## Abstract

A variational model that describes the interactive buckling of a thin-walled equal-leg angle strut under pure axial compression is presented. A formulation combining the Rayleigh–Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Solving the equations using numerical continuation reveals progressive cellular buckling (or snaking) arising from the nonlinear interaction between the weak-axis flexural buckling mode and the strong-axis flexural–torsional buckling mode for the first time—the resulting behaviour being highly unstable. Physical experiments conducted on 10 cold-formed steel specimens are presented and the results show good agreement with the variational model.

## 1. Introduction

Thin-walled metallic structural components are well known to suffer from a variety of different elastic instability phenomena due to the combination of local and global properties [1–5]. One of the most classic problems is perhaps the buckling of struts or columns [6], namely Euler buckling when flexure about the weak axis of bending of the cross section occurs once the corresponding theoretical critical buckling load is reached. In this work, a strut under axial compression made from a linearly elastic material with a mono-symmetric equal-leg angle section is studied using an analytical approach based on variational principles. Owing to the limited resistance to warping, equal-leg angle sections are highly susceptible to torsional buckling where the cross section rotates about its shear centre. Allen & Bulson [7] showed that for this type of section, torsional buckling is always coupled with strong-axis flexural buckling whereas weak-axis flexural (Euler) buckling may still occur as an independent mode. Moreover, since equal-leg angle sections are effectively made from two identical rectangular plate elements, an alternative approach to describe the torsional component of the flexural–torsional buckling mode is to consider buckling of the individual plate elements that are pin-jointed at the common longitudinal edge, whereas the opposite edges remain free [7–9]—a point that is discussed in detail later.

In recent decades, the buckling phenomena of angle sections have been studied extensively by various researchers through experimental and numerical approaches [10–13]. Popovic *et al.* [14] conducted a series of compression tests on equal-leg angle struts made from cold-formed steel and found that a number of test specimens underwent interaction between Euler buckling and the strong-axis flexural–torsional buckling modes. Dinis *et al.* [9] also found this type of mode interaction using a combination of generalized beam theory and finite-element analysis; other works that revealed mode interaction in angle sections include [15,16]. It is well-known that when two or more buckling modes occur simultaneously, the post-buckling behaviour can be far more unstable than when they are triggered individually [17–21]. Hitherto, detailed studies on the nonlinear mode interaction between the weak-axis Euler buckling and the strong-axis flexural–torsional buckling in equal-leg angle sections have only been conducted using semi-analytical and numerical methods [22]. As far as the authors are aware, a mathematical model that describes such nonlinear interactive buckling behaviour has not been forthcoming.

‘Cellular buckling’ [23] or ‘snaking’ [24], represented by snapback instabilities on equilibrium paths, showing sequential destabilization and restabilization, and a progressive formation of new peaks and troughs in the deformation profile, has been found in mechanical systems such as in the post-buckling of cylindrical shells [25], stiffened panels [26], thin-walled I-section columns [27] and beams [28]. Encouraged by the findings of Dinis *et al.* [9], where numerical results revealed one snapback on the equilibrium path and the switch from one ‘half-wave’ to three ‘half-waves’ in the torsional buckling profiles, cellular buckling may also be expected to exist in equal-leg angle struts undergoing nonlinear mode interaction. Moreover, the snap-through behaviour of equal-leg angle test specimens exhibiting interactive buckling found in the experiments by Popovic *et al.* [14] is also an indication of the potential existence of cellular buckling phenomena. This article presents the development of a variational model of a perfect equal-leg angle strut. A system of nonlinear ordinary differential equations (ODEs) subject to boundary and integral conditions is derived and solved using the numerical continuation package Auto [29].

For validation purposes, a series of physical tests were conducted on 10 equal-leg angle specimens made from cold-formed steel. The results from the variational model and the experiments compare excellently in terms of the mechanical destabilization and the post-buckling deformation. The full cellular buckling behaviour is captured by the variational model. Moreover, the predicted progressive decrease in the initial periodic torsional buckling wavelength is captured by the experiments. A brief discussion on potential further studies is presented before final conclusions are drawn.

## 2. Variational model

The thin-walled equal-leg angle strut of length *L*, leg width *b* and thickness *t* is made from a linear elastic, homogeneous and isotropic material with Young’s modulus *E* and Poisson’s ratio *ν*. The coordinate systems and cross-section properties are shown in figure 1*a*. Note that the origins of both coordinate systems are located at the shear centre of the cross section *b*≫*t*, where *i*={1,2} throughout the current article, as shown in figure 1*a*. The strut is loaded by an axial force *P* that is applied at the centroid of the cross section, as shown in figure 1*b*, and it is assumed that the force is transferred uniformly across the entire cross section. The strut is simply supported by ‘cylindrically pinned’ end conditions where the cross section is free to rotate about the weak-axis but restricted about the strong axis, at both ends of the strut.

