The strength limit state for this relatively short I-section is local buckling. Note the repeating wave-like patterns in the flanges and web. The mode shape depends on the relative stiffness of the flanges (see models P4 and P5) and the web (see models P1 and P2). The member is being uniformly compressed with fixed-fixed end restraints.
The strength limit state for this short to intermediate length cruciform shape is torsional (twist) buckling. Note the wave-like pattern does not repeat along the length of the member. Each leg of the section is essentially failing in plate buckling (see model P4). The member is being uniformly compressed with fixed-fixed end restraints.
By increasing the length of the member, the strength limit sate of the cruciform shape transitions from torsional buckling (model C2) to flexural buckling (as shown). With a symmetrical section, it is interesting to think about which buckling axis controls (enjoy!). The member is being uniformly compressed with fixed-fixed end restraints.
The strength limit state for this relatively short square HSS-section is local buckling. Note the repeating wave-like patterns in the walls. The length of the waves depends on the relative stiffness of the walls (see models P1 and P2). With the corners acting like stiffeners, buckling initiates along the centerline of each wall where the resistance to bending is the lowest. The member is being uniformly compressed with fixed-fixed end restraints.
By increasing the length of the member, the strength limit state of this square HSS-section transitions from local buckling (model C4) to flexural buckling (as shown). With a symmetrical section, it is interesting to think about which buckling axis controls (enjoy!). The member is being uniformly compressed with fixed-fixed end restraints.
The strength limit state for this relatively short circular HSS-section is local buckling. Note the repeating wave-like patterns in the walls. The number of waves around the circumference of this section depend on the length of the member (compare to model C7). Local buckling of circular section is often quite dramatic (stand on a soda can, and have a friend tap on the side). The member is being uniformly compressed with fixed-fixed end restraints.
The strength limit state for this intermediate (but still relatively short) circular HSS-section continues to be local buckling, but with much fewer circumferential and longitudinal waves when compared to a shorter length member (model C6). The member is being uniformly compressed with fixed-fixed end restraints.
By increasing the length of the member, the strength limit state of this circular HSS-section transitions from local buckling (models C6 and C7) to flexural buckling (as shown). With a symmetrical section, it is interesting to think about which buckling axis controls (enjoy!). The member is being uniformly compressed with fixed-fixed end restraints.
When the unbraced length of a simply-supported I-shape member is assumed small, with the controlling strength limit state being local buckling of the compression flange and/or the compression portion of the web. In this case, wave-like patterns are observed in the compression portions of the section. Equal and opposite major-axis moments are applied to the ends of member, thereby resulting in uniform moment with compression in the top flange.
By increasing the unbraced length of the member, the strength limit state of this simply-supported I-shape transitions from local buckling (model B1) to lateral-torsional buckling (LTB, as shown). Both lateral (out-of-plane) movement of the compression flange and twist is observed in this mode. Equal and opposite major-axis moments are applied to the ends of member, thereby resulting in uniform moment with compression in the top flange; the fork supports (vertical posts) provided at the ends of the member are intended to restrain twist and out-of-plane movement.
With the addition of compressive axial force, the mode for the lateral-torsional buckling strength limit (model B2) now has an accentuated amount of out-of-plane movement in the bottom flange (as shown). Depending on the relative amount of applied moment and axial force at the member ends, the failure mode will be between LTB (no axial force) and flexural buckling (no moment, see model I1) Axial compressive force and equal and opposite major-axis moments are applied to the ends of member; the fork supports (vertical posts) provided at the ends of the member are intended to restrain twist and out-of-plane movement.
The strength limit state for this I-shape column is minor-axis flexural buckling; a strength limit state often referred to as Euler buckling. Be sure to learn about many of the amazing scientific contributions made by Leonard Euler!. With pinned-pinned supports, the failure mode is a half of a sinusoidal wave (see model I5 for the full sine wave). Why does it fail about the minor axis and not the major axis of the cross section? By understanding the several assumptions that Euler made in determining a solution to this problem, you can understand the various factors that influence the strength of a column.
By decreasing the unbraced length of the Euler column, the strength limit state of this I-shape transitions from flexural buckling (model I1) to local buckling in the flanges and/or web (as shown here and in model C1). The member is being uniformly compressed with fixed-fixed end restraints.
By changing the support conditions from pinned-pinned to fixed-fixed, the unbraced length of the Euler column (model I1) can be increased by approximately 40% without a reduction in the elastic buckling strength. The buckling mode is still about the minor-axis, but with no rotation at the fixed support. Careful observation of the model will reveal a dimple (at the inflection point) that shows the Euler column (have fun comparing 3D printouts of this model and model I1).
By changing the support conditions from pinned-pinned to fixed-fixed, the unbraced length of the Euler column (model I1) can be increased by a factor of 2 without a reduction in the elastic buckling strength. The buckling mode is still about the minor-axis, with no rotation at the fixed supports. Careful observation of the model will reveal a pair of dimples (at the inflection points) that show the Euler column (have fun comparing 3D printouts of this model and model I1).
