Source: Roberto Leon, Department of Civil and Environmental Engineering, Virginia Tech, Blacksburg, VA
In the design of civil works, it is important to deliver structures that are not only safe under unexpected loads, but also provide excellent performance under everyday loads at a reasonable economic cost. The latter is often tied to minimum use of materials, ease of fabrication, and rapid construction in the field. Structures made of steel members can be very economical because of the great strength of the material and the extensive prefabrication of their members and connections, which help maximize the speed of construction on site. Generally, the skeleton of a steel structure will be very slender as compared to a reinforced concrete one. While its behavior in tension is governed primarily by the strength of the material, steel in compression is governed by another failure mode common to all materials- buckling. This behavior is easily demonstrated by pressing down on a slender wooden ruler, which under a compressive load will suddenly move sideways and lose load carrying capacity. This phenomenon will occur in any slender member of a structure. In this lab, we will measure the buckling capacity of a series of slender aluminum columns to illustrate this failure mode, which over time has led to many catastrophic failures including that of the Quebec River Bridge, which was erected in 1918.
Plot the results from the table as buckling stresses vs. slenderness (kL/r), along with the curve given by Eq. 9. Compare your results with the predicted values. The experimental results shows two distinct regions. When the columns are relatively long, the critical load is given by multiplying Eq. 9 by the area of the column. As the columns begin to get shorter, the critical load begins to approach the strength of the material. At this point the behavior shifts from a purely elastic one to a partial inelastic one that approaches asymptotically the squash load of the column. When a column buckles elastically, deformation can become every large suddenly and trigger failures either in the buckled member or in adjacent ones that become overloaded as the buckled member sheds its loads. Thus, in design it is important to prevent elastic buckling failures in primary structural members.
This experiment demonstrated the validity of the Euler approach for calculating local buckling loads for simple columns. Although the problem becomes far more complicated if either the boundary conditions are not well known, the member is not prismatic, or if the material does not exhibit a bi-linear stress-strain curve, the solution of the problem follows the same general process. In many practical cases, it will not be possible to solve the resulting differential equations exactly, but there are many numerical techniques that can be applied to approximate the solution to those problems. The importance of buckling is recognized in the construction industry aphorism that holds that the successful design of steel structures is predicated on a good grasp of buckling issues, while successful design of reinforced concrete structures is based on good detailing.
Economy in design requires that the volume of material be minimized. This detail is particularly true for metal building and bridge structures, where the materials costs are a significant portion of the total structural cost. In general, minimizing cost boils down to getting the lowest L/r. For a fixed L, this means obtaining the largest possible r (or largest I for a given A), leading to the widespread use of W-shaped members. For a fixed r, this means decreasing L, which entails the use of bracing members. For a W-shape, there will be both an Ix and Iy, and corresponding (kL/r)x and (kL/r)y; for optimum design, both of these values should be close to one another, which is often obtained by providing more bracing in the y-direction. Another way of preventing buckling is to add stiffeners, which reduce the buckling lengths in plates; examples of these include stiffeners in bridge plate girders and stiffening lips in cold-form structural members.
Buckling phenomenon is of critical importance in designing structures that are safe under unexpected loads and also provide excellent performance under everyday loads at a reasonable cost.
Due to the material’s strength, the skeleton of a steel structure is very slender when compared to brick or reinforced concrete. The prefabrication of steel components increases the onsite construction speed and makes steel structures more economical than other building materials.
Under a load, the structural elements are subjected to tension or compression forces. Under tension, steel behavior is governed primarily by the strength of the material. Under compression, steel is subjected to buckling. This phenomenon occurs in any slender structure indifferent of material.
Buckling consists of a sudden sideway deflection of the column. A small increase in the applied load can lead to a sudden and catastrophic collapse of the structure. The collapse of the Quebec River Bridge due to the buckling of the lower cord members of the structure is an example of such catastrophic failure. This video will discuss the buckling failure mode and show how to determine the buckling capacity of slender columns.
A column under an axial compressive load will buckle, or suddenly move sideways, and lose load carrying capacity. Euler, a Swiss mathematician, was the first to provide the solution to the buckling load by reasoning that a perfectly straight column could be an equilibrium in two configurations: an undeformed one and a deformed one.
Euler postulated that at the equilibrium in a slightly deformed configuration, the internal moments M are balanced by the external moments given by the load P acting at an eccentricity y. The second derivative of the lateral displacement y is the curvature of the member. This quantity is proportional with the internal resistance or to the internal moment divided by the bending stiffness.
In this equation, E is the modulus of elasticity, and I is the moment of inertia, a geometrical property of the section. By substituting the first equation into the second equation, we get the differential equation of buckling, where k is a substitution variable.
