Short Notes: Force Method of Analysis | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

 Page 1


 
 
FORCE METHOD OF ANALYSIS 
1. Energy methods: 
Energy methods are based on linear elastic behaviour of material and conservation of energy 
i.e. work done by external forces is equal to the energy stored in the structure under the load. 
Strain Energy in various cases is given by following expressions. 
In Axial tension or compression, ?? =
?? 2
?? 2????
 
In Bending, ?? =
?? 2
?? 2????
 
In Torsion, ?? =
?? 2
?? 2????
 
2. Castigliano’s Method 
As per Castigliano’s theory 
?=
????
????
 
And, 
?? =
????
????
 
This relation can also be used in finding deflection in the beams as explained in the following 
example. 
Example: Find rotation and deflection at free end in the beam shown in the figure below: 
 
(a) Rotation at the free end: 
Bending Moment at a distance x from the free end ?? ?? = -?? 
So, the strain energy stored in the beam 
?? = ?
?? 2
????
2????
?? 2
/
0
+ ?
?? 2
????
4????
?? ?? 2
/
 
So, rotation at the free end, 
?? ?? =
????
????
= ?
?????? ????
?? 2
/
0
+ ?
?????? 2????
?? ?? 2
/
=
3????
4????
 
(b) Deflection at free end: 
Applying vertical load P at the free end 
 
Page 2


 
 
FORCE METHOD OF ANALYSIS 
1. Energy methods: 
Energy methods are based on linear elastic behaviour of material and conservation of energy 
i.e. work done by external forces is equal to the energy stored in the structure under the load. 
Strain Energy in various cases is given by following expressions. 
In Axial tension or compression, ?? =
?? 2
?? 2????
 
In Bending, ?? =
?? 2
?? 2????
 
In Torsion, ?? =
?? 2
?? 2????
 
2. Castigliano’s Method 
As per Castigliano’s theory 
?=
????
????
 
And, 
?? =
????
????
 
This relation can also be used in finding deflection in the beams as explained in the following 
example. 
Example: Find rotation and deflection at free end in the beam shown in the figure below: 
 
(a) Rotation at the free end: 
Bending Moment at a distance x from the free end ?? ?? = -?? 
So, the strain energy stored in the beam 
?? = ?
?? 2
????
2????
?? 2
/
0
+ ?
?? 2
????
4????
?? ?? 2
/
 
So, rotation at the free end, 
?? ?? =
????
????
= ?
?????? ????
?? 2
/
0
+ ?
?????? 2????
?? ?? 2
/
=
3????
4????
 
(b) Deflection at free end: 
Applying vertical load P at the free end 
 
 
 
Bending Moment at a distance x from free end ?? ?? = -?? - ?? ?? 
So, Strain energy stored in the beam 
?? = ?
(-?? - ???? )
2
????
2????
?? 2
/
0
+ ?
(-?? - ???? )
2
????
4????
?? ?? 2
/
 
So, Deflection at free end, 
?
?? =
????
????
|
?? =0
= ?
(?? + ???? )?????? ????
?? 2
/
0
+ ?
(?? + ???? )??????
2????
?? ?? 2
/
|
?? =0
 
? ?
?? = ?
???????? ????
?? 2
/
0
+ ?
???????? 2????
=
5?? ?? 2
16????
?? ?? 2
/
 
3. Unit Load Method 
Deflection at a point as per unit load method is given by 
?= ?
?? ?? ?? ?? ????
????
 
Where, 
Mx is the bending moment due to external loading. 
mx is the bending moment due to virtual unit load. 
EI is the flexural rigidity of the beam. 
The application of unit load method is explained using the example given below. 
4. Maxwell Law of Reciprocal Theorem 
This law states that in a linearly elastic structure, the deflection at any point A due to loading 
at some point B will be equal to deflection at B due to loading at A. 
Betti’s Theorem: This is a generalised case of Maxwell reciprocal theorem. As per this 
theorem the virtual work done by P system of forces in going through the deformation of Q 
system of forces is equal to virtual work done by Q system of forces in going through the 
deformation of P systems of forces. 
    
