = absolute value of moment at centerline of unbraced segment
• \(M_C\) = absolute value of moment at three-quarter point of unbraced segment

Conservative value: \(C_b = 1.0\)

For uniform moment: \(C_b = 1.0\)

Range: \(1.0 \leq C_b \leq 3.0\) (typically)

Compact Section Requirements

For flanges of I-shaped sections bent about major axis:

\[\frac{b_f}{2t_f} \leq \lambda_{pf} = 0.38\sqrt{\frac{E}{F_y}}\]

For webs of I-shaped sections in flexure:

\[\frac{h}{t_w} \leq \lambda_{pw} = 3.76\sqrt{\frac{E}{F_y}}\]

Variables:

• \(b_f\) = flange width (in or mm)
• \(t_f\) = flange thickness (in or mm)
• \(h\) = clear distance between flanges less fillet (in or mm)
• \(t_w\) = web thickness (in or mm)
• \(\lambda_{pf}\) = limiting slenderness for compact flange
• \(\lambda_{pw}\) = limiting slenderness for compact web

Shear Strength of Beams

Nominal shear strength:

\[V_n = 0.6 F_y A_w C_{v1}\]

For webs of rolled I-shaped members:

\[A_w = d t_w\]

Variables:

• \(A_w\) = area of web (in² or mm²)
• \(d\) = overall depth of member (in or mm)
• \(C_{v1}\) = web shear coefficient (dimensionless)

Web Shear Coefficient

When \(\frac{h}{t_w} \leq 2.24\sqrt{\frac{E}{F_y}}\):

\[C_{v1} = 1.0\]

When \(2.24\sqrt{\frac{E}{F_y}} < \frac{h}{t_w}="" \leq="">

\[C_{v1} = \frac{2.24\sqrt{\frac{E}{F_y}}}{\frac{h}{t_w}}\]

When \(\frac{h}{t_w} > 2.45\sqrt{\frac{E}{F_y}}\):

\[C_{v1} = \frac{5.34E}{\left(\frac{h}{t_w}\right)^2 F_y}\]

Design Flexural and Shear Strength

LRFD (Flexure): \(\phi_b = 0.90\)

\[\phi_b M_n\]

ASD (Flexure): \(\Omega_b = 1.67\)

\[\frac{M_n}{\Omega_b}\]

LRFD (Shear): \(\phi_v = 0.90\) (for \(h/t_w \leq 2.24\sqrt{E/F_y}\)), \(\phi_v = 1.0\) otherwise

\[\phi_v V_n\]

ASD (Shear): \(\Omega_v = 1.67\) (for \(h/t_w \leq 2.24\sqrt{E/F_y}\)), \(\Omega_v = 1.50\) otherwise

\[\frac{V_n}{\Omega_v}\]

Deflection Limits

Typical maximum deflection criteria (not code-mandated, project-specific):

• Roof members (live load): \(L/240\) or \(L/360\)
• Floor members (live load): \(L/240\) or \(L/360\)
• Members supporting plastered ceilings (live load): \(L/360\)
• Total load deflection: varies by application

Combined Stresses

Members Subject to Combined Axial and Flexure

When \(\frac{P_r}{P_c} \geq 0.2\):

\[\frac{P_r}{P_c} + \frac{8}{9}\left(\frac{M_{rx}}{M_{cx}} + \frac{M_{ry}}{M_{cy}}\right) \leq 1.0\]

When \(\frac{P_r}{P_c} <>

\[\frac{P_r}{2P_c} + \left(\frac{M_{rx}}{M_{cx}} + \frac{M_{ry}}{M_{cy}}\right) \leq 1.0\]

Variables:

• \(P_r\) = required axial strength (LRFD) or \(P_a\) (ASD)
• \(P_c\) = available axial strength (LRFD: \(\phi_c P_n\), ASD: \(P_n/\Omega_c\))
• \(M_{rx}\) = required flexural strength about x-axis
• \(M_{ry}\) = required flexural strength about y-axis
• \(M_{cx}\) = available flexural strength about x-axis (LRFD: \(\phi_b M_{nx}\), ASD: \(M_{nx}/\Omega_b\))
• \(M_{cy}\) = available flexural strength about y-axis (LRFD: \(\phi_b M_{ny}\), ASD: \(M_{ny}/\Omega_b\))

