CIE 136: STRUCTURAL
THEORY
Week 2
LESSON OBJECTIVES
At the end of the lesson, you should be able to:
1. Identify different types of Trusses
2. Solve for the static determinacy of plane trusses.
3. Know the analysis of truss by method of joints and
method of section
INTRODUCTION
If all the members of a truss and the applied
loads lie in a single plane, the truss is called a
plane truss. Plane trusses are commonly used
for supporting decks of bridges and roofs of
buildings. The figure on the right shows a typical
framing system for a roof supported by plane
trusses. In this case, two or more trusses are
connected at their joints by beams, termed
purlins, to form a three-dimensional framework.
If all the members of a truss and the applied
loads lie in a single plane, the truss is called a
plane truss.
INTRODUCTION
Although a great majority of trusses can
be analyzed as plane trusses, there are
some truss systems, such as
transmission towers and latticed domes
(as shown in the figure), that cannot be
treated as plane trusses because of their
shape, arrangement of members, or
applied loading. Such trusses, which are
called space trusses, are analyzed as
three-dimensional bodies subjected to
three-dimensional force systems. Geodesic Dome Climatron
Showing Glass-and-
Aluminum Geodesie
Design Missouri Botanical
Garden, St. Louis,
Missouri
ASSUMPTIONS FOR ANALYSIS OF TRUSSES
The analysis of trusses is usually based on the following simplifying assumptions:
1. All members are connected only at their ends by frictionless hinges in plane
trusses and by frictionless ball-and-socket joints in space trusses.
2. All loads and support reactions are applied only at the joints.
3. The centroidal axis of each member coincides with the line connecting the
centers of the adjacent joints.
TYPICAL TRUSSES
Roof Trusses
TYPICAL TRUSSES
Bridge Trusses
COMPONENTS OF TRUSS
top chord
joints (nodes)
web
king post
e
ac
pitch
br
al
on
run
ag
di
rise
bottom chord
Tension and compression in truss members
Tension – A pulling force that
stretches the member
compression
Compression – A pushing force
that squeezes the member
tension
Load
Tension and compression in truss members
F F
F2 compression
joint
F3 tension
F1 compression
F F
tension compression
Basic Truss Element
The simplest internally stable (or rigid) plane truss can be formed by
connecting three members at their ends by hinges to form a triangle, as
shown in figure. This triangular truss is called the basic truss element. Note
that this triangular truss is internally stable in the sense that it is a rigid body
that will not change its shape under loads. In contrast, a rectangular truss
formed by connecting four members at their ends by hinges, is internally
unstable because it will change its shape and collapse when subjected to a
general system of coplanar forces.
Basic Truss Element
Three bars joined by pin
Four or more bars joined
at their ends (rigid
by pin at their ends
frame).
(non-rigid frame)
Simple Trusses
Simple Trusses
A simple truss is formed by enlarging the basic truss element, which contains
three members and three joints, by adding two additional members for each
additional joint, so the total number of members m in a simple truss is given
by:
expresses the requirement of the minimum
m = 3 + 2(j-3) = 2j -3 number of members that a plane truss of j joints
must contain if it is to be internally stable.
where:
j = total number of joints ( including those attached to the supports)
m= the total number of members
Simple Trusses
Compound Trusses
Compound trusses are constructed by
connecting two or more simple trusses
to form a single rigid body. To prevent
any relative movement between the
simple trusses, each truss must be
connected to the other(s) by means of
connections capable of transmitting at
least three force components, all of
which are neither parallel nor
concurrent.
Internal Stability
m < 2j – 3 internally unstable
m ≥ 2𝑗 – 3 internally stable
It is very important to realize that although the foregoing criterion for internal
stability is necessary, it is not sufficient to ensure internal stability. A truss must
not only contain enough members to satisfy the 𝑚 ≥ 2𝑗 – 3 condition, but the
members must also be properly arranged to ensure rigidity of the entire truss.
Recall from our discussion of simple and compound trusses that in a stable truss,
each joint is connected to the rest of the structure by at least two nonparallel
members, and each portion of the truss must be connected to the remainder of
the truss by connections capable of transmitting at least three nonparallel and
nonconcurrent force components.
STATIC DETERMINACY, INDETERMINACY, AND STABILITY
OF PLANE TRUSSES
We consider a truss to be statically determinate if the forces in all its members,
as well as all the external reactions, can be determined by using the equations of
equilibrium.
