1. Chapter 6 A four-node quadrilateral element is shown in the figure. 1.Write the (x,y) coordinates in the elements in terms of (s,t) of the reference element. Using Eq. 6.62 and 6.63 we get 2. What are the (s,t) coordinates in the reference element of nodes 2 and 3? From the definition of the reference element these are (1,-1) and (1,1). 3. Calculate from (1) the (s,t) coordinates of (0.5,0.5) : (-1/3,0) 4. As a result of heating, the element expands so that all the coordinates are multiplied by 1.01. For example, the coordinates of point 2 are (1.01,1.02). Write the displacement vector for the element. u=[001, 0.01, 0, 0], v=[0,0.01,0.02,0] 5. What are the displacements at (0.5,0.5)? u=0.01x=0.005, v=0.01y=0.005 6. Calculate the first column of the Jacobian matrix? [0, -0.5]’

2. Chapter 6 continued Consider the integral o What is the highest n that you can integrate exactly with 3 Gauss integration points? n=5. o Perform the integration both ways to check your answer. 10.667 What significance do the integration points have for the quadrilateral finite element? Why is the element called isoparametric?

3. Chapter 7 What considerations dictate where you should refine your mesh and how you transition from a region of a coarse mesh to a region of a fine mesh? The convergence rate for a finite element mesh is known to be 1. For your initial mesh, the stress was 39MPa, and when the number of elements was quadrupled the stress was 41MPa. Give an estimate of the answer if you quadruple again the number of elements. Why is it only an estimate? 42

4. Chapter 8 A truss where all members of a cross section of 2x2” has three members with stresses of 50, 60, and 70 ksi. The failure stress of the material is 200 ksi. What is the safety margin of the three elements? 150, 140, 130. If the truss is to be designed with a safety factor of 2, calculate the cross-sectional area of the three members if you wish to attain fully-stressed design? 2, 2.4, 2.8 in 2. For what kind of trusses will you attain a fully stressed state for the three members with this change? Write the formulation of the optimization problem for the truss you will use if you want to use Solver instead of FSD. Assume that you have a function s(A) that gives you a vector s of stresses in the elements given a vector A of cross sectional areas. The length of the elements is in a vector l, and they are all made of the same material.

5. Chapter 5 Consider a 2m long circular tapered conducting rod that is kept at 100 o C at one side and 0 o C at the other, while being insulated around. The cross sectional area is 2cm 2 at the hot end and 1cm 2 at the cold end, and the thermal conductivity k=10 W/m o C. Calculate the heat flux using a single element with average area 1.5cm 2 ) 0.075W Calculate the heat flux and the midpoint temperature using two elements and the average areas (1.75cm 2 and 1.25cm 2 ). 0.0729W, 58.3 o C What type of boundary conditions we have here, essential or natural?