Computational Statics and Dynamics by Andreas Öchsner

Author:Andreas Öchsner
Language: eng
Format: epub
Publisher: Springer Singapore, Singapore

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Partial differential equation

A single-equation description for the Timoshenko beam can be obtained under the assumption of constant material (E, G) and geometrical properties: Rearranging and two-times differentiation of Eq. (4.34) gives:

(4.35)

(4.36)

One-time differentiation of Eq. (4.33) gives:

(4.37)

Inserting Eq. (4.35) into (4.37) and consideration of (4.36) gives finally the following expression:

(4.38)

The last equation reduces for shear-rigid beams, i.e. , to the classical Bernoulli formulation as given in Table 3.​5.

For the derivation of analytical solutions, the system of coupled differential equations as summarized in Table 4.4 has to be solved. Through the use of computer algebra systems (CAS) for the symbolic calculation of mathematical expressions,4 the general solution of the system as given in Eqs. (4.33) and (4.34) results for constant , AG, and in:

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