Chapter 11 · Section 5

Stability of Compound Columns

Compound columns consist of chords + lacings (or batten plates) — they offer high axial rigidity and low self-weight, widely used in steel truss bridges, industrial columns, cranes, towers, etc.. However, when they buckle about the virtual axis, shear deformation significantly reduces the critical load. In design, the equivalent slenderness ratio $\lambda_{0}$ replaces $\lambda$ in Euler's formula.

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11.5.1

Classification · Real & virtual axes

Classification

Two common forms of compound columns — laced and battened. Chords (and sometimes more than two) are connected by lacings (pin-jointed, truss-like) or batten plates (rigidly jointed, frame-like). This section focuses on two-chord compound columns.

Anim. 11.5.1-a

Laced Laced / Spread

Anim. 11.5.1-b
  • Chords and diagonal / transverse lacings are pin-connected — the column acts like an open-web truss;
  • lacings carry mostly axial force;
  • low shear deformation, so higher critical load.

Battened Battened

Anim. 11.5.1-c
  • Chords are rigidly connected to batten plates — the column acts like a closed frame;
  • battens carry bending + shear;
  • more shear deformation, so lower critical load.
Real axis and virtual axis

In a two-chord compound section (see Animation 11.5.1-c): the $y$-$y$ axis passing through the chord material is the real axis; the $x$-$x$ axis passing through the "hollow center" is the virtual axis.

Buckling about the real axis (in the $x$-$z$ plane) — same as a solid column, use Euler's formula;
Buckling about the virtual axis (in the $y$-$z$ plane) — despite a large moment of inertia for the gross section, overall shear deformation is large, and the critical load is substantially reduced. This is the central issue of compound-column stability.

11.5.2

Effect of shear deformation on critical load

Shear Deformation

When an axially compressed column buckles, in addition to axial force and bending moment, there is also a shear force. The associated shear deformation adds to the lateral deflection and thus reduces the critical load.

Anim. 11.5.2-a

① Decomposition of total deflection

Let $y_{1}$ be the bending-deflection contribution and $y_{2}$ the additional shear-deformation contribution; the actual deflection is

$$ y \;=\; y_{1} \;+\; y_{2} $$
(11.5-1)

The shear-deformation-induced additional rotation on a segment is $\mathrm d y_{2}/\mathrm d x$ — equal to the average shear angle $\gamma_{0}$:

$$ \gamma_{0} \;=\; k\,\dfrac{F_{\mathrm Q}}{GA} \qquad \Rightarrow \qquad \dfrac{\mathrm d y_{2}}{\mathrm d x} \;=\; \dfrac{k}{GA}\cdot \dfrac{\mathrm d M}{\mathrm d x} $$
(11.5-2)

② Deflection ODE with shear

Total curvature = bending + shear:

$$ \dfrac{\mathrm d^{2} y}{\mathrm d x^{2}} \;=\; -\dfrac{M}{EI} \;+\; \dfrac{k}{GA}\,\dfrac{\mathrm d^{2} M}{\mathrm d x^{2}}$$
(11-19)

For a pinned-pinned column $M = F_{\mathrm P}\, y$, so (11-19) becomes

$$ EI\!\left(1 - \dfrac{k F_{\mathrm P}}{GA}\right) y'' \;+\; F_{\mathrm P}\, y \;=\; 0$$
(11-20)
Same as the no-shear equation except for the factor $(1 - kF_{\mathrm P}/GA)$ multiplying $y''$

③ Critical load

Let $\alpha^{2} = F_{\mathrm P}/\left[EI(1 - kF_{\mathrm P}/GA)\right]$; general solution $y = A\cos\alpha x + B\sin\alpha x$. With pinned boundary conditions $y(0)=y(l)=0$, we get $\sin \alpha l = 0$, smallest positive root $\alpha l = \pi$:

$$ F_{\mathrm{Pcr}} \;=\; \dfrac{\pi^{2} EI/l^{2}}{1 + (\pi^{2} EI/l^{2})\cdot k/GA} \;=\; \dfrac{F_{\mathrm{Pe}}}{1 + F_{\mathrm{Pe}} \cdot k/GA}$$
(11-21)
$F_{\mathrm{Pe}} = \pi^{2} EI/l^{2}$ = Euler's critical load for a pinned-pinned solid column; the parenthetical denominator encodes shear effects
11.5.3

Laced compound column · Unit shear-angle derivation

Laced Column

For a laced compound column, we first derive the shear angle $\bar{\gamma}$ induced by a unit shear force $\bar{F}_{\mathrm Q} = 1$, and then plug $\bar{\gamma}$ into $k/GA$ in Eq. (11-21).