 
Virtual work done by P system of forces due to the displacements caused by Q system of 
forces = ?? 1
?? 1?? + ?? 2
?? 2?? 
Similarly, 
Virtual work done by Q system of forces due to the displacements caused by P system of 
forces = ?? 1
?
1?? + ?? 2
?
2?? 
As per Maxwell-Betti’s Theorem 
?? 1
?? 1?? + ?? 2
?? 2?? = ?? 1
?
1?? + ?? 2
?
2?? 
Page 3


 
 
FORCE METHOD OF ANALYSIS 
1. Energy methods: 
Energy methods are based on linear elastic behaviour of material and conservation of energy 
i.e. work done by external forces is equal to the energy stored in the structure under the load. 
Strain Energy in various cases is given by following expressions. 
In Axial tension or compression, ?? =
?? 2
?? 2????
 
In Bending, ?? =
?? 2
?? 2????
 
In Torsion, ?? =
?? 2
?? 2????
 
2. Castigliano’s Method 
As per Castigliano’s theory 
?=
????
????
 
And, 
?? =
????
????
 
This relation can also be used in finding deflection in the beams as explained in the following 
example. 
Example: Find rotation and deflection at free end in the beam shown in the figure below: 
 
(a) Rotation at the free end: 
Bending Moment at a distance x from the free end ?? ?? = -?? 
So, the strain energy stored in the beam 
?? = ?
?? 2
????
2????
?? 2
/
0
+ ?
?? 2
????
4????
?? ?? 2
/
 
So, rotation at the free end, 
?? ?? =
????
????
= ?
?????? ????
?? 2
/
0
+ ?
?????? 2????
?? ?? 2
/
=
3????
4????
 
(b) Deflection at free end: 
Applying vertical load P at the free end 
 
 
 
Bending Moment at a distance x from free end ?? ?? = -?? - ?? ?? 
So, Strain energy stored in the beam 
?? = ?
(-?? - ???? )
2
????
2????
?? 2
/
0
+ ?
(-?? - ???? )
2
????
4????
?? ?? 2
/
 
So, Deflection at free end, 
?
?? =
????
????
|
?? =0
= ?
(?? + ???? )?????? ????
?? 2
/
0
+ ?
(?? + ???? )??????
2????
?? ?? 2
/
|
?? =0
 
? ?
?? = ?
???????? ????
?? 2
/
0
+ ?
???????? 2????
=
5?? ?? 2
16????
?? ?? 2
/
 
3. Unit Load Method 
Deflection at a point as per unit load method is given by 
?= ?
?? ?? ?? ?? ????
????
 
Where, 
Mx is the bending moment due to external loading. 
mx is the bending moment due to virtual unit load. 
EI is the flexural rigidity of the beam. 
The application of unit load method is explained using the example given below. 
4. Maxwell Law of Reciprocal Theorem 
This law states that in a linearly elastic structure, the deflection at any point A due to loading 
at some point B will be equal to deflection at B due to loading at A. 
Betti’s Theorem: This is a generalised case of Maxwell reciprocal theorem. As per this 
theorem the virtual work done by P system of forces in going through the deformation of Q 
system of forces is equal to virtual work done by Q system of forces in going through the 
deformation of P systems of forces. 
    
 
Virtual work done by P system of forces due to the displacements caused by Q system of 
forces = ?? 1
?? 1?? + ?? 2
?? 2?? 
Similarly, 
Virtual work done by Q system of forces due to the displacements caused by P system of 
forces = ?? 1
?
1?? + ?? 2
?
2?? 
As per Maxwell-Betti’s Theorem 
?? 1
?? 1?? + ?? 2
?? 2?? = ?? 1
?
1?? + ?? 2
?
2?? 
 
 
5. THEOREM OF LEAST WORK 
This is a special case of Castigliano’s theorem. This theorem states that for any statically 
indeterminant structure, the redundant should be such that strain energy of the system is 
minimum. 
Thus, 
????
????
= 0 
Where, 
U = Strain energy stored in the system 
R = Redundant force 
6. DEFLECTION OF STATICALLY DETERMINATE TRUSSES 
Two methods mainly used to calculate deflection in trusses are 
(i) Castigliano’s Method 
(ii) Unit load method 
6.1. Castigliano’s Method  
For getting the deflection in case of truss, there are two theorems. According to these 
theorem deflection and slope can be determined as follows. 
(i) Castigliano’s I
st
 theorem:  
?? =
????
????
 
Here,  
w = load 
?u = change in strain energy 
?d = variation in deflection. 
(ii) Castigliano’s II
nd
 theorem  
It states, that the first partial derivative of total strain energy with respect to a load at 
any point in the structure gives deflection at that point in the direction of load. 
?? =
????
????
     ?? =
????
????
 
Application of Castigliano’s theorem: 
(i) To find absolute deflection of a joint in a truss. 
 
