FINAL INDEPENDENT PROJECT | To numerate design parameters according to Theories of Failure.

AIM : To numerate design parameters according to Theories of Failure.

OBJECTIVE :

1. To calculate principle stresses and maximum shear stress for 3D loading,
2. To calculate factor of safety for different theories of failure and determinig which is suitable.

INTRODUCTION : The measurement and study of the stress-strain-strength behavior of any design components in general stress states involving principal

stress rotation are necessary and valuable. The mechanics of deformable solids is more concerned with the internal forces and associated changes in the

geometry of the components involved. Of particular importance are the properties of the materials used, the strength of which will determine whether the

components fail by breaking in service, and the stiffness of which will determine whether the amount of deformation they suffer is acceptable. Therefore, the

subject of mechanics of materials or strength of materials is central to the whole activity of engineering design. Usually the objectives in analysis here will be

the determination of the stresses, strains, and deflections produced by loads.

This project helps us to determine the values of stresses applied on a body if the load acting on it and it's cross-sectional area is known (e.g. for students and

teachers in the som practical laboratory) and to determine which theories of failure is suited for a particular material.

THEORY : Let us introduce the concept of stress as we know that the main problem of engineering mechanics of material is the investigation of the internal

resistance of the body, i.e. the nature of forces set up within a body to balance the effect of the externally applied forces.

The externally applied forces are termed as loads. These externally applied forces may be due to any one of the reason.

(i) due to service conditions

(ii) due to environment in which the component works

(iii) through contact with other members

(iv) due to fluid pressures

(v) due to gravity or inertia forces.

As we know that in mechanics of deformable solids, externally applied forces acts on a body and body suffers a deformation. From equilibrium point of view,

this action should be opposed or reacted by internal forces which are set up within the particles of material due to cohesion.

These internal forces give rise to a concept of stress. Therefore, let us define a stress Therefore, let us define a term stress

Stress: Let us consider a rectangular bar of some cross – sectional area and subjected to some load or force (in Newtons )

Let us imagine that the same rectangular bar is assumed to be cut into two halves at section XX. The each portion of this rectangular bar is in equilibrium under

the action of load P and the internal forces acting at the section XX has been shown

Now stress is defined as the force intensity or force per unit area. Here we use a symbol to represent the stress.

Where A is the area of the X – section.

The basic units of stress in S I units i e (International system) are N / m (or Pa)

The basic units of stress in S.I units i.e. (International system) are N / m (or Pa)

Stress Tensor Matrix : In three dimensions, in each orthogonal direction X, Y & Z, there could be one normal and two shear stresses. Thus the most generalized

state stress at a point in 3D is as shown below. It is also conveniently described by a stress tensor as follows:

Theories of failure : Theories of failure are those theories which help us to determine the safe dimensions of a machine component when it is subjected to

combined stresses due to various loads acting on it during its functionality.

Some examples of such components are as follows:

1. I.C. engine crankshaft
2. Shaft used in power transmission
3. Spindle of a screw jaw
4. Bolted and welded joints used under eccentric loading

Theories of failure are employed in the design of a machine component due to the unavailability of failure stresses under combined loading conditions.

Theories of failure play a key role in establishing the relationship between stresses induced under combined loading conditions and properties obtained from

tension test like ultimate tensile strength (Sut) and yield strength (Syt).

Various Theories of Failure

1. Maximum Principal Stress theory also known as RANKINE’S THEORY
2. Maximum Shear Stress theory or GUEST AND TRESCA’S THEORY
3. Maximum Principal Strain theory also known as St. VENANT’S THEORY
4. Total Strain Energy theory or HAIGH’S THEORY
5. Maximum Distortion Energy theory or VONMISES AND HENCKY’S THEORY

