5.2
Truss Connections
Trusses
Figure 5 – 7: A two-force member
Figure 5 – 6: Example of a space truss
The two-force member does not carry moments at either of its
ends. The member remains in equilibrium by an internal
reaction that passes through the line between the two ends of
the member, as shown in Fig. 5 – 71. The internal reaction is a
stretching force. The stretching force can be compressive or
tensile. Most frequently, the end of a two-force member is
represented by a pin connection or a rolling connection. The
pinned connection is created by welds, bolts, or rivets. They act
like a pinned connection when the centerlines of the structural
members intersect at the center of the joint. By design, the
moments they carry are relatively small (See Fig. 5 – 8). In
practice, the rolling connection is frequently created by a roller,
a rocker, or an expansion joint.
The truss is a very popular type of structure. It’s used
in roof systems, bridges and towers and it’s an aesthetic feature
in many buildings (See Fig. 5 – 6). The popularity of the truss
lies both in its simplicity and its efficiency.
A truss is a structure composed exclusively of
two-force members. The external forces acting on a truss, both
the external reactions and the applied forces, act at the joints of
the members. In those cases in which an applied force does not
actually act at either end of a structural member, the applied
force can be replaced with an equivalent force at each end of
the member. For example, the weight load is a distributed force
that acts along the length of a structural member. It can be
replaced with an equivalent pair of point forces at each end of
the member. This procedure creates a truss system that can be
analyzed using the methods developed in this section.
Trusses are designed for either strength, stiffness,
action, or a combination thereof. Forces need to be determined
in trusses designed for strength and displacements need to be
determined in trusses designed for stiffness. The next two subsections describe three ways of finding the forces in a planar
truss: the method of joints, the method of sections, and the
sequential method. Finally, it is shown how to find the
displacements in a planar truss.
Section Objectives
Sub-Section
Truss Connections
The Method of Joints and the
Method of Sections
The Sequential Method
Finding Displacements in Planar
Trusses
Statically Indeterminate Trusses
Figure 5 – 8: The ends of a structural member are often
welded or riveted to a plate, called the gusset plate. The
connection is considered to be a pin connection if the
center lines of the members intersect the center of the
gusset plate.
Objective
To describe the types of connections used in trusses
To describe how and when to use the Method of Sections and the Method of
Joints
To describe how and when to use the Sequential Method
To develop the displacement equations used to find displacements in planar
trusses
To point out that statically indeterminate truss problems can be solved by
the methods described in this section but that the number of equations is
large. It’s best to solve these problems by computer.
1
The two-force member was also discussed in Section 3.4 in the subsection entitled “Two-Force and Three-Force Members.”
1
In many trusses, the structural members, internal reactions,
external reactions and applied forces all lie in one plane. These
are called planar trusses. In general, though, the trusses are
space trusses. The truss shown in Fig. 5 – 6 is an example of a
space truss.
The Method of Joints and the Method of
Sections
The unknown forces in a truss problem consist of
stretching forces in members and the force reactions at joints. In
some situations the interest lies in finding all of the unknown
forces and in other situations that interest lies in finding selected
forces of interest. The method of joints refers to a problemsolving process that makes use of all of the 2n force equations
described earlier. The method of joints is used in two situations:
A fully constrained planar truss is fixed by at least three
external reactions. The external reactions constrain the planar
truss’s two translational degrees of freedom and its rotational
degree of freedom. When the number of external reactions and
the number of structural members are “minimal” the truss
becomes statically determinate. The statically determinate truss
can freely expand or contract when the temperature changes.
(1) When the truss is made up of just two or three members
(2) When the problem is solved by computer
There is a simple relationship between the number of
members and the number of joints in a statically determinate
truss. Denote the number of members by m, the number of
joints by n, and the number of external reactions by p. The
entire truss is maintained in static equilibrium if each of the
truss’s joints is maintained in static equilibrium. Since the
forces acting on a joint are concurrent, static equilibrium of the
truss is maintained by the resultant forces acting on the truss’s
joints. The sum of the moments is not needed. In the case of a
planar truss, the resultant forces acting at the joints in the x and
y directions are summed to zero to produce 2n force equations.
The number of unknowns is m + p, so for the statically
determinate planar truss
When only a few of the stretching forces or force reactions are
sought and the truss consists of more than three members, then
it becomes more convenient to find the stretching forces and the
force reactions by “cutting” the truss into sections, called body
sections. A body section is a rigid body in itself that must be
maintained in static equilibrium. Therefore, the resultant forces
and the resultant moment acting on the body section are zero,
which produces three equilibrium equations. Finding stretching
forces and member reactions from the equations governing the
equilibrium of one or more body sections is called the method
of sections.
The Method of Joints
(5 – 1)
2n = m + p.
