Buckling load

A polypropylene bar with a square section (10 mm x 10 mm) is 225 mm long. It is pinned at both ends and an axial compressive load of 140 N is applied. How long would it be before buckling would occur. The relationship between the buckling load, Fc, and the bar geometry is... 

The analysis to find the fiber buckling load in each mode is based on the energy method described by Timoshenko and Gere [3-31], The buckling criterion is that the change in strain energy for the fiber, AUf, and for the associated matrix material, AUf, is equated to the work done by the fiber force, AW, during deformation to a buckled state, that is,... 

Laminated composite plates under in-plane tensile loading exhibit deformation response that is both like a ductile metal plate under tension and iike a metai plate that buckles. That is, a composite plate exhibits progressive faiiure on a layer-by-layer basis as in Figure 4-34. Of course, a composite plate in compression buckles in a manner similar to that of a metal plate except that the various failures in the compressive loading version of Figure 4-34 could be lamina failures or the various plate buckling events (more than one buckling load occurs). 

A plate buckles when the in-plane compressive load gets so large that the originally flat equilibrium state is no longer stable, and the plate deflects into a nonflat (wavy) configuration. The load at which the departure from the flat state takes place is called the buckling load. The flat equilibrium state has only in-plane forces and undergoes only ex-... 

The buckling load will be determined for plates with various laminations specially orthotropic, symmetric angle-ply, antisymmetric cross-ply, and antisymmetric angle-ply. The results for the different lamination types will be compared to find the influence of bend-twist coupling and bending-extension coupling. As with the deflection problems in Section 5.3, different simply supported edge boundary conditions will be used in the several problems addressed for convenience of illustration. 

The smallest value of obviously occurs when n = 1, so the buckling load expression further reduces to... 

For other materials, other families of curves such as in Figure 5-22 are obtained with correspondingly different buckling loads and different points of change from one buckling mode to another. 

Figure 5-23 Which Plate Has the Highest Buckling Load ...

Figure 5-24 Buckling Loads for Rectangular Symmetric Angle-Ply Plates under Uniform Compression, Nj, (After Whitney [5-1])...

Note that if Bn is zero, then T13 and T23 are also zero, so Equation (5.81) reduces to the specially orthotropic plate solution. Equation (5.65), if D11 =D22- Because Tn, T12, and T22 are functions of both m and n, no simple conclusion can be drawn about the value of n at buckling as could be done for specially orthotropic laminated plates where n was determined to be one. Instead, Equation (5.81) is a complicated function of both m and n. At this point, recall the discussion in Section 3.5.3 about the difference between finding a minimum of a function of discrete variables versus a function of continuous variables. We have already seen that plates buckle with a small number of buckles. Consequently, the lowest buckling load must be found in Equation (5.81) by a searching procedure due to Jones involving integer values of m and n [5-20] and not by equating to zero the first partial derivatives of N with respect to m and n. 

One of the major complications in the plate buckling solution is the need to investigate the influence of buckle mode shape on the buckling load itself. That is, the plate buckling load in Equation (5.81) is a function... 

However, because we are usualiy interested oniy in the lowest buckling load for a column, m is always one. For plates, both m and n enter the buckling equation as well as the plate aspect ratio, a/b, so the lowest buckling load does not typically occur for m = 1 and n = 1. Thi, we must find the absolute minimum of the values of the buckling load, N, or more generally, X, for a wide range of m and n values. 

Figure 5-25 Determination of the Absolute Minimum Buckling Load...

As for the deflection problem in Section 5.3.3, the effect of the number of layers on the buckling load is found by dividing a constantthickness, equal-weight cross-ply laminate into more and more laminae as in Figure 5-12. Results for graphite-epoxy antisymmetric cross-ply laminated plates for which Ei/E2 = 40, Gi2/E2 = - - v,2 = -25 are... 

Figure 5-26 Buckling Loads for Anfisymmefric Cross-Ply Laminated Plates under Uniform Uniaxial Compression (After Jones [5-19])...

Note that if B g and 825 are zero, then and T23 are also zero, so Equation (5.92) reduces to the specially orthotropic plate solution. Equation (5.65). The character of Equation (5.92) is the same as that of Equation (5.81) for antisymmetric cross-ply laminated plates, so the remarks on finding the buckling load in Section 5.4.3 are equally applicable here. 

Note that the vibration frequency reductions are far less than the buckling load reductions. That this conclusion must be reached is clear from the fact that Equation (5.109) involves the square of the natural frequency, whereas Equation (5.81) Involves the buckling load to the first power. Thus, the square root of the differences represented by the right-hand sides of Equations (5.81) and (5.109) is smaller than the differences themselves. 

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