Exploring Engineering
Chapter 12
Civil Engineering
The Art and Engineering of Bridge
Design
We Cover These Topics
• Free-Body Diagrams in Static Equilibrium
• Structural Elements
• Efficient Structures
• The Method of Joints
• Solution of Large Problems
• Designing with Factors of Safety
Free-Body Diagrams in Static
Equilibrium
A FBD is a picture you draw of a system that
shows it isolated from its environment, with all the
connections, supports, and weights on the system
replaced by the forces they exert on the system.
FBD Example
200. lbf
FBD:
200. lbf
A
B
5.00 ft
11.3o
11.3o
y
50.0 ft
Figure 1: The rope bridge
x
 5.00 
tan-1 
 = 11.3o
 25.0 
Figure 2: Free-body
diagram of the rope
For this system, there are two equations governing static equilibrium
(What are they?)
These equations are special cases of Newton’s Law of Motion for a
system is at rest. Thus each force component, if not zero, is balanced by
at least one other force.
Structural Elements
• Bridges are an assembly of different types of
structural elements. Each element is
characterized by its geometry, the forces
acting upon it, or both.
• There are many different types of such
elements, ranging from plates and shells to
springs.
• Here, we will consider three different types of
elements: beams,
compression members, and
tension members.
A Beam
FBD:
A
C1
C2
C3
C4
B
The dashed line represents the deflected shape.
A Pier is an Example of a
Compression Member
FBD:
D
D
A Beam Supported Using
Tension Members
F
FBD:
F
What Kinds of Structural
Elements are Shown Here?
Efficient Structures
• The goal in designing an efficient structure is
to satisfy the requirements of the design at
minimum cost.
• For example, this means building a bridge
using the least amount of material.
• A bridge designer will employ the following
elements to achieve this goal.
Beams, Trusses, and Arches
Beams
Equal area beam cross-sections can have
very different efficiencies. From least
efficient to most efficient: (a) square crosssection, (b) I-beam, (c) truss bridge (end
view).
Increasing resistance to bending
N
S
(a)
N
S
(b)
N
S
(c)
Trusses
A structure composed of many connected members has
to meet the following two conditions to qualify as a
truss: (1) it must be composed entirely of tension and
compression members and (2) it must be fully
triangulated, meaning that every open space within the
structure is triangular in shape. In order to meet the first
condition, both the live load (due to traffic) and the
dead load (due to the weight of the members) should
be applied at the joints.
Some Examples of Truss
Bridges
Arches
An arch is a curved beam that is highest at midspan and lowest at the ends where it touches
ground. Because of the curvature, the
transmission of force from the top of the arch to
the ground supports is much more direct than in
a straight beam, where the live load effectively
has to make a 90 degree turn to reach the ends
of the beam. As a result, arches are stronger in
bending than straight beams with the same
cross-section.
Some Examples of Arches in
Bridges
The Method of Joints
• The method of joints is a way to find unknown forces
in a truss structure. The principle behind this method
is that all forces acting on a joint must add to zero if
the joint is to remain stationary.
• It assumes that all the members are made of
frictionless pins, making them two force members.
• Equations of static equilibrium can then be written for
each pinned joint, and the set of equations can be
solved simultaneously to find the forces acting in the
members.
Three-Hinged Arch
TOP VIEW
pin
First, draw a free
body diagram.
1.00 105 lbf
6.5 ft
1.00 105 lbf
A
B
4.27o
C
4.27o
y
B
FAB
174 ft
x
FBC
FAB
FBC
FAB
FBC
Ay
FAB
A
4.27o
Cy
4.27o FBC
Ax
 6.50 
where tan-1 
 = 4.27o
87
.
0


C
Cx
Three-Hinged Arch
Next, write the equilibrium force equations
for each joint.
Then, solve these equations for the
unknown forces.
lbf
Solution of Large Problems
 For
structures with more than two
members, solution of the equations by
hand is impractical. The require a matrix
solution that is best done by a computer.
 Computer programs for truss analysis
are readily available on the Internet.
Designing with Factors of
Safety

A Factor of Safety is the ratio of the maximum
strength of a part to the maximum load to be applied
to it.
 Factors of Safety are based on the accuracy of load,
strength, and wear estimates, the consequences of
engineering failure, and the cost.
 Components whose failure could result in substantial
financial loss, serious injury or death usually have a
safety factor of four or higher (often ten).
 Buildings commonly use a factor of safety of 2.0 for
each structural member.
Factors of Safety Example
Calculate the overall factor of safety of a truss
4.20 feet in length, a cross-sectional area of
0.500 in2 that is subjected to a force of 12,300
lbf. Assume SY = 36. ×106 lbf/in2, E = 29. ×106
lbf/in2 with a square cross-section.
Solution:
σ = F/A = 12,300/0.500 = 24,600 lbf/in2
FOS = Sy/σ = 36,000/24,600 = 1.46
Summary
A procedure for designing a statically determinate truss was outlined.
The steps in this procedure are:
 Estimate the live load.
 Establish a target value for the structure’s overall factor of safety.
 Choose a material, and look up its yield strength (SY) and elastic
modulus (E) in a property table.
 Propose an efficient, fully triangulated member topology and assign
enough dimensions to uniquely locate all of the joints.
 Apply the Method of Joints to determine the forces on the members.
 Calculate the required cross-sectional areas of the members.
If instead of design, the goal is to evaluate an existing statically determinate
truss with a known live load, the following steps should be taken:
 Apply the Methods of Joints to determine the forces on the members.
 Calculate the factor of safety of each member.
 Take the smallest of the factors of safety computed in step 2 to be the
overall factor of safety of the structure.