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

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 (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.

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

The vertical column shown in the picture has both endsfixed. The elastic modulus E is uniform. Its cross-sectional area A(x) is uniform in the upper part and varieslinearly in the lower part. It carries its own weight, so thatthe load is proportional to the cross-sectional area:

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 1: For the bar subjected to the linear varying axial load shown in Figure, deter- mine the nodal displacements and axial stress distribution using (a)three equal-length elements and (b) five equal-length elements. Let A = 2 in.² and E = 30 x 106 psi. Compare the finite element solution with an exact solution. 60 in. T₁= 10x lb/in.

Q2. (10 points) The nodal coordinates of the triangular element are shown below. At the interior point P, the Y coordinate is 2.7 and N₂ = 0.4. Determine N₁, N3 at point P. Then determine the X coordinate at point P. Nodal coordinates are from nodes 1 to 3: (5,1) (2,6) (-2,-2)

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)

: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.