BRIDGE

The Grand Valley Bridge Project

Adam Miller
Tony Kuipers
EGR 209
November 4, 2002

Introduction and Objective
This project was used to build a truss bridge that would span a 40 meter
gap. The bridge was not allowed to exceed a maximum height of 14.5 meters above the road bed nor could it be more than 12.5 meters below the road bed. The bridge must be 5 meters wide with joints at 4 meter intervals along the deck. The roadbed must consist of 15 centimeters of concrete and 5 centimeters of asphalt. The bridge can be constructed of either standard solid bar or hollow tubing made of either carbon steel, high strength low alloy steel, or quenched and tempered steel and must support the weight of the members themselves, the deck, an additional 12.0 kN at each deck joint to account for cross members, and the weight of a moving H20-44 truck. The goal of this project was to create a bridge that would safely support the required loading while maintaining a minimum cost. Also, the internal member forces of the bridge were analyzed while the truck was in any of three different positions. The first position was with the truck directly over the second deck joint, the second was with the truck directly over the third deck joint, and the third was with the truck directly centered on the bridge. Also, factors of safety were calculated for each member in each of the three cases. Based upon the results of the internal force analysis, a minumum standard bolt diameter was calculated which could be used to construct the bridge.

THE GRAND VALLEY BRIDGE PROJECT RESULTS AND ANALYSIS

The bridge design chosen was that of an arch bridge. The arch bridge allows for minimal structural cost while still being able to safely support the required loading. The bridge chosen also allowed the use of only five different types of members which reduced production cost. The bridge design can be seen in appendix A-1.

The first step in analyzing the bridge was to calculate the forces induced at each bridge joint due to the weight of the concrete, asphalt, deck members, structural members, and truck. Based on the 1994 American Association of State Highway and Transportation Officials (AASHTO) the loading at each bridge joint will be given by the following formula.

EQUATION #1 LOADING

Q=Total Load
DC=Dead Load of Structural Components
DW=Dead Load of Wearing Surface (Asphalt)
LL=Live Load (Truck Weight)

(The weight of the truck must also be increased by a factor or 1.33 to account for dynamic loading.)

Given EQUATION #1, the next step was to calculate the weight of the concrete, asphalt, and structural members. This was done using the density of each material. The volume of each material was calculated and then multiplied by its density to find the weight in kg. This was then multiplied by 9.81 to convert kg to Newtons.

Density of Concrete = 2400 kg per cubic meter
Density of Asphalt = 2250 kg per cubic meter
Density of Steel = 7850 kg per cubic meter

Weight of Concrete

Weight of Asphalt

Since each half of the bridge will support half the deck weight, the total weight of the concrete and asphalt is first divided in half. Also, each outside deck joint will support half as much area as that of each inside deck joint. To take this into account, half the weight of the concrete and asphalt is divided by one less than the number of deck joints. This is the force supported by each inner deck joint, while the force supported by each outside deck joint will be half that.

Force Per Inside Deck Joint Due to Concrete

Force Per Outside Deck Joint Due to Concrete

Force Per Inside Deck Joint Due to Asphalt

Force Per Outside Deck Joint Due to Asphalt

The next step was to calculate the force on each deck joint due to the weight of the truck. Since the wheel base of the truck is 4 meters long, the axles will always lie directly between two joints in any of the three loading cases. This means that the forward most joint will support half the weight of the front axle, the rearward most joint will support half the weight of the rear axle, and the middle joint will support the remaining weight. Also, we assumed that half the weight of the truck would be supported by each side of the bridge. To account for this we divided the axle weights of the truck in half. The calculated forces will then be multiplied by a factor of 1.33 to account for dynamic loading.
(See Figure 1-1)

FIGURE 1-1 Truck Weight Distribution

Force Per Front Joint Due To Truck

Force Per Rear Joint Due To Truck

Force Per Middle Joint Due To Truck

The final loading comes from the weight of the structural members themselves. After the bridge was designed, The bridge was broken down into is individual joints and each structural member and joint was labeled. For a diagram of this labeling see appendix A-2. To calculate the member weight supported by each joint, the length and cross sectional area of each beam at a joint were multiplied by the density of steel, added together, and then divided by two. The weight is divided by two since each beam is supported by the two joints that support it. A complete table of member properties is given in appendix A-3.

Member Strengths

Also necessary for analysis is the calculated tensile and compressive strengths of each bridge member. The AASHTO standard for the strength of a member in tension is given by the following equation.

EQUATION #2 TENSILE STRENGTH

=factored tensile strength

=resistance factor for tension yielding (0.95)

=cross sectional area of member

For example, the calculated tensile strength of member AB would be as follows...

Calculation of compressive strength is also defined by the AASHTO.

EQUATION #3 COMPRESSIVE STRENGTH

= factored compressive load

= resistance factor for compression (0.95)

=yield stress of the member

=cross sectional area of the member

= dimensionless parameter that defines the boundary between inelastic and elastic buckling

L=Length of the member

I=moment of inertia of member
(given by West Point Bridge Designer)

For example, the calculation of the compressive strength of member AB would be as follows...

Since l is greater than 2.25, the compressive strength is...

A complete table of member properties can be seen in appendix A-3

The next step in analysis is calculating the internal member forces due to the truck being over the second bridge joint. A summary of the joint loading for case #1, as calculated by equation #1, is given in the following table.

Table 1-Case #1 Joint Loading

From this table we were able to draw free body diagrams for each joint and apply the proper loads. Next we wrote equations at each joint for the sum of the vertical and the sum of the horizontal forces. The coefficients of these equations then went into a 40x40 matrix. The applied loads were also arranged into a matrix. The coefficient matrix was then inverted and multiplied by the force matrix to obtain the internal member forces. A negative internal force meant that the member was in compression while a positive member force meant that the member was in tension. Based on the members being in tension or compression, the tensile or compressive strength of the members was divided by the members internal force to calculate the members' factors of safety. A complete view of the F.B.D's and calculations can be seen in Appendix B-1.