This study focuses on the nonlinear mode interaction between weak-axis flexural buckling and flexural–torsional buckling (strong-axis flexure). Figure 1*c* shows the decomposition of the interactive buckling mode where the strut exhibits a weak-axis flexural (global) displacement, a strong-axis flexural (global) displacement and a torsional displacement. It is worth noting that Euler buckling of the strut is represented by the weak-axis flexural displacement alone, whereas the strong-axis flexural and torsional displacements in combination represent the flexural–torsional buckling mode. An alternative description of the torsional displacement is to consider each leg as an individual buckled plate element with the common longitudinal edge simply supported and the opposite edges being free [7]. For the current type of rectangular plate it was shown in [30] that the buckling eigenmode has a linear distribution of displacement across the width of the plate. Since the two plate elements are identical, buckling of both plates is triggered simultaneously, and hence the plate buckling displacement profiles essentially resemble the rotation of the entire cross section about its shear centre. Moreover, for a cruciform (+) cross section it was shown in [7] that the critical buckling load determined from plate buckling theory is more accurate than that obtained from torsional buckling theory. Therefore, the current work considers the torsional component of the flexural–torsional buckling mode as a result of the plate elements buckling that represent the angle legs. For the purposes of clarity, the corresponding displacement profile, as shown by figure 1*c*(iii), is referred to hereafter as the plate buckling displacement, instead of the torsional displacement.

Timoshenko beam theory is assumed, implying that flexural shear strains are included [27,28]. Two generalized coordinates, *q*_{sw} and *q*_{tw}, defined as the amplitudes of the degrees of freedom known as ‘sway’ and ‘tilt’, are introduced to describe the weak-axis global mode [2], as shown in figure 1*d*. Similarly, two generalized coordinates *q*_{ss} and *q*_{ts} are also introduced to describe the strong-axis flexural component of the flexural–torsional mode. Note that the cross-section rotation about the strong axis is restricted at both ends of the strut. Also note that, throughout the current article, the second subscript ‘w’ or ‘s’ corresponds to weak- or strong-axis flexure, respectively. Thus, the lateral displacements *W*_{w} and *W*_{s}, and the cross-section rotations *θ*_{w} and *θ*_{s} are given by the following expressions:
*q*_{sw}=*q*_{tw} and *q*_{ss}=*q*_{ts}.

For the modal description of individual plate buckling, representing the torsional component of the flexural–torsional mode, lateral displacement functions *w*^{ⓘ}(*x*_{i},*z*) are introduced for legs *i*, as shown in figure 1*c*, where encircled superscripts ① and ② correspond to the angle legs in the *x*_{1} and *x*_{2} directions, respectively. Note that a linear distribution of displacement is assumed in both *x*_{i} directions. A pair of in-plane displacements *u*^{ⓘ}(*x*_{i},*z*) is also introduced with transverse distributions being linear in *x*_{i}, as shown in figure 2. This assumption is in fact a further consequence of using Timoshenko beam theory where plane sections are assumed to remain plane. Note that figure 2 represents a case where the in-plane displacement from individual plate buckling is not restricted at either end of the strut. However, real physical boundaries are often associated with rigid plates that are connected to the strut ends, which of course implies that *u*^{ⓘ}(−*b*,0)=0. Nevertheless, such conditions do not affect the current definition of the in-plane displacement functions. Furthermore, a rigid rotation of the cross section implies that *w*^{①}=*w*^{②} when *x*_{1}=*x*_{2}, hence the out-of-plane displacements at the free edges of both legs *w*^{①}(−*b*,*z*) and *w*^{②}(−*b*,*z*) are identical, and are therefore replaced by a single displacement function *w*(*z*). The same condition applies for *u*^{ⓘ}, thus *w*^{ⓘ}(*x*_{i},*z*)=−*x*_{i}*w*(*z*)/*b* and *u*^{ⓘ}(*x*_{i},*z*)=−*x*_{i}*u*(*z*)/*b*. The transverse in-plane displacements *v*^{ⓘ}(*x*_{i},*z*) are assumed to be small and are hence neglected for the current case; this reflects the findings from Koiter & Pignataro [31] for rectangular plates with one longitudinal edge being pinned and one being free.

