Finite Element Analysis

Q1. (10 points) The element shown has 3 linear sides and one quadratic side. Determine the five shape functions N₁ (x, y). Dimensions are 2a by 2b. Hint: Use either method 1 or method 3 (demonstrated for the quadratic triangular element) described in the class notes, which are discussed in the attached pages similar to Ch 6 of the Cook textbook and in Handout_Ch04supp.pdf (Section 6.11).

Problem Statement: Consider the model of nonlinear dynamics defined by an initial value problem governed by the following ordinary differential equation: ü−e(1-rû²)ù+α(1+ ßu² ) u = F₁ cost with parameters a, ß, e, %, and Fo, along with initial conditions (0) = 4, and (0)=%- Investigate the character of this problem and its solutions using both analytical and numerical methods, including those discussed in Module 2 of the course. Several subsets of the general problem, formed by setting some of the parameters to zero, may be investigated as part of your study. An emphasis on interesting aspects of the problem is encouraged. Prepare a report detailing your methods, approaches and findings, while citing all references used. Submit your report as a single pdf file through the UB Learns system. You may work individually or in a group of two students. For project groups, please submit only a single report, but indicate both names on the cover page.

Consider the plane quadrilateral element shown below. (a) Numerically evalute the strains

Sketch a quadrilateral with corners properly letterd and r,s axes properly oriented if shape functions

Q3. Sketch a four-node isoparametric element for which the determinant J is a function of r but not of s.

Consider a 4 node rectangular finite element given earlier in the class notes with x dimension a=3 cm and y dimension b-4 cm, and a thickness of 1 cm. For steel which has a mass density of 8 g/cm³, find the first row of the consistent mass matrix (8 terms) for the element and the first diagonal entry of the lumped mass matrix using the row-sum technique.

Consider a thin plate shown in figure above, which is loaded at its tip: a. Using 2 triangular elements, determine the deflection and stress on the plate. b. Comment on the plate stress, if a single element is used instead of 2 elements.

Q2. Using Hermite cubic shape functions derive the element force vector for a distributed load of constant intensity fo (namely, a constant transverse applied load, in units of force/length). The solution (take h² = l) is:

(10 points) Using the nondimensionalized shape functions listed in equation (10.46) on the next page, only calculate

For the plane truss structure, the force P = 15,000 N, length L = 1 m, elastic modulus of the material E= 180 GPa, and cross section area A = 4,000 mm². All the bar members have the same material and cross section. Find the followings by APDL .Take screenshots and embed them into a words or pdf. a) Attach your code b) Print the displacements of all the nodes c) Plot the deformed structure d) Print the reaction forces at the supports

Determine the nodal displacements and the element stresses for a thin plate subjected to the loading shown. The thickness of the plate is 3 cm. The material properties are given as E = A

а.Draw the element in s-t coordinate scale if element's dimensions are 2b = 4 and 2a = 3.Then, calculate the area of the element. \text { b. Show how values in the bottom row of the stiffness matrix [kGG } \left.{ }_{G}^{(e)}\right] \text { of the rectangular } \text { element are obtained from equation }\left[k_{G}^{(e)}\right]=\int_{A} G \cdot[N]^{T} \cdot[N] \cdot d A \text {. The resultant } \text { matrix equation is equation }\left[k_{G}^{(\theta)}\right]=\frac{G-A}{36} \cdot\left[\begin{array}{lll} 4 & 21 & 2 \\ 2 & 42 & 1 \\ 1 & 24 & 2 \\ 2 & 14 & 4 \end{array}\right] \text {. } \text { Multiphy }\left[\begin{array}{lll} 4 & 21 & 2 \\ 2 & 42 & 1 \\ 1 & 24 & 2 \\ 2 & 14 & 4 \end{array}\right] \frac{G-A}{36} \text { if } G=3 \text {. You must obtain value of A to complete this }

Problem 4: Determine the unknown nodal displacements for beam shown in the \text { figure } \left.\left(y_{2}, y_{3} \text {, and } y_{4}\right) \text {. (Use El=1 } \times 10^{8} N_{4} \cdot m^{2}\right)

Problem 7: Explain how problem setup and solution of Problem 3 change if element would be in state of plane strain. Start by explaining the difference between states of plain stress and plain strain. Then discuss matrix [D}. Th end is cuss Poisson ratio. Then discuss how problem should be modified and solved.

