Exploring Engineering

Chapter 12

Civil EngineeringThe 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

50.0 ft

5.00 ft

200. lbf

Figure 1: The rope bridge

200. lbf

A B

11.3o11.3o

FBD:

x

y

tan-1 = 11.3o

0.25

00.5

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.

• 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.

Structural Elements

A Beam

A B

C1 C2 C3 C4FBD:

The dashed line represents the deflected shape.

A Pier is an Example of a Compression Member

D

D

FBD:

A Beam Supported Using Tension Members

F

FFBD:

What Kinds of Structural Elements are Shown Here?

• 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

Efficient Structures

BeamsEqual area beam cross-sections can have very different efficiencies. From least efficient to most efficient: (a) square cross-section, (b) I-beam, (c) truss bridge (end view).

N S N SN S

Increasing resistance to bending

(a) (b) (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

ArchesAn arch is a curved beam that is highest at mid-span 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 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.

The Method of Joints

Three-Hinged Arch

First, draw a free body diagram.

AB C

TOP VIEW

pin

174 ft

6.5 ft

1.00î 105 lbf

1.00î 105 lbf

FBCFAB

B

4.27o4.27o

A

FAB

Ax

Ay 4.27o

C

FBCCy4.27o

Cx

FBC

FBC

FAB

FAB

x

y

where tan-1 = 4.27o

0.87

50.6

Three-Hinged Arch

Next, write the equilibrium force equations for each joint.

Then, solve these equations for the unknown forces.

lbf

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.

Solution of Large Problems

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.

Designing with Factors of Safety

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

SummaryA 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.