Short columns: axial load and bending

Columns carry the floors above them down to the foundations. Almost every column carries some bending as well as axial load, from unbalanced floor spans, frame action under wind, or simply the unavoidable eccentricity of construction. This lesson develops the strength of a short column, one in which the extra moment from lateral deflection is negligible, under combined axial load and bending. The tool for this is the interaction diagram, built by the same strain-compatibility method used for beams in lesson 4. The example is the ground-floor interior column of the office building.

Types of columns

Most concrete columns are tied columns: longitudinal bars held in position by rectangular ties at regular spacing. Spiral columns have their bars arranged in a circle and enclosed by a continuous helix of closely spaced wire or bar. The spiral confines the core concrete so well that after the cover spalls off the column can carry its load at large deformations, which is why spiral columns get a higher resistance factor and a higher maximum axial load. Composite columns encase a structural steel shape in concrete or fill a steel tube with concrete.

Columns are also classed by slenderness. A short column fails by material failure, crushing of the concrete or yielding of the steel, at the load given by its cross-section. A slender column deflects laterally under load, the axial load acting on that deflection adds moment, and the column fails at a lower load. Most building columns braced against sway are short; lesson 11 covers slender ones.

Axial load capacity

Under concentric axial load, the concrete and steel shorten together. The concrete reaches its capacity at a strain of about 0.002, at which point Grade 400 or Grade 60 steel has just yielded, so the strength is the sum of the two contributions:

The factor 0.85 reflects the lower strength of concrete in a column compared with a cylinder, caused by the size, the casting position and the sustained nature of the load. No real column is perfectly concentric, so ACI 318 caps the design axial strength at \(\phi P_{n,max} = 0.80\phi P_o\) for tied columns (with \(\phi = 0.65\)) and \(0.85\phi P_o\) for spiral columns (with \(\phi = 0.75\)). That cap allows for an accidental eccentricity of roughly 10 percent of the column depth.

The steel ratio \(\rho_g = A_{st}/A_g\) must be between 1 and 8 percent. The lower limit gives some resistance to bending and controls creep, which slowly shifts load from the concrete to the steel. The upper limit avoids congestion; in practice 4 percent is a sensible maximum where bars are lap-spliced, because the laps double the steel at the splice. Rectangular and circular tied columns need at least four bars, and spiral columns at least six.

Axial load with bending

When a column carries a moment \(M\) together with an axial load \(P\), the load acts as if it were applied at an eccentricity \(e = M/P\) from the centroid. A small eccentricity puts the whole section in compression and the column fails by crushing. A large eccentricity puts the far face in tension, the far steel yields, and the column behaves more like a beam. Between these lies the balanced condition, in which the concrete reaches its crushing strain at the same moment that the extreme tension steel reaches its yield strain.

For each assumed position of the neutral axis there is one strain profile, one set of bar stresses and one pair of resultants \((P, M)\). Sweeping the neutral axis from far outside the section (pure compression) to near the compression face (tension) traces out a curve: the interaction diagram. Any combination of factored load and moment that falls inside the curve is safe.

Here \(C_c = 0.85f'_c ab\), \(y_i\) is the depth of bar layer \(i\) from the compression face, \(f_{si} = E_s\varepsilon_{si}\) limited to \(\pm f_y\) and taken positive in compression, and the concrete displaced by compression bars inside the stress block is subtracted. Each nominal point \((P_n, M_n)\) is then multiplied by \(\phi\), which depends on the net tensile strain \(\varepsilon_t\) of that point: 0.65 for compression-controlled points, 0.90 for tension-controlled points, and a linear transition between. That variation produces the characteristic kink in the ACI design diagram.

The diagram has a recognizable shape. From the capped top, increasing moment lowers the axial capacity in the compression-controlled region. At the balanced point the moment capacity is a maximum (for the factored diagram of a symmetrical section). Below balanced, in the tension-controlled region, a little axial compression actually increases the moment capacity, because it delays yielding of the tension steel. This last fact matters: a column with low axial load and high moment is unconservative to check with the largest axial load, so designers check the combinations with minimum axial load too, such as 0.9D with wind.

The office column

The ground-floor interior column supports two floors and the roof over a 25 ft × 25 ft tributary area. From the floor and roof loads and the column self-weight, Owen finds service loads of \(D = 380\) kips and \(L = 160\) kips (with the live load reduction ASCE 7 allows for large tributary areas). The frame analysis of CIV-321 gives a factored moment of 90 kip·ft from unbalanced live load on the adjacent spans. The factored axial load is \(P_u = 1.2(380) + 1.6(160) = 712\) kips, rounded up to 720 kips to allow for the weight of finishes added later. Priya has proposed 20 × 20 in. columns with 8 #8 bars (three per face), \(f'_c = 5{,}000\) psi.

Computing a full diagram by hand is tedious, so Sofia extends the strain-compatibility routine of lesson 4 to a net axial force and sweeps the neutral axis depth. She also interpolates the moment capacity at the factored axial load.

