Buckling of Columns
Stability of slender compression members: the Euler load, effective lengths, eccentric loading, and column design.
Chapter Objectives
- Explain stable, neutral, and unstable equilibrium and the concept of the critical (bifurcation) load.
- Derive the Euler buckling load for pin-ended columns and extend it to other supports via effective length.
- Apply the secant formula to eccentrically loaded columns and the tangent-modulus idea to intermediate columns.
- Outline how design codes treat concentrically and eccentrically loaded columns.
13.1Critical Load
Columns and Buckling
Long, slender members under compression are columns; the sudden lateral deflection they can undergo is buckling — often a dramatic, catastrophic failure mode. Beyond strength and deflection, such members must be checked for stability. The largest axial load a column can carry on the verge of buckling is its critical load \(P_{cr}\).
The Spring-and-Bars Model of Stability
Two rigid bars pinned together with a restoring spring (stiffness k) at the joint capture the essence of buckling. Displace the joint slightly: the spring's restoring force fights the load's disturbing components. Comparing them (small angles, with θ cancelling):
\[ P < \frac{kL}{4}: \text{stable} \qquad P = \frac{kL}{4} = P_{cr}: \text{neutral} \qquad P > \frac{kL}{4}: \text{unstable} \]The transition point on the load–rotation plot is the bifurcation point: at \(P_{cr}\), equilibrium holds for any small displacement — the system neither restores nor diverges. A real column past its critical load similarly requires large deflections to find equilibrium, which structures cannot tolerate.
13.2Ideal Column with Pin Supports
An ideal column is perfectly straight, homogeneous, linear elastic, and loaded through its centroid. Writing the elastic-curve equation for the slightly buckled shape, \(EI\,v'' = -Pv\), and solving the resulting harmonic equation with \(v = 0\) at both ends gives nontrivial solutions only at discrete loads \(P = n^2\pi^2EI/L^2\); the smallest (n = 1, a single half-sine wave \(v = C_1\sin(\pi x/L)\)) is the critical load:
- \(E\) = modulus of elasticity; \(I\) = least moment of inertia of the cross section; \(L\) = unsupported length
- \(P_{cr}\) depends on stiffness, not strength: a high-strength steel column buckles at the same load as a mild-steel one (same E). The buckled amplitude \(C_1\) is indeterminate in this linearized theory.
- The column buckles about the weakest principal axis (think of a meter stick) — so efficient columns balance \(I_x \approx I_y\) and push material outward: tubes are excellent columns.
- \(r\) = smallest radius of gyration; \(L/r\) = the slenderness ratio, the measure of a column's flexibility that classifies it as short, intermediate, or long.
- Valid only while \(\sigma_{cr} \le \sigma_Y\) (the proportional limit): the curve is a hyperbola cut off at yield. For structural steel (σY = 36 ksi) the limit falls at L/r ≈ 89; below it, the material yields before it can buckle elastically and Euler's formula does not apply.
13.3Columns Having Various Types of Supports
For a column fixed at its base and free at its top, the same elastic-curve analysis (now with the nonhomogeneous equation \(v'' + (P/EI)v = (P/EI)\delta\)) yields \(P_{cr} = \pi^2EI/4L^2\) — only one quarter of the pin-ended value. All support cases are unified by the effective length: the distance between the buckled shape's points of zero moment.
- \(L_e = KL\), with the effective-length factor K from the end conditions:
- Pinned–pinned: K = 1 · Fixed–free: K = 2 · Fixed–fixed: K = 0.5 · Fixed–pinned: K = 0.7
- \(KL/r\) = effective slenderness ratio. Fixing both ends quadruples the capacity (1/0.5² = 4); a flagpole-style column has only a quarter of it (1/2² = ¼).
13.4The Secant Formula
Eccentric Loading
Real loads rarely pass exactly through the centroid. An eccentricity \(e\) bends the column from the start — deflection grows continuously with load rather than bifurcating. Solving the elastic-curve equation with end moments \(M' = Pe\) gives the midspan deflection and the maximum moment \(M = Pe\sec\left(\sqrt{P/EI}\,L/2\right)\); superposing axial and bending stress yields:
- \(e\) = eccentricity; \(c\) = distance from the centroidal axis to the extreme compression fiber; \(r\) = radius of gyration about the bending axis; \(L_e\) = effective length
- \(ec/r^2\) = the eccentricity ratio. Setting \(\sigma_{max} = \sigma_Y\) gives the limiting load — but P appears inside the secant, so the equation is solved by trial and error.