Amplification Factors

For moment magnification in frames (second-order effects):

Non-sway frames:

\[B_1 = \frac{C_m}{1 - \frac{P_r}{P_{e1}}} \geq 1.0\]

Sway frames:

\[B_2 = \frac{1}{1 - \frac{\sum P_r}{\sum P_{e2}}} \geq 1.0\]

Variables:

• \(C_m\) = coefficient for unequal end moments (0.6 to 1.0)
• \(P_{e1}\) = Euler buckling strength for non-sway = \(\frac{\pi^2 EI}{(K_1 L)^2}\)
• \(P_{e2}\) = Euler buckling strength for sway = \(\frac{\pi^2 EI}{(K_2 L)^2}\)
• \(K_1\) = effective length factor for non-sway
• \(K_2\) = effective length factor for sway

Coefficient \(C_m\) for members without transverse loading:

\[C_m = 0.6 - 0.4\left(\frac{M_1}{M_2}\right)\]

where \(\frac{M_1}{M_2}\) is the ratio of smaller to larger moment at ends (positive for reverse curvature)

Connections

Bolted Connections - Bearing Type

Nominal shear strength per bolt:

\[R_n = F_{nv} A_b\]

Variables:

• \(F_{nv}\) = nominal shear stress (ksi or MPa)
• \(A_b\) = nominal unthreaded body area of bolt (in² or mm²)

For A325 and A490 bolts:

• A325-N (threads not excluded from shear plane): \(F_{nv} = 54\) ksi
• A325-X (threads excluded from shear plane): \(F_{nv} = 68\) ksi
• A490-N (threads not excluded from shear plane): \(F_{nv} = 68\) ksi
• A490-X (threads excluded from shear plane): \(F_{nv} = 84\) ksi

Nominal Tensile Strength per Bolt

\[R_n = F_{nt} A_b\]

For A325 and A490 bolts:

• A325: \(F_{nt} = 90\) ksi
• A490: \(F_{nt} = 113\) ksi

Bearing Strength at Bolt Holes

When deformation at service load is a consideration:

\[R_n = 1.2 L_c t F_u \leq 2.4 d t F_u\]

When deformation at service load is not a consideration:

\[R_n = 1.5 L_c t F_u \leq 3.0 d t F_u\]

Variables:

• \(L_c\) = clear distance from edge of hole to edge of adjacent hole or edge of material in direction of force (in or mm)
• \(t\) = thickness of connected material (in or mm)
• \(d\) = nominal bolt diameter (in or mm)
• \(F_u\) = specified minimum tensile strength of connected material (ksi or MPa)

Slip-Critical Connections

Available slip resistance per bolt:

\[R_n = \mu D_u h_f T_b n_s\]

Variables:

• \(\mu\) = mean slip coefficient (Class A = 0.30, Class B = 0.50)
• \(D_u\) = 1.13 (multiplier for standard holes)
• \(h_f\) = 1.0 (factor for fillers)
• \(T_b\) = minimum bolt tension (kips or kN)
• \(n_s\) = number of slip planes

Minimum bolt pretension:

• A325: 3/4 in = 28 kips, 7/8 in = 39 kips, 1 in = 51 kips
• A490: 3/4 in = 35 kips, 7/8 in = 49 kips, 1 in = 64 kips

Design Strength of Bolts

LRFD:

• Shear: \(\phi = 0.75\)
• Tension: \(\phi = 0.75\)
• Bearing: \(\phi = 0.75\)
• Slip-critical: \(\phi = 1.00\) (serviceability), \(\phi = 0.85\) (strength)

ASD:

• Shear: \(\Omega = 2.00\)
• Tension: \(\Omega = 2.00\)
• Bearing: \(\Omega = 2.00\)
• Slip-critical: \(\Omega = 1.50\) (serviceability), \(\Omega = 1.76\) (strength)

Combined Tension and Shear in Bolts

For bearing-type connections:

\[F'_{nt} = 1.3 F_{nt} - \frac{F_{nt}}{F_{nv}} f_{rv} \leq F_{nt}\]