The method of joints is a systematic method of determining
all of the reactions and the stretching forces in a truss. In the
transition step, a free body diagram is drawn for each joint. It’s
helpful to either a) label the forces as though they’re all in
tension (the compressive members later turn out to be negative)
or to b) label the force in each member depending on whether
you expect the member to be in compression or in tension. Both
approaches help you keep track of the senses of the forces. In
the equation step, write down the force equations, namely, Fx
= 0 and Fy = 0 for each joint. In the answer step, a
simultaneous set of linear algebraic equations needs to be
solved. In the knowledge step, the signs of the unknown forces
are checked. Some of the members are expected to act in
tension and others to act in compression. Also inspect the
magnitudes of the forces. Some truss members will be loaded
more than others. The examples at the end of the section
demonstrate the method of joints.
Equations (5 – 1) is used in the equation step to check a truss’s
static determinacy.
Space Trusses
In the case of a space truss, the resultant force acting at
the joints in the x, y, and z directions are summed to zero to
produce 3n force equations. For the statically determinate space
truss
(5 – 2)
3n = m + p.
2
are found using the method of sections by letting the body
section be the entire truss. The unknown forces are found by
looking at the force equations sequentially, moving from joint
to joint. At each successive joint, at most two forces are
unknown. This way, the unknown forces can be found by
summing forces in the x and y directions at the joint. Then, you
move on to the next joint and find the unknown forces at that
joint. These steps are repeated until you’ve covered the entire
truss. The following describes the problem-solving steps.
The Method of Sections
The method of sections is used when the focus is on a
few of the structural members. In the transition step, the truss is
cut along a section containing the force that is sought. (Don’t
cut directly through a joint.) The body that remains after the cut
is called the body section. The body section is a rigid body in
itself. Draw the free body diagram of the body section.
Remember that only the forces that are external to the body
section need to be included in the free body diagram. It’s
important to cut the truss where the number of unknown
reactions is no greater than three (See Fig. 5 – 9).

Set-up
To begin, check whether or not the structure is a planar
truss. It’s a planar truss if
(a) all of the members and forces lie in a plane, and
(b) the structural members are each two-force members.
The truss is a space truss if (a) does not hold but (b) does hold.
It’s not a truss if (b) does not hold.

Use Eq. (5 – 1) to check whether the planar truss is statically
determinate.
Transition
If the structure is not a truss because an external force(s)
does not act at a joint, then the structure can be reduced to a
truss by replacing the external force(s) acting on the given
member with equivalent forces that act at each end of the given
member. The stretching forces are accurate representations of
the forces in all of the members except in the given member.

Figure 5 – 9: The truss below is cut through line A-A. This
creates the body section shown. Notice that the body section
has three unknown forces.

Inspect the structure at each of the joints to determine
whether there are any unknown forces that can be eliminated in
advance. Unknown forces can be eliminated in advance in the
following three situations (Refer to Fig. 5 – 10):
In the equation step, write down for the body section Fx = 0,
Fy = 0, and  = 0. The number of equations and the number
of unknowns should be three.
(a) Members A and B are pinned at a single joint. First assume
that member A is aligned with member B. It follows that the
stretching force PA in member A and the stretching force PB in
member B are equal. This eliminates one unknown. Also, the
stretching forces in the two members need to act in tension.
Otherwise, these two members become unstable. Next, assume
that member A is not aligned with member B. Then PA = PB =
0. This eliminates both unknowns.
Note that you’ve already been using the method of
sections in this course even though the term “method of
sections” was not used, per say. In the simplest case, when
external forces and moments acting on a body are found by
summing forces and moments acting on the body, the method of
sections is being used. The body section is the entire body itself.
The entire body is being cut away from the external boundary
leaving the external reactions and the applied forces as the only
external forces acting on the body.
(b) Members A, B, and C are pinned at a single joint. Assume
that member A is aligned with member B. The stretching force
PC in member C is then zero. This eliminates one unknown.
But, with PC = 0, the joint now
looks
like
case
(a).
Therefore, use case (a) to evaluate the remaining stretching
forces in members A and B.
The examples at the end of the section demonstrate the
method of sections.
The Sequential Method
In many truss problems that are solved by hand, the
stretching forces and the force reactions can be found more
easily if the method of sections and the method of joints are
used in combination. The method of sections is used first to find
the truss’s external reactions and the method of joints is then
used to find the truss’s stretching forces. The external reactions
(c) Members A, B, C, and D are pinned at a single joint.
Assume that members A and B are aligned and that members C
and D are aligned. Then, the stretching force in A is equal to the
stretching force in B and the stretching force in C is equal to the
stretching force in D, that is, PA = PB and PC = PD.
3
Equation
Write down Fx = 0, Fy = 0, and  = 0 for the body
section. Then, write down the force equations in sequence, that
is Fx = 0 and Fy = 0 for each joint.
