Strength of Materials Formulas for GATE ME Exam | Strength of Materials (SOM) - Mechanical Engineering PDF Download

 Page 2


 
 
 
 
 
Brinell Hardness Number  
(BHN) 
 
 
 
 
Elastic constants: 
 
 
where, P = Standard load, D = Diameter of steel ball, and d = Diameter of the indent. 
 
 
Page 3


 
 
 
 
 
Brinell Hardness Number  
(BHN) 
 
 
 
 
Elastic constants: 
 
 
where, P = Standard load, D = Diameter of steel ball, and d = Diameter of the indent. 
 
 
 
 
 
 
 
Axial Elongation of Bar Prismatic Bar Due to External Load 
                                                             ?=
????
????
  
 
 
 
Elongation of  Prismatic Bar Due to Self  Weight 
?=
????
?????? =
?? ?? ?? ????
 
Where ?? is specific weight 
Elongation of Tapered Bar  
• Circular Tapered 
?=
?????? ?? ?? ?? ?? ?? ?? 
 
 
• Rectangular Tapered 
?=
???? ?????? ?? (
?? 2
?? 1
)
?? . ?? (?? 2
- ?? 1
)
 
 
Stress Induced by Axial Stress and Simple Shear 
• Normal stress 
  
• Tangential stress 
 
  
Principal Stresses and Principal Planes 
• Major principal stress 
 
 
• Major principal stress 
 
 
 
 
Page 4


 
 
 
 
 
Brinell Hardness Number  
(BHN) 
 
 
 
 
Elastic constants: 
 
 
where, P = Standard load, D = Diameter of steel ball, and d = Diameter of the indent. 
 
 
 
 
 
 
 
Axial Elongation of Bar Prismatic Bar Due to External Load 
                                                             ?=
????
????
  
 
 
 
Elongation of  Prismatic Bar Due to Self  Weight 
?=
????
?????? =
?? ?? ?? ????
 
Where ?? is specific weight 
Elongation of Tapered Bar  
• Circular Tapered 
?=
?????? ?? ?? ?? ?? ?? ?? 
 
 
• Rectangular Tapered 
?=
???? ?????? ?? (
?? 2
?? 1
)
?? . ?? (?? 2
- ?? 1
)
 
 
Stress Induced by Axial Stress and Simple Shear 
• Normal stress 
  
• Tangential stress 
 
  
Principal Stresses and Principal Planes 
• Major principal stress 
 
 
• Major principal stress 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Principal Strain  
 
 
 
 
 
 
Mohr’s Circle- 
 
 
 
 
 
 
 
STRAIN ENERGY 
Energy Methods: 
(i) Formula to calculate the strain energy due to axial loads (tension): 
U  = ?  P ² / ( 2AE)dx limit 0 toL 
 
Where, P = Applied tensile load, L =  Length of the member ,  A = Area of the member, and 
E = Young’smodulus. 
(ii) Formula to calculate the strain energy due tobending: 
U  = ?   M ² /  ( 2EI) dx limit 0 toL 
 
Where, M = Bending moment due to applied loads, E = Young’s modulus, and I = Moment of 
inertia. 
(iii) Formula to calculate the strain energy due totorsion: 
U  = ?  T ² / ( 2GJ)  dx limit 0 toL 
Page 5


 
 
 
 
 
Brinell Hardness Number  
(BHN) 
 
 
 
 
Elastic constants: 
 
 
where, P = Standard load, D = Diameter of steel ball, and d = Diameter of the indent. 
 
 
 
 
 
 
 
Axial Elongation of Bar Prismatic Bar Due to External Load 
                                                             ?=
????
????
  
 
 
 
Elongation of  Prismatic Bar Due to Self  Weight 
?=
????
?????? =
?? ?? ?? ????
 
Where ?? is specific weight 
Elongation of Tapered Bar  
• Circular Tapered 
?=
?????? ?? ?? ?? ?? ?? ?? 
 
 
• Rectangular Tapered 
?=
???? ?????? ?? (
?? 2
?? 1
)
?? . ?? (?? 2
- ?? 1
)
 
 
Stress Induced by Axial Stress and Simple Shear 
• Normal stress 
  
• Tangential stress 
 
  
Principal Stresses and Principal Planes 
• Major principal stress 
 
 
• Major principal stress 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Principal Strain  
 
 
 
 
 
 
Mohr’s Circle- 
 
 
 
 
 
 
 
STRAIN ENERGY 
Energy Methods: 
(i) Formula to calculate the strain energy due to axial loads (tension): 
U  = ?  P ² / ( 2AE)dx limit 0 toL 
 
Where, P = Applied tensile load, L =  Length of the member ,  A = Area of the member, and 
E = Young’smodulus. 
(ii) Formula to calculate the strain energy due tobending: 
U  = ?   M ² /  ( 2EI) dx limit 0 toL 
 
Where, M = Bending moment due to applied loads, E = Young’s modulus, and I = Moment of 
inertia. 
(iii) Formula to calculate the strain energy due totorsion: 
U  = ?  T ² / ( 2GJ)  dx limit 0 toL 
 
 
 
 
 
 
Where, T = Applied Torsion , G = Shear modulus or Modulus of rigidity, and J = Polar 
moment ofinertia 
(iv) Formula to calculate the strain energy due to pureshear: 
U  =K  ?  V ² / ( 2GA)  dx limit 0 to L 
Where, V= Shearload 
G = Shear modulus or Modulus of rigidity 
A = Area of cross section. 
K = Constant depends upon shape of cross section. 
 
(v) Formula to calculate the strain energy due to pure shear, if shear stress isgiven: 
U  =  t ² V / ( 2G ) 
 
Where, t = ShearStress 
 
G = Shear modulus or Modulus of rigidity 
V = Volume of the material. 
 
(vi) Formula to calculate the strain energy , if the moment value isgiven: 
U  =  M ² L / (2EI) 
 
Where, M = Bending moment 
L = Length of the beam 
E = Young’smodulus 
I = Moment ofinertia 
 
(vii) Formula to calculate the strain energy , if the torsion moment value isgiven: 
U= T ²L /  ( 2GJ) 
 
Where, T = AppliedTorsion 
 
L = Length of the beam 
 
G = Shear modulus or Modulus of rigidity 
J = Polar moment of inertia