Question

Problem 1.1 Consider the beam bending problem below 2 Po Consider the beam to be homogenous and linearly elastic, with length L, stiffness E, and moment of inertia I. The beam is cantilevered at x = 0 an d is supported by a linear spring of stiffness k at x-L. A uniformly distributed transverse load po (N/m) is applied to the upper surface a) Write and solve the GDE to obtain the exact solution for the deflection w(x) of this beam. Put this solution into a non-dimensional form and plot the solution for various values of the non-dimensional spring stiffness. Comment on this solution, how does the spring stiffness impact the deflection? b) Knowing that the expression of the potential energy for this beam is EI use the RRM to get an approximate expression for the beam deflection using 1. The simplest polynomial basis function 2. The first two simplest polymomial basis functions Compare your two "RRM attempts" to the exact solution (for various values of the spring deflection) and comment on your solutions

Answer 1

"By general differential eg for the given configuration EIw�� = M(x) = -P_0 L^2/2 i.e EIw� = -P_0 L^2 x/2 - P_0 x^3/24 + C_1 i.e EIw = -P_0 L^2 x^""/2 - P_0 x^4/24 + C_1 x + C_2 - (1) because w�(0) = 0, C_1 = 0 and w(L) = P_0 L^4/48EI, C_2 = 11P_0 L^4/24 substituting in equation (1) we get w(x) = -P_0/24EI (1x^4 + 12x^2 L^2 - 11L^4) Now, let us assume, take x = 0 and x = L at points A and B. Then deflection w(x = 0) = 11/24 P_0 L^4/EI and w(x = L) = -P_0 L^4/EI (1/12) Now let P_0, L, E and I = 1 Then w(x = 0) = 11/24 and w(x = 1) = -1/12 = -0.08 and at(x = 0.5L)w = -P_0/24EI ~(0.5L)^4 - 11L^4 + 12(0.5L)^2 L^2] i.e w(x = 0.5L) = -1/24 [0.0625L^4 - 11L^4 + 3L^4] = 0.331 putting these values we get, \r\n we can see that deflection (w) "