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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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FAQs on The Force Method of Analysis: Trusses - 1 - Structural Analysis - Civil Engineering (CE)
| 1. What is the Force Method of Analysis for trusses? |  |
Ans. The Force Method of Analysis is a structural analysis technique used in civil engineering to determine the internal forces and displacements of trusses. It involves analyzing each individual member of the truss by considering the equilibrium of forces at each joint and applying the method of joints or method of sections.
| 2. How is the Force Method different from other analysis methods? | |
Ans. The Force Method differs from other analysis methods, such as the displacement method, by focusing on the forces within the truss members rather than the displacements or deformations. It provides a more detailed understanding of the internal forces and allows for more accurate analysis and design of truss structures.
| 3. What are the steps involved in using the Force Method of Analysis for trusses? | |
Ans. The steps involved in using the Force Method of Analysis for trusses are as follows:
1. Identify the external forces and reactions acting on the truss.
2. Choose a joint as the starting point and determine the unknown forces at that joint using the method of joints.
3. Move to the next joint and repeat the process until all the unknown forces in the truss are determined.
4. Analyze the stability of the truss by checking for zero-force members or redundant members.
5. Calculate the displacements of the truss members based on the determined forces and apply compatibility equations to ensure compatibility between the member displacements.
6. Verify the results by checking the equilibrium of forces at each joint and ensuring that the displacements satisfy the compatibility equations.
| 4. What are the advantages of using the Force Method of Analysis for trusses? | |
Ans. The advantages of using the Force Method of Analysis for trusses include:
- It provides a detailed understanding of the internal forces within truss members, allowing for more accurate design and analysis.
- It is applicable to both simple and complex truss structures.
- It can be used to analyze statically indeterminate trusses, which cannot be easily analyzed using other methods.
- It allows for the identification of zero-force members and redundant members in the truss, which can lead to more efficient designs.
- It is a versatile method that can be easily applied to various types of trusses, including both plane and space trusses.
| 5. Are there any limitations or drawbacks of using the Force Method of Analysis for trusses? | |
Ans. While the Force Method of Analysis is a powerful technique, it does have some limitations and drawbacks:
- It requires a good understanding of structural mechanics and the ability to determine the equilibrium of forces at each joint accurately.
- It can be time-consuming and labor-intensive, especially for complex truss structures with many members and joints.
- It assumes that the truss members are idealized as perfectly rigid and that the joints are frictionless, which may not be entirely accurate in real-world situations.
- It does not consider the effects of geometric nonlinearity or material nonlinearity, which may be significant in certain cases.
- It is primarily suited for truss structures and may not be suitable for analyzing other types of structures, such as beams or frames.
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