Finite Element Analysis

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

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.

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

A hot surface at 125°C is to be cooled by attaching 8 cm long, 0.8 cm in diameter aluminum pin fins(k = 237 W/m-°C and a = 97.1 X 106 m²/s) to it. The temperature of the surrounding medium is 17°C, and the heat transfer coefficient on the surfaces is 38 W/m²-°C. Initially, the fins are at a uniform temperature of 29°C, and at time t=0, the temperature of the hot surface is raised to 125°C.Assuming one-dimensional heat conduction along the fin and taking the nodal spacing to be Ax = 2.0 cm and a time step to be At = 1.5 s, determine the nodal temperatures after 5 minutes by using the explicit finite difference method. Also determine how long it will take for steady conditions to be reached.

Consider a 2-D region shown below, where the boundary conditions are shown as symbols on the figure, and their corresponding values are shown in the table. Using the finite difference mesh shown: Write the finite-difference equations for each node. (22 nodes) b) Find the steady state temperatures of all the nodes. (22 nodes)

: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.

A thin, flat component with a step is to be examined, see Fig. 1. The task can be solved with Abaqus or DAEdalon. E = 210000 MPa und v = 0.3 can be used Make a selection for the amount of loading and the geometry of the heel. Investigate the voltage increase factor K, for at least two networks and compare it with the analytical value. Use Eq. (1) and the corresponding coefficients from Table 1. K_{t}=\frac{\sigma_{\max }}{\sigma_{0}}=C_{1}+C_{2}\left(\frac{2 h}{D}\right)+C_{3}\left(\frac{2 h}{D}\right)^{2}+C_{4}\left(\frac{2 h}{D}\right)^{3} \quad \text { mit } h=\frac{D-d}{2}

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.

A single four-node element in its initial position is considered as a section of a component in the plane stress state. The nodal coordinates of the initial position as well as the nodal displacements due to the loading of the entire component are shown in Fig. 1 given. E =76000 MPa and v= 0.25 should be assumed as material values Fig1: Node coordinates undand displacements u ef a planar four-node element (inmm) a) Determine the element stiffness matrix for the element shown. To do this, specify a self-selected node connectivity. Work at the numeric Integration with a Gauss point. b) Which stresses and distortions arise at the integration point?