Let’s assume that the column deformation is given by the following function. We also assume that the column has pinned ends that do not displace laterally with respect to one another. Then, the boundary condition at Z equals zero and Z equals L is given by the lateral displacement y equals zero. As a consequence, kL equals N pi. Here, N is an integer, and its lowest value is one which is the elastic buckling load P critical. For a column with pinned ends, P critical is given by the Euler buckling load.
The critical load is the minimum load that may cause the column to buckle. Note that this equation does not contain any terms related to the strength of the material, only to its stiffness and dimensions. In order to increase the value of the critical load for a column, we can maximize the moment of inertia.
Let’s consider a W-shaped section. Its moment of inertia with respect to the centroid of the section is given by the summation of the moment of inertia for each rectangle. For each rectangle, the total moment has two components. The moment of inertia of the individual rectangle, plus its area, times its distance to the centroid of the entire section. In consequence, the value of I can be increased significantly by putting most of the material as far away from the centroid as possible.
The relationship between the moment of inertia I and area A is defined by the radius of gyration r. The buckling capacity is sometimes expressed as a critical stress, Fcr, by dividing the critical load by the area. Keep in mind that there are some limitations inherent in the derivation of buckling capacity with Euler theory, since we assume: purely elastic behavior, load applied at the centroid of the column, the column is initially perfectly straight, a deflected shape which gives an exact solution, idealized boundary conditions, the absence of any residual stresses.
These limitations are generally treated as imperfections, and their magnitudes are key to established construction tolerance. The limitations related to the boundary conditions can be treated by introducing in the expression of Euler buckling capacity an effective length factor, k. The denominator is known as the slenderness of the column. A low value of this factor, for example less than 20, is synonymous with a stocky column. While a large value, for example higher than 100, is synonymous with a slender column very susceptible to buckling.
Let’s plot now the critical stress as a function of the effective slenderness lambda. The critical stress is capped by the yield strength of the material. Meaning that for any given steel strength, there will be a value of the slenderness below which buckling will not occur. Euler formulation indicates that as the axial load reaches its critical value, buckling will occur suddenly. However, because of structural imperfections, there is a transition between the elastic buckling stress and the squash load. As a result, in real life there will be a smooth transition between the elastic buckling curve and the yield limit states.
Now that you understand Euler Buckling Theory, let’s use this to analyze the buckling capacity of slender metal columns.
Have a set of testing specimens manufactured from one inch by a quarter inch aluminum bar cut to lengths ranging from eight inches to 72 inches. Machine both ends of each specimen to a radius of 1/8 of an inch. Measure the dimensions, length, width, and thickness, of each specimen to the nearest 0.02 inches.
Manufacture a testing fixture for the specimens from two small blocks of steel approximately two inches on a side. Machine a very smooth, half-inch circular groove along one side to mate with the specimens. On the sides opposite the groove, an insert should be provided for fixing to the universal testing machine. Before you begin testing, familiarize yourself with the machine and all safety procedures. Insert the steel blocks into the testing machine with a specimen and ensure that everything is carefully aligned to eliminate eccentricities.
In the test software, set the machine to deflection control and have both load and axial deformations recorded. Program the machine to slowly apply to deformation of up to 0.2 inches and then begin the test. This limit can be varied with specimen length, but the test should be stopped when the load has stabilized or before it drops more than 20% from the maximum capacity.
When the test is complete, record the maximum load reached for this specimen. Then reset the machine and repeat the testing procedure for the remaining specimens. After all of the specimens have been tested, you are ready to look at the results.
First, calculate the slenderness parameter lambda, and then using Euler’s formula, calculate the buckling stress for each specimen. Next, use the material strength to calculate the characteristic slenderness below which buckling will not occur.
Plot the ratio between the buckling stress and the material strength as a function of the slenderness ratio. On the same graph, also plot for all specimens the measured buckling load normalized with the material strength. Now compare the measured values with the calculated values.
The experimental results show two distinct regions. When the columns are relatively long, the data follow the Euler buckling curve. As the columns begin to get shorter, the critical load begins to approach the strength of the material. At this point, the behavior shifts from a purely elastic one to a partial inelastic one that approaches asymptotically the squash load of the column.
The importance of buckling is well-recognized in the construction industry where the design of steel structures is predicated on a good grasp of buckling issues.
Economy and design requires that the volume of material be minimized while also preventing buckling instabilities. In bridge structures, this is achieved by the widespread use of W-shaped members, and by adding stiffeners in the bridge plate girders to reduce the buckling lengths in plates.
A structural system is said to be imperfection sensitive if its load carrying capacity is substantially less than that of the perfect system. While columns are imperfection insensitive, spheres and cylinders are sensitive to imperfections and, as a result, much care must be given during construction of shells; for example, domes, cooling towers, and storage tanks, and other such structures to obtain the correct geometry.
You have just watched JoVE’s introduction to buckling of steel columns. You should now understand how to apply Euler’s Theory of Buckling to determine the buckling capacity of slender metal members.
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