# Interaction diagram for a tied column, ACI 318 (design strengths phi*Pn, phi*Mn)
fc, fy, Es, ecu = 5000.0, 60000.0, 29e6, 0.003
b = h = 20.0
layers = [(3 * 0.79, 2.5), (2 * 0.79, 10.0), (3 * 0.79, 17.5)]          # 8 #8
b1 = min(0.85, max(0.65, 0.85 - 0.05 * (fc - 4000) / 1000))
Ag = b * h; Ast = sum(A for A, _ in layers)
Po = 0.85 * fc * (Ag - Ast) + fy * Ast
phiPnmax = 0.80 * 0.65 * Po
ety = fy / Es
dt = max(y for _, y in layers)

def point(c):
    a = min(b1 * c, h)
    Cc = 0.85 * fc * a * b
    P = Cc; M = Cc * (h / 2 - a / 2)
    for As, y in layers:
        es = ecu * (c - y) / c
        fs = max(-fy, min(fy, Es * es))
        f = As * fs - (0.85 * fc * As if y < a else 0)
        P += f; M += f * (h / 2 - y)
    et = ecu * (dt - c) / c
    phi = 0.90 if et >= ety + 0.003 else (0.65 if et <= ety else 0.65 + 0.25 * (et - ety) / 0.003)
    return phi, phi * P / 1000, phi * M / 12000, et

print(f"rho = {Ast/Ag*100:.2f}%, Po = {Po/1000:.0f} kips, phi*Pn,max = {phiPnmax/1000:.0f} kips")
cb = ecu / (ecu + ety) * dt
print("   c (in.)   eps_t     phi   phiPn (kips)  phiMn (kip-ft)")
pts = []
for c in (24, 20, 16, 14, cb, 8, 6, 4.5, 3):
    phi, P, M, et = point(c)
    pts.append((P, M))
    tag = "  balanced" if c == cb else ""
    print(f"{c:8.2f} {et:9.4f} {phi:6.3f} {min(P, phiPnmax/1000):11.0f} {M:12.0f}{tag}")
lo, hi = 0.1, 10.0
for _ in range(60):
    c = (lo + hi) / 2
    lo, hi = (lo, c) if point(c)[1] > 0 else (c, hi)
print(f"pure bending: c = {c:.2f} in., phiMn = {point(c)[2]:.0f} kip-ft")
Pu, Mu = 720, 90
pts.sort()
for (P1, M1), (P2, M2) in zip(pts, pts[1:]):
    if P1 <= Pu <= P2:
        Mcap = M1 + (M2 - M1) * (Pu - P1) / (P2 - P1)
print(f"at Pu = {Pu} kips the diagram gives phiMn = {Mcap:.0f} kip-ft vs Mu = {Mu} -> {'OK' if Mu <= Mcap and Pu <= phiPnmax/1000 else 'not OK'}")

rho = 1.58%, Po = 2052 kips, phi*Pn,max = 1067 kips
   c (in.)   eps_t     phi   phiPn (kips)  phiMn (kip-ft)
   24.00   -0.0008  0.650        1067           70
   20.00   -0.0004  0.650        1027          191
   16.00    0.0003  0.650         810          274
   14.00    0.0008  0.650         692          302
   10.36    0.0021  0.650         454          335  balanced
    8.00    0.0036  0.774         387          371
    6.00    0.0058  0.900         256          375
    4.50    0.0087  0.900         136          314
    3.00    0.0145  0.900           1          234
pure bending: c = 2.99 in., phiMn = 233 kip-ft
at Pu = 720 kips the diagram gives phiMn = 295 kip-ft vs Mu = 90 -> OK

At \(P_u = 720\) kips the column provides 295 kip·ft against a demand of 90 kip·ft, so the section is controlled by axial load rather than moment. Priya considers reducing the bars to 8 #7, but keeps 8 #8 because the same column size continues to the roof, where moments from the roof frame are larger relative to the axial load, and because a uniform bar size simplifies splices.

The table of points also shows why the interaction diagram matters even for a lightly loaded column. Near pure bending the capacity is lowest; adding axial load up to the balanced point increases it. If the roof column carries only a small axial load with a large wind moment, its capacity comes from the lower part of the curve, and that is where it may govern.

Biaxial bending

Corner columns and columns in two-way frames carry moments about both axes. The exact solution needs a three-dimensional interaction surface. A simple, widely used approximation is the Bresler reciprocal load equation, which estimates the strength under biaxial bending from the uniaxial strengths:

where \(P_{rx}\) and \(P_{ry}\) (or \(P_{nx}\), \(P_{ny}\)) are the axial capacities at the actual eccentricity about each axis alone. The equation is accurate when axial load is significant, above about a tenth of the pure axial capacity. For lower loads, the section is checked as a beam in biaxial bending, or a computer program integrates the stresses over the section for an inclined neutral axis.

Practical column design

• Size columns for axial load first: with 1.5 to 2 percent steel, a tied column resists a factored axial load of roughly \(0.5f'_c A_g\) at its cap, so \(A_g \approx 2P/f'_c\) is a good first guess; then check moments with the interaction diagram.
• Keep column sizes constant over several floors and vary the steel and concrete strength, so formwork can be reused. Higher-strength concrete in lower stories is common.
• Columns from 300 mm (12 in.) upward are common; very small columns are hard to cast and congested at beam-column joints.
• Symmetrical reinforcement is used because moments usually reverse under wind and pattern loads.
• Load combinations with minimum axial load must be checked, because they may govern in the tension-controlled region.