Beam-columns and combined forces

Few members carry pure axial load or pure bending. A column in a moment frame bends as the frame resists wind; a wall column carries roof load and spans between floors to resist wind on the cladding; a truss top chord carries joist loads between panel points. These are beam-columns. Their design combines the column rules of lesson 4 and the beam rules of lessons 5 and 6 in interaction equations, and it must account for the extra moment that the axial load produces as the member deflects. This lesson develops the interaction checks and applies them to a wall column of the maintenance bay.

Second-order moments

Consider a pinned column with an axial load \(P\) and a uniform lateral load that bends it. The lateral load alone gives a first-order moment \(M_o\) and a deflection \(\delta_o\). With the axial load acting on the deflected shape, each section has an additional moment \(P\delta\), which increases the deflection, which increases the moment again. The process converges to an amplified moment:

This member effect, called \(P\)-\(\delta\), is significant only when \(P\) is a noticeable fraction of the Euler load of the member in the plane of bending. The factor \(C_m\) accounts for the shape of the first-order moment diagram. For a member bent in single curvature by end moments, or by a transverse load, the deflected shape adds to the existing moment where it is largest, and \(C_m\) is close to 1.0. For a member bent in reverse curvature by end moments, the maximum first-order moment is at an end, where the deflection is zero, and the amplification is smaller.

A second effect, \(P\)-\(\Delta\), arises when a whole storey sways: the gravity loads acting on the displaced frame add overturning moment to every column. It belongs to the frame, not to one member, and lesson 11 treats it. A column in a braced frame, such as the wall column in this lesson, has only the \(P\)-\(\delta\) effect.

Interaction of axial load and bending

A short member that is fully yielded under axial load and moment together has an interaction curve between \(P/P_y = 1\) and \(M/M_p = 1\). For a W shape bent about its strong axis, that curve is nearly a straight line from \((0, 1)\) to about \((0.15, 1)\) and then down to \((1, 0)\): a small axial load barely reduces the plastic moment, because the web carries the axial force near the neutral axis. For longer members, buckling under axial load and lateral-torsional buckling under bending reduce both end points, and the second-order amplification bends the curve inward.

S16 replaces these curves with linear interaction equations that are checked three times, for three ways the member can fail. For class 1 and 2 I-shaped members:

The three checks use different resistances. (a) Cross-sectional strength uses the squash load \(C_r = \phi AF_y\) and the full moment resistance; it governs short members. (b) Overall member strength uses the compressive resistance for buckling in the plane of bending and the full moment resistance; it governs members that buckle in the plane of bending. (c) Lateral-torsional buckling strength uses the compressive resistance for the governing axis and the lateral-torsional moment resistance \(M_r\) from lesson 6, with \(U_{1x}\) not less than 1.0. For class 3 sections and other members not covered by the 0.85 coefficient, the coefficient on the moment term is 1.0. Each moment term must also satisfy \(M_f \le M_r\) on its own. The factor \(\beta = 0.6 + 0.4\lambda_y \le 0.85\) applies to weak-axis bending.

The amplification factor U1

The factor \(U_1\) amplifies the first-order moment for the \(P\)-\(\delta\) effect. \(C_e = \pi^2EI/L^2\) is the Euler load in the plane of bending. The factor \(\omega_1\) plays the role of \(C_m\): \(\omega_1 = 0.6 - 0.4\kappa \ge 0.4\) for members with end moments only, where \(\kappa\) is the ratio of the smaller to the larger end moment, positive for double curvature; \(\omega_1 = 1.0\) for members with distributed or concentrated transverse loads between supports. In a braced frame, the moments \(M_f\) from a first-order analysis are amplified by \(U_1\) inside the interaction equation.

The maintenance bay wall column

The end-wall columns of the maintenance bay are 6.0 m high, pinned at the base and at the roof, and the roof and wall bracing prevent sway. Each column supports the end reaction of a roof truss, \(D = 69.3\) kN and \(S = 177.3\) kN, and spans vertically between base and roof to resist wind on the wall cladding, delivered by horizontal girts. The columns are 7.5 m apart and the net specified wind pressure is 1.0 kPa, so the wind load is 7.5 kN/m and its moment is \(7.5(6.0)^2/8 = 33.8\) kN·m. The girts at mid-height brace the column about its weak axis, but under wind suction the inner flange is in compression and is unbraced over the full 6.0 m.

Two load cases compete. Case 3, with snow principal and wind companion, gives the largest axial load: \(C_f = 1.25(69.3) + 1.5(177.3) = 353\) kN with \(M_f = 0.4(33.8) = 13.5\) kN·m. Case 4, with wind principal and snow companion, gives the largest moment: \(C_f = 175\) kN with \(M_f = 1.4(33.8) = 47.2\) kN·m. Both are checked, together with 0.9D + 1.4W.

