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Finite Element Analysis

2 (15%) Consider a tapered bar of circular cross section. The length of the bar is 1m, and the radius varies as r(x)= 0.05–0.04x, where r and x are in meters. Assume Young's modulus E = 100 MPa. Both ends of the bar are fixed and F = 10,000 N is applied at the center. Determine the displacements, axial force distribution and the wall reactions using four elements of equal length.

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Question 44373

Finite Element Analysis

For the planar truss below, determine the nodal displacements in the Global Coordinate system using the finite element direct method.
Assume all the truss members are of the same Young's modulus E = 65x10° N/m?. Element 1 and element 3 have the same cross-sectional area of 0.01 m and the cross-sectional area of element 2 is0.02 m. Do not rename the nodes or elements.

Finite Element Analysis

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.

Finite Element Analysis

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.

Finite Element Analysis

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.

Finite Element Analysis

: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.

Finite Element Analysis

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)

Finite Element Analysis

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

Finite Element Analysis

Problem 2: Determine the displacements and rotations in the beam shown in the following figure. The beam is fixed at both ends and was discretized into four elements as shown below. (EI = (101º) N-cm² throughout the beam length).

Finite Element Analysis

Problem 1: Calculate the y-component of the displacement of node 4 and the axial force in each element for the three-bar truss shown in the figure. Let E = 2 (107)N/cm? and A = 5 cm? for each element.

Finite Element Analysis

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.