BRIDGES Christopher Rego October 28, 2006 Revised June 2003 SECME – M-DCPS Division of Mathematics and Science Education FIU

BRIDGESBRIDGES

Christopher Rego

October 28, 2006Revised June 2003

SECME – M-DCPS Division of Mathematics and Science EducationFIU

• History of Bridge Development

• How Bridges Work

• Basic Concepts

• Types of Bridges

• Concepts Associated with Bridge Engineering

• Truss Analysis

• Tips for Building Bridges

• Bridge Construction

Work Plan Work Plan

700 A.D. Asia700 A.D. Asia

100 B.C. Romans100 B.C. Romans

Natural BridgesNatural Bridges

Clapper Bridge

Tree trunkStone

The ArchNatural Cement

Roman Arch Bridge

History of Bridge DevelopmentHistory of Bridge Development

Great Stone Bridge in China

Low BridgeShallow Arch

1300 A.D. Renaissance1300 A.D. Renaissance

Strength of Materials

Mathematical Theories

Development of Metal

First Cast-Iron Bridge

Coalbrookdale, England

1800 A.D.1800 A.D.

History of Bridge DevelopmentHistory of Bridge Development

Britannia Tubular Bridge

1850 A.D.1850 A.D.

Wrought Iron

Truss BridgesMechanics of Design

Suspension BridgesUse of Steel for the suspending cables

1900 A.D.1900 A.D.

1920 A.D.1920 A.D.

Prestressed ConcreteSteel

2000 A.D.2000 A.D.

Every passing vehicle shakes the bridge up and down, making waves that can travel at hundreds of kilometers per hour. Luckily the bridge is designed to damp them out, just as it is designed to ignore the efforts of the wind to turn it into a giant harp. A bridge is not a dead mass of metal and concrete: it has a life of its own, and understanding its movements is as important as understanding the static forces.

How Bridges Work?How Bridges Work?

Compression Tension

Basic Concepts Basic Concepts

Span - the distance between two bridge supports, whether they are columns, towers or the wall of a canyon.

Compression - a force which acts to compress or shorten the thing it is acting on.

Tension - a force which acts to expand or lengthen the thing it is acting on.

Force - any action that tends to maintain or alter the position of a structure


Beam - a rigid, usually horizontal, structural element

Pier - a vertical supporting structure, such as a pillar

Cantilever - a projecting structure supported only at one end, like a shelf bracket or a diving board

Beam

Pier

Load - weight distribution throughout a structure


Truss - a rigid frame composed of short, straight pieces joined to form a series of triangles or other stable shapes

Stable - (adj.) ability to resist collapse and deformation; stability (n.) characteristic of a structure that is able to carry a realistic load without collapsing or deforming significantly

Deform - to change shape

To dissipate forces is to spread them out over a greater area, so that no one spot has to bear the brunt of the concentrated force.

To transfer forces is to move the forces from an area of weakness to an area of strength, an area designed to handle the forces.


Buckling is what happens when the force of compression overcomes an object's ability to handle compression. A mode of failure characterized generally by an unstable lateral deflection due to compressive action on the structural element involved.

Snapping is what happens when tension overcomes an object's ability to handle tension.

The type of bridge used depends on various features of the obstacle. The main feature that controls the bridge type is the size of the obstacle. How far is it from one side to the other? This is a major factor in determining what type of bridge to use.

The biggest difference between the three is the distances they can each cross in a single span.

Types of BridgesTypes of BridgesBasic Types:

•Beam Bridge•Arch Bridge•Suspension Bridge

Types of BridgesTypes of Bridges

Beam BridgeBeam Bridge

Consists of a horizontal beam supported at each end by piers. The weight of the beam pushes straight down on the piers. The farther apart its piers, the weaker the beam becomes. This is why beam bridges rarely span more than 250 feet.

Forces

When something pushes down on the beam, the beam bends. Its top edge is pushed together, and its bottom edge is pulled apart.


Beam BridgeBeam Bridge

Truss BridgeTruss Bridge

Forces

Every bar in this cantilever bridge experiences either a pushing or pulling force. The bars rarely bend. This is why cantilever bridges can span farther than beam bridges


Arch BridgesArch Bridges

The arch has great natural strength. Thousands of years ago, Romans built arches out of stone. Today, most arch bridges are made of steel or concrete, and they can span up to 800 feet.


Forces

The arch is squeezed together, and this squeezing force is carried outward along the curve to the supports at each end. The supports, called abutments, push back on the arch and prevent the ends of the arch from spreading apart.


Arch BridgesArch Bridges

Suspension BridgesSuspension Bridges

This kind of bridges can span 2,000 to 7,000 feet -- way farther than any other type of bridge! Most suspension bridges have a truss system beneath the roadway to resist bending and twisting.


Forces

In all suspension bridges, the roadway hangs from massive steel cables, which are draped over two towers and secured into solid concrete blocks, called anchorages, on both ends of the bridge. The cars push down on the roadway, but because the roadway is suspended, the cables transfer the load into compression in the two towers. The two towers support most of the bridge's weight.


Suspension BridgesSuspension Bridges

The cable-stayed bridge, like the suspension bridge, supports the roadway with massive steel cables, but in a different way. The cables run directly from the roadway up to a tower, forming a unique "A" shape. Cable-stayed bridges are becoming the most popular bridges for medium-length spans (between 500 and 3,000 feet).


Cable-Stayed BridgeCable-Stayed Bridge

How do the following affect your structure?ForcesLoadsMaterialsShapes

Let’s try it: http://www.pbs.org/wgbh/buildingbig/lab/forces.html

The bridge challenge at Croggy Rock:http://www.pbs.org/wgbh/buildingbig/bridge/index.htmlbridge/index.html

Interactive Page Interactive Page

Congratulations!

