The Force Method of Analysis: Trusses - 1 Civil Engineering (CE) Notes | EduRev

Structural Analysis

Civil Engineering (CE) : The Force Method of Analysis: Trusses - 1 Civil Engineering (CE) Notes | EduRev

 Page 1


Instructional Objectives 
After reading this chapter the student will be able to 
1. Calculate degree of statical indeterminacy of a planar truss 
2. Analyse the indeterminate planar truss for external loads 
3. Analyse the planar truss for temperature loads 
4. Analyse the planar truss for camber and lack of fit of a member. 
 
 
10.1 Introduction  
The truss is said to be statically indeterminate when the total number of reactions 
and member axial forces exceed the total number of static equilibrium equations. 
In the simple planar truss structures, the degree of indeterminacy can be 
determined from inspection. Whenever, this becomes tedious, one could use the 
following formula to evaluate the static indeterminacy of static planar truss (see 
also section 1.3). 
 
j r m i 2 ) ( - + =    (10.1) 
 
where j m, and rare number of members, joints and unknown reaction 
components respectively. The indeterminacy in the truss may be external, 
internal or both. A planar truss is said to be externally indeterminate if the 
number of reactions exceeds the number of static equilibrium equations available 
(three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to 
be internally indeterminate if it has exactly three reaction components and more 
than members. Finally a truss is both internally and externally 
indeterminate if it has more than three reaction components and also has more 
than members. 
( 3 2 - j )
) ( 3 2 - j
 
The basic method for the analysis of indeterminate truss by force method is 
similar to the indeterminate beam analysis discussed in the previous lessons. 
Determine the degree of static indeterminacy of the structure. Identify the number 
of redundant reactions equal to the degree of indeterminacy. The redundants 
must be so selected that when the restraint corresponding to the redundants are 
removed, the resulting truss is statically determinate and stable. Select 
redundant as the reaction component in excess of three and the rest from the 
member forces. However, one could choose redundant actions completely from 
member forces. Following examples illustrate the analysis procedure. 
 
Example 10.1  
Determine the forces in the truss shown in Fig.10.1a by force method. All the 
members have same axial rigidity. 
 
 
Page 2


Instructional Objectives 
After reading this chapter the student will be able to 
1. Calculate degree of statical indeterminacy of a planar truss 
2. Analyse the indeterminate planar truss for external loads 
3. Analyse the planar truss for temperature loads 
4. Analyse the planar truss for camber and lack of fit of a member. 
 
 
10.1 Introduction  
The truss is said to be statically indeterminate when the total number of reactions 
and member axial forces exceed the total number of static equilibrium equations. 
In the simple planar truss structures, the degree of indeterminacy can be 
determined from inspection. Whenever, this becomes tedious, one could use the 
following formula to evaluate the static indeterminacy of static planar truss (see 
also section 1.3). 
 
j r m i 2 ) ( - + =    (10.1) 
 
where j m, and rare number of members, joints and unknown reaction 
components respectively. The indeterminacy in the truss may be external, 
internal or both. A planar truss is said to be externally indeterminate if the 
number of reactions exceeds the number of static equilibrium equations available 
(three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to 
be internally indeterminate if it has exactly three reaction components and more 
than members. Finally a truss is both internally and externally 
indeterminate if it has more than three reaction components and also has more 
than members. 
( 3 2 - j )
) ( 3 2 - j
 
The basic method for the analysis of indeterminate truss by force method is 
similar to the indeterminate beam analysis discussed in the previous lessons. 
Determine the degree of static indeterminacy of the structure. Identify the number 
of redundant reactions equal to the degree of indeterminacy. The redundants 
must be so selected that when the restraint corresponding to the redundants are 
removed, the resulting truss is statically determinate and stable. Select 
redundant as the reaction component in excess of three and the rest from the 
member forces. However, one could choose redundant actions completely from 
member forces. Following examples illustrate the analysis procedure. 
 
Example 10.1  
Determine the forces in the truss shown in Fig.10.1a by force method. All the 
members have same axial rigidity. 
 