### (a) Total potential energy

The total potential energy *V* is determined with the main contributions being the total plate bending energy *U*_{b}, the membrane energy *U*_{m} and the work done by the load *i* are determined thus:
*D*=*Et*^{3}/[12(1−*ν*^{2})] being the individual plate flexural rigidity. Integration yields identical expressions for *U*^{①}_{b} and *U*^{②}_{b} and so:
*z*. The membrane strain energy (*U*_{m}) comprises the contributions from the direct strains (*U*_{d}) and the shear strains (*U*_{s}) in both legs thus:
*ε* and *γ* are the direct and shear strains, respectively; the shear modulus *G* is given by *E*/[2(1+*ν*)] for a homogeneous and isotropic material. The transverse components of direct strain *ε*_{z} has to be modelled separately for angle legs *i*. The in-plane displacements from the tilt components of the weak-axis global mode and the strong-axis flexural component of the flexural–torsional mode are given by *a* and *b*, respectively; hence:
*ε*_{zw} and *ε*_{zs} are the contributions from the global flexural displacements and, therefore, correspond to the components along the principal axes. The direct strain components in leg 1 (*x*_{1}*z* plane) and leg 2 (*x*_{2}*z* plane) can be determined. Considering arbitrary points *n*^{①}(*x*_{1n},0) and *n*^{②}(0,*x*_{2n}) on leg 1 and 2, respectively, the coordinates of the corresponding projections on axes *n*^{ⓘ}_{w} and *n*^{ⓘ}_{s}) can be determined from the geometry, as shown in figure 3*c*, thus *c*, the total direct strains in legs *i*, denoted as

where *ε*^{ⓘ}_{zw} and *ε*^{ⓘ}_{zs} are the direct strain components from weak- and strong-axis bending, respectively. The direct strain energy *U*_{d}, assuming that *b*≫*t*, is thus:
*x*_{1}*z* plane) are given by *x*_{2}*z* plane) are given by *i*, including the contributions from the plate buckling displacements are given by
*P* is given by *u* and the out-of-plane displacement *w* are considered. The corresponding components in the *u*^{T}(*z*)=*u*(*z*) and *a* and *b*, respectively. Thus, the longitudinal displacement from plate buckling—measured at the centroid where the axial load *P* is applied—is given by

As shown in figure 4*b*, the plate buckling configuration also represents a rigid rotation of the cross section about its shear centre, which causes a lateral displacement *W*_{s,plate} in the *α* and the distance between the shear centre and the centroid in the *w*/*b* and *P* is given by
*q*_{ss} and *q*_{sw} and *q*_{ss} or between *q*_{sw} and *V* is assembled thus:

### (b) Equilibrium equations and critical buckling

The governing equilibrium equations are obtained by performing the calculus of variations on the total potential energy *V* following a well-established procedure that has been detailed in [2]. The integrand of the total potential energy *V* can be expressed as the Lagrangian (*V* , which is denoted as *δV* , is given by
*V* must be stationary, which requires *δV* to vanish for any small change in *w* and *u*. Since *w* and *u*; these comprise a nonlinear fourth-order ODE in terms of *w* and a nonlinear second-order ODE in terms of *u*. Since the equations are nonlinear, they are solved numerically using the continuation package Auto; the system variables need to be rescaled with respect to the non-dimensional spatial coordinate *w*/*L* and *u*/*L*, respectively. The non-dimensional differential equations for *ϕ*=*L*/*b* and dots now represent differentiation with respect to *V* with respect to the generalized coordinates *q*_{sw}, *q*_{tw}, *q*_{ss}, *q*_{ts} and Δ, i.e. ∂*V*/∂*q*_{sw}=∂*V*/∂*q*_{tw}=∂*V*/∂*q*_{ss}=∂*V*/∂*q*_{ts}=∂*V*/∂Δ=0. These essentially provide five integral equations, which, again are presented in non-dimensional form:
*V* :

Recalling figure 2, the formulation is based on the assumption such that the angle leg plates are free to exhibit in-plane displacements at the ends of the strut. However, in engineering practice, structural components in compression are usually connected with cross section, relatively stiff end plates that transfer the axial compression across the entire cross-section, and therefore, the in-plane displacements would be fully restrained at the strut ends. This implies that the term

Linear eigenvalue analysis is conducted to determine the buckling load for the weak-axis global mode by invoking the condition that the Hessian matrix of *V* is singular. Note that the cross terms between the generalized coordinates of weak-axis global buckling (*q*_{sw}, *q*_{tw}) and the strong-axis flexural component (*q*_{ss}, *q*_{ts}) are zero in the total potential energy functional *V* . Hence the buckling loads for weak-axis global buckling *P*^{C}_{w} can be determined by considering the determinant of the 2×2 Hessian matrix thus:
*q*_{sw}=*q*_{tw}=*q*_{ss}=*q*_{ts}=*w*=*u*=0. Hence, the buckling load for weak-axis global buckling is:

It should be emphasized that there is another solution of *P*, namely *q*_{ss} and *q*_{st} being singular. The expression for *P*^{C}_{s}, however, does not represent an actual buckling load and is, in fact, an artefact of the hybrid formulation with plate displacements being modelled as continuous functions without specific generalized coordinates. Hence, the effective quadratic cross-terms between *q*_{ss} and *w* and *u* are introduced in the current system to represent the torsional component of flexural–torsional buckling.

