Question

Question 1. Substitution of given form of solution and hyperbolic functions. The non-linear ordinary differential equation describing the smooth shape of a structural arch of constant thickness in mechanical equilibrium under its own weight per unit length w, and a horizontal compressive force T, is (y")2 = k2(1 + (y')"). Here k is a known constant and y(x) is the vertical height of the arch at position x, the horizontal distance from a given reference point. (a) Using hyperbolic function properties (see Appendix A.2 of the printed notes), show that there is a solution satisfying y'(x)sinh(kr +C). Here C is a yet-to-be-determined constant. (b) Hence find a solution y(x). (As well as C, it will contain another undetermined constant.) Question 2. Boundary conditions (and more on hyperbolic functions) Consider an arch of the type described above, positioned on a horizontal surface. Let us take as our reference point the left base of the arch (where the left side of the arch makes contact with the supporting surface). The right base is 2L metres away from the left base. (a) Sketch this situation and mark on your diagram all information. (b) Write down a boundary condition involving y(0). Also, given that the arch is smooth and symmetric about its centre, write down a boundary condition for y'() at-L (c) Use your boundary conditions from (b) above to find all unknown constants in your answer to Question 1(b)

Answer 1

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