Variables:

• \(F'_{nt}\) = nominal tensile stress modified for shear (ksi or MPa)
• \(f_{rv}\) = required shear stress (ksi or MPa)

Welded Connections - Fillet Welds

Nominal strength per unit length:

\[R_n = F_w A_{we}\]

Effective throat area per unit length:

\[A_{we} = t_e \times 1.0\]

Effective throat thickness:

\[t_e = 0.707 w\]

Variables:

• \(w\) = leg size of fillet weld (in or mm)
• \(t_e\) = effective throat thickness (in or mm)
• \(F_w\) = nominal stress of weld metal (ksi or MPa)

Nominal stress of weld metal:

\[F_w = 0.60 F_{EXX} \times 1.0\]

Variables:

• \(F_{EXX}\) = electrode classification strength (60, 70, 80, 90, 100, 110 ksi)

Minimum Fillet Weld Size

Based on thicker part joined:

• Material thickness ≤ 1/4 in: minimum weld size = 1/8 in
• Material thickness > 1/4 to 1/2 in: minimum weld size = 3/16 in
• Material thickness > 1/2 to 3/4 in: minimum weld size = 1/4 in
• Material thickness > 3/4 in: minimum weld size = 5/16 in

Maximum fillet weld size:

• Along edges of material < 1/4="" in="" thick:="" thickness="" of="">
• Along edges of material ≥ 1/4 in thick: thickness minus 1/16 in

Design Strength of Welds

LRFD: \(\phi = 0.75\)

\[\phi R_n\]

ASD: \(\Omega = 2.00\)

\[\frac{R_n}{\Omega}\]

Groove Welds

Complete penetration groove welds:

• Strength = strength of base metal
• Treated as continuous base metal in design

Partial penetration groove welds:

• Effective throat = depth of chamfer for J- and U-grooves
• Effective throat = depth of chamfer minus 1/8 in for bevel and V-grooves

Block Shear Rupture

Nominal strength:

\[R_n = 0.6 F_u A_{nv} + U_{bs} F_u A_{nt} \leq 0.6 F_y A_{gv} + U_{bs} F_u A_{nt}\]

Variables:

• \(A_{nv}\) = net area subject to shear (in² or mm²)
• \(A_{nt}\) = net area subject to tension (in² or mm²)
• \(A_{gv}\) = gross area subject to shear (in² or mm²)
• \(U_{bs}\) = 1.0 for uniform tension, 0.5 for nonuniform tension

LRFD: \(\phi = 0.75\)

ASD: \(\Omega = 2.00\)

Plate Girders

Flexural Strength

Nominal flexural strength for plate girders with slender webs:

\[M_n = R_{pc} M_y\]

where \(M_y = F_y S_{xc}\) for compact or noncompact flanges

Web plastification factor:

\[R_{pc} = \frac{M_p}{M_y}\]

For sections with \(\frac{h_c}{t_w} > \lambda_{pw}\), reduction factors apply.

Shear Strength with Tension Field Action

When tension field action is permitted:

\[V_n = 0.6 F_y A_w \left[C_v + \frac{1 - C_v}{\sqrt{1.15 + \frac{1}{a^2/h^2}}}\right]\]

Variables:

• \(a\) = clear distance between transverse stiffeners (in or mm)
• \(h\) = clear distance between flanges (in or mm)

Transverse Stiffener Requirements

Moment of inertia requirement:

\[I_{st} \geq a t_w^3 \left(\frac{h}{a}\right)^2 j\]

Where:

\[j = 2.5\left(\frac{h}{a}\right)^2 - 2 \geq 0.5\]

Composite Members

Composite Beam Strength

For full composite action with concrete slab:

Positive moment capacity (plastic neutral axis in slab):

\[M_n = A_s F_y \left(d + t + \frac{t_s}{2} - \frac{a}{2}\right)\]

Variables:

• \(A_s\) = area of steel beam (in² or mm²)
• \(d\) = depth of steel beam (in or mm)
• \(t\) = thickness of steel deck (if present) (in or mm)
• \(t_s\) = thickness of concrete slab (in or mm)
• \(a\) = depth of compression block in concrete (in or mm)