U = Strain energy in all members 
2
Pl
U
2AE
?
= 
Page 4


 
 
FORCE METHOD OF ANALYSIS 
1. Energy methods: 
Energy methods are based on linear elastic behaviour of material and conservation of energy 
i.e. work done by external forces is equal to the energy stored in the structure under the load. 
Strain Energy in various cases is given by following expressions. 
In Axial tension or compression, ?? =
?? 2
?? 2????
 
In Bending, ?? =
?? 2
?? 2????
 
In Torsion, ?? =
?? 2
?? 2????
 
2. Castigliano’s Method 
As per Castigliano’s theory 
?=
????
????
 
And, 
?? =
????
????
 
This relation can also be used in finding deflection in the beams as explained in the following 
example. 
Example: Find rotation and deflection at free end in the beam shown in the figure below: 
 
(a) Rotation at the free end: 
Bending Moment at a distance x from the free end ?? ?? = -?? 
So, the strain energy stored in the beam 
?? = ?
?? 2
????
2????
?? 2
/
0
+ ?
?? 2
????
4????
?? ?? 2
/
 
So, rotation at the free end, 
?? ?? =
????
????
= ?
?????? ????
?? 2
/
0
+ ?
?????? 2????
?? ?? 2
/
=
3????
4????
 
(b) Deflection at free end: 
Applying vertical load P at the free end 
 
 
 
Bending Moment at a distance x from free end ?? ?? = -?? - ?? ?? 
So, Strain energy stored in the beam 
?? = ?
(-?? - ???? )
2
????
2????
?? 2
/
0
+ ?
(-?? - ???? )
2
????
4????
?? ?? 2
/
 
So, Deflection at free end, 
?
?? =
????
????
|
?? =0
= ?
(?? + ???? )?????? ????
?? 2
/
0
+ ?
(?? + ???? )??????
2????
?? ?? 2
/
|
?? =0
 
? ?
?? = ?
???????? ????
?? 2
/
0
+ ?
???????? 2????
=
5?? ?? 2
16????
?? ?? 2
/
 
3. Unit Load Method 
Deflection at a point as per unit load method is given by 
?= ?
?? ?? ?? ?? ????
????
 
Where, 
Mx is the bending moment due to external loading. 
mx is the bending moment due to virtual unit load. 
EI is the flexural rigidity of the beam. 
The application of unit load method is explained using the example given below. 
4. Maxwell Law of Reciprocal Theorem 
This law states that in a linearly elastic structure, the deflection at any point A due to loading 
at some point B will be equal to deflection at B due to loading at A. 
Betti’s Theorem: This is a generalised case of Maxwell reciprocal theorem. As per this 
theorem the virtual work done by P system of forces in going through the deformation of Q 
system of forces is equal to virtual work done by Q system of forces in going through the 
deformation of P systems of forces. 
    
 
Virtual work done by P system of forces due to the displacements caused by Q system of 
forces = ?? 1
?? 1?? + ?? 2
?? 2?? 
Similarly, 
Virtual work done by Q system of forces due to the displacements caused by P system of 
forces = ?? 1
?
1?? + ?? 2
?
2?? 
As per Maxwell-Betti’s Theorem 
?? 1
?? 1?? + ?? 2
?? 2?? = ?? 1
?
1?? + ?? 2
?
2?? 
 
 
5. THEOREM OF LEAST WORK 
This is a special case of Castigliano’s theorem. This theorem states that for any statically 
indeterminant structure, the redundant should be such that strain energy of the system is 
minimum. 
Thus, 
????
????
= 0 
Where, 
U = Strain energy stored in the system 
R = Redundant force 
6. DEFLECTION OF STATICALLY DETERMINATE TRUSSES 
Two methods mainly used to calculate deflection in trusses are 
(i) Castigliano’s Method 
(ii) Unit load method 
6.1. Castigliano’s Method  
For getting the deflection in case of truss, there are two theorems. According to these 
theorem deflection and slope can be determined as follows. 
(i) Castigliano’s I
st
 theorem:  
?? =
????
????
 
Here,  
w = load 
?u = change in strain energy 
?d = variation in deflection. 
(ii) Castigliano’s II
nd
 theorem  
It states, that the first partial derivative of total strain energy with respect to a load at 
any point in the structure gives deflection at that point in the direction of load. 
?? =
????
????
     ?? =
????
????
 
Application of Castigliano’s theorem: 
(i) To find absolute deflection of a joint in a truss. 
 
