Formula Sheet: Strength of Materials (SOM) | Strength of Materials (SOM) - Mechanical Engineering PDF Download

 Page 1


                                                       
 
  
 
 
 
I. SIMPLE STRESS AND STRAIN 
 
1. SIMPLE STRESS AND STRAIN FORMULAE HAND BOOK FOR STRENGTH OF 
MATERIALS    
 
N 
Stress(s) ?? =
?? ?? 
Where, 
           s =stress in N/mm
2
 
            P= Load in N 
            A=Area in mm
2
 
Strain (e) ?? =
??? ?? 
Where, 
           e =strain 
          ?L= Change in length or elongation in mm 
            L=Original  length or gauge length in mm 
Youngs 
modulus(E) 
?? =
?? ?? 
Where, 
           E= Youngs modulus or modulus of elasticity in N/mm
2 
           s =stress in N/mm
2
 
           e =strain 
Factor of 
safety(FoS) 
?????? =
?? ?? ?? 
Where, 
           s
u
 =Ultimate stress in N/mm
2
 
           s =Working stress in N/mm
2
 
 
Area(A)
 
?? = ?? /4 × ?? 2
 
?? = ?? /4 × (?? 2
- ?? 2
 
?? = ?? × ?? 
 
Where, 
         A=area in mm
2
 
         D= Major diameter or outer diameter in mm 
         d= Minor diameter or inner diameter in mm 
         b= breadth or wide in mm 
         t= Thickness in mm 
 
2.  UTM (TENSILE TEST) 
 
Yield Stress(sy) 
?? ?? =
?? ?? ?? 
Where, 
           s
y
 = Yield stress in 
N/mm
2
 
            P
y
= Yield Load in N 
            A=Area in mm
2
 
Ultimate Stress(su) 
???? =
???? ?? 
Where, 
           s
u
= Ultimate stress in 
N/mm
2
 
            P
u
= Ultimate Load in 
N 
            A=Area in mm
2
 
Page 2


                                                       
 
  
 
 
 
I. SIMPLE STRESS AND STRAIN 
 
1. SIMPLE STRESS AND STRAIN FORMULAE HAND BOOK FOR STRENGTH OF 
MATERIALS    
 
N 
Stress(s) ?? =
?? ?? 
Where, 
           s =stress in N/mm
2
 
            P= Load in N 
            A=Area in mm
2
 
Strain (e) ?? =
??? ?? 
Where, 
           e =strain 
          ?L= Change in length or elongation in mm 
            L=Original  length or gauge length in mm 
Youngs 
modulus(E) 
?? =
?? ?? 
Where, 
           E= Youngs modulus or modulus of elasticity in N/mm
2 
           s =stress in N/mm
2
 
           e =strain 
Factor of 
safety(FoS) 
?????? =
?? ?? ?? 
Where, 
           s
u
 =Ultimate stress in N/mm
2
 
           s =Working stress in N/mm
2
 
 
Area(A)
 
?? = ?? /4 × ?? 2
 
?? = ?? /4 × (?? 2
- ?? 2
 
?? = ?? × ?? 
 
Where, 
         A=area in mm
2
 
         D= Major diameter or outer diameter in mm 
         d= Minor diameter or inner diameter in mm 
         b= breadth or wide in mm 
         t= Thickness in mm 
 
2.  UTM (TENSILE TEST) 
 
Yield Stress(sy) 
?? ?? =
?? ?? ?? 
Where, 
           s
y
 = Yield stress in 
N/mm
2
 
            P
y
= Yield Load in N 
            A=Area in mm
2
 
Ultimate Stress(su) 
???? =
???? ?? 
Where, 
           s
u
= Ultimate stress in 
N/mm
2
 
            P
u
= Ultimate Load in 
N 
            A=Area in mm
2
 
                               
 
  
 
 
 
Breaking Stress(sB) 
???? =
???? ?? 
Where, 
           s
B
= Breaking stress 
in N/mm
2
 
            P
B
= Breaking Load 
in N 
            A=Area in mm
2
 
% of Elongation (%?L) 
%??? =
(???? - ???? )
???? × 100 
Where, 
           L
F
= Final length in 
mm 
            L
I
= Initial length in 
mm 
           
% of Reduction area 
(%?A) 
%??? =
?? - ?? 
?? × 100 
Where, 
          A= Initial area in mm
2 
           a= Final  area or area 
of neck in mm
2
 
           
 
3. VOLUMETRIC STRAIN 
 
Volumetric strain 
(e
v
) 
?? ?? = ??? /?? = ???????? × (1 - 2
× 1/?? ) 
Where, 
          e
v
 = Volumetric strain 
         ?V= Change in volume in mm
3 
           V=Original  volume in mm
3
 
            1/m = Poisson’s ratio 
       e
lin
= Linear strain
 
Volume (V) ?? = ?? × ?? 
Where, 
                  A= Area in mm2 
                 L= Length  in mm 
 
Poisson’s ratio (1/m) 
1/?? = ?? ?????? / ?? ?????? 
Where, 
       e
Lat
= Linear strain 
       e
Lin
= Linear strain or Longitudinal strain 
 
Lateral strain (e
Lat)
 
???????? = ??? /?? (???? ) ??? /?? (???? ) ??? /?? 
Where, 
           ?d= Change in diameter in mm 
              d= Original diameter in mm 
           ?b = Change in breadth or width in mm 
              b= Original breadth or width in mm 
             ?t = Change in thickness in mm 
               t = Original thickness in mm 
Linear strain (e
Lin)
        ?? ?????? = ??? /?? 
    Where, 
     ?L= Change in length or elongation in mm 
       L=Original  length or gauge length in mm 
 
 
Page 3


                                                       
 
  
 
 
 
