Finite Element Analysis

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.

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

Determine the nodal displacements and reaction forces using the finite element direct method for the1-D bar elements connected as shown below. Do not rename the nodes or elements. Assume that bars can only undergo translation in x (1 DOF at each node). Nodes 1 and 4 are fixed.Elements 1, 2 and 3 have Young's Modulus of Ei=300 Pa, E2=200 Pa, E3=200 Pa. All elements have length of 1 m and cross-sectional area of 1 m?. There is an applied external force acting at Node 2 of20 N.

\left(T_{m-1}+T_{m-1 \Omega}\right)+2 \frac{h \Delta x}{k} T_{=-2}\left(\frac{h \Delta x}{k}+1\right) T_{m=}=0 Node at an external corner with convection

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)

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.

2\left(T_{m-1, n}+T_{m+1+1}\right)+\left(T_{m+1, n}+T_{m a-1}\right) +2 \frac{h \Delta x}{k} T_{m}-2\left(3+\frac{h \Delta x}{k}\right) T_{m=}=0 Node at an internal corner with convection

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.