Following the analysis with the truck in the first position on the bridge, the bridge was analyzed with the truck in the second and third positions. The loading for each joint was calculated by EQUATION #1 and can be seen in Appendix C-1 while the F.B.D.'s and force calculations can be seen in Appendix C-2.

The next step in the bridge analysis was to calculate the internal member forces and factors of safety for the bridge with the truck directly over the center joint. The loading for each joint was again calculated from EQUATION #1. The Loading, F.B.D.'s and calculations can be seen in Appendices D-1 and D-2.

Minimum Bolt Diameter Calculations

In order to calculate the minimum required bolt diameter for the bridge, we assumed the worst case scenario in which the member with the maximum force was pinned to the outside of a joint. This would provide the maximum shear force. (See Figure 2)

Figure 2- Maximum Bolt Shear

From The Nutty Company Inc., the ultimate tensile strength of a grade 8 bolt is 150000psi. The shear strength can be considered half that of the ultimate tensile strength, therefore the shear strength is 75000 psi. The formula for shear stress is...

EQUATION #4 SHEAR STRESS

Our maximum internal force was...

Solving for the minimum bolt diameter is as follows...

The minimum bolt diameter is 1.8 inches, and the next largest standard bolt is 1.875 inches.

There were several expectations that we had before completing the calculations on the bridge. The main expectation was that all the factors of safety should be greater than 1. This implies that each member will be strong enough to support the load induced upon it. It was also expected that the internal force in member FP would be exactly that of the weight force of the members at joint P. This is because FP is the only member in the vertical direction at that joint. Another expectation that we had was that the members along the arch of the bridge would all be in compression, or have a negative force resultant. Also, the members along the base of the bridge were expected to be in tension, or have a positive internal force resultant. It was also expected that the total weight of the bridge would be the sum of the calculated vertical external forces and the horizontal external forces would be zero.
Our bridge analysis revealed that all of our expectations were true, excluding small errors due to the rounding error. For example, our horizontal external force was slightly greater than zero. All of our factors of safety were greater than 1 which implies that the bridge is strong enough to support the load placed upon it. The force in FP was equal to the structural weight of joint P, as expected. Our objectives were met in that our bridge can safely support the weight of the truck, and yet it was constructed with minimal cost.
Analyzing this bridge was similar to many of the homework problems done in class. The only difference was the increased difficulty of calculating loads and the increased number of members. Organization was a key to successfuly being able to calculate data. The organization of our data was the greatest strength we had in completeing this project. The most significant error we made was to not carry enough significant figures in the angles between members. This caused much of our error in calculations. If done again, more accuracy could be obtained by carrying more significant figures.
In designing our bridge we kept a few things in mind. Members that would be in compression were made from hollow tubing because hollow tubing has a smaller moment of inertia than solid bar. This increases the critical load that can be placed upon it before it buckles. Members that would be in tension were made from solid bars. Also, constructing members from higher strength steel alloys made the yield strength higher, which reduces the cross sectional area of the member, and thus reduces the weight of the bridge itself. However, stronger steel is more expensive and iterative design was used to find the dividing line between cost and strength.
When experimenting with the West Point Bridge Designer, it was discovered that an arch bridge would be the cheapest bridge that safely supported the loading placed on it. The first trial bridge desigs were Warren through trusses and Pratt deck trusses. The cost of these bridges did not meet our expectations, so alternative truss desings were created. It was thought that an arch bridge would transmit the loading smoothly throughtout the arch of the bridge. When creating the arch bridge design, we attempted to reduce the number of different members in the bridge which kept production cost down. Also, higher strength steel was used in high compression members and along the bridge deck to keep the weight of the bridge down.

This experiment is significant because we found an optimal truss design that safely meets all of the design specifications at the lowest cost. This is very important because our first goal as engineers is safety followed by cost, and these were both big points in the experiment. The bridge we constructed was under the maximum height limit at six meters. By finding all of the internal forces involved we found the maximum forces during the three truck loading cases. The maximum force for the first case was 6.1x10^5 N. The maximum force for the second case was 7.2x10^5 N. The maximum force for the last case was 8.5x10^5 N. We used the highest force to find the minimum bolt diameter that is required to safely support our bridge. The bolt diameter was 1.8 inches. By going on Nutty.com we chose a nominal bolt size 1.875 inches in diameter. Our goal of making a safe and cheap bridge was successful because the bridge we designed cost $7,011 and safely supported a moving weight. All of our results seem quite reasonable. Our member forces seemed reasonable in that the members that would appear to be in tension were also calculated to be in compression. Also, a bolt diameter of 1.875 inches seems reasonable in constructing a bridge.
The calculations in the experiment proved to be our toughest problem. Organizing and understanding the equations took a lot of time and brainstorming. We feel like this experiment did a good job of combining our class learning with a real situation. In performing this experiment again we would try to be more careful with our calculations so that they are easier to follow through.

Ressler, Colonel Stephen. "West Point Bridge Designer 2002." Department of Civil and Mechanical
Engineering,United States Military Academy, West Point, New York.

Nutty Company Inc. Web Site. Internet. (9/19/02) Available:
http://www.nutty.com

Reffeor, Wendy. "Truss Design Project": Handout

Ressler, Colonel Stephen. "Structural Analysis of a Typical Truss Bridge".:West Point Bridge Designer 2002. Department of Civil and Mechanical Engineering,United States Military Academy, West Point, New York.