CODING SECTION

% A program to calculate principle stresses for 3D loading 
close all
clear all
clc
% Subscript for different parameters %
% N_x = normal stress along x-axis | t_xy = shear stress along x-y plane %
% N_y = normal stress along y-axis | t_yz = shear stress along y-z plane %
% N_z = normal stress along z-axis | t_zx = shear stress along z-x plane %
% s1 = principle stress 1 | T_xy = max shear stress along x-y plane
% s2 = principle stress 2 | T_yz = max shear stress along y-z plane
% s3 = principle stress 3 | T_zx = max shear stress along z-x plane
% fos = factor of safety %
% Syt = strength in tension %
% Syc = strength in compression( Assumption Syt=Syc) %
% Ssy = strength in shear %
% mu = Poission's ratio %

% Inputting data stresses acting on body -
N_x = input('Please enter normal stess along x-axis (in MPa) = ');
N_y = input('Please enter normal stess along y-axis (in MPa) = ');
N_z = input('Please enter normal stess along z-axis (in MPa) = ');
t_xy = input('Please enter shear stress along x-y plane (in MPa) = ');
t_yz = input('Please enter shear stress along y-z plane (in MPa) = ');
t_zx = input('Please enter shear stress along z-x plane (in MPa) = ');
% Calculation of principle stresses -
syms I1 I2 I3 S ;
I1 = N_x + N_y + N_z;
A = [N_x , t_xy ;t_xy , N_y];
a = det(A);
B = [N_y , t_yz ;t_yz , N_z];
b = det(B);
C = [N_x , t_zx ;t_zx , N_z];
c = det(C);
I2 =a+b+c;
E = [N_x, t_xy, t_zx ; t_xy, N_y, t_yz ; t_zx , t_yz , N_z];
I3 = det(E);
equation_1 = S.^3 - I1.*S.^2 + I2.*S -I3 == 0;
Ans = vpasolve(equation_1,S);
sigma1 = Ans(1 , :);
sigma2 = Ans(2 , :);
sigma3 = Ans(3 , :);
fprintf('The principle stresses are %.3f MPan',sigma1)
fprintf('The principle stresses are %.3f MPan',sigma2)
fprintf('The principle stresses are %.3f MPan',sigma3)
% Calculation of Max and Min Shear stresses -
if sigma1>sigma2 && sigma1> sigma3
 if sigma2>sigma3
 T1 = (sigma1-sigma3)/2;
 t1 = (sigma2-sigma3)/2;
 fprintf('The Maximum Principle stress is %.3f MPan',sigma1)
 fprintf('The Minimum Principle stress is %.3f MPan',sigma3)
 fprintf('The Maximum Shear stress is %.3f MPan',T1)
 fprintf('The Minimum Shear stress is %.3f MPan',t1)
 else
 T1 = (sigma1-sigma2)/2;
 t1 = (sigma3-sigma2)/2;
 fprintf('The Maximum Principle stress is %.3f MPan',sigma1)
 fprintf('The Minimum Principle stress is %.3f MPan',sigma2)
 fprintf('The Maximum Shear stress is %.3f MPan',T1)
 fprintf('The Minimum Shear stress is %.3f MPan',t1)
 end
end
if sigma2>sigma1 && sigma2> sigma3
 if sigma1>sigma3
 T2 = (sigma2-sigma3)/2;
 t2 = (sigma1-sigma3)/2;
 fprintf('The Maximum Principle stress is %.3f MPan',sigma2)
 fprintf('The Minimum Principle stress is %.3f MPan',sigma3)
 fprintf('The Maximum Shear stress is %.3f MPan',T2)
 fprintf('The Minimum Shear stress is %.3f MPan',t2)
 else
 T2 = (sigma2-sigma1)/2;
 t2 = (sigma3-sigma1)/2;
fprintf('The Maximum Principle stress is % 3f MPan' sigma2)