## 3. Physical experiments

### (a) Test specimens

A series of physical experiments was conducted on equal-leg angle struts fabricated from thin-walled cold-formed steel sheets. Some key material properties provided by the manufacturer were the Young’s modulus *E*=206 kN mm^{−2}, the yield stress *σ*_{y}=235 N mm^{−2} and Poisson’s ratio *ν*=0.3. The cross-section properties were chosen to be 64 and 2 mm for the angle leg width *b* and thickness *t*, respectively. The chosen strut lengths ranged approximately from 1400 to 1800 mm with a 100 mm increment. Two identical specimens were fabricated for each length accounting for a total of 10 specimens. The precise geometry of the cross section includes the corner radii *r* of approximately 4 mm and therefore the flat width of each leg is given by *b*_{f}=*b*_{t}−*r*, where *b*_{t} is the total width, as shown in figure 5*a*. For each specimen, the leg width *b*_{t} was measured at the mid-span of both legs. An average value was then calculated for the twin specimens and summarized in table 1. The corresponding rounded-up values for the flat width *b*_{f} and the average strut length of each set of the twin specimens are also summarized in table 1 and later used in the variational model for validation.

### (b) Test set-up

The experimental set-up shown in figure 5 was used for all test specimens. Square end plates of thickness 10 mm were welded on to both ends of each specimen where the principal axes of the cross section coincided with the lines joining the midpoints of opposite edges of the end plates. The component was then bolted onto specially designed hinge assemblies that resemble the cylindrically pinned end conditions, as shown in figure 5*b*–*d*. The overall assembly ensured that the axial force was approximately applied at the centroid of the cross section and the applied stress was uniformly distributed across the cross section. The loading was applied with a hand-operated hydraulic jack with displacement control and measured by a load cell with an output range of 100 kN, which was connected to a data logger with a maximum recording frequency of 100 Hz. Five linear variable differential transformers (LVDTs), labelled as *T*1, *T*2, *B*1, *B*2 and *M*1, were used to measure the rotation of the end cross sections and the weak-axis global buckling displacement at the mid-span of the strut; two cable-extension transducers, labelled as *L*1 and *L*2, were used to measure the individual plate displacements and the strong-axis flexural displacement at the strut mid-span, as shown in figure 5.

## 4. Results and validation

### (a) Numerical example from the variational model

The full system of the equilibrium equations (2.16) subject to integral equations (2.17) and boundary conditions (2.18) and (2.19) were solved numerically using the continuation and bifurcation package Auto-07p [29] that has been shown to be an ideal tool for the current type of mechanical system [2,27,28]. An example set of cross-section and material properties was chosen as the following: angle leg width *b*=60 mm, thickness *t*=2 mm, total cross-section area *A*=240 mm^{2}, Young’s modulus *E*=206 kN mm^{−2} and Poisson’s ratio *ν*=0.3. Recall that the weak-axis global buckling load *σ*^{C}_{ft} is determined numerically using Auto. To demonstrate the coupling effect of the strong-axis flexural displacement and the individual plate displacement, a simplified model representing the buckling of a single rectangular plate with the same width, thickness and boundary conditions is also analysed in Auto. An estimate of the plate buckling critical stress *σ*^{C}_{l} can also be obtained using the well-known plate buckling formula *k*_{p}=0.426 for the current plate boundary conditions (three edges simply supported and one edge free), assuming that the plates are relatively long [30]. Table 2 summarizes the buckling loads computed by Auto and calculated using the theoretical formulae for a series of struts with different lengths. It is seen that for *L*=1.8 m and *L*=1.4 m, the buckling stress of weak-axis global buckling *σ*^{C}_{w} is greater than the flexural–torsional buckling stress *σ*^{C}_{ft} so flexural–torsional buckling is critical, whereas for *L*=2.2 m, *σ*^{C}_{l} computed by Auto, it is clear that *σ*^{C}_{l} calculated by the plate buckling formula (approx. 4% lower than the Auto value) provides a better estimate to the flexural–torsional buckling stress for short to intermediate length struts than for long length struts.

