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A Fixed Free Cantilever Beam Deflection and Stress Analysis with Matlab

This tutorial is related to analyzing the deflection and stress distribution of a cantilever beam. A fixed-free cantilever beam, (Young’s Modulus 𝐸, 𝑏×ℎ cross-section, length 𝐿) is supported at its left-hand end. A Force 𝐹 is applied at a distance 𝑎 from the left-hand end. We will first calculate the shape of the deflected beam and plot the result of deflection, shear and bending moment diagram. Next, we will generate a 2D contour map of the stress distribution along the beam.

A fixed-free cantilever beam

Length of the beam, L = 1000 mm

Width of the beam, b = 50 mm

Height of the beam, h = 155 mm

Distance of the force location, a = 800 mm

Modulus of elasticity, E = 200 Gpa

Applied force, F = 25 KN

The shape of the deflected beam is calculated as [2-4],













The moment of inertia is calculated as,

I = 1/12 bh^3



Cantilever beam deflection
















Cantilever beam deflection
















The following Matlab codes are used to calculate the beam deflection and plot the results.

close all
clear 
clc
 
% Parameters from the question
E = 200*10^9;
a = 0.8;
b = 0.05;
h = 0.155;
I = (1/12)*(b*h^3);
f = 25000;
 
x = 0:0.01:0.8;
delta_1 = - (f*x.^2)./(6.*E.*I).*(3.*a - x);  % for 0 <= x <= a
delta_2 = - (f*a.^2)./(6.*E.*I).*(3.*x - a);  % for a <= x <= L
 
figure(1)
plot(x, delta_1)
xlabel('Length (m)')
ylabel('Beam Deflection (m)')
title('Cantilever Beam Deflection for 0 <= x <= a')
 
figure(2)
plot(x, delta_2)
xlabel('Length (m)')
ylabel('Beam Deflection (m)')
title('Cantilever Beam Deflection for a <= x <= L')


The shear force and bending moment diagrams for the fixed free cantilever beam is shown below:

Shear force and bending moment diagram















The Matlab codes for the calculation of shear and bending moments as well as plotting these diagrams are provided below.


MATLAB Codes

close all
clear
clc
 
L = 1; % Unit in meters
n = 2; % Number of point loads on the beam
 
for i = 1:n
fprintf('Enter the point load and distance where load acts for %d node\n',i)
Wc(i)=input('Enter the load in Newton\n');
Lc(i) = input('Enter the distance of the point load from the fixed end\n');
            
if Lc(i)>L || Lc(i)<0                
    error('Please check the Length')            
end
end
 
NL = zeros(1,n);        
NW = zeros(1,n);
       
for i=1:n
    [a,b]=max(Lc);            
    NL(i)=a;           
    NW(i)=Wc(b);          
    Lc(b)=[];           
    Wc(b)=[];        
end
 
NL(n+1) = 0;
        
% Shear force diagram
figure(1)
Ra = sum(NW);
if NL(1)==L
X1 = L;
F1 = NW(1);
elseif i==1
X1 = [L NL(1) NL(1) 0];
F1 = [0 0 Ra Ra];
else
X1 = [L NL(1)];
F1 = [0 0];
end
S=[]; X=[]; F=[];
        if n>=2
            for i=2:n+1
                x = [NL(i-1) NL(i)];
                S= [S NW(i-1)];
                f = sum(S);
                X= [X x];
                F = [F f f];
            end
        end
subplot(2,1,1)
        
plot([X1 X],[F1 F],'b')
xlim([0 L]);
hline = refline(0,0);
hline.Color = 'k';
legend('Shear Force','Reference')          
title('Shear Force Diagram of Cantilever Beam')
xlabel('Length of the Beam in Meter')
ylabel('Shear Force in Newton')
  
hold on
grid on
fprintf('Reaction Force =%d N\n',Ra)
    
% Bending moment diagram
subplot(2,1,2)
if NL(1)<L
X1=NL(1): L;
M1 = zeros(size(X1));
plot(X1,M1,'r')
hold on 
grid on
end
 
Xm=[]; 
Mm=[];
    
for i=1:n
X = NL(i+1):0.1:NL(i);
M = -(NW(1,1:i)*((NL(1,1:i))'-newX(X,i))) ;
Xm =[X Xm];
Mm =[M Mm];
end    
    
plot(Xm,Mm,'r'
xlim([0 L]);
hline = refline(0,0);
hline.Color = 'k';
legend('Bending Moment','Reference')  
    
title('Bending Moment Diagram of Cantilever Beam')
xlabel('Length of the Beam in Meter')
ylabel('Bending Moment in Newton-Meter')
hold off
grid on
 
function x = newX(X,i)                                                
[~,d] = size(X);                                                 
x = zeros(i,d);                                                
for j=1:i                                                        
x(j,1:d) = X;                                                   
end                                                             
end    

The bending stress in the beam is calculated as [2-4],





Where, δ is calculated as,








In the following, the stress distributions for the cantilever beam are shown. The Matlab codes are appended after the results.

Mesh plot for cantilever beam deflection
















Contour plot for cantilever beam deflection
















Mesh plot for cantilever beam deflection
















Contour plot for cantilever beam deflection
















MATLAB CODES

close all
clear 
clc
 
% Parameters from the question
L = 1;
E = 200*10^9;
a = 0.8;
b = 0.05;
h = 0.155;
I = (1/12)*(b*h^3);
f = 25000;
M1 = f*a;
M2 = f*(L-a);
 
x = 0:0.01:0.8;
y = 0:0.01:0.8;
 
delta_1 = - (f*x.^2)./(6.*E.*I).*(3.*a - x);  % for 0 <= x <= a
delta_2 = - (f*a.^2)./(6.*E.*I).*(3.*x - a);  % for a <= x <= L
 
sigma_1 = (M1.*delta_1)./I;
 
x = rand(100,1)*4-2;
y = rand(100,1)*4-2;
z = (M1.*((f*x.^2)./(6.*E.*I).*(3.*a - x)))./I;
z1 = (M2.*((f*a.^2)./(6.*E.*I).*(3.*x - a)))./I;
 
F = TriScatteredInterp(x,y,z);
ti = -1:.25:1;
[qx,qy] = meshgrid(ti,ti);
qz = F(qx,qy);
 
figure(1)
scatter3(x,y,z)
title('Mesh Plot for 0 <= x <= a')
hold on
mesh(qx,qy,qz)
 
figure(2)
contour(qx,qy,qz)
title('Contour Plot for 0 <= x <= a')
 
F = TriScatteredInterp(x,y,z1);
ti = -1:0.25:1;
[qx,qy] = meshgrid(ti,ti);
qz = F(qx,qy);
 
figure(3)
scatter3(x,y,z)
title('Mesh Plot for a <= x <= L')
hold on
mesh(qx,qy,qz)
 
figure(4)
contour(qx,qy,qz)
title('Contour Plot for a <= x <= L')



REFERENCES

[1] MATLAB 2020b Academic Version from MathWorks (https://www.mathworks.com/)

[2] Budynas-Nisbett, "Shigley's Mechanical Engineering Design," 8th Ed.

[3] Gere, James M., "Mechanics of Materials," 6th Ed.

[4] Lindeburg, Michael R., "Mechanical Engineering Reference Manual for the PE Exam," 13th Ed.

[5] Solid Mechanics Part I: An Introduction to Solid Mechanics by ‪Piaras Kelly‬‬



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