 
 
 
The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The 
truss is externally determinate the reactions can be evaluated from the 
equations of statics alone. Select the bar force in member 
. .e i
AD
F ADas the 
redundant. Now cut the member AD to obtain the released structure as shown in 
Fig. 10.1b. The cut redundant member ADremains in the truss as its 
deformations need to be included in the calculation of displacements in the 
released structure. The redundant ( )
AD
F consists of the pair of forces acting on 
the released structure. 
 
 
Page 3


Instructional Objectives 
After reading this chapter the student will be able to 
1. Calculate degree of statical indeterminacy of a planar truss 
2. Analyse the indeterminate planar truss for external loads 
3. Analyse the planar truss for temperature loads 
4. Analyse the planar truss for camber and lack of fit of a member. 
 
 
10.1 Introduction  
The truss is said to be statically indeterminate when the total number of reactions 
and member axial forces exceed the total number of static equilibrium equations. 
In the simple planar truss structures, the degree of indeterminacy can be 
determined from inspection. Whenever, this becomes tedious, one could use the 
following formula to evaluate the static indeterminacy of static planar truss (see 
also section 1.3). 
 
j r m i 2 ) ( - + =    (10.1) 
 
where j m, and rare number of members, joints and unknown reaction 
components respectively. The indeterminacy in the truss may be external, 
internal or both. A planar truss is said to be externally indeterminate if the 
number of reactions exceeds the number of static equilibrium equations available 
(three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to 
be internally indeterminate if it has exactly three reaction components and more 
than members. Finally a truss is both internally and externally 
indeterminate if it has more than three reaction components and also has more 
than members. 
( 3 2 - j )
) ( 3 2 - j
 
The basic method for the analysis of indeterminate truss by force method is 
similar to the indeterminate beam analysis discussed in the previous lessons. 
Determine the degree of static indeterminacy of the structure. Identify the number 
of redundant reactions equal to the degree of indeterminacy. The redundants 
must be so selected that when the restraint corresponding to the redundants are 
removed, the resulting truss is statically determinate and stable. Select 
redundant as the reaction component in excess of three and the rest from the 
member forces. However, one could choose redundant actions completely from 
member forces. Following examples illustrate the analysis procedure. 
 
Example 10.1  
Determine the forces in the truss shown in Fig.10.1a by force method. All the 
members have same axial rigidity. 
 
 
 
 
The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The 
truss is externally determinate the reactions can be evaluated from the 
equations of statics alone. Select the bar force in member 
. .e i
AD
F ADas the 
redundant. Now cut the member AD to obtain the released structure as shown in 
Fig. 10.1b. The cut redundant member ADremains in the truss as its 
deformations need to be included in the calculation of displacements in the 
released structure. The redundant ( )
AD
F consists of the pair of forces acting on 
the released structure. 
 
 
 
 
Evaluate reactions of the truss by static equations of equilibrium. 
 
5kN (downwards)
5kN (downwards)
15 kN(upwards)
Cy
Cx
Dy
R
R
R
= -
=-
=
    (1) 
 
 
Please note that the member tensile axial force is taken as positive and 
horizontal reaction is taken as positive to the right and vertical reaction is taken 
as positive when acting upwards. When the member cut ends are displaced 
towards one another then it is taken as positive. 
 
 
The first step in the force method is to calculate displacement ( )
L
? corresponding 
to redundant bar force in the released structure due to applied external 
loading. This can be readily done by unit-load method.    
AD
F
 
 
Page 4


Instructional Objectives 
After reading this chapter the student will be able to 
1. Calculate degree of statical indeterminacy of a planar truss 
2. Analyse the indeterminate planar truss for external loads 
3. Analyse the planar truss for temperature loads 
4. Analyse the planar truss for camber and lack of fit of a member. 
 
 
10.1 Introduction  
The truss is said to be statically indeterminate when the total number of reactions 
and member axial forces exceed the total number of static equilibrium equations. 
In the simple planar truss structures, the degree of indeterminacy can be 
determined from inspection. Whenever, this becomes tedious, one could use the 
following formula to evaluate the static indeterminacy of static planar truss (see 
also section 1.3). 
 
j r m i 2 ) ( - + =    (10.1) 
 
where j m, and rare number of members, joints and unknown reaction 
components respectively. The indeterminacy in the truss may be external, 
internal or both. A planar truss is said to be externally indeterminate if the 
number of reactions exceeds the number of static equilibrium equations available 
(three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to 
be internally indeterminate if it has exactly three reaction components and more 
than members. Finally a truss is both internally and externally 
indeterminate if it has more than three reaction components and also has more 
than members. 
( 3 2 - j )
) ( 3 2 - j
 