#### (i) Flexural–torsional buckling being critical

The strut with length *L*=1.8 m is first chosen as an example, where flexural–torsional buckling is critical. The numerical continuation process was initiated from zero load with the principal parameter *P* being varied. After the first bifurcation point *C* was reached, the post-buckling path was computed using the branch switching facility within the software, whereupon interactive buckling solutions were computed. In this case, a symmetric mode solution (figure 7) was found for the out-of-plane displacement *w*. Another numerical continuation was performed by switching branch from the second bifurcation point *S* on the fundamental path with the corresponding critical stress being approximately 3% higher than *σ*^{C}_{ft}. In this second case, an antisymmetric mode solution was found for *w*.

Figure 6 shows a series of equilibrium diagrams for both the symmetric and antisymmetric cases. Considering first the symmetric case, it is found that the plate buckling displacements, together with the strong-axis flexural displacements are triggered at *p*=0.92. Weak-axis global buckling displacements are also triggered automatically, but a significant increase in the magnitude is only observed after the limit point where a relatively large plate displacement has already developed, as shown in figure 6*a*,*b*,*d*. For the antisymmetric case, interactive buckling is triggered at *p*=0.95, which is immediately followed by an unstable post-buckling path, as opposed to the initial stable post-buckling path exhibited in the symmetric case. It is observed for both cases that *q*_{sw} and *q*_{tw} start to deviate as equilibrium progresses, implying that the shear strain is relatively significant, as shown in figure 6*b*. For *w* being symmetric, the strong-axis flexural displacement amplitudes *q*_{ss} and *q*_{ts} are found to be very small compared to *q*_{sw}, *q*_{tw} and *w*. As loading progresses, the amplitudes of the strong-axis flexural component change from positive to negative and then back to positive, as shown in figures 6*c*,*e*. On the other hand, *q*_{ss} and *q*_{ts} are essentially zero for *w* being antisymmetric, as shown in figure 6*c*.

A sequence of snapbacks is observed on all equilibrium paths shown in figure 6, implying that cellular buckling is captured in both the symmetric and antisymmetric cases. The interactive post-buckling path is effectively divided into four individual parts (or cells), denoted as *C*1–*C*4. Each cell corresponds to the formation of a new out-of-plane displacement peak or trough in both plate elements. Figure 7 shows, for both cases, the corresponding progression of the numerical solutions for the plate buckling displacement functions *w* and *u* from *C*1 to *C*4. It is clear in figure 7*a* that the initial ‘half-sine wave’ in *w* quickly transformed to three ‘half-sine waves’ at *C*2. Interactive buckling ensues with the sequential destabilization and restabilization as the wave number of *w* continues to increase, until a total number of seven half-sine waves are formed at *C*4. In contrast, figure 7*b* shows an initial antisymmetric ‘full-sine wave’ developing into four ‘full-sine waves’ at *C*4. Figure 8 shows a series of three-dimensional representations of the deflected strut that comprise the components of weak-axis global buckling *W*_{w} and *θ*_{w}, strong-axis flexural buckling *W*_{s} and *θ*_{s} with plate buckling *w* and *u*, directly corresponding to the load level shown in figure 7.

Although the antisymmetric mode is triggered at a higher load, the post-buckling paths of the symmetric and antisymmetric cases show very close correlation. A distinct feature on the equilibrium paths is that the snapbacks for the symmetric and antisymmetric cases appear to be ‘out-of-phase’. Similar behaviour may be found in the problem of a strut on a nonlinear asymmetric foundation where the maximum deflection is either positive or negative [23]. Moreover, figure 6*f* shows almost identical post-buckling paths, despite the fact that the snapbacks occur in turn. Since the end shortening *w* being symmetric or antisymmetric.

Despite the restabilization paths in the snapbacks, the overall post-buckling behaviour is clearly unstable and the load drops significantly as equilibrium paths progress to *C*4. For both the symmetric and antisymmetric cases, at *p*=0.65 the maximum out-of-plane displacements of the plates reach approximately 4.6 mm that is more than twice the leg thickness *t*, and the maximum direct strain is approximately 0.09%, which is well below the yield strain of structural steels; the maximum direct strain is observed at the outermost fibre of the free edge on the concave side of the strong-axis flexure, whereas it is marginally smaller on the convex side. Further progression of the equilibrium state would be expected to cause more snapbacks and hence new peaks and troughs to form until the buckling deformation becomes extremely large and begins to be affected by the boundaries. The model solutions at that advanced stage may not be practically meaningful since the material may be beyond its elastic range.