① Pin-jointed truss idealization

Treat the lacings as pin-jointed: under unit shear, diagonal lacings carry axial forces and transverse lacings transmit the shear. By the virtual-work principle:

$$ \delta_{11} \;=\; \sum \dfrac{\bar{F}_{\mathrm N}^{2}\, l}{EA} \qquad (\text{d}) $$
(11.5-3)

② Substitute section parameters

The chord cross-section is much larger than the lacings, so only the lacings' axial deformation matters. Let one pair of diagonal lacings have area $A_{1x}$, one pair of transverse lacings area $A_{2x}$, diagonal inclination $\alpha$, panel length $d$:

$$ \delta_{11} \;=\; \dfrac{d}{E}\!\left(\dfrac{1}{A_{1x}\, \sin\alpha \cos^{2}\alpha} \;+\; \dfrac{1}{A_{2x}\, \tan\alpha}\right) $$
(11.5-4)

Substituting yields the shear angle per unit shear:

$$ \bar{\gamma} \;=\; \dfrac{1}{E}\!\left(\dfrac{1}{A_{1x}\, \sin\alpha \cos^{2}\alpha} \;+\; \dfrac{1}{A_{2x}\, \tan\alpha}\right) \qquad (\text{e}) $$
(11.5-5)

③ Plug into (11-21) to get the critical load

$$ F_{\mathrm{Pcr}} \;=\; \dfrac{F_{\mathrm{Pe}}}{1 \;+\; \dfrac{F_{\mathrm{Pe}}}{E}\!\left(\dfrac{1}{A_{1x} \sin\alpha \cos^{2}\alpha} + \dfrac{1}{A_{2x} \tan\alpha}\right)}$$
(11-22)
11.5.4

Laced · Equivalent slenderness

Equivalent Slenderness

Using $I_{x} = A\, i_{x}^{2}$, $\lambda_{x} = l/i_{x}$, Eq. (11-22) rearranges to:

$$ F_{\mathrm{Pcr}} \;=\; \dfrac{F_{\mathrm{Pe}}}{1 \;+\; \dfrac{\pi^{2}}{\lambda_{x}^{2}}\!\left(\dfrac{A}{A_{1x}}\cdot \dfrac{1}{\sin\alpha \cos^{2}\alpha} \;+\; \dfrac{A}{A_{2x}}\cdot \dfrac{1}{\tan\alpha}\right)}$$
(11-23)
Anim. 11.5.4-a

Approximation

The transverse lacings' deformation is usually much smaller than that of the diagonal lacings and can be neglected:

$$ F_{\mathrm{Pcr}} \;=\; \dfrac{F_{\mathrm{Pe}}}{1 \;+\; \dfrac{\pi^{2}}{\lambda_{x}^{2}}\cdot \dfrac{A}{A_{1x}}\cdot \dfrac{1}{\sin\alpha \cos^{2}\alpha}}$$
(11-24)

In practice the diagonal inclination $\alpha \in 40^{\circ} \sim 70^{\circ}$, so

$$ \dfrac{\pi^{2}}{\sin\alpha\, \cos^{2}\alpha} \;\approx\; 27 $$
(11.5-6)

Substituting:

$$ F_{\mathrm{Pcr}} \;=\; \dfrac{F_{\mathrm{Pe}}}{1 + \dfrac{27\, A}{\lambda_{x}^{2}\, A_{1x}}} \;=\; \dfrac{\pi^{2} EI}{(\mu l)^{2}}$$
(11-25)
$$ \mu \;=\; \sqrt{\,1 \;+\; \dfrac{27\, A}{\lambda_{x}^{2}\, A_{1x}}\,}$$
(11-26)
Effective-length factor of a pinned-pinned laced column about the virtual axis
$$ \boxed{\;\lambda_{0} \;=\; \mu\, \lambda_{x} \;=\; \sqrt{\,\lambda_{x}^{2} \;+\; 27\,\dfrac{A}{A_{1x}}\,}\;}$$
(11-27)
The equivalent slenderness formula for laced two-chord compound columns used in steel-design codes
Engineering meaning

In design, $\lambda_{0}$ replaces $\lambda$ for a solid column in Euler's formula or the empirical curves; this yields the critical stress and hence the capacity — "treat the laced compound column as a solid one with slenderness $\lambda_{0}$".