Depth of compression block:

\[a = \frac{A_s F_y}{0.85 f'_c b}\]

Variables:

• \(f'_c\) = specified compressive strength of concrete (ksi or MPa)
• \(b\) = effective width of concrete flange (in or mm)

Effective Width of Concrete Flange

Effective width is the minimum of:

• 1/4 of beam span
• Center-to-center spacing of beams
• Distance to edge of slab

Shear Connectors

Nominal strength of one stud shear connector:

\[Q_n = 0.5 A_{sc} \sqrt{f'_c E_c} \leq R_g R_p A_{sc} F_u\]

Variables:

• \(A_{sc}\) = cross-sectional area of stud shear connector (in² or mm²)
• \(E_c\) = modulus of elasticity of concrete = \(w_c^{1.5} \sqrt{f'_c}\) (psi or MPa)
• \(w_c\) = unit weight of concrete (lb/ft³ or kg/m³); typically 145 pcf for normal weight
• \(R_g\) = group effect factor
• \(R_p\) = position effect factor
• \(F_u\) = tensile strength of stud (typically 60 ksi)

Number of Shear Connectors Required

For full composite action between point of maximum moment and zero moment:

\[N = \frac{V'_h}{Q_n}\]

Horizontal shear force:

\[V'_h = \min(A_s F_y, \; 0.85 f'_c b t_s)\]

Design Strength of Shear Connectors

LRFD: \(\phi_{sc} = 0.75\)

ASD: \(\Omega_{sc} = 2.00\)

Column Base Plates

Required Base Plate Area

For concentric axial load:

\[A_1 = \frac{P_u}{0.85 f'_c \phi_c}\]

where \(\phi_c = 0.65\) for concrete bearing (LRFD)

For ASD:

\[A_1 = \frac{P_a \Omega_c}{0.85 f'_c}\]

where \(\Omega_c = 2.31\) for concrete bearing

Bearing Strength on Concrete

\[P_p = 0.85 f'_c A_1 \sqrt{\frac{A_2}{A_1}} \leq 1.7 f'_c A_1\]

Variables:

• \(A_1\) = bearing area of base plate (in² or mm²)
• \(A_2\) = maximum area of supporting concrete that is geometrically similar to and concentric with loaded area (in² or mm²)

Base Plate Thickness

Required thickness based on cantilever bending:

\[t_{req} = \ell \sqrt{\frac{2 f_p}{0.9 F_y}}\]

Variables:

• \(\ell\) = critical cantilever dimension of plate (in or mm)
• \(f_p\) = bearing pressure on concrete (ksi or MPa)

Bracing and Stability

Lateral Bracing Strength Requirements

Required strength for nodal bracing:

\[P_{br} = 0.004 P_r\]

Variables:

• \(P_r\) = required compressive strength of member being braced

Relative Bracing Stiffness

Required stiffness for nodal bracing of columns:

\[\beta_{br} = \frac{2 P_r}{L_b \phi}\]

Variables:

• \(L_b\) = length between braces (in or mm)
• \(\phi\) = stability reduction factor

Fatigue

Fatigue Stress Range

Allowable stress range for constant amplitude loading:

\[F_{sr} = \left(\frac{C_f}{N}\right)^{1/3}\]

Variables:

• \(C_f\) = constant from AISC Table (depends on detail category A through E')
• \(N\) = number of stress range cycles

Detail categories and constants (ksi³):

• Category A: \(C_f = 250 \times 10^8\)
• Category B: \(C_f = 120 \times 10^8\)
• Category C: \(C_f = 44 \times 10^8\)
• Category D: \(C_f = 22 \times 10^8\)
• Category E: \(C_f = 11 \times 10^8\)
• Category E': \(C_f = 3.9 \times 10^8\)

Constant Amplitude Fatigue Threshold

Threshold stress range (for N > 2 million cycles, infinite life):

• Category A: \(F_{TH} = 24\) ksi
• Category B: \(F_{TH} = 16\) ksi
• Category C: \(F_{TH} = 10\) ksi
• Category D: \(F_{TH} = 7\) ksi
• Category E: \(F_{TH} = 4.5\) ksi
• Category E': \(F_{TH} = 2.6\) ksi