#### (ii) Weak-axis global buckling being critical

The strut with length *L*=2.2 m is now analysed where weak-axis global (flexural) buckling is critical. Figure 9 shows the plots of equilibrium diagrams that correspond directly to figure 6. The numerical continuation process was initiated from zero load with *P* being varied. The first bifurcation point *C* was detected at *P*/*P*^{C}_{w}=1, implying that weak-axis global buckling was indeed critical. A weakly stable equilibrium path was then detected after the branch switch where *p* remains at 1 as *q*_{sw} and *q*_{tw} increase. The strong-axis flexural and the plate buckling displacements remained zero, implying that the strut was only exhibiting the weak-axis global buckling, as shown in figure 9*a*–*e*. A series of secondary bifurcation points were then detected along the weakly stable path, where both the strong-axis flexural displacements and the plate buckling displacements were triggered simultaneously. The symmetric and antisymmetric solutions of *w* were found after branch switches from the secondary bifurcation points *S*_{1} and *S*_{2}, respectively. Interactive buckling was triggered and unstable post-buckling paths were found for both *w* being symmetric and antisymmetric. Similar to the case where flexural–torsional buckling was critical, the post-buckling equilibrium paths for the symmetric and antisymmetric case were almost identical with the snapbacks occurring out-of-phase, despite the fact that *S*_{2} was found with a larger *q*_{sw}. The emergence of the buckling cells in sequence is very similar to the case where flexural–torsional buckling is critical. Nevertheless, the current case shows that a relatively long strut may first exhibit pure weak-axis global buckling once the theoretical buckling load *P*^{C}_{w} is reached, and flexural–torsional buckling may be triggered as a secondary instability.

Mechanical systems suffering from cellular buckling are shown to exhibit sequential destabilization and restabilization. In the current case, the destabilization is primarily caused by the interaction of the global flexural and plate buckling instabilities whereas the restabilization is introduced by the stretching of the buckled plates as they bend into double curvature [27]. The current results resonate with the findings in [9], where the numerical results from a finite element model revealed one snapback on the equilibrium path as the plate buckling profile switches from a single half-sine wave to three half-sine waves and the strong-axis flexural displacement varies from a positive to a negative value. Moreover, this mode jumping phenomenon is usually associated with strut lengths that are close to the ‘transition length’ where the critical mode changes from flexural–torsional buckling to weak-axis global buckling [32]. Indeed, this is where the strut is most susceptible to the interaction between both types of instabilities.

### (b) Experimental results and validation

Compression tests were conducted on all test specimens. For each specimen, it was observed that the measured load reached a peak value and began to drop, with the end displacement being continuously applied by the hydraulic jack; testing was manually stopped when a significant drop in the load measurement was observed. All test specimens exhibited unstable post-buckling responses due to the mode interaction of weak-axis global and flexural–torsional buckling, as shown by the example in figure 10, where the buckled strut *L*1400_2 exhibited both plate buckling displacements and weak-axis global buckling displacements, shown in figure 10*a* and *b*, respectively. Note that the strong-axis global buckling displacements are negligible when compared to the plate and weak-axis global displacements, and so are not clearly observed in the photographs. For all test specimens, a relatively large plate buckling displacement was first observed, and followed by a significant increase in the weak-axis global displacement as interactive buckling was introduced. This is consistent with the predictions in table 2 showing that flexural–torsional buckling is critical for the current choice of cross-section properties for strut lengths shorter than 1.8 m. Cellular buckling was observed; however, it was found in all experiments that the plate buckling profile transformed from an initial symmetric half-wave to an antisymmetric full-wave, instead of three half-waves as found in the symmetric case of the current variational model and in [9].

Figure 11 shows a series of photographs for struts *L*1600_2 and *L*1800_2, representing two typical but slightly different behaviours observed during the tests. For strut *L*1600_2, the initial plate buckling profile was approximately a half-sine wave; as the load approached the limit point and the weak-axis global displacements began to increase, the plate buckling wavelength started to reduce and the profile became approximately a half-sine-squared wave, as shown in the second row of figure 11*a*. At this stage, the load had dropped significantly because of the unstable post-buckling behaviour. As loading was continued, further reduction in the plate buckling wavelength was observed until an antisymmetric buckling profile was fully developed, as shown in the final row of figure 11*a*. This behaviour represents a typical case where a smooth transformation in the plate buckling profile was observed. Strut *L*1800_2 represents another typical case where snap-through behaviour was found as the initial symmetric half-sine wave switched to the opposite side. This usually happened very soon after weak-axis global buckling was triggered. The photograph in the second row of figure 11*b* captured the moment of snap-through and after this point the plate buckling wavelength began to decrease and the profile continued to develop into the antisymmetric full-wave, as found in *L*1600_2.