fprintf( The Maximum Principle stress is %.3f MPan ,sigma2)
 fprintf('The Minimum Principle stress is %.3f MPan',sigma1)
 fprintf('The Maximum Shear stress is %.3f MPan',T2)
 fprintf('The Minimum Shear stress is %.3f MPan',t2)
 end
end
if sigma3>sigma1 && sigma2 < sigma3
 if sigma1>sigma2
 T3 = (sigma3-sigma2)/2;
 t3 = (sigma1-sigma2)/2;
 fprintf('The Maximum Principle stress is %.3f MPan',sigma3)
 fprintf('The Minimum Principle stress is %.3f MPan',sigma2)
 fprintf('The Maximum Shear stress is %.3f MPan',T3)
 fprintf('The Minimum Shear stress is %.3f MPan',t3)
 else
 T3 = (sigma3-sigma1)/2;
 t3 = (sigma2-sigma1)/2;
 fprintf('The Maximum Principle stress is %.3f MPan',sigma3)
 fprintf('The Minimum Principle stress is %.3f MPan',sigma1)
 fprintf('The Maximum Shear stress is %.3f MPan',T3)
 fprintf('The Minimum Shear stress is %.3f MPan',t3)
 end
end
% Entering a set of data for different materials 
fprintf('Choose from the following materials :n')
fprintf('1. Mild_Steeln2. Cr_V_Steeln3. HDPEn4. Stainless_Steeln5. Al_alloy6061n')
Mild_Steel = 250;
Cr_V_Steel = 620;
HDPE = 30;
Stainless_Steel = 500;
Cast_Iron = 130;
Al_alloy6061 = 240;
Sy = [Mild_Steel,Cr_V_Steel,HDPE,Stainless_Steel,Cast_Iron,Al_alloy6061];
m = [0.27,0.3,0.46,0.31,0.22,0.32];
list = {'Mild_Steel','Cr-V_Steel','HDPE','Stainless_Steel','Cast_Iron','Al_alloy6061'};
[indx,tf] = listdlg('ListString',list);
% Call different functions depending on user's selection
switch indx
 case 1
 Mild_Steel;
 Syt = Sy(:,1);
 mu = m(:,1); 
 case 2
 Cr_V_Steel;
 Syt = Sy(:,2);
 mu = m(:,2); 
 case 3
 HDPE;
 Syt = Sy(:,3);
 mu = m(:,3); 
 case 4
 Stainless_Steel;
 Syt = Sy(:,4);
 mu = m(:,4); 
 case 5
 Cast_Iron;
 Syt = Sy(:,5);
 mu = m(:,5); 
 case 6
 Al_alloy6061;
 Syt = Sy(:,6);
 mu = m(:,6); 
end
 
% Computing design parameters according to Theories of Failure considering-
% max and min principle stresses and max shear stress in case of 2D loading
% Max Principle Stress Theory//RANKINE THEORY(based for brittle material)
fprintf('Maximum Principle Stress Theoryn')
if sigma1>sigma2 && sigma1>sigma3
 sigma1 = k1;
end
if sigma2>sigma1 && sigma2>sigma3
 sigma2 = k1;
end
if sigma3>sigma1 && sigma2= 1.2 
 fprintf('* The design conditions are safe according to Maximum Principle Stress Theory n')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 
% Max Shear Stress Theory//COULOMB & TRESCA THEORY(based for brittle material)
fprintf('Maximum Shear Stress Theoryn')
if sigma1>sigma2 && sigma1> sigma3
 if sigma2>sigma3
 fos_2 = Syt/(sigma1-sigma3);
 fprintf('Factor of safety is %.3fn',fos_2)
 Ssy = 0.5*Syt;
 if fos_2 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Shear Stress Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 else
 fos_2 = Syt/(sigma1-sigma2);
 fprintf('Factor of safety is %.3fn',fos_2)
 Ssy = 0.5*Syt;
 if fos_2 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Shear Stress Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 end
end
 if sigma2>sigma1 && sigma2> sigma3
 if sigma1>sigma3 
 fos_2 = Syt/(sigma2-sigma3);
 fprintf('Factor of safety is %.3fn',fos_2)
 Ssy = 0.5*Syt;
 if fos_2 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Shear Stress Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 else
 fos_2 = Syt/(sigma2-sigma1);
 fprintf('Factor of safety is %.3fn',fos_2)
 Ssy = 0.5*Syt;
 if fos_2 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Shear Stress Theory n')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 
 end
 end 
if sigma3>sigma1 && sigma2 < sigma3
 if sigma1>sigma2
 fos_2 = Syt/(sigma3-sigma2);
 fprintf('Factor of safety is %.3fn',fos_2)
 Ssy = 0.5*Syt;
 if fos_2 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Shear Stress Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 else
 fos_2 = Syt/(sigma3-sigma1);
 fprintf('Factor of safety is %.3fn',fos_2)
 Ssy = 0.5*Syt;
 if fos_2 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Shear Stress Theory n')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 
 end
end 
 