This model depicts a long I-section subjected to axial compression with pinned supports at both ends. The member undergoes flexural buckling about its minor (weak) axis, as indicated by the horizontal dowels representing the pinned conditions. At this length, the column buckles in two half-sine waves, with an inflection point located at the midpoint—six inches from each end. A separate minor-axis buckling model with a total length of six inches could align perfectly between the dowels, illustrating how buckling behavior changes with increased member length.
This model shows a long I-section undergoing major-axis flexure under axial compression with simple (pinned) supports. The dowels at either end signify the axis in which the member is bending about—perpendicular to the flanges. At this length, the member buckles in a single smooth curve from one end restraint to the other, only bending laterally with no twist. In general, a cross-section is stiffer about its major axis, meaning it requires a higher critical load to induce buckling in this direction.
With adequate bracing provided to restrain the column from buckling about either its minor- or major-axis, the strength limit state of the column could be controlled by torsional buckling, as shown in this model. In this case, the column end supports s are fixed-fixed.
This model shows the torsional buckling of an equal leg angle fixed at both ends. Notice that the cross-sectional geometry of an equal leg section is exactly a quarter of a cruciform shape. Since it is open and symmetric along its diagonal axis, it lacks sufficient torsional stiffness at a specific length. As the member is compressed, both legs of the cross-section begin to twist rather than deforming through a combination of bending and torsion, which occurs in longer members.
By increasing the length of the member, the strength limit state of this equal-leg angle transitions from local buckling (model A1) to flexural-torsional buckling (as shown). Due to its asymmetric geometry, the angle does not bend within a single plane at a medium length. Instead, the member twists near the end supports and deflects laterally, also bending in flexure. Unlike doubly symmetric sections, equal angles have an uneven distribution of stiffness which makes them especially susceptible to lateral and torsional instability if unbraced. The result is an interesting buckling behavior that is influenced by both flexure and torsion, as demonstrated by this member.
By increasing the length of the member, the strength limit state of this equal-leg angle transitions from flexural-torsional buckling (model A2) to flexural buckling (as shown). Due to its long length, the member buckles primarily in flexure along its diagonal axis of symmetry, without any torsional component. Unlike more symmetric cross-sections, angle sections are inherently prone to out-of-plane instability due to their geometric asymmetry, leading to distinctive buckling behavior not commonly observed in other structural shapes.
For this relatively short lipped C-section, the controlling strength limit state is local buckling. The repeating wave-like patterns in the flanges and web are related to the plate buckling shown in models P1 and P2, while the lip (stiffening portion) is shown in models P4 and P5. The member is being uniformly compressed with fixed-fixed end restraints.
By increasing the length of the member, the buckling mode transitions from local buckling (model LC1) to distortional buckling (as shown). In contrast to local buckling in which the corners of the shape do not (or barely) translate; the flanges, however, move upward (or downward) as shown in plate buckling models P4 and P5. The member is being uniformly compressed with fixed-fixed end restraints.
By further increasing the length for distortional buckling, a member buckling mode of flexural-torsional buckling is observed, which includes both bending and twisting deformation. A key contributor to this behavior is the differing locations of the section’s centroid and the shear center. The member is being uniformly compressed with fixed-fixed end restraints.
Eventually, the strength limit state of a long lipped-channel member will be flexural buckling about the section’s minor axis. The member is being uniformly compressed with fixed-fixed end restraints.
The strength limit state for this relatively short plate is local buckling. This model represents a plate subject to compression that is simply supported (pinned) at every edge. Pin supports provide some restraint against lateral or vertical translation but allow rotation, which creates single, smooth curves in both lateral and longitudinal directions. The plate's rectangular shape creates two curves parallel to the long edges.
By changing the support conditions from simple-simple to fixed-fixed, the strength limit state of this plate transitions from local buckling (model P1) to local buckling with increased capacity (as shown). This model shows the buckling of a plate fixed along its long edges and pinned along its short edges. When compressed, the fixed edges are unable to rotate or translate in any direction, creating a smooth curve between ends with two inflection points. Because the plate is rectangular, two curves are developed from one short edge to the other with no restriction in rotation.
By changing the support conditions from fixed-fixed to fixed-simple, the strength limit state of this plate transitions from local buckling with increased capacity (model P2) to local buckling with intermediate capacity (as shown). This model shows a plate fixed-fixed along its long edges and pinned along its short edges. The long, fixed edge offers significant restraint against both rotation and translation, while the opposing pinned edge is free to rotate but restrained laterally. This asymmetric boundary condition creates an asymmetric line of curvature, with the location of greatest deformation closer to the pinned edge. Two curves (pinned at the ends) form parallel to the long edges due to the plate's rectangular shape.
By changing the support conditions from fixed-simple to simple-free, the strength limit state of this plate transitions from local buckling with intermediate capacity (model P3) to local buckling with reduced capacity (as shown). This model shows the buckling of a plate that is simply supported (pinned) along its short edges and pinned-free along its long edges. The resulting deformation is a single long curve that extends from one short edge to the other along the unrestrained free edge. The pinned edges only allow rotation while the free edge is completely unrestrained. This results in the most deformation occurring in the middle of the long, free edge.