In addition to the evidence provided visually, evidence of cellular buckling was also observed when the experimental equilibrium paths were plotted. Figure 12 shows a series of detailed comparisons between the experimental and the variational model results for (*a*–*c*) *L*1600_1 and (*d*–*f*) *L*1500_2. The strong-axis global displacements were negligible, hence not presented for brevity. Note that the variational model is now solved with boundary conditions (2.20) and the cross-section properties summarized in table 1. Similar to the results in figure 6, it is observed in figure 12 that the symmetric and antisymmetric paths are extremely close with snapbacks on each curve occurring out-of-phase. This provides a rational explanation for the experimental observation of the plate buckling profile transforming from a symmetric to an antisymmetric mode. Since the symmetric mode is always associated with a lower critical load, without any restraints, flexural–torsional buckling of the strut would therefore be triggered with the plate buckling profile being symmetric, i.e. a single half wave. However, since the symmetric and the antisymmetric paths are almost identical, as shown in figure 12*c*,*f* where the total end shortening

As shown in figure 12*a*,*b*,*d*,*e*, the plate out-of-plane displacement *w* begins to exhibit snapback instabilities as a significant weak-axis global buckling displacement *W*_{w} begins to increase, signifying the reduction of the plate buckling wavelength. It needs to be emphasized that since the experimental data of *w* were collected at the mid-span of the strut, a perfect antisymmetric mode implies that *w*=0 at *z*=*L*/2. Therefore, the equilibrium paths obtained from the variational model in figure 12*b*,*e* only show the post-buckling response for the symmetric case where *w* is maximum at *z*=*L*/2. In the experiments, however, owing to geometric imperfections, the inflexion point rarely occurred at mid-span, hence *w* was found to decrease to either a positive (*L*1600_1) or a negative (*L*1500_2) value depending on to which side of the measuring point the inflexion point shifted. Then *w* would increase again owing to the continuing reduction in the buckling wavelength.

It is also worth noting the close correlation between the ultimate loads found by the experiments and the variational model. The post-buckling equilibrium paths of load versus plate buckling displacements are found to be approximately parallel between the two models, whereas for load versus weak-axis global displacements, a less stable post-buckling behaviour is observed in the experiments, as the restabilization paths are not fully captured in the *P* versus *W*_{w} graphs. This is because, during the experiments, the axial compression was continuously applied by the displacement-control loading device, which was not capable of automatic unloading, i.e. it was incapable of releasing the applied end displacement in the reverse direction without manual operation. Under such a condition, the equilibrium path was likely to snap to a new state in the next cell of the equilibrium path at a constant end displacement, instead of undergoing the full destabilization and restabilization process. Nevertheless, the progressive shortening of the plate buckling wavelength and the transformation in the plate buckling profile from a single half-sine wave to a full-sine wave is clearly observed in the equilibrium paths of *P* versus *w* and from the visual evidence shown in figure 11.

Figure 13 shows the axial load *P* versus the weak-axis lateral displacement *W*_{w} for all test specimens with the symmetric and antisymmetric paths from the variational model superimposed. A close correlation between the ultimate load of the experiments *P*^{u}_{e} and the flexural–torsional buckling load *P*^{C}_{ft} is observed for all cases, except for *L*1400_1 where a slightly larger error is found. Table 3 summarizes *P*^{u}_{e}, *P*^{u}_{a}, *P*^{C}_{ft} and their relative ratios for each test specimen. As mode interaction advanced during the experiments, the equilibrium path progressed from a stable to an unstable state and the strut exhibited either a smooth or an abrupt failure, represented by a gradual or a snap-through transformation in the plate buckling profile, respectively. The failure behaviour is summarized in the final column of table 3. This is because although the hinge assemblies were designed to represent an axial loading condition, there could always be some degree of eccentric loading caused by manufacturing tolerances; an eccentricity in the positive *d* where strut *L*1500_1 exhibited a negative *W*_{w} as the loading was applied; as *P* reached the limit point an abrupt failure occurred and *W*_{w} began to increase in the positive direction. The abrupt failure usually caused the strut to vibrate, represented by some disrupted measurements observed particularly on the equilibrium paths of *L*1800_2 and *L*1400_2 in figure 13*a* and *e*, respectively. It is also worth mentioning that a negative eccentricity normally implies a higher ultimate load as the initial weak-axis bending compensates for the compressive strain caused by the plate buckling displacements in the vulnerable parts of the angle legs. After the limit point, the equilibrium paths approach asymptotically to the cases where a positive eccentricity exists, and become relatively parallel to the equilibrium paths found by the variational model, as clearly observed in the cases of *L*1700_1, *L*1500_1 and *L*1400_2. Nevertheless, an average percentage error of 13% is found between the experiments and the variational model in terms of the ultimate load, showing reasonably good agreement.