The basic method for the analysis of indeterminate truss by force method is 
similar to the indeterminate beam analysis discussed in the previous lessons. 
Determine the degree of static indeterminacy of the structure. Identify the number 
of redundant reactions equal to the degree of indeterminacy. The redundants 
must be so selected that when the restraint corresponding to the redundants are 
removed, the resulting truss is statically determinate and stable. Select 
redundant as the reaction component in excess of three and the rest from the 
member forces. However, one could choose redundant actions completely from 
member forces. Following examples illustrate the analysis procedure. 
 
Example 10.1  
Determine the forces in the truss shown in Fig.10.1a by force method. All the 
members have same axial rigidity. 
 
 
 
 
The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The 
truss is externally determinate the reactions can be evaluated from the 
equations of statics alone. Select the bar force in member 
. .e i
AD
F ADas the 
redundant. Now cut the member AD to obtain the released structure as shown in 
Fig. 10.1b. The cut redundant member ADremains in the truss as its 
deformations need to be included in the calculation of displacements in the 
released structure. The redundant ( )
AD
F consists of the pair of forces acting on 
the released structure. 
 
 
 
 
Evaluate reactions of the truss by static equations of equilibrium. 
 
5kN (downwards)
5kN (downwards)
15 kN(upwards)
Cy
Cx
Dy
R
R
R
= -
=-
=
    (1) 
 
 
Please note that the member tensile axial force is taken as positive and 
horizontal reaction is taken as positive to the right and vertical reaction is taken 
as positive when acting upwards. When the member cut ends are displaced 
towards one another then it is taken as positive. 
 
 
The first step in the force method is to calculate displacement ( )
L
? corresponding 
to redundant bar force in the released structure due to applied external 
loading. This can be readily done by unit-load method.    
AD
F
 
 
To calculate displacement , apply external load and calculate member forces 
as shown in Fig. 10.1b and apply unit virtual load along  and calculate 
member forces ( (see Fig. 10.1c). Thus, 
(
L
?)
()
i
P
AD
F
)
i v
P
 
()
AE
AE
L
P P
i
i v i L
03 . 103
=
= ?
?
    (2) 
 
 
In the next step, apply a real unit load along the redundant and calculate 
displacement by unit load method. Thus, 
AD
F
11
a
 
()
AE
E A
L
P a
i i
i
i v
142 . 24
2
11
=
=
?
      (3) 
 
 
 
The compatibility condition of the problem is that the relative displacement 
L
? of 
the cut member AD due to external loading plus the relative displacement of the 
member AD caused by the redundant axial forces must be equal to zero  . .e i
 
Page 5


Instructional Objectives 
After reading this chapter the student will be able to 
1. Calculate degree of statical indeterminacy of a planar truss 
2. Analyse the indeterminate planar truss for external loads 
3. Analyse the planar truss for temperature loads 
4. Analyse the planar truss for camber and lack of fit of a member. 
 
 
10.1 Introduction  
The truss is said to be statically indeterminate when the total number of reactions 
and member axial forces exceed the total number of static equilibrium equations. 
In the simple planar truss structures, the degree of indeterminacy can be 
determined from inspection. Whenever, this becomes tedious, one could use the 
following formula to evaluate the static indeterminacy of static planar truss (see 
also section 1.3). 
 
j r m i 2 ) ( - + =    (10.1) 
 
where j m, and rare number of members, joints and unknown reaction 
components respectively. The indeterminacy in the truss may be external, 
internal or both. A planar truss is said to be externally indeterminate if the 
number of reactions exceeds the number of static equilibrium equations available 
(three in the present case) and has exactly ( ) 3 2 - j members. A truss is said to 
be internally indeterminate if it has exactly three reaction components and more 
than members. Finally a truss is both internally and externally 
indeterminate if it has more than three reaction components and also has more 
than members. 
( 3 2 - j )
) ( 3 2 - j
 
The basic method for the analysis of indeterminate truss by force method is 
similar to the indeterminate beam analysis discussed in the previous lessons. 
Determine the degree of static indeterminacy of the structure. Identify the number 
of redundant reactions equal to the degree of indeterminacy. The redundants 
must be so selected that when the restraint corresponding to the redundants are 
removed, the resulting truss is statically determinate and stable. Select 
redundant as the reaction component in excess of three and the rest from the 
member forces. However, one could choose redundant actions completely from 
member forces. Following examples illustrate the analysis procedure. 
 