- The σ–P relation is nonlinear: superposition of loads does not apply, and a factor of safety must be applied to the load, not the stress.
13.5Inelastic Buckling
Column Classification and the Tangent Modulus
Columns are classified by their failure stress: long columns buckle elastically (Euler); short columns ("posts") simply yield or fracture; intermediate columns — the most common in practice — buckle inelastically, at stresses above the proportional limit. Engesser's fix (1889): at buckling, the material's incremental stiffness is the tangent modulus \(E_t = \Delta\sigma/\Delta\epsilon\), the local slope of the stress–strain curve, so
\[ \sigma_{cr} = \frac{\pi^2 E_t}{(KL/r)^2} \]Since \(E_t < E\) above the proportional limit, the inelastic curve falls below the Euler hyperbola, joining the short-column strength smoothly. Because \(E_t\) itself depends on the (unknown) stress, the equation is solved by trial and error from the stress–strain diagram.
13.6Design of Columns for Concentric Loading
Why Codes Use Empirical Formulas
Real columns are never ideal: crookedness, residual stresses from nonuniform cooling, imperfect supports, and uncertain load placement all reduce capacity. Tests show an "accidental" eccentricity ratio of roughly 0.1–0.6 even for nominally concentric columns. Design codes therefore curve-fit large test databases with empirical formulas by slenderness range:
Steel (AISC pattern): an inelastic (parabolic/exponential) formula for intermediate slenderness, switching to an Euler-based formula with a fixed factor of safety beyond a limiting KL/r; a maximum slenderness (typically 200) is imposed.
Aluminum (AA pattern): three ranges — constant allowable stress for short columns, a straight line for intermediate, Euler-based for long.
Timber (NFPA pattern): similar range-based formulas, with slenderness expressed as KL/d (d = least cross-section dimension) and capped near 50.
Whatever the code, the workflow is the same: compute the governing slenderness about the weak axis, pick the formula for that range, and obtain σ_allow.
13.7Design of Columns for Eccentric Loading
Two Code Approaches
Allowable stress method: require the peak combined stress to stay within the column's allowable value,
\[ \frac{P}{A} + \frac{Mc}{I} \le \sigma_{allow} \]— conservative and simple, with σ_allow usually taken from the concentric-column formula at the governing slenderness.
Interaction method: recognize that axial and bending capacity trade off, and require the sum of the utilization ratios not to exceed unity:
\[ \frac{\sigma_a}{(\sigma_a)_{allow}} + \frac{\sigma_b}{(\sigma_b)_{allow}} \le 1 \]where \(\sigma_a = P/A\) is checked against the column (buckling) allowable and \(\sigma_b = Mc/I\) against the bending allowable. The interaction format is the basis of most modern code checks for beam-columns; codes refine it with moment-amplification factors.
Physical meaning
Buckling is a stability loss, not an average-stress calculation. A tiny lateral displacement creates a bending moment from the compressive force; near the critical load the member has almost no restoring stiffness. The force remains compressive while the resulting curvature and fiber stress can grow rapidly.
Problem-solving strategy
- Draw the member free-body diagram, identify the unsupported segment, and locate the weak principal bending axis.
- Classify the loading as centered or eccentric and convert actual restraints to an effective length only when their rotational restraint is justified.
- Compute both radius of gyration and effective slenderness, then decide whether elastic Euler theory is admissible.
- Use the appropriate stability or beam-column relation and separately check yielding, section-local buckling, connection behavior, and serviceability.
- Audit units and sensitivity: capacity changes with the inverse square of effective length, so uncertain restraint is consequential.
Recognition strategy
The cues are a slender member, axial compression, pin joints, and a requested instability threshold. With no eccentricity specified, Euler theory is the appropriate first elastic-stability model, followed by a stress-range check.
