Principles of Piping Analysis by David Burgreen
By David Burgreen
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The equations are homogeneous simultaneous equations, and the value of k, and thus the critical load Pcr, are found by setting the determinant equal to zero. Mathematical names for the types of problems described here are eigenvalue problems, or characteristic value problems. 5 FIVE FUNDAMENTAL CASES shown next: 0 0 0 Àk2 0 Àk2 1 Â L sin kL cos kL À 0 Â 1 sin kL cos kL 0 Àk2 sin kL Àk2 cos kL 0 Àk2 sin kL Àk2 cos kL 0 0 0 0 0 Àk2 þ 0 Â 1 L sin kL ¼ 0 cos kL À 1 Â 1 L 0 0 Àk2 sin kL 0 0 Àk2 cos kL Only the first determinant is not equal to zero, and its decomposition leads to Lk4 sin kL ¼ 0.
The unexpected occurrence of such a snap-through event during the test of a full-sized column in the laboratory is something not easily forgotten. 8b), v is the deflection at location e, P is the axial force, and EI is the product of the elastic modulus and the moment of inertia, respectively. 29) sin kL The maximum deflection occurs at the center at z ¼ L=2 and it is equal to: ! 30) When P ¼ PE ¼ p2 EI p kL p e ; k ¼ ; cos ¼ cos ¼ 0; vðL=2Þ ¼ ! 1 2 L L 2 2 0 Thus, the deflection is infinite when P equals the elastic buckling load, as expected.
Following we will consider a number of applications of this differential equation. 2, we derived the Euler buckling equation using a derivation based on equilibrium of the deformed cross-section and small displacement theory. In this section, we will use the general equation of planar flexure and discuss the results in greater detail. Columns with other boundary conditions will be measured against this fundamental element through the device of the effective length concept that is familiar to structural engineering students and professionals.