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AEE/CEE/EGM/MEE 546 Finite Element Analysis I Assignment 8 1) The figure below shows the geometry of a 0

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AEE/CEE/EGM/MEE 546 Finite Element Analysis I
Assignment 8
1) The figure below shows the geometry of a 0.25-in. thick specimen used in the tensile test of a plate with a circular hole. The specimen is designed to have uniform stress ?xx of 10 ksi along Section A-A and Section C-C due to the applied load, F. You are running the analysis to determine the stress distribution around the hole.
For this specimen, sketch the finite element model you would use to analyze the stress field due to the applied load, F. Be sure you include and clearly label:
a) Symmetry, if appropriate
b) Boundary conditions (No rigid body motions are permitted!)
c) Loading
d) The element type that should be used (for example, 1D bar, plane stress quad, 8-node brick, etc.)
e) Any special meshing accommodations (regions you would refine, etc.).
2) The figure on the right shows a proposed six-node, 2D quadrilateral element. This element is isoparametric and, as pictured, uses a second order interpolation on one axis and a linear interpolation on the other. Answer the following questions regarding this element:
a) What are the appropriate shape functions for each node? Recall, shape functions must equal 1 at the node with which it is associated, and zero at all other nodes.
OVER, PLEASE!
??
??
b) What is the form of the displacement approximation for this element? (? = ?0 +
?1? + ?)
c) Is it possible for the sides of the physical element to be curved? Explain
3) A four-node quadrilateral element possesses two edges that are collinear, as shown in the sketch below.
??
a) Compute the Jacobian matrix [?] = [??
??
??
??] at a generic position (ξ,η)
??
within the element.
b) Will the determinant, det J, be zero or negative anywhere in the element?
Recall:
4) Answer each of the following
a) What are the primary differences between beam and bar elements, and why are the shape functions different?
b) True or False. The maximum stresses within a fully integrated element occur at the element’s integration points. Explain.
c) What is the modeling error in the sketch below if the two forces are not equal? How would you correct it?
d) Set up the equation to evaluate the following integral using 2x2 Gaussian Quadrature. Do not perform the actual integration. Will the solution be exact?
1 1
? = ∫ ∫(?2 − 2? + 9)?2????
−1 −1
5) Consider the differential equation
??
??
Assume an approximation:
+ 2? = 1, 0 < ? < 1, ?(0) = 1
? = ? + ?? + ??2
in which a, b, and c are constants.
a) Using the Galerkin Method, set up the integral that can be used to solve for a, b, and c.
b) What two derivatives must be calculated, once the integral is set, to solve for a, b, and c.