Given information
- Take \(K=1.0\), \(L=3.00 \text{m}\), and use the supplied area and second moment about the weaker axis; the desired output is \(P_{cr}\).
Model and interpretation
The free-body has equal and opposite centered end forces. Let lateral displacement be positive in either transverse direction; the circularized supplied inertia is treated as the weak-axis value, and material behavior is linear elastic only to the stated limit.
Verification
Units: Because \(E I/L^2\) has units \(\text{N}\), multiplying by \(\pi^2\) correctly leaves the reported critical load in newtons.
Signs and direction: Compression is represented by negative internal \(N\) in the chosen convention, while \(P_{cr}\) is stated as its positive magnitude; either sideways buckling direction is equally possible in this ideal model.
Plausibility: The calculated \(154 \text{MPa}\) is less than \(180 \text{MPa}\), so elastic behavior is plausible; doubling length would reduce the Euler load to one quarter, as the inverse-square relation requires.
Engineering interpretation
This is the ideal onset of lateral instability, not an allowable mast load. Imperfections, wind-induced bending, base flexibility, local tube buckling, and required code load factors can all reduce the usable resistance.
Recognition strategy
The end restraints change the buckled inflection spacing, so the effective-length form of Euler’s equation is required. The requested assessment means the computed critical stress must be compared with the supplied elastic range rather than accepted automatically.
Given information
- Use \(L=5.00 \text{m}\), \(K=0.70\), \(I=8.00\times10^{-6} \text{m}^4\), \(A=0.00320 \text{m}^2\), and the stated proportional limit.
Model and interpretation
The member free-body is concentrically compressed and the weak axis controls. The value \(K=0.70\) is an assumed restraint model; no claim is made that the joint is perfectly fixed in an installed frame.
Verification
Units: The expression \(E I/L_e^2\) reduces to \(\text{N}\), and division by area gives \(\text{Pa}\); \(403 \text{MPa}\) is dimensionally consistent.
Signs and direction: The load magnitude is positive although the internal axial resultant is compressive. A negative sign would only identify compression in a section convention, not a negative capacity.
Plausibility: Since \(403 \text{MPa}>300 \text{MPa}\), the member enters inelastic response before the Euler bifurcation; an elastic Euler value that exceeds the material range is not an admissible resistance.
Engineering interpretation
The result is useful precisely because it rejects a method: inelastic column provisions and the governing design standard are needed. Yielding is material failure, whereas buckling is instability; this strut may experience both interacting mechanisms.
Recognition strategy
A known eccentricity creates first-order bending, and the question asks for peak fiber stress rather than a bifurcation load. The secant formula captures elastic moment amplification for a prismatic, initially straight column with the stated effective length.
Given information
- Use \(P=180 \text{kN}\), \(e=8.0 \text{mm}\), and convert all geometry to metres before evaluating the dimensionless secant argument.
Model and interpretation
The axial free-body gives a compressive resultant \(N=-P\); eccentricity produces moment \(Pe\). The reported stress is a positive compressive magnitude at the extreme fiber, assuming linear elastic material and no local yielding.
Verification
Units: The factor inside brackets is dimensionless, so (P/A) supplies stress units; the final \(77.3 \text{MPa}\) follows directly from \(\text{N}/\text{m}^2\).
Signs and direction: The result is a compression magnitude at the fiber on the eccentric-load side. Reversing the eccentricity swaps the critical fiber but does not change this magnitude in the symmetric idealization.
Plausibility: The average stress is \(60.0 \text{MPa}\); amplification to \(77.3 \text{MPa}\) is credible and exceeds the first-order value, as expected. Approaching the secant singularity would signal loss of applicability.
Engineering interpretation
Even a small installation offset raises peak stress by about \(29\%\) here. A real frame also requires second-order analysis, connection and foundation stiffness, residual-stress effects, local-section checks, and code-prescribed interaction verification.
Chapter recap
- Identify instability from a compression-member free-body and use the weak-axis stiffness.
- Choose \(KL/r\) only after defending the end restraint, then check whether elastic Euler theory is admissible.
- For eccentric compression, distinguish average stress from amplified extreme-fiber stress and recognize the limits of the secant model.
- Before moving on: verify units, material range, local and global instability modes, and code-based design requirements.