Figure 5 – 10: Situations that eliminate unknown
reactions
Use Eq.(5 – 1) to check that the planar truss is statically
determinate.

Answer
The equations governing the equilibrium of the truss are
linear algebraic equations. Solve the equations of the body
section first. Then solve the equations for each joint in the order
of the sequence.
Knowledge
Check the signs of the unknown forces. Some of the
members are expected to act in tension and others to act in
compression. Also inspect the magnitudes of the forces. Some
truss members will be loaded more than others.
(a) Two members are pinned at the joint
The examples at the end of the section demonstrate the
sequential method.
Finding Displacements in Planar Trusses
When designing for stiffness, it becomes important to
keep a truss’s joint displacements within specified tolerances. It
can even be important when the joint displacements are so small
that they’re not easy to see. For example, the back truss on a
large microwave antenna dish needs to be extremely stiff to
maintain the alignment of electrical signals transmitted or
received by the antenna.
(b) Three members are pinned at the joint
The joint displacements of a structure are caused by
the deformations of its members. A truss member deforms
along its axis. The deformation depends on the stretching force
and the material characteristics of the member. Assume that the
stretching force in the member has already been found by one of
the methods developed earlier in this section. As shown below,
the unknowns will consist of a stretch in each member and
displacements of each joint. They’ll be found from
displacement equations. The displacement equations are of
three types, called member displacement equations, joint
displacement equations and member-joint displacement
equations.
(c) Four members are pinned at the joint
Select a body section that has at most three unknown forces
(Don’t count the unknowns that are internal to the body section)
and determine the sequence of joints needed to solve the
problem. Remember that at most two forces (stretching forces
and reactions) are unknown at a joint when the equations of the
joint are solved.
Displacement Equations =
Member Displacement Equations +
Joint Displacement Equations +
Member-Joint Displacement Equations
Draw the free body diagram of the body section and the free
body diagrams of the truss’s joints in sequential order.
4
where the subscript s refers to the member. Equations (5 – 5)
are m member displacement equations. They describe the
relationship between the stretch in the members and the
stretching forces in the members.
First consider the member displacement equations. The
question arises as to the relationship between the known
stretching force and the unknown stretch in the member. Figure
5 – 11 shows an element of a truss member of length dx acted
on by a stretching force P.
Next, consider the joint displacement equations.
Assume that the truss has n joints numbered from 1 to n. The
axes of the coordinate system for the truss are denoted by X and
Y, respectively. The location of a given joint before the load is
applied is designated as point A. After the load is applied, the
joint moves to point A’. The joint displacement vector is
U A  [U X UY ], (r  1, 2, , n) as shown in Fig. 5 – 12. There
are 2n joint displacements.
Figure 5 – 11: The length dx of a truss member stretches
an amount du.
The stretch du of the element divided by the original length dx
of the element is called the strain of the element. The
stretching force of the member is linearly proportional to
the strain of the element, written
P  ku
(5 – 3)
Figure 5 – 12: The r-th joint displacement
du
dx
where the proportionality constant ku is called the stretching
modulus of the member. The dimension of the stretching
modulus ku is [F]. The stretching modulus depends on the
geometry of the cross-section and the material properties of the
member. Assume that the cross-sectional properties, both the
geometry and the material properties, are the same anywhere
along the length of the member. This causes ku to be constant
throughout the member. Both sides of Eq. (5 – 3) can be
multiplied by dx and then integrated to get
L
The joint displacements are constrained in two ways.
They’re constrained by force reactions and they’re constrained
by member displacements. The joint displacement equations
express the constraining of the joint displacements by the force
reactions. For example, if there is a force reaction RX acting at a
given external joint, then the displacement UX at that joint is
zero (See Fig. 5 – 13). Since there are p force reactions, there
are p corresponding joint displacement equations.
u
 0 Pdx   0 ku du .
Since P and ku are constant, it follows that
u
(5 – 4)
P
ku L
where u denotes the stretch of the member. Let the truss be
composed of m members. Then, Eq. (5 – 4) is rewritten as
(5 – 5)
us 
Figure 5 – 13: When a force reaction acts at a
joint in a given direction, the displacement is
zero at the joint in that direction.
Ps
, ( s  1, 2, , m),
kus Ls
5
Next, consider the member-joint displacement equations, which
express the constraining of the joint displacements by the
member displacements. Figure 5 – 14 shows a typical member.
Before the load is applied, the left end of the member is located
at point A and the right end is located at point B. After the load
is applied, the left end of the member is located at point A’ and
the right end is located at point B’. The joint displacement of the
left end is UA and the joint displacement of the right end is UB.
