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

## 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
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 ( )
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
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 ( )
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
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
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 ( )
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
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
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,
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
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 ( )
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
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
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,
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
= + ?
F a     (4)

103.03
24.142
4.268 kN(compressive)
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
i
P
Forces in
the
released
truss due
to unit
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
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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