EN75: Advanced Mechanics of Solids Homework 6: Energy methods, Implementing FEA. School of Engineering Brown University. The figure shows a eam with clamped ends sujected to a point force at its center. Its potential energy is L d vx ( ) Π= EI dx Pv( L / ) dx Due Friday Nov 9, 8 x Where v is the downward deflection of the eam.. Start y re-writing the potential energy in dimensionless form, y defining dimensionless measures of deflection and position as follows ˆ vei ˆ x ˆ Π v= x= Π= EI 3 L 3 PL P L Show that the dimensionless form for potential energy is independent of loading, geometry, or material properties. Sustituting for the relevant variales gives 3 3 3 ˆ P L PL d vˆ ( x) PL Π = EI Ldxˆ P vˆ( L / ) EI EIL dxˆ EI ˆ d vx ˆ( ) Π= dxˆ vˆ( L / ) dxˆ The normalized expression has no parameters so it is universal. [ POINTS]. Use a displacement field of the form N ˆ ˆ n vx ( ) = ax n n= to otain a Rayleigh-Ritz approximation to the deflected shape of the eam. Write a MATLAB script (you can modify the one from class you will only need to change a few lines in the code) to calculate vx ˆ( ˆ) for an aritrary numer of terms in the approximation. Plot a graph of the solution with 5, and 5 terms in the series. Hand in a copy of your plot, there is no need to sumit MATLAB code The plot is shown elow
[5 POINTS]. Modify the Matla code given in class to solve prolems involving plane stress deformation instead of plane strain (this should require a change to only one line of the code).. Check the modified code y solving the prolem shown in the figure. Assume that the lock has unit length in oth horizontal and e vertical directions, use Young s modulus and Poisson s ratio.3, and take the magnitude of the distriuted load to e (all in aritrary units). Compare the predictions of the FEA analysis with the exact solution (report your comparison as a tale that gives your FEA predictions for the displacements at the 4 nodes, along with the exact solution). Please hand in a description of the line(s) of code you modified as a solution to this prolem, and the tale. There is no need to sumit MATLAB code. To do this you need to () Change the elastic constant matrix (line 4) to Dmat = [[,nu,];[nu,,];[,,(-nu)/]]*e/((-nu^)); () Edit the input file to enforce the correct oundary conditions e 4 3
The predicted displacement field is elow To get the exact solution note that the material is in a state of uniaxial tension; the strain is therefore /=.; the vertical displacement of the top will e.; the lateral displacement of nodes and 4 will e νε L =.3. So the FEA solution is exact. [3 POINTS]. Repeat the test done in class to calculate the stress field in plate containing a central hole, with Poissons ratio.499 (you can download a copy of the input file for this simulation of the homework page of the wesite). Plot the stress contours, and find the predicted maximum stress concentration factor (you can just search for the element with the maximum stress). Compare the solution with your hand calculations using the Airy function in Homework 5. Upload a copy of your contour plot along with the hand calculation. There is no need to sumit MATLAB code. The stress contours are shown. There is no locking with plane stress elements.
Matla reports the maximum stress as 4. units. The remote stress can e calculated from the displacement applied to the oundary (.), which gives a strain of./6; the modulus is so the remote stress is /6. This gives a stress concentration of.5 - a it lower than the analytical solution predicts, ut the mesh is extremely coarse. [ POINTS] 3. In this prolem you will develop and apply a finite element method to calculate the shape of a tensioned, inextensile cale sujected to transverse loading (e.g. gravity or wind loading). The cale is pinned at A, and passes over a frictionless pulley at B. A tension T is applied to the end of the cale as shown. A (nonuniform) distriuted load q(x) causes the cale to deflect y a distance w(x) as shown. For w<<l, the potential energy of the system may e approximated as L L T dw Π= dx qwdx dx e e 3 q(x 3 ) w(x) x 3 3 4 5 6 7 A s T B To develop a finite element scheme to calculate w, divide the cale into a series of -D finite elements as shown. Consider a generic element of length l with nodes a, at its ends. Assume that the load q is uniform over the element, and assume that w varies linearly etween values w, w at the two nodes. a w a (a) l w ()
3. Write down an expression for w at an aritrary distance s from node a, in terms of w a, w, s and l. (assume a linear variation) w= w ( s/ l) + w s/ l 3. Deduce an expression for dw / dx within the element, in terms of w a, w and l a dw / dx = ( w w )/ l a [ POINTS] [ POINT] 3.3 Hence, calculate an expression for the contriution to the potential energy arising from the element shown, and show that element contriution to the potential energy may e expressed as elem T [ ] / l T, / l wa ql [, ] / V = wa w wa w T / l T / l w ql/ (assume that q is uniform over the segment) We can compute the contriution from one element to the two terms in the potential energy l T dw T T T T / l T / l wa dx= ( w wa) = wa( wa w) + w( w wa) = [ wa, w] dx l l l T / l T / l w l ql ql / qwdx = ( wa + w) = [ wa, w] ql / [3 POINTS] 3.4 Write down expressions for the element stiffness matrix and force vector. By inspection T / l T / l [ kel ] = T / l T / l ql / [ rel ] = ql / [ POINT]
3.5 Consider the finite element mesh shown in the figure. The loading q is uniform, and each element has the same length. The cale tension is T. Calculate the gloal stiffness matrix and residual vectors for the mesh, in terms of T, L, and q (efore any constraints are applied). Following the procedure discussed in class, we add contriutions from neighoring elements to each node. The two central nodes are shared y two elements, so 3 3 T 3 6 3 [ K] = L 3 6 3 3 3 ql r = 6 [ POINTS] A q 3 L 4 B 3.6 Show how the gloal stiffness matrix and residual vectors must e modified to enforce the constraints w = w = 4 We need to replace the first and last equations to enter the two constraint equations T 3 6 3 [ K] = L 3 6 3 ql r = 3 [ POINTS] 3.7 Hence, calculate values of w at the two intermediate nodes. The equation system can e solved to give w ql = w = 9T 3 [ POINTS]
3.8 Write a simple MATLAB script to solve the prolem for an aritrary numer of elements with equal spacing etween nodes (you can just use the pattern you see in 3.6; you don t need to write a fancy code). Compare the FEA predictions with the exact solution. q w= x3( L x3) T Please upload your MATLAB code as a solution to this prolem; make your code plot a graph comparing the exact and FEA solutions. The plot is shown elow The FEA solution (at the nodes) turns out to e exact regardless of the numer of elements! [5 POINTS]