Before the load is applied, the length of the member is L and the
member acts in the n direction where n is the unit vector
directed along the length of the unstretched member. After the
The stretch in each member is unknown and the two
displacements of each joint are unknown so there are a total of
m + 2n unknowns. As seen above the number of member
displacement equations is m, the number of joint displacement
equations is p, and the number of member-displacement
equations is m, so for a statically determinate planar truss
(5 – 10)
2m + p = m + 2n.
An example at the end of this section illustrates step-by-step
how to find the displacements in statically determinate planar
trusses.
Space Trusses
In the case of a space truss, there are 3n joint
displacements instead of 2n so the total number of unknowns is
m + 3n. For a statically determinate space truss
(5 – 11)
Figure 5 – 14: The joint displacements and the
member displacements
m + p = 3n.
Space trusses are considered further in Section 5.5.
Statically Indeterminate Trusses
load is applied, the member displaces and the length of the
member becomes L + u where u is the stretch of the member.
The displaced member acts in the nS direction. From Fig. 5 – 14
(5 – 6)
Recall that it was pointed out in section 3.6 in the
discussion on determinacy that the forces in a statically
determinate system can not be found without considering the
system’s displacements. In the previous sub-section, a
procedure was presented for finding displacements. Does this
mean that we can now solve statically indeterminate truss
problems? The answer is yes.
U A  ( L  u)n S  Ln  U B ,
Multiply each side of Eq. (5 – 6) by n· to get
(5 – 7)
u(n S  n)  L(n S  n  1)  (U B  U A )  n.
In a statically determinate truss, the force equations
were solved first, after which displacement equations were
solved. All of the equations were linear algebraic equations. In
statically indeterminate trusses, it’s not possible to first solve
force equations and then displacement equations. But this
doesn’t mean that the equations can’t be solved. It’s simply not
possible to solve the equations in that order because the
equations become coupled, meaning only that they need to be
solved together. This does not present a problem when solving
the equations by computer. But when solving the equations by
hand, the number of simultaneous equations becomes large,
except in simple cases, and therefore considerably more
difficult to solve by hand.
In Eq. (5 – 7) n S  n = cos where is the angle of rotation of
the member (See Fig. 5 – 14). Since the angle of rotation is
small, cosEquation (5 – 7) reduces to
(5 – 8)
u  (U B  U A )  n.
Given that the truss is composed of m members, Eq. (5 – 8),
applied to each member, becomes
(5 – 9)
us  (U B  U A )  n s , (s  1, 2,  m).
Equations (5 – 9) are m member-joint displacement equations.
6
Let’s look at the number of equations and the number of
unknowns in a statically indeterminate planar truss. The number
of equations and the number of unknowns in the different types
of equations considered in this section are tabulated below.
Equation
Type
Force
Displacement
Total
Number of
Equations
2n
2m + p
2m + 2n + p
Key Terms
Body Section; Displacement Equations; Force Equations;
Gusset Plate; Joint Displacement Equations; Member
Displacement Equations; Member-Joint Displacement
Equations; Method of Joints; Method of Sections; Planar
Truss; Sequential Method; Space Truss; Strain; Stretching
Modulus, Two-Force Members; Truss;
Number of
Unknowns
m+p
m + 2n
2m + 2n + p
Review Questions
In general, the truss can be partially constrained or fully
constrained. If partially constrained, the truss has free degrees
of freedom creating a system that has no unique solution. In
practice, the truss problem needs to be set up so that the truss is
fully constrained. In the Table we see that the total number of
equations is the same as the total number of unknowns. This
table shows that the statically indeterminate truss problem can
be solved.
1. Define a truss structure.
2. The weight load of a structural member acts throughout the
length of a truss member, which violates the definition of a
truss. How is this overcome?
3. What is a planar truss? What is a space truss?
4. State the relationship between the number of joints, the
number of members, and the number of external reactions in a
planar truss and in a space truss.
5. Describe the method of joints.
6. Describe the method of sections.
7. Describe the method of sequence.
8. Which truss method is used to solve space truss problems by
computer?
9. Define strain.
10. Name the three types of displacement equations.
11. Name the different types of equations needed to solve
statically indeterminate truss problems.
Note that the conclusions arrived at above also apply to
space trusses. For space trusses the number of equations and the
number of unknowns are tabulated as:
Equation
Type
Force
Displacement
Total
Number of
Equations
3n
2m + p
2m + 3n + p
Number of
Unknowns
m+p
m + 3n
2m + 3n + p
Again, the total number of equations is equal to the total
number of unknowns. Therefore, the forces and displacements
in fully constrained space trusses can also be solved using the
equations developed in this section.
Statically indeterminate truss problems are generally solved
by computer because of the relatively large number of equations
involved. As the table shows, a statically indeterminate planar
truss consisting of only n = 3 joints, m = 2 members, and p = 3
reactions has 2m + 2n + p = 13 equations. Examples illustrating
how to solve statically indeterminate truss problems are
deferred to section 5.5 entitled Computer Analysis.
7