11.5.5

Battened compound column · Equivalent slenderness

Battened Column

A battened two-chord compound column can be idealized as a single-span multi-story frame — the bending deformation of the chords under shear produces inflection points at the midpoints between adjacent joints. Under a unit shear $\bar{F}_{\mathrm Q} = 1$ (see Animation 11.5.5-a), the chord's top/bottom moments vanish and the shear is split evenly between the two chords.

Anim. 11.5.5-a

① Unit shear angle (moment-diagram multiplication)

Drawing the unit moment diagram of and using the graphical-integration (moment-area) method:

$$ \delta_{11} \;=\; \sum\!\int\! \dfrac{M^{2}}{EI}\,\mathrm d s \;=\; \dfrac{d^{3}}{24\, EI_{1}} \;+\; \dfrac{b\, d^{2}}{12\, EI_{\mathrm h}} \qquad (\text{f}) $$
(11.5-7)
$I_{1}$ = inertia of a single chord about its own centroidal axis; $I_{\mathrm h}$ = total inertia of a pair of battens
$$ \bar{\gamma} \;=\; \dfrac{d^{2}}{24\, EI_{1}} \;+\; \dfrac{b\, d}{12\, EI_{\mathrm h}} \qquad (\text{g}) $$
(11.5-8)

② Critical load

Plug $\bar{\gamma}$ into Eq. (11-21):

$$ F_{\mathrm{Pcr}} \;=\; \dfrac{F_{\mathrm{Pe}}}{1 \;+\; F_{\mathrm{Pe}}\!\left(\dfrac{d^{2}}{24\, EI_{1}} + \dfrac{b\, d}{12\, EI_{\mathrm h}}\right)}$$
(11-28)
First term = chord deformation; second term = batten deformation

Battens are usually much stiffer than chords in bending, so neglect batten deformation:

$$ F_{\mathrm{Pcr}} \;\approx\; \dfrac{F_{\mathrm{Pe}}}{1 \;+\; \dfrac{\pi^{2} d^{2}\, I_{x}}{24\, l^{2}\, I_{1}}}$$
(11-29)

③ Equivalent slenderness

Using $I_{x} = A\, i_{x}^{2}$, $I_{1} = \tfrac{1}{2} A\, i_{1}^{2}$, $\lambda_{x} = l/i_{x}$, $\lambda_{1} = d/i_{1}$ and approximating the coefficient $0.82 \approx 1$:

$$ F_{\mathrm{Pcr}} \;=\; \dfrac{\lambda_{x}^{2}}{\lambda_{x}^{2} + \lambda_{1}^{2}}\, F_{\mathrm{Pe}}$$
(11-31)
$$ \mu \;=\; \sqrt{\dfrac{\lambda_{x}^{2} + \lambda_{1}^{2}}{\lambda_{x}^{2}}}$$
(11-32)
$$ \boxed{\;\lambda_{0} \;=\; \mu\, \lambda_{x} \;=\; \sqrt{\,\lambda_{x}^{2} + \lambda_{1}^{2}\,}\;}$$
(11-33)
Equivalent-slenderness formula for battened two-chord compound columns used in steel-design codes
Section summary
  1. Compound columns are either laced (truss-like) or battened (frame-like);
  2. Buckling about the real axis follows solid-column rules; buckling about the virtual axis must include shear effects;
  3. With shear correction $F_{\mathrm{Pcr}} = F_{\mathrm{Pe}} / (1 + F_{\mathrm{Pe}}\, \bar{\gamma})$ — Eq. (11-21);
  4. Laced: $\lambda_{0} = \sqrt{\lambda_{x}^{2} + 27\, A/A_{1x}}$; battened: $\lambda_{0} = \sqrt{\lambda_{x}^{2} + \lambda_{1}^{2}}$;
  5. In design, use $\lambda_{0}$ in place of $\lambda$ in Euler's formula to get the critical capacity.
§11-6 Frame Stability →