5%) Bar with cross-sectional area A and length L is inclined at angle a. Determine the element force vector {f}=[k]{d} for the following nodal displacements: (a) rigid body translation along the x-direction, (b) rigid body rotation.

Problem 6: Consider rectangular element with length = 8 along x-axis and width =6 along y-axis. If Dx = Dy = 1 and G = Q = 0.75, determine stiffness matrix and force vector of the element, as shown in Chapter 7.

:Consider the triangular element with heat source, where: Heat source, Q*= 80 W, is located at point (1,3) Determine the total heat at each node by allocating Q* to each node.

Problem 3: For the plane stress elasticity element shown in the figure, Calculatethe element stiffness matrix and the element stress vector using the nodal displacements (in cm) listed below. The thickness of the element is 2 cm and thematerial matrix is U_{2 k-1}=0.012 \quad U_{21}=0 \quad U_{2 f-1}=0.01 \quad U_{2 j}=-0.003 \quad U_{2 k-1}=0.008 \quad U_{2 k}=0

The bar of length 2L (see the figure) is loaded along its axes with the load q(x)=q_{0} x / L The left end is fixed. At the right end,the spring of stiffness k connects the bar to the fixed support. It is fully relaxed when no external load exists. The elastic modulus E is uniform, and the cross-sectional area varies as A(x)=A_{0}(3-x / L) \text { The governing equation of the problem is } \frac{d}{d x}\left(E A(x) \frac{d u(x)}{d x}\right)+q(x)=0 \text {. } (a) (20%) Write the boundary conditions in terms of displacement and its derivative. Which one is the essential and which one is the natural boundary condition? [Hint: Watch the signs!] (b) (40%) Define the one condition that must be satisfied by the otherwise arbitrary test function w(x). Then derive the weak form of the problem. (c) (40%) Discretize the problem using the 2 finite elements (el and e2) and three nodes (seethe figure). The interpolation functions are piecewise linear and are illustrated in the figure. Derive the finite element equations for the two unknowns: \left[\begin{array}{ll} K_{22} & K_{23} \\ K_{32} & K_{33} \end{array}\right]\left\{\begin{array}{l} u_{2} \\ u_{3} \end{array}\right\}=\left\{\begin{array}{l} F_{2} \\ F_{3} \end{array}\right\} \text { Compute the coefficients } K_{22}, K_{23}, K_{33}, F_{2} \text {, and } F_{3} \text { in terms of known quantities } k, q_{0}, A_{0}, E \text { and } L

8. A simply supported steel beam is subjected to two point-loads as shown in the figure. Thegoverning differential equation for the deflection curve is E I \frac{d^{2} q}{d x^{2}}-M(x)=0 b) Redo part a using 8 elements (each element is 50 cm long). c) Briefly discuss the results in parts (a) and (b).

9.For the non circular shaft shown in the figure, solve for the nodal values of o, and calculate the maximum shear stress at node three. Use (2g0 = 1396)

Discuss the factors that influence on the accuracy of the finite element method. O Figure Q4 shows an element with three nodes. Plot the profile of strain and stress along the element for every one meter interval. Given E = 70kN/mm².

2. Use numerical methods to solve for total deflection with 1, 2, 3 and 4 elements. Do calculations by hand and show all work! May treat as springs in series. Graph deflection vs. number of elements to show convergence on analytical solution. d 14 F

For the beam below with fixed-fixed boundary conditions, do the following calculations by hand. Let k₁ = 480 #/in, k₂ = 310 # /in, k3 = 70 #/in, k4 = 90 # /in, k5 = 10 # /in, F₂ = 200 # and F3 = 500 #. 1. Solve for the displacements at nodes 1 through 4 (U₁, U₂, U3, and 4). 2. Solve for the reaction forces at both ends (R₁ and R4). R 1 T www F 2 2 кг www 42 F 3 3 K3 k 4. 4 mi K5 из R 4 4