The final plate buckling wavelength of the test specimens, *Λ*_{e}, was measured by first finding the inflexion point, from which point straight lines were projected parallel to the longitudinal axis *z*. The plate buckling wavelength is defined by the distance between the points where these straight lines meet the edge of the angle leg plate, as shown by the example in figure 14*a*. The plate buckling wavelength of the antisymmetric case from the variational model *Λ*_{a} is determined by the numerical solutions of *w* when *q*_{sw}≈0.01, as shown in figure 14*b*. Note that owing to the cellular buckling behaviour, *q*_{sw}≈0.01 may correspond to two separate equilibrium states and the one with the higher load is examined and compared against the experimental results. Table 4 summarizes the comparison of the plate buckling wavelength between the experiments and the variational model. The measurements of *Λ*_{e} of the twin specimens are similar, therefore an average value is calculated and presented for each strut length. It is worth noting that practically it was difficult to measure *Λ*_{e} according to one specific state during the experiments; instead it was measured at the end of each test and hence may represent a different equilibrium state and therefore a different level of localization in the plate buckling profile. Nevertheless, the final plate buckling profile was found to be an antisymmetric full wave in all tests, as described previously. The ratio between *Λ*_{e} and *Λ*_{a} is fairly consistent in all cases except for *L*1800. The average error in the plate buckling wavelength is found to be 9% with the experimental wavelengths being generally larger, implying a lower degree of localization in the test specimens, when compared with the plate buckling profile from the variational model.

Thin-walled structural components in compression exhibiting nonlinear mode interaction are known to be highly imperfection sensitive [17]. The combination of a global and a local type of imperfection would reduce the ultimate load dramatically and local imperfections associated with plate deflections with various forms (including different wavelengths and degrees of localization) may also lead to different levels of reduction in ultimate load [33]. Moreover, geometrical imperfections would account not only for the lower ultimate load, but also for a lower residual load in the post-buckling range [18]. It is found currently that the geometrical imperfections are primarily in local form where an initial out-of-plane displacement may already exist in the angle legs prior to testing. The loading eccentricity would cause the strut to bend as soon as the test begins—an effect that is similar to a global type of imperfection. Clearly, further works are required to investigate the imperfect behaviour including full measurements of the geometrical imperfections in the test specimens and the development of a variational model representing the imperfect case. Nevertheless, the current experimental results compare favourably with the variational model of the ‘perfect’ system in terms of the mechanical destabilization and post-buckling deformation with the cellular buckling phenomenon being captured in both models. Therefore, in the opinion of the authors, the variational model describing the nonlinear interactive buckling behaviour in thin-walled equal-leg angle struts has been validated successfully.

## 5. Concluding remarks

A nonlinear model based on variational principles has been presented for perfect thin-walled mono-symmetric equal-leg angle struts under axial compression that is applied on the centroid of the cross section. An important and potentially dangerous interaction between different types of instabilities including weak-axis flexural buckling, strong-axis flexural buckling and plate buckling of the angle legs (or torsional buckling of the cross section) is identified. For the first time, a theoretically based model predicts highly unstable cellular buckling through a series of snapback instabilities where a progressive formation of new peaks and troughs is observed in the plate buckling profile in equal-leg angle struts. A series of physical experiments was conducted for a range of strut lengths made from thin-walled cold-formed steel. The experimental results clearly demonstrate cellular buckling through the changing plate buckling profile as the initial symmetrical half-sine wave shifts to an anti-symmetrical full-sine wave. Comparisons with the variational model results are very encouraging and essentially validate the model. Extending the modelling approach to include initial geometrical imperfections would allow further investigations on real structural components and provide a more profound understanding of the underlying phenomena. This, in turn, would provide valuable information about the imperfection sensitivity of this type of thin-walled structural component, for the benefit of more economical and efficient structural design.

## Data accessibility

The experimental data in the current work can be accessed at https://dx.doi.org/10.6084/m9.figshare.c.3906091.

## Authors' contributions

L.B. contributed to the development of the mathematical model, interpretation of the computational results, design and execution of the experiments and writing the paper; F.W. contributed to the design and execution of the experiments, interpretation of the experimental results and drafting the manuscript; M.A.W. contributed to the interpretation of the numerical results and revising the manuscript critically for important intellectual content; J.Y. contributed to the design of the experiments and revising the manuscript critically for important intellectual content. Furthermore, each author certifies that this material or similar material has not been and will not be submitted to or published in any other publication before its appearance in Proceedings A. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

The work was funded by the Science and Technology Commission of Shanghai Municipality through project grant no. 16YF1405900. J.Y. also thanks the National Natural Science Foundation of China for the support through grant no. 51378308.

## Acknowledgements

We express our gratitude to all technicians from Shanghai Jiao Tong University Civil Engineering Experiment Centre for helping with the experimental set-up.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3906091.

- Received August 22, 2017.
- Accepted September 28, 2017.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.