Understanding the Application of the
Equations of Equilibrium
Procedure for Analysis
1
Free-Body Diagrams
• Disassemble the structure and draw a freebody diagram of each member.
• Recall that reactive forces common to two
members act with equal magnitudes but
opposite directions on the respective free-body
diagrams of the members.
• All two-force members should be
identified. These members, regardless of
their shape, have no external loads on
them, and therefore their free-body diagrams
are represented with equal but opposite
collinear forces acting on their ends.
• In many cases it is possible to tell by
inspection the proper arrowhead sense of
direction of an unknown force or couple
moment
2
Equations of Equilibrium
• Count the total number of unknowns to make
sure that an equivalent number of equilibrium
equations can be written for solution.
• Many times, the solution for the unknowns will be
straightforward if the moment equation is applied
about a point (©MO = 0) that lies at the
intersection of the lines of action of as many
unknown forces as possible.
• When applying the force equations and ©Fx = 0
©Fy = 0, orient the x and y axes along lines that will
provide the simplest reduction of the forces into their
x and y components.
• If the solution of the equilibrium equations
yields a negative magnitude for an unknown force or
couple moment, it indicates that its arrowhead sense
of direction is opposite to that which was assumed on
the free-body diagram.
Sample Problems
Determine the Reactions of the Beam Shown:
Regular Figure
Solution:
FBD
Sample Problems
Determine the Reactions of the Beam Shown:
Solution:
Sample Problems
Determine the reactions on the beam. Assume A is a pin and the support at B is a roller (smooth surface).
Solution:
Analysis of Statically Determinate Trusses
TRUSS
A truss is a structure composed of slender members joined together at their end points.
Bridge Trusses
Roof Trusses
are often used as part of an
industrial building frame
Are often used to
support bridges
Analysis of Statically Determinate Trusses
ASSUMPTIONS FOR THE DESIGN
1
The members are joined together
by smooth pins. In cases where
bolted or welded joint
connections are used, this
assumption is generally
satisfactory provided the center
lines of the joining members
are concurrent at a point
2
All loadings are applied at the
joints. In most situations, such as
for bridge and roof trusses, this
assumption is true. Frequently in
the force analysis, the weight of
the members is neglected, since
the force supported by the
members is large in comparison
with their weight.
Classification of Coplanar Trusses
SIMPLE TRUSS
Simple Trusses are made with Stable
Elements such as
shown in the figure, commonly,
triangular members. For this method
of construction, however, it is
important to realize that simple
trusses do not
have to consist entirely of triangles.
Analysis of Statically Determinate Trusses
COMPOUND TRUSS
A compound truss is formed by
connecting two or more simple
trusses together. Quite often this
type of truss is used to support
loads acting over a large span, since
it is cheaper to construct a
somewhat lighter compound truss
than to use a heavier single simple
truss.
Analysis of Statically Determinate Trusses
COMPLEX TRUSS
A complex truss is one that cannot
be classified as being either simple
or compound.
METHOD OF JOINTS
If a truss is in
equilibrium, then each of
its joints must also be in
equilibrium. Hence, the
method of joints consists
of satisfying the
equilibrium conditions and
for the forces exerted Fx = 0
Fy = 0 on the pin at each
joint of the truss.
METHOD OF JOINTS
Procedure for Analysis
METHOD OF JOINTS
Determine the force at joint A, G and B of the roof truss shown in the photo. The dimensions and
loadings are shown. State whether the members are in tension or compression.
Solution:
At Joint A
At Joint G
At Joint B
METHOD OF JOINTS
Determine the force at the scissor roof truss shown in the photo. The dimensions and loadings
are shown. State whether the members are in tension or compression.
METHOD OF SECTION
If the forces in only a few members of a
truss are to be found, the method of
sections generally provides the most
direct means of obtaining these forces.
The method of sections consists of
passing an imaginary section through
the truss, thus cutting it into two parts.
Provided the entire truss is in equilibrium,
each of the two parts must also be in
equilibrium; and as a result, the three
equations of equilibrium may be applied
to either one of these two parts to
determine the member forces at the “cut
section.”
METHOD OF SECTION
Procedure for Analysis
METHOD OF SECTION
Determine the force in members GJ and CO of
the roof truss shown in the photo. The
dimensions and loadings are shown. State
whether the members are in tension or
compression. The reactions at the supports
have been calculated.
MEMBER GC
MEMBER CF
METHOD OF SECTION
Determine the force in members GF and GD of the truss shown. State whether
the members are in tension or Compression. The reactions at the supports
have been calculated.
CABLES AND ARCHES
Cables subjected to Concentrated Load
When a cable of negligible
weight
supports
several
concentrated loads, the cable
takes the form of several
straight-line segments, each of
which is subjected to a constant
tensile force.
CABLES AND ARCHES
Example: Determine the tension in each segment of the cable shown. Also, what is the dimension h?
Solution:
From Figure:
@ Point C:
@ Point B:
CABLES AND ARCHES
Cables subjected to Uniformly Distributed Load
𝑻𝑨𝒚
𝑻𝑨
𝑻𝑩𝒚
𝑻𝑨𝒙
h1
𝑻𝑩
A
B
x
L-x
L
W (Kn/m)
h2
𝑻𝑩𝒙
CABLES AND ARCHES
Example: Solve for the value of the Tension at A, Slope of the cable at A and Tension at the lowest
Point C. The Cable shown supports a horizontal uniform load of 10 Kn/m.
12
A
B
30
6
CABLES AND ARCHES
The Three-Hinged Arch is subjected to the two concentrated forces as shown, find the Reaction
at A, B, C.
20 kN
B
25
15
8 kN
6
C
A
23
7.6
23
Thank You!