FINAL INDEPENDENT PROJECT
AIM : To compute design parameters according to Theories of Failure.
OBJECTIVE :
- To calculate principle stresses and maximum shear stress for 3D loading,
- 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 `sigma` to represent the stress.
`sigma = P/A`σ=PA">
Where A is the area of the X – section.
The basic units of stress in S.I units i.e. (International system) are N / m2 (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:
`[sigma_(ij)]` = `[[sigma_x,tau_(xy),tau_(xz)],[tau_(yx),sigma_y,tau_(yz)],[tau_(zx),tau_(zy),sigma_z]]`
Calculation of Principle stress in 3D loading-
`sigma^3-I_1sigma^2+I_2sigma-I_3=0`
where `I_1 = sigma_x+sigma_y+sigma_z`I1=σx+σy+σz">
`I_2=`determinant of `[[sigma_x,tau_(xy)],[tau_(yx),sigma_y]] +[[sigma_y,tau_(yz)],[tau_(yz),sigma_y]]+[[sigma_z,tau_(xz)],[tau_(zx),sigma_z]]`
`I_3 =`determinant of `[[sigma_x ,tau_(xy),tau_(xz)],[tau_(yx),sigma_y,tau_(yz)],[tau_(zx),tau_(zy),sigma_z]]`
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 THEOR
1. Maximum Principal Stress theory - According to this theory, condition for failure is,
Maximum principal stress (`sigma_1`) > failure stresses (Syt or Sut ) and Factor of safety (F.O.S) = 1
Condition for safe design,
Factor of safety (F.O.S) > 1
Maximum principal stress (`sigma_1`) ≤ Permissible stress (`sigma_(per`)
where permissible stress = Failure stress`/`Factor of safety = Syt`/`N or Sut`/`N
`sigma_1`≤ Syt`/`N or Sut`/`N
2. Maximum Shear Stress theory - According to this theory, condition for failure, Maximum shear stress induced at a critical Yield strength in shear under tensile point under triaxial combined stress test
Absolute `tau_(max)`> (Ssy) or Syt`/`2 ,
Condition for safe design,
Maximum shear stress induced at a critical ≤ Permissible shear stress (`tau_(per)`) tensile point under triaxial combined stress where,
Permissible shear stress = Yield strength in shear under tension test`/`Factor of safety = (Ssy)`/`N = Syt`/`2N
Absolute `tau_(max)` ≤ (Ssy)`/`N or Syt`/`2N
3. Maximum Principal Strain theory - According to this theory, condition for failure,
Maximum Principal strain (`epsilon_1`) > Yielding strain under tensile test (`epsilon_(y.p.)`)
`epsilon_1`>`epsilon_(y.p.)` or Syt`/`E
where E is Young’s Modulus of Elasticity.
Condition for safe design,
Maximum Principal strain ≤ Permissible strain
where Permissible strain = Yielding strain under tensile test`/`Factor of safety = Syt`/`EN
`sigma_1-mu(sigma_2+sigma_3) =` Syt`/`N
4. Total Strain Energy theory - According to this theory, condition for failure,
Total Strain Energy per unit volume (T.S.E. /vol) > Strain energy per unit volume at yield point under tension test (S.E /vol) Y.P.
Condition for safe design,
Total Strain Energy per unit volume ≤ Strain energy per unit volume at yield point under tension test,
`σ_1^2 + σ_2^2 + σ_3^2 - 2μ (σ_1 σ_2 + σ_2 σ_3 +σ_3 σ_1) ≤ ( (Syt) /N )^2`
5. Maximum Distortion Energy Theory - According to this theory, condition for failure,
Maximum Distortion Energy/volume > Distortion energy/volume at yield point under tension
(M.D.E/vol) test (D.E/vol) Y.P.
Condition for safe design,
Maximum Distortion Energy/volume ≤ Distortion energy/volume at yield point
under tension test
`[(σ_1 - σ_2)^2 + (σ_2 - σ_3)^2 + (σ_3 - σ_1)^2] ≤ 2 ((Syt)/N )^2`
SOLUTION STEPS :
- First user will be asked to enter normal stress acting on a body in respective directions which can be found experimentally. This was done using input command.
- Then principle stresses are calculated using above given formulas and maximum shear stress is also calculated and displayed in the command window.
- From the given set of materials, user can choose respective material and factor of safety will be calculated based on different theories of failure.
- In this way, one can find out factor of safety for any design components for given stress value acting on a body.
MATLAB CODE :
% 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 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')
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')
endRESULTS :
- First user will enter data of his/her choice, for our case had taken data from a question of our curriculum course, as shown here. Then user will choose desired material from a list of options shown on screen and all the results will be calculated in the command window.
- All the theories of the failure will give the same result when one of the principal stresses is very large as compared to the other principal stresses.
- For pure shear state of stress, all the theories of failure will give the different result.
(a) MDET and MSST will be used under pure shear state of stress.
(b) MDET will be preferred over MSST. - Maximum Shear Stress theory is suitable for ductile materials and gives oversafe design i.e. safe and uneconomic design.
- Maximum Distortion energy theory is best theory of failure for ductile material. It gives safe and economic design.
- Maximum Principle stress theory is suitable for the safe design of machine components made of brittle materials under all loading conditions (tri-axial, biaxial etc.) because brittle materials are weak in tension. This theory is not suitable for the safe design of machine components made of ductile materials because ductile materials are weak in shear.
REFERENCES :
- NPTEL pdfs ,
- Introduction to Machine Design by V.B Bhandari ,
- Wikipedia ,
- Matlab forum and help page.