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HW-8 - ASE-321K Howework M.E Mear Spring 2006 Problem 1...

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Unformatted text preview: ASE-321K _ Howework M.E. Mear Spring 2006 Problem 1: Consider an axial bar which is constrained between rigid walls and sub- jected to a uniform distributed load q as shown in the figure. The bar has axial stiffness AE and length L. Use the finite element method (with linear elements) to obtain an approximate solution for the displacement field. Employ an element—by- element assembly process with a) two elements of equal length and b) three elements of equal length. Plot the two approximate solutions along with the exact solution. (For purposes of plotting the results, normalize the displacements in the same way you did for the last homework assignment). Problem 2: The axial bar shown below has axial stiffness AE and length L, and it is subjected to a concentrated force Q at its right end. The bar is on a “foundatiOn” which is modeled using a continuous distribution of springs having spring constant k. Equilibrium of the bar dictates that dgu The boundary conditions are u(0) = O and AE(du/d:r)|x:L = Q. 0<I<L For algebraic simplicity, now take AE = 1, L : 1, k = 1 and Q = 1 (with some appropriate units which you need not be concerned with). A symmetric, weak form statement of the boundary value problem is as follows. Let E du‘ dz d2: W*(u;u*) E PL u'(0) + PRu'(L) — f:[ + inf] dz: in which PL E —AE(du/d:r)|z=g is the wall reaction and (from the natural boundary condition) PR E AE(du/d$)|$=;, = Q 2 1. The actual displacement field u(:1:) is the one that satisfies the essential boundary condition 21(0) = 0 and is such that W‘ (u;u*) = 0 for every u‘ (x). _We wish to adopt the finite element method to obtain an approximate solution uh(m). (a) First, consider an arbitrary mesh with M elements and N = M + 1 nodes. The final system of linear algebraic equations can be expressed as ELI Kijuj = P,- (for i = 1, 2, ..., N) in which Kij are the entries of the stiffness matrix, uj are the nodal displacements, and R: are the entries of the (total) load vector. (Note: If we also had a distributed load q, then there would be a contribution to the total load vector associated with this load — in class we denoted this contribution E.) Showing’all relevant details, establish an expression for the enties of the stiffness matrix, Kg. In your development let the (piecewise-linear) nodal baSe functions _be denoted 95,-. Also statethe entries of the total load vector, R. (b) Develop the 2 x 2 element stifiness matrix It”. Carry out your development for an element of arbitrary length h. (d) Consider now a mesh comprised of three elements of equal length. Use the element stiffness matrix (along with connectivity) to construct the final system of equations ELI Kijuj = P,- (for i = 1, 2,3,4). Solve the system of equations for the nodal displacements and for the wall reaction. ...
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