U = Strain energy in all members 
2
Pl
U
2AE
?
= 
 
 
Where, P1, P2 … Pn = force in members due to applied load w.  
and l1, l2 …. Ln = length of each member. 
From Castigliano’s II theorem  
?  ?? ?? =
????
????
= ?? 2?? ????
????
?? 2????
= ?? ?????? ????
 
Where, k = k1, k2 = 
12
PP
,
ww
??
=
??
 force in all members due to unit load applied at a point 
where we have to find deflection (d). 
 
If we want to find relative displacement of any two joints B and E, apply unit loads at B 
and E in the direction BE. Find forces in all members due to this load then relative 
displacement of two joints B and E is 
BE
Pkl
AE
? = ? 
Where,  
P = P1, P2 etc forces in all member due to applied loads unit loads 
K = forces in all members due to unit loads applied at B and E. 
If we want to find rotation of any member FG, apply unit couple at G and F (these two 
forces form unit couple i.e., 1/a × a = 1). Find forces in all members due to these two 
loads, then rotation of member is given as 
GF
Pkl
AE
? = ? 
Where, P1, P2 … Pn = force in all members. 
k = forces in all member due to unit couple applied at G and F. 
6.2. Unit Load Method 
This method is based on method of virtual work. From virtual work principle external 
work done on a body is equal to internal work done by the body. 
If a unit virtual load produces internal stresses ui in the member and the real 
displacement of the i
th
 member is dli then the internal virtual work done is equal to S?? ?? ????
?? . 
If the virtual load at any point is 1and the displacement at that point due to external 
forces is ? then, 
1 × ? = S?? ?? ????
?? 
(i) Due to external loading: 
Deflection of truss due to external loading is given by 
Page 5


 
 
FORCE METHOD OF ANALYSIS 
1. Energy methods: 
Energy methods are based on linear elastic behaviour of material and conservation of energy 
i.e. work done by external forces is equal to the energy stored in the structure under the load. 
Strain Energy in various cases is given by following expressions. 
In Axial tension or compression, ?? =
?? 2
?? 2????
 
In Bending, ?? =
?? 2
?? 2????
 
In Torsion, ?? =
?? 2
?? 2????
 
2. Castigliano’s Method 
As per Castigliano’s theory 
?=
????
????
 
And, 
?? =
????
????
 
This relation can also be used in finding deflection in the beams as explained in the following 
example. 
Example: Find rotation and deflection at free end in the beam shown in the figure below: 
 
(a) Rotation at the free end: 
Bending Moment at a distance x from the free end ?? ?? = -?? 
So, the strain energy stored in the beam 
?? = ?
?? 2
????
2????
?? 2
/
0
+ ?
?? 2
????
4????
?? ?? 2
/
 
So, rotation at the free end, 
?? ?? =
????
????
= ?
?????? ????
?? 2
/
0
+ ?
?????? 2????
?? ?? 2
/
=
3????
4????
 
(b) Deflection at free end: 
Applying vertical load P at the free end 
 
 
 
Bending Moment at a distance x from free end ?? ?? = -?? - ?? ?? 
So, Strain energy stored in the beam 
?? = ?
(-?? - ???? )
2
????
2????
?? 2
/
0
+ ?
(-?? - ???? )
2
????
4????
?? ?? 2
/
 
So, Deflection at free end, 
?
?? =
????
????
|
?? =0
= ?
(?? + ???? )?????? ????
?? 2
/
0
+ ?
(?? + ???? )??????
2????
?? ?? 2
/
|
?? =0
 
? ?
?? = ?
???????? ????
?? 2
/
0
+ ?
???????? 2????
=
5?? ?? 2
16????
?? ?? 2
/
 
3. Unit Load Method 
Deflection at a point as per unit load method is given by 
?= ?
?? ?? ?? ?? ????
????
 
Where, 
Mx is the bending moment due to external loading. 
mx is the bending moment due to virtual unit load. 
EI is the flexural rigidity of the beam. 
The application of unit load method is explained using the example given below. 
4. Maxwell Law of Reciprocal Theorem 
This law states that in a linearly elastic structure, the deflection at any point A due to loading 
at some point B will be equal to deflection at B due to loading at A. 
Betti’s Theorem: This is a generalised case of Maxwell reciprocal theorem. As per this 
theorem the virtual work done by P system of forces in going through the deformation of Q 
system of forces is equal to virtual work done by Q system of forces in going through the 
deformation of P systems of forces. 
    