% Max Distortion Energy Theory//VON MISES & HENCKY ( based for ductile
% material)
fprintf('Maximum Distortion Energy Theoryn')

p gy y
deno_term_1 = (sigma1-sigma2)^2;
deno_term_2 = (sigma2-sigma3)^2;
deno_term_3 = (sigma1-sigma3)^2;
deno_term_4 = (deno_term_1+deno_term_2+deno_term_3)*0.5;
deno_term_5 = deno_term_4^0.5;
fos_3 = Syt/deno_term_5;
Ssy = 0.577*Syt;
fprintf('Factor of safety is %.3fn',fos_3)
 if fos_3 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Distortion Energy Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
% Max Principle Strain Theory//ST. VENANAT THEORY ( based for brittle
% material)
fprintf('Maximum Principle Strain Theoryn')
if sigma1>sigma2 && sigma1> sigma3
 term_1 = sigma2 + sigma3;
term_2 = mu*term_1;
fos_4 = Syt/(sigma1-term_2);
fprintf('Factor of safety is %.3fn',fos_4)
 if fos_3 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Principle Strain Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
 end
if sigma2>sigma1 && sigma2> sigma3
 
term_1 = sigma1 + sigma3;
term_2 = mu*term_1;
fos_4 = Syt/(sigma2-term_2);
fprintf('Factor of safety is %.3fn',fos_4)
 if fos_3 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Principle Strain Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
end
if sigma3>sigma1 && sigma2< sigma3
 
term_1 = sigma1 + sigma2;
term_2 = mu*term_1;
fos_4 = Syt/(sigma3-term_2);
fprintf('Factor of safety is %.3fn',fos_4)
 if fos_3 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Principle Strain Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end
end
% Max Strain Energy Theory//HAIGH&BELTRAMI THEORY ( based for ductile material)
fprintf('Maximum Strain Energy Theoryn')
term_3 = sigma1*sigma2 + sigma2*sigma3 + sigma3*sigma1;
term_4 = 2*mu*term_3;
term_5 = sigma1^2 + sigma2^2 + sigma3^2;
term_6 = (term_5-term_4)^0.5;
fos_5 = Syt/term_6;
fprintf('Factor of safety is %.3fn',fos_5)
 if fos_3 > 1.2 
 fprintf('* The design conditions are safe according to Maximum Strain Energy Theoryn')
 else 
 fprintf('* The design conditions are not safe and prone to failuren')
 end

Output:

1. Maximum Shear Stress theory is suitable for ductile materials and gives oversafe design i.e. safe and uneconomic design.
2. Maximum destortion energy theory is best theory of failure for ductile material. It gives safe and economic design.
3. Maximum Principle stress theory is suitable for the safe design of machine components made of brittle materials under all loading conditions (tri-axial) etc.
4. because brittle materials are weak in tension. This theory is not suitable for the safe design of machine components made of ductile
5. materials because ductile materials are weak in shear.

Reference

1. Google.com
2. Wikipedia
3. Nptel
4. Research gate