Plastic Design

Plastic Hinge Formation

Required plastic moment capacity:

\[M_p = Z F_y\]

Shape factor:

\[f = \frac{Z}{S}\]

Variables:

• \(f\) = shape factor (typically 1.10-1.15 for I-shapes)
• \(Z\) = plastic section modulus (in³ or mm³)
• \(S\) = elastic section modulus (in³ or mm³)

Mechanism Analysis

Virtual work equation:

\[\sum W_u \delta = \sum M_p \theta\]

Variables:

• \(W_u\) = factored load (kips or kN)
• \(\delta\) = virtual displacement (in or mm)
• \(M_p\) = plastic moment (kip-in or kN-m)
• \(\theta\) = plastic rotation (radians)

Ponding

Ponding Flexibility Constants

Primary flexibility constant:

\[C_p = \frac{32 L_s^4}{10^7 I_p}\]

Secondary flexibility constant:

\[C_s = \frac{32 S^4}{10^7 I_s}\]

Variables:

• \(L_s\) = column spacing parallel to primary member (ft)
• \(S\) = column spacing perpendicular to primary member (ft)
• \(I_p\) = moment of inertia of primary member (in⁴)
• \(I_s\) = moment of inertia of secondary member (in⁴)

Ponding Stability Check

\[0.25 + 0.9 C_p C_s \leq 1.0\]

If this inequality is not satisfied, the roof system is unstable under ponding conditions.

Torsion

Uniform Torsion (St. Venant Torsion)

Shear stress due to uniform torsion:

\[\tau = \frac{T t}{2 A_o}\]

Variables:

• \(T\) = applied torque (kip-in or N-mm)
• \(t\) = thickness of element (in or mm)
• \(A_o\) = area enclosed by centerline of walls (in² or mm²)

Warping Torsion

Normal stress due to warping:

\[\sigma_w = \frac{E W_n}{L^2} \frac{d\theta}{dx}\]

Variables:

• \(W_n\) = normalized warping function (in⁴ or mm⁴)
• \(\theta\) = angle of twist (radians)

Material Properties

Common Steel Grades

ASTM A36:

• \(F_y = 36\) ksi (250 MPa)
• \(F_u = 58\) ksi (400 MPa)

ASTM A992 (W-shapes):

• \(F_y = 50\) ksi (345 MPa)
• \(F_u = 65\) ksi (450 MPa)

ASTM A572 Grade 50:

• \(F_y = 50\) ksi (345 MPa)
• \(F_u = 65\) ksi (450 MPa)

ASTM A500 Grade B (HSS):

• Round: \(F_y = 42\) ksi (290 MPa), \(F_u = 58\) ksi (400 MPa)
• Rectangular: \(F_y = 46\) ksi (315 MPa), \(F_u = 58\) ksi (400 MPa)

ASTM A500 Grade C (HSS):

• Round: \(F_y = 46\) ksi (315 MPa), \(F_u = 62\) ksi (425 MPa)
• Rectangular: \(F_y = 50\) ksi (345 MPa), \(F_u = 62\) ksi (425 MPa)

Elastic Properties

• Modulus of elasticity: \(E = 29,000\) ksi (200,000 MPa)
• Shear modulus: \(G = 11,200\) ksi (77,200 MPa)
• Poisson's ratio: \(\nu = 0.3\)
• Coefficient of thermal expansion: \(\alpha = 6.5 \times 10^{-6}\) per °F (11.7 × 10^-6 per °C)

Serviceability

Camber

Deflection due to uniform load on simply supported beam:

\[\Delta = \frac{5 w L^4}{384 E I}\]

Variables:

• \(w\) = uniform load (kip/ft or N/m)
• \(L\) = span length (ft or m)

Vibration

Natural frequency of simply supported beam:

\[f_n = \frac{\pi}{2L^2} \sqrt{\frac{EI g}{w}}\]

Variables:

• \(g\) = acceleration due to gravity = 386 in/s² (9810 mm/s²)
• \(w\) = weight per unit length (kip/in or N/mm)