 
Virtual work done by P system of forces due to the displacements caused by Q system of 
forces = ?? 1
?? 1?? + ?? 2
?? 2?? 
Similarly, 
Virtual work done by Q system of forces due to the displacements caused by P system of 
forces = ?? 1
?
1?? + ?? 2
?
2?? 
As per Maxwell-Betti’s Theorem 
?? 1
?? 1?? + ?? 2
?? 2?? = ?? 1
?
1?? + ?? 2
?
2?? 
 
 
5. THEOREM OF LEAST WORK 
This is a special case of Castigliano’s theorem. This theorem states that for any statically 
indeterminant structure, the redundant should be such that strain energy of the system is 
minimum. 
Thus, 
????
????
= 0 
Where, 
U = Strain energy stored in the system 
R = Redundant force 
6. DEFLECTION OF STATICALLY DETERMINATE TRUSSES 
Two methods mainly used to calculate deflection in trusses are 
(i) Castigliano’s Method 
(ii) Unit load method 
6.1. Castigliano’s Method  
For getting the deflection in case of truss, there are two theorems. According to these 
theorem deflection and slope can be determined as follows. 
(i) Castigliano’s I
st
 theorem:  
?? =
????
????
 
Here,  
w = load 
?u = change in strain energy 
?d = variation in deflection. 
(ii) Castigliano’s II
nd
 theorem  
It states, that the first partial derivative of total strain energy with respect to a load at 
any point in the structure gives deflection at that point in the direction of load. 
?? =
????
????
     ?? =
????
????
 
Application of Castigliano’s theorem: 
(i) To find absolute deflection of a joint in a truss. 
 
U = Strain energy in all members 
2
Pl
U
2AE
?
= 
 
 
Where, P1, P2 … Pn = force in members due to applied load w.  
and l1, l2 …. Ln = length of each member. 
From Castigliano’s II theorem  
?  ?? ?? =
????
????
= ?? 2?? ????
????
?? 2????
= ?? ?????? ????
 
Where, k = k1, k2 = 
12
PP
,
ww
??
=
??
 force in all members due to unit load applied at a point 
where we have to find deflection (d). 
 
If we want to find relative displacement of any two joints B and E, apply unit loads at B 
and E in the direction BE. Find forces in all members due to this load then relative 
displacement of two joints B and E is 
BE
Pkl
AE
? = ? 
Where,  
P = P1, P2 etc forces in all member due to applied loads unit loads 
K = forces in all members due to unit loads applied at B and E. 
If we want to find rotation of any member FG, apply unit couple at G and F (these two 
forces form unit couple i.e., 1/a × a = 1). Find forces in all members due to these two 
loads, then rotation of member is given as 
GF
Pkl
AE
? = ? 
Where, P1, P2 … Pn = force in all members. 
k = forces in all member due to unit couple applied at G and F. 
6.2. Unit Load Method 
This method is based on method of virtual work. From virtual work principle external 
work done on a body is equal to internal work done by the body. 
If a unit virtual load produces internal stresses ui in the member and the real 
displacement of the i
th
 member is dli then the internal virtual work done is equal to S?? ?? ????
?? . 
If the virtual load at any point is 1and the displacement at that point due to external 
forces is ? then, 
1 × ? = S?? ?? ????
?? 
(i) Due to external loading: 
Deflection of truss due to external loading is given by 
 
 
? = ?
?????? ????
 
Where, 
P = Force in the member due to external loading 
u = Force in the member due to unit load applied in the direction at the point where 
deflection is to be calculated after removal of external loading 
L = Length of the member 
AE = Axial rigidity of the member 
(ii) Temperature Change Case: 
Due to temperature change, 
?? ?? ?? = ?? ?? ?? ?? ??? ?? 
So, 
?= S?? ?? ?? ?? ?? ?? ??? ?? 
Where, 
? = Deflection of truss due to temperature change 
ui
 
= Forces in the member due to unit load at the point where deflection is to be 
computed. 
li = length of the member 
ai = Coefficient of linear expansion 
Ti = Temperature increment  
(iii) Fabrication Error (Lack of Fit Case): If the member is shorter or longer in length, 
it will induce stresses in truss. In this case, the joint deflection is calculated as 
?= S?? ?? ?? ?? ?? 
Where, 
? = Deflection of truss due to fabrication error 
ui
 
= Forces in the member due to unit load at the point where deflection is to be 
computed. 
dli = Fabrication error