Determine the temperature distribution by using two equal

Question

Question 1

Given the following tapered fin, determine the temperature distribution by using two equal length elements.

You may assume that the cross-sectional area A‾ = (A_i + A_j)/2 where A_i and A_j are the cross-sectional areas of element e at nodes i and j respectively.

For a tapered element:

                 [3p_i + p_j       p_i + p_j]
K_h = hL/12
                  [p_i + p_j         p_i + 3p_j]

Question 2

(a) Consider a beam element of length L. Write down the cubic spline functions N_i(x) for this element.

(b) Given the following beam with E = 21 x 10¹⁰N/m² and I = 1.9 x 10^-6m⁴, discretize the beam into 3 elements and determine the load variation w(i)(x), for i = 1,2,3.

(c) Given that the load vectors due to the distributed loads are given by
                Xe+1
w_i^(e) =      ∫   N_i^(e) w⁽ⁱ⁾(x)dx
               xe
Obtain the load vectors w₁, w₂, w₃.

(d) By taking moments to be positive in the counterclockwise direction and downward forces to be positive, compute the element stiffness matrices and assemble the global stiffness matrix.

(e) Solve for the unknown transverse nodal displacements v_i and rotations Φ_i

Note that for the given sign convention:

Question 3

a) Derive the shape functions for the following four-noded rectangular element

b) Given that the x,y coordinates may be expressed in terms of curvilinear coordinates by x = x(s, t) and y = y(s, t) and that for the rectangular case, s = x/b, and t = y/a, where s, t replace the x and y global coordinates. Rewrite the bilinear shape functions obtained in (a), in the natural coordinate system.

c) Find the values for ∂N₁/∂x and ∂N₁/∂y at x = 3 and y = 3 for the element shown below, given that the x and y transformations are expressed in the form:

x = 2s + 4
y = 2t + 4