Since the two methods of analysis presented in the following sections can be
used to analyze only statically determinate trusses, it is important for the student
to be able to recognize statically determinate trusses before proceeding with the
analysis.
STATIC DETERMINACY, INDETERMINACY, AND STABILITY
OF PLANE TRUSSES
m + r < 2j statically unstable truss
m + r = 2j statically determinate truss
m + r > 2j statically indeterminate truss
The first condition, for the static instability of trusses, is both necessary and sufficient in
the sense that if m < 2j – r, the truss is definitely statically unstable. However, the remaining
two conditions, are necessary but not sufficient conditions. In other words, these two
equations simply tell us that the number of members and reactions is sufficient for stability.
They do not provide any information regarding their arrangement. A truss may have a
sufficient number of members and external reactions but may still be unstable due to
improper arrangement of members and/ or external supports.
STATIC DETERMINACY, INDETERMINACY, AND STABILITY
OF PLANE TRUSSES
STATIC DETERMINACY, INDETERMINACY, AND STABILITY
OF PLANE TRUSSES
Analysis of Plane
Trusses
Identification of Zero-Force Members
Because trusses are usually designed to support several different loading conditions, it
is not uncommon to find members with zero forces in them when a truss is being
analyzed for a particular loading condition. Zero-force members are also added to
trusses to brace compression members against buckling and slender tension members
against vibrating. The analysis of trusses can be expedited if we can identify the zero-
force members by inspection. Two common types of member arrangements that result
in zero-force members are the following:
1. If only two noncollinear members are connected to a joint that has no external loads
or reactions applied to it, then the force in both members is zero.
2. If three members, two of which are collinear, are connected to a joint that has no
external loads or reactions applied to it, then the force in the member that is not
collinear is zero.
Identification of Zero-Force Members
Identification of Zero-Force Members
Identification of Zero-Force Members
Method of Joints
SAMPLE PROBLEM
Using the method of joints, determine the force in each member of the
truss shown. State whether each member is in tension or compression.
945 lb
9ft
C
B
12 ft 3.75 ft
SAMPLE PROBLEM
Using the method of joints, determine the force in each member of the
truss shown. State whether each member is in tension or compression.
945 lb
Solution:
A
9ft
C
B
12 ft 3.75 ft
RB RC
SAMPLE PROBLEM
Using the method of joints, determine the force in each member of the
truss shown. State whether each member is in tension or compression.
945 lb
Solution: @ joint B:
A FBA
Solve for :
FBC
9ft
225 lb
C
B
12 ft 3.75 ft compression
RB = 225 lb RC = 720 lb
SAMPLE PROBLEM
Using the method of joints, determine the force in each member of the
truss shown. State whether each member is in tension or compression.
945 lb
Solution: @ joint B:
A FBA
FBC
9ft
tension
225 lb
C
B
12 ft 3.75 ft
RB = 225 lb RC = 720 lb
SAMPLE PROBLEM
Using the method of joints, determine the force in each member of the
truss shown. State whether each member is in tension or compression.
945 lb
Solution: @ joint C:
A FCA
Solve for :
9ft FBC= 300 lb
720 lb
C
B
12 ft 3.75 ft compression
RB = 225 lb RC = 720 lb
Sample Problem #1: Determine the force in each member of the Warren truss shown in
the figure by method of joints.
Problem #1: Use the method of joints to calculate the force in each members
of the given truss.
40 KN 20 KN
B C D
4m
A
H G F E
3m 3m 3m
3m
Problem #2: Use the method of joints to calculate the force in each members
of the given truss.
Method of Sections
Analysis of Plane Trusses by the Method of Sections
The method of joints, presented in the preceding section, proves to be very
efficient when forces in all the members of a truss are to be determined.
However, if the forces in only certain members of a truss are desired, the
method of joints may not prove to be efficient, because it may involve
calculation of forces in several other members of the truss before a joint is
reached that can be analyzed for a desired member force. The method of
sections enables us to determine forces in the specific members of
trusses directly, without first calculating many unnecessary member
forces, as may be required by the method of joints.
SAMPLE PROBLEM
Determine the forces in member CD,DG and GH of the truss shown using
method of sections
SAMPLE PROBLEM
Determine the forces in members CJ and IJ of the truss shown in
the figure by the method of sections
Problem #3: Determine the force in members ED,BD,and BC and state whether
they are in tension or compression.