Example 10.1  
Determine the forces in the truss shown in Fig.10.1a by force method. All the 
members have same axial rigidity. 
 
 
 
 
The plane truss shown in Fig.10.1a is statically indeterminate to first degree. The 
truss is externally determinate the reactions can be evaluated from the 
equations of statics alone. Select the bar force in member 
. .e i
AD
F ADas the 
redundant. Now cut the member AD to obtain the released structure as shown in 
Fig. 10.1b. The cut redundant member ADremains in the truss as its 
deformations need to be included in the calculation of displacements in the 
released structure. The redundant ( )
AD
F consists of the pair of forces acting on 
the released structure. 
 
 
 
 
Evaluate reactions of the truss by static equations of equilibrium. 
 
5kN (downwards)
5kN (downwards)
15 kN(upwards)
Cy
Cx
Dy
R
R
R
= -
=-
=
    (1) 
 
 
Please note that the member tensile axial force is taken as positive and 
horizontal reaction is taken as positive to the right and vertical reaction is taken 
as positive when acting upwards. When the member cut ends are displaced 
towards one another then it is taken as positive. 
 
 
The first step in the force method is to calculate displacement ( )
L
? corresponding 
to redundant bar force in the released structure due to applied external 
loading. This can be readily done by unit-load method.    
AD
F
 
 
To calculate displacement , apply external load and calculate member forces 
as shown in Fig. 10.1b and apply unit virtual load along  and calculate 
member forces ( (see Fig. 10.1c). Thus, 
(
L
?)
()
i
P
AD
F
)
i v
P
 
()
AE
AE
L
P P
i
i v i L
03 . 103
=
= ?
?
    (2) 
 
 
In the next step, apply a real unit load along the redundant and calculate 
displacement by unit load method. Thus, 
AD
F
11
a
 
()
AE
E A
L
P a
i i
i
i v
142 . 24
2
11
=
=
?
      (3) 
 
 
 
The compatibility condition of the problem is that the relative displacement 
L
? of 
the cut member AD due to external loading plus the relative displacement of the 
member AD caused by the redundant axial forces must be equal to zero  . .e i
 
0
11
= + ?
AD L
F a     (4) 
 
103.03
24.142
4.268 kN(compressive)
AD
F
-
=
=-
  
 
Now the member forces in the members can be calculated by method of 
superposition. Thus, 
 
( )
i v AD i i
P F P F + =     (5) 
 
The complete calculations can be done conveniently in a tabular form as shown 
in the following table.  
 
Table 10.1 Computation for example 10.1 
Member Length 
i
L 
Forces in 
the 
released 
truss due 
to applied 
loading 
i
P 
Forces in 
the 
released 
truss due 
to unit 
load ( )
i v
P 
 
()
AE
L
P P
i
i v i
 
 
()
i i
i
i v
E A
L
P
2
 
 
()
i v AD i
i
P F P
F
+
=
 
 m kN kN m m/kN kN 
AB 5 0 
2 / 1 - 
0 AE 2 / 5 3.017 
BD 5 -15 
2 / 1 - AE 2 / 75
 
AE 2 / 5 -11.983 
DC 5 0 
2 / 1 - 
0 AE 2 / 5 3.017 
CA 5 0 
2 / 1 - 
0 AE 2 / 5 3.017 
CB 
2 5 2 5 
1 AE / 50 
AE / 2 5 
2.803 
AD 
2 5 
0 1 0 
AE / 2 5 
-4.268 
   Total 
AE
03 . 103
 
AE
142 . 24
 
 
 
Example 10.2  
Calculate reactions and member forces of the truss shown in Fig. 10.2a by force 
method. The cross sectional areas of the members in square centimeters are 
shown in parenthesis. Assume . 
2 5
N/mm 10 0 . 2 × = E
 
 
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