I. SIMPLE STRESS AND STRAIN 
 
1. SIMPLE STRESS AND STRAIN FORMULAE HAND BOOK FOR STRENGTH OF 
MATERIALS    
 
N 
Stress(s) ?? =
?? ?? 
Where, 
           s =stress in N/mm
2
 
            P= Load in N 
            A=Area in mm
2
 
Strain (e) ?? =
??? ?? 
Where, 
           e =strain 
          ?L= Change in length or elongation in mm 
            L=Original  length or gauge length in mm 
Youngs 
modulus(E) 
?? =
?? ?? 
Where, 
           E= Youngs modulus or modulus of elasticity in N/mm
2 
           s =stress in N/mm
2
 
           e =strain 
Factor of 
safety(FoS) 
?????? =
?? ?? ?? 
Where, 
           s
u
 =Ultimate stress in N/mm
2
 
           s =Working stress in N/mm
2
 
 
Area(A)
 
?? = ?? /4 × ?? 2
 
?? = ?? /4 × (?? 2
- ?? 2
 
?? = ?? × ?? 
 
Where, 
         A=area in mm
2
 
         D= Major diameter or outer diameter in mm 
         d= Minor diameter or inner diameter in mm 
         b= breadth or wide in mm 
         t= Thickness in mm 
 
2.  UTM (TENSILE TEST) 
 
Yield Stress(sy) 
?? ?? =
?? ?? ?? 
Where, 
           s
y
 = Yield stress in 
N/mm
2
 
            P
y
= Yield Load in N 
            A=Area in mm
2
 
Ultimate Stress(su) 
???? =
???? ?? 
Where, 
           s
u
= Ultimate stress in 
N/mm
2
 
            P
u
= Ultimate Load in 
N 
            A=Area in mm
2
 
                               
 
  
 
 
 
Breaking Stress(sB) 
???? =
???? ?? 
Where, 
           s
B
= Breaking stress 
in N/mm
2
 
            P
B
= Breaking Load 
in N 
            A=Area in mm
2
 
% of Elongation (%?L) 
%??? =
(???? - ???? )
???? × 100 
Where, 
           L
F
= Final length in 
mm 
            L
I
= Initial length in 
mm 
           
% of Reduction area 
(%?A) 
%??? =
?? - ?? 
?? × 100 
Where, 
          A= Initial area in mm
2 
           a= Final  area or area 
of neck in mm
2
 
           
 
3. VOLUMETRIC STRAIN 
 
Volumetric strain 
(e
v
) 
?? ?? = ??? /?? = ???????? × (1 - 2
× 1/?? ) 
Where, 
          e
v
 = Volumetric strain 
         ?V= Change in volume in mm
3 
           V=Original  volume in mm
3
 
            1/m = Poisson’s ratio 
       e
lin
= Linear strain
 
Volume (V) ?? = ?? × ?? 
Where, 
                  A= Area in mm2 
                 L= Length  in mm 
 
Poisson’s ratio (1/m) 
1/?? = ?? ?????? / ?? ?????? 
Where, 
       e
Lat
= Linear strain 
       e
Lin
= Linear strain or Longitudinal strain 
 
Lateral strain (e
Lat)
 
???????? = ??? /?? (???? ) ??? /?? (???? ) ??? /?? 
Where, 
           ?d= Change in diameter in mm 
              d= Original diameter in mm 
           ?b = Change in breadth or width in mm 
              b= Original breadth or width in mm 
             ?t = Change in thickness in mm 
               t = Original thickness in mm 
Linear strain (e
Lin)
        ?? ?????? = ??? /?? 
    Where, 
     ?L= Change in length or elongation in mm 
       L=Original  length or gauge length in mm 
 
 
                      
 
  
 
 
 
 
 
 
4. ELASTIC CONSTANTS (E,G &K) 
 
Young’s modulus (E) Bulk modulus (K) 
Rigidity modulus (C or N or 
G) 
?? =
?? ?? 
?? =
?? ?? ?? ?? 
?? = ?? = ?? =
?? ?? ?? ?? 
Where, 
           E= Youngs modulus 
or modulus of elasticity in 
N/mm
2 
 
           s =stress in N/mm
2
 
           e =strain 
Where, 
          K= Bulk modulus or 
in N/mm
2 
 
           s
d
 = Direct stress in 
N/mm
2
 
           e
V
 = Volumetric 
strain 
Where, 
C=N=G=Modulus of rigidity  
or shear modulus in N/mm
2 
e
s
= Shear strain 
Relationship between E,G,K 
?? = 2?? × (1 + 1/?? ) 
?? = 3?? × (1 - 2 × 1/?? ) 
?? = 9???? / (3?? + ?? ) 
 
5. OMPOSITE BAR 
 
Condition (i) 
P= P
1
+ P
2 
P= s
1
A
1
+ s
2
A
2
 
Where, 
        P=Total load acting on the composite in N
 
      s
1
=Stress induced in Material -1 
      s
2
=Stress induced in Material -2 
       A
1
= Mareial-1 cross sectional area in mm
2
 
       A
2
= Mareial-2 cross sectional area in mm
2
 
Condition (ii) 
?? 1 = ?? 2
 
?? 1/?? 1= ?? 2/?? 2 
Where, 
        s
1
=Stress induced in Material -1 
        s
2
=Stress induced in Material -2 
          E
1
= Young’s modulus of Material -1 
          E
2
= Young’s modulus of Material -2
 
 
 
II. SHEAR FORCE AND BENDING MOMENT DIAGRAM 
 
1. DIAGRAM SHORT CUT (SFD AND BMD) 
 
LOAD SFD BMD 
Point Horizontal Inclination 
U.D.L Inclination Parabola 
U.V.L Parabola Parabola