Week 4 Challenge- Shear Forces, Bending Moments and Stiffness/Flexibility Matrix

1) First assume a cantilever beam and develop flexibility matrix as below -

2) Use the equations developed in problem 1 and write a matlab module to calculate coefficients in a flexibility matrix for a cantilever beam

Ans :-

Progamme for above problem -

 function [beam]= beam_problem(node_xy,elements,E,b,h,fy)
no_elements = size(elements,1);
A = b*h;
I = b*h^3/12;
node_x=node_xy(:,1);
node_y=node_xy(:,2);
x_co=node_x(2,1)-node_x(1,1);
y_co=node_y(2,1)-node_y(1,1);
L=sqrt(x_co*x_co+y_co*y_co);
Mz2 = L*fy
v2 =[(fy*(L^3))/(3*E*I)+(Mz2*L^2)/(2*E*I)]
theta = [(fy*(L^2))/(2*E*I)+(Mz2*L^2)/(2*E*I)]
end

Explanation of codes -

function - to create function for cantilever beam flexibility matrix.

size - to calculate the number of elements.

A and I - to calculate area of cross section and moment of inertia resp.

L and Mz2 - to calculate length of beam and moment at z.

v2 and theta - coefficients for flexibility matrix.

end - to end the function.

3) Using the matlab module developed in Problem 2 solve the following beam problem Calculate the Bending Moment and shear force diagram

Ans:- using function generated in solution of problem 2 we get following results.

a) For the cantilever beam subjected to uniformly distributed load as shown below compute the displacement and rotation at end B

Result :-

b) For the cantilevers subjected to point loads as shown below compute the displacement and rotation at end B

Result:-

4) Assume a simply supported beam Develop stiffness matrix for the same using slope-deflection equations

Ans :-

Using equilibrium equattion the stiffness matrix generated is

5) Using equations developed in problem 4 develop a matlab module to calculate coefficients in a stiffness matrix for simply supported beam

Ans :-

Programme for above -

 function [frame_stiffness]=Frame_siffness_matrix(node_xy,elements,E,b,h)
no_elements = size(elements,1);
A = b*h;
I = b*h^3/12;
node_x=node_xy(:,2);
node_y=node_xy(:,3);
x_co=node_x(2,1)-node_x(1,1);
y_co=node_y(2,1)-node_y(1,1);
length_element=sqrt(x_co*x_co+y_co*y_co);
frame_stiffness = zeros(6,6)

for n = 1:no_elements
    k11 = A*E/length_element;
    k22 = 12*E*I/length_element^3;
    k23 = 6*E*I/length_element^2;
    k33 = 4*E*I/length_element;
    k36 = 2*E*I/length_element;

    frame_stiffness(:,:) = [k11 0 0 -k11 0 0;0 k22 k23 0 -k22 k23;0 k23 k33 0 -k23 k36;-k11 0 0 k11 0 0;0 -k22 -k23 0 k22 -k23;0 k23 k36 0 -k36 -k33]
end
end

Explanations of code -

function - to create function for cantilever beam flexibility matrix.

size - to calculate the number of elements.

A and I - to calculate area of cross section and moment of inertia resp.

zeros - to create zero matrix.

for - to loop the equation over no of elemets.

frame_stiffness - to create 6 by 6 matrix.

end - to end the function.

6) Using the matlab module developed in problem 5 solve the following beam problem Calculate bending moment and shear force diagram

Ans :-