By changing the support conditions from simple-free to fixed-free, the strength limit state of this plate transitions from local buckling with reduced capacity (model P4) to local buckling with increased capacity (as shown). This model shows a buckled plate fixed-free along its long edges and simply supported along its short edges. The fixed end prohibits all rotation and lateral deflection while the free edge is entirely unrestrained and allowed to deflect upward/downward. When looking at either short edge, it is apparent that these sides deflect similarly to a cantilever beam, in a single curve with the maximum deformation occuring at the free edge. Since the plate is rectangular, two curves are formed parallel to the long edges, free to rotate at the pins.
The strength limit state for this relatively short member with a circular hollow section subject to torsion (twist) is local shear buckling. Torsion of a circular section produces shear stresses within the cross-section; these shear stresses can be resolved into perpendicular tension and compression fields (think Mohr’s circle) that act on a helical path along the length of the member. With the compression field producing buckling of the walls, the wave-like pattern also wraps around the member in a helical path. The member is being twisted by prescribing a rotation at one of the fixed-fixed end restraints.
By increasing the length of the member, the buckling mode transitions from local shear buckling (model T1) to helical buckling (as shown). Twist the ends of a lengthy piece of hose, such as a garden hose, and helical buckling will most likely be observed. The member is being twisted by prescribing a rotation at one of the fixed-fixed end restraints.
The strength limit state for this relatively short member with a square hollow section subject to torsion (twist) is local shear buckling. Torsion of the member produces shear stresses within the walls of the cross-section; these shear stresses can be resolved into perpendicular tension and compression fields (think Mohr’s circle) that act on a helical path along the length of the member. With the compression field producing buckling of the walls, the wave-like pattern also wraps around the member in a helical path. The edges of the square section do provide additional resistance to this buckling mode when compared to a member with a hollow circular section; there is also some amount cross section warping restraint, but this is difficult to observe in this model. The member is being twisted by prescribing a rotation at one of the fixed-fixed end restraints.
Similar to the member with a hollow circular section (model T2), increasing the length of the member will result in a transition from local shear buckling to helical buckling. The member is being twisted by prescribing a rotation at one of the fixed-fixed end restraints.
A strength limit state of deep girder with a relatively slender web is often shear buckling. The resulting shear stresses through the depth of the section can be resolved into perpendicular tension and compression fields (think Mohr’s circle). The compression field produces buckling waves observed in the model. By sketching the moment diagram for this member, it will become clear why only portions of the top and bottom flanges are also buckling.
The length of compression field can be significantly reduced by providing a vertical transverse stiffener at mid-span. An interesting feature in this model is an observation of the tension field (normal to the compression field) that develops and provides the girder significant post-buckling strength. As with model G1, the moment diagram explains the flange buckling. However, the use of a truss analogy with diagonals representing the tension field can explain why the stiffener is buckling (pretty cool!). In design practice, it is ideal to space stiffeners such that the resulting web panels are roughly square—where the distance between stiffeners equals the distance between the flanges—for optimal shear resistance.
Similar to elements in a cross-section buckling and member buckling, frames can also buckle when subject to significant gravity (vertical) load. In this case, the resistance to the eventual sidesway buckling (horizontal movement) shown is provided by the major-axis flexural stiffness of the beams and columns. The base supports shown are pinned connections that do not provide rotational restraint., and most. This model portrays the buckling of a frame system subject to compression with pinned end connections.
By replacing the pinned connections in model F1 with the fixed base connections shown, the accompanying increase in this rotational restraint results in the columns providing more bending resistance, and in an increase in the buckling capacity of this system. Bending of the both the beams and the columns is observed; in other words, the relative stiffness of the beams and columns determines the effective buckling length of the columns (think alignment charts!).
To illustrate the impact that the relative beam to column stiffness has on the buckling shape of the columns, this model illustrates the extrema of having rigid (infinite flexural stiffness) beams. The system buckling is at its greatest, with inflection points observed at mid-height of the columns. In other words, the columns’ effective length factors are unity.
At the other extreme, the beams could have little to no resistance to bending (wet spaghetti!), and the sidesway stability of the system is provided by only the cantilevered columns. In this case, the system buckling capacity is much smaller than that of frames F2 and F3, with effective length factors equal to or exceeding 2.0.
With the columns oriented for minor axis bending, the sidesway buckling resistance of the frame is significantly reduced when compared to that given by model F2. An increase in the buckling (Euler) length of the column is observed, with very little bending in the relatively stiff beams.
This model of a lipped channel is similar to the local buckling behavior observed model LC1. This model was created using MSASect2 and not the high-fidelity finite element software used to prepare almost all of the models provided in this gallery.
This model illustrates that complex geometries comprised of many elements can have one of many local buckling failure modes; in this case, the mode resembles plate buckling model P4. Structural members of such cross sections are often produced by extruding aluminum through a multicomponent die. Similar to model SFSM1, this buckling model was prepared using MSASect2.