The program checks all three load combinations, so that no combination is missed when loads change. It also computes the wind deflection.

import math
# Wall column W200x46 (class 3 in bending): axial load from the truss plus wind, CSA S16 clause 13.8
E, G, Fy, phi, n = 200000.0, 77000.0, 345.0, 0.90, 1.34
A, Ix, Sx, Iy, rx, ry, J, Cw = 5860, 45.5e6, 448e3, 15.4e6, 88.1, 51.2, 223e3, 142e9
L, Ly, Lb = 6000.0, 3000.0, 6000.0     # height; weak axis braced by girts; inner flange unbraced

D, S, Wm = 7.7 * 9, 19.7 * 9, 7.5 * 6.0**2 / 8      # truss reactions (kN) and wind moment (kN.m)
cases = {"Case 3: 1.25D + 1.5S + 0.4W": (1.25 * D + 1.5 * S, 0.4 * Wm),
         "Case 4: 1.25D + 1.4W + 0.5S": (1.25 * D + 0.5 * S, 1.4 * Wm),
         "Case 4: 0.9D + 1.4W":         (0.9 * D, 1.4 * Wm)}

def Cr(KL, r):
    lam = KL / r * math.sqrt(Fy / (math.pi**2 * E))
    return phi * A * Fy * (1 + lam**(2 * n))**(-1 / n) / 1e3

My = Sx * Fy / 1e6
Mu = 1.13 * math.pi / Lb * math.sqrt(E * Iy * G * J + (math.pi * E / Lb)**2 * Iy * Cw) / 1e6
Mr_ltb = min(1.15 * phi * My * (1 - 0.28 * My / Mu), phi * My) if Mu > 0.67 * My else phi * Mu
Mr_cs = phi * My
Cr0, Crx, Cry = phi * A * Fy / 1e3, Cr(L, rx), Cr(Ly, ry)
Cr_c = min(Crx, Cry)
Cex = math.pi**2 * E * Ix / L**2 / 1e3
print(f"Cr: section {Cr0:.0f}, x-axis {Crx:.0f}, y-axis {Cry:.0f} kN; Mr: section {Mr_cs:.1f}, LTB {Mr_ltb:.1f} kN.m (Mu = {Mu:.0f}); Cex = {Cex:.0f} kN")

for name, (Cf, Mf) in cases.items():
    U1 = 1.0 / (1 - Cf / Cex)                      # omega1 = 1.0 for the distributed wind load
    a = Cf / Cr0 + U1 * Mf / Mr_cs
    b = Cf / Crx + U1 * Mf / Mr_cs
    c = Cf / Cr_c + max(U1, 1.0) * Mf / Mr_ltb
    print(f"{name}: Cf = {Cf:5.1f} kN, Mf = {Mf:4.1f} kN.m, U1 = {U1:.3f} -> "
          f"(a) {a:.2f}  (b) {b:.2f}  (c) {c:.2f}")

# Wind deflection with the specified wind load (7.5 kN/m)
d = 5 * 7.5 * L**4 / (384 * E * Ix)
print(f"Wind deflection = {d:.1f} mm = L/{L/d:.0f}")

Cr: section 1820, x-axis 1196, y-axis 1342 kN; Mr: section 139.1, LTB 117.8 kN.m (Mu = 164); Cex = 2495 kN
Case 3: 1.25D + 1.5S + 0.4W: Cf = 352.6 kN, Mf = 13.5 kN.m, U1 = 1.165 -> (a) 0.31  (b) 0.41  (c) 0.43
Case 4: 1.25D + 1.4W + 0.5S: Cf = 175.3 kN, Mf = 47.2 kN.m, U1 = 1.076 -> (a) 0.46  (b) 0.51  (c) 0.58
Case 4: 0.9D + 1.4W: Cf =  62.4 kN, Mf = 47.2 kN.m, U1 = 1.026 -> (a) 0.38  (b) 0.40  (c) 0.46
Wind deflection = 13.9 mm = L/431

The column is used at about 60 per cent of its capacity. A lighter W200 would satisfy strength, but the team kept the W200x46 because it matches the mezzanine columns, simplifies purchasing and keeps the wind deflection small, 14 mm or span/430 under the specified wind, which suits the metal cladding.

Other beam-column cases

• Truss chords with loads between panel points: purlins or joists that bear on the top chord between joints bend it locally. The chord is a beam-column, continuous over the panel points, with the local moment amplified by the chord's axial compression.
• Columns with eccentric beam reactions: a beam bearing on a column cap or a seat at some distance from the column axis applies a moment equal to the reaction times the eccentricity. In simple construction a nominal eccentricity, typically the distance to the face of the column plus 100 mm (4 in.), is taken and split between the column lengths above and below.
• Moment frame columns: the moments come from the frame analysis and include sway effects. Lesson 11 shows how the amplification changes.
• Biaxial bending: corner columns and columns with eccentric loads in both directions add a weak-axis term. The weak-axis moment resistance of a W shape is about 1.5 times \(S_y F_y\) and the weak-axis term often adds noticeably to the interaction.

Tension with bending

Members in tension and bending, such as a truss bottom chord carrying a hoist load between panel points, are checked with \(T_f/T_r + M_f/M_r \le 1.0\), and also for lateral-torsional buckling with the tension allowed to reduce the compression in the critical flange. Tension straightens the member, so no amplification is needed.