Pythagorean Theorem

Basic math and science conceptsBasic math and science concepts

Bridge EngineeringBridge Engineering

a

c

b

c2=b2+a2



Fundamentals of Statics

Fy = R1+R2-P = 0

Fx = 0

F

R1 R2

x

y



Fundamentals of Mechanics of Materials

Modulus of Elasticity (E):

E

E= StressStrain

F/AL/Lo

=

Where:

F = Longitudinal Force

A = Cross-sectional Area

L = Elongation

Lo = Original Length

Lo

F

F

To design a bridge like you need to take into account the many forces acting on it :

•The pull of the earth on every part

•The ground pushing up the supports

•The resistance of the ground to the pull of the cables

•The weight of every vehicle

Then there is the drag and lift produced by the wind

•The turbulence as the air rushes past the towers




Density 163 ± 10 kg/m³

low density 4.7 MPamedium density 12.1 MPahigh density 19.5 MPa

low density 7.6 MPamedium density 19.9 MPahigh density 32.2 MPaElastic Modulus - Compression 460 ± 71 MPaElastic Modulus - Tension 1280 ± 450 MPa

Compressive Strength¤

Tensile Strength¤


Balsa Wood Information

Truss AnalysisTruss Analysis


Structural Stability Formula

K = 2J - R

Where:K = The unknown to be solvedJ = Number of JointsM = Number of MembersR = 3 (number of sides of a triangle)

K Results Analysis:

If M = K Stable Design

If M < K Unstable Design

If M > K Indeterminate Design



Structural Stability Formula (Example)

Joints

J=9

Members

M=15

K = 2 (9) – 3 = 15

15 = M = K then The design is stable

http://www.jhu.edu/virtlab/bridge/truss.htm

West Point Bridge Software:

http://bridgecontest.usma.edu/



Tips for building a bridgeTips for building a bridge

1. Commitment - Dedication and attention to details. Be sure you understand the event rules before designing your prototype.

1) Draw your preliminary design

2) ALL joints should have absolutely flush surfaces before applying glue.

Glue is not a "gap filler", it dooms the structure!

3) Structures are symmetric.

4) Most competitions require these structures to be weighed. Up to 20% of the structure's mass may be from over gluing.

Stresses flow like water.

Where members come together there are stress concentrations that can destroy your structure.

Here is a connection detail of one of the spaghetti bridges.

The Importance of ConnectionsThe Importance of Connections

Tacoma Narrows FailureTacoma Narrows Failure

Basswood Bridge Assignment

• Brainstorm & decide on a truss configuration.– Use Appendix A for

help

• Create the structural model.

• Check static determinacy and stability.

• Draw plans.• Create a schedule of

truss members.• Build the bridge.

Decide on Truss Configuration

• Are you going for aesthetics or strength or ease of construction?

• Only basswood sized at 3/32 inch square cross-section is allowed.

• Any commonly available adhesive is allowed. • Basswood may be notched, cut, sanded or

laminated.• Bridge may not be stained, painted or coated in

any fashion with any foreign substance.

Truss Configuration•Mass shall be no greater than 25grams

•Bridge must span a gap (S) of 300 mm

•Maximum length (L) of 400 mm

•Maximum width (W) of 80 mm

•Maximum height (H) of 200 mm above the support surfaces

•No portion of the bridge may extend below the top of the support surfaces.

•The loading plane (P) shall be horizontal and shall lie on the physical top of the bridge between

100 mm -200 mm above the support surfaces.

•Bridge must support loading plate

without rod attached

•Clearance allowed for loading eyebolt to be attached to the plate and hang down vertically through the bridge below the loading point.

Truss Configuration

• The load will be applied downward, from below, by means of a 50.0 mm square plate

• The plate will be 6.35 mm (1/4 inch) thick

• The eyebolt will have a 9.53 mm (3/8 inch) diameter

• Loading points are 50 mm on either side of the center of the 300 mm span.

Structural Model

• A pair of identical two-dimensional trusses.

• Draw the geometry of one main truss

• Indicate the dimensions of the member centerlines.

• Identify joints with the letters A -Z.

Check Static Determinacy and Stability

• We must first verify that it is statically determinate and stable (slide #27)

• j is the number of joints

• m is the number of members

• K=2J-R K Results Analysis:

If M = K Stable Design

If M < K Unstable Design

If M > K Indeterminate Design

Draw Plans

FULL SCALE LAYOUT

• Now you are smart enough to do the drawing yourself!

• Big sheet of paper

• Metric Ruler

• Pencil

• Good Eraser

• T-Square

• Triangles

• Tape

ACCURACY COUNTS!!!

• Graph paper works well

• Parallel and Perpendicular lines should be just that!!!

Draw Plans• Lay Out Centerlines• Draw one main truss• Drawing must be exactly full size• Use the dimensions decided on your structural

model to ensure that all joints and members are at their correct positions.

• Label the joints with letters• Use only a single line for each member. • These lines are the centerlines of the members.

Draw Plans•All member centerlines must intersect precisely at the joints.

•If centerlines do not all intersect at a common point, the members will bend when the truss is loaded.

•Bending may cause some members to fail at a lower load than they were designed for.

•Performance depends heavily on the accuracy of the member centerlines!•Draw with Care!!!!

Documents

BRIDGES Christopher Rego October 28, 2006 Revised June 2003 SECME – M-DCPS Division of Mathematics and Science Education FIU