Radius of the beam r 2 cm and elastic modulus E 210 GPa As

Radius of the beam r = 2 cm, and elastic modulus E = 210 GPa. Assume the beam is undergoing a small linearly elastic deformation. Using MATLAB, analytically compute the deflection at the free end of the beam using the following equation: delta = Px^2/6EI(3L - x) Using the finite element method (FEM) with five beam elements of equal length: Generate the individual stiffness matrices K_e for each element of the beam and assemble them into the global stiffness matrix K_c. Print each of the individual matrices and the global stiffness matrix. Calculate and print the displacements and rotations (in degree) for each node of the beam. On the same figure, plot the vertical deflection throughout the beam for both analytical and FEM solution. Include title, axis labels, legends, and grid Compare and discuss the two solutions. Discuss how you would increase the bending stiffness of the beam when the cross-sectional area and material must be kept the same.

Solution

a)   I = pi*r^4/4 = pi * 0.02^4/4 = 1.25664 *10^-7 m^4

     deflection = 2*P*l^3 / 3*E *I

                     = 2 * 100 * 1^3 / (3 * 210 * 10^9 * 1.25664 * 10^-7)

                     = 0.002526 m

Matlab Code

P = 100;
l = 1;
r = 0.02;
Em = 210E9;
I = pi * r^4/4 ;
x = linspace(0,l,100);
deflection_freeend = P * x.^2 .* (3*l-x) / (3*Em*I);
plot(x,deflection_freeend)

d) Bending stiffness of a cantilever beam is given by

    k = 3 * E * I / l^3

    for a circular beam   I = pi * r^4 /4    (where r is the radius of cross section)

    E & Area are constant , therefore   E and r is a constant

    The only way to increase the stiffness is to reduce the length of the beam.