Problem 6: Beam equations  The simplest model for the deflection w of a beam in terms of an applied load q is given by the Euler-Bernoulli beam equation:  c12 = q(x) • dx- Here E is Young's modulus and I is related the beam's cross sectional area

Problem 6: Beam equations 
The simplest model for the deflection w of a beam in terms of an applied load q is given by the Euler-Bernoulli beam equation: 
c12 = q(x) • dx-
Here E is Young's modulus and I is related the beam's cross sectional area. Often, the product El (known as the flexural rigidity) is a constant and the equation can be simplified. In the case when one also wishes to model shear deformation and rotation bending effects, for example in the case of thick or composite beams, the Timoshenko beam theory is usually applied instead. This replaces the Euler-Bernoulli equation with the following pair of ODEs: 
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d2 (L-1°) '1(0 -thc2 dx , 7= vco- —d (El '19 KAG dr ) 
Here A is the cross-sectional area of the beam, G is the shear modulus and h is the Timoshenko shear coefficient, which depends on the cross-sectional geometry. This system is equivalent to the Euler-Bernoulli equation if the final term in the second equation is negligible. This project will study the formulation of suitable boundary value problems for these beam equations, and their solution using both analytic and numerical methods. The conditions under which the Euler-Bernoulli model is valid will also be investigated in order to determine when the more complex Timoshenko theory is necessary.