TRUSS BRIDGE DESIGN
Objective
Engineers have been building bridges since the beginning of time. The simple task of crossing a waterway presents a challenge rooted in the most fundamental concept of physics and mathematics. This academic exercise was the subject of my final project for ENGR0135: Statics and Mechanics of Materials I in the Fall 2020 semester. We were placed into teams of three and assigned the task of constructing a balsa wood truss bridge that could safely accommodate 100 pounds of force. We had to decide the length, angulation, and general arrangement of the beams supported by proper analyses. However, this project was completed during the height of the COVID-19 pandemic, so the bridge was never physically constructed. The virtual bridge was a complete SolidWorks CAD model.
The Global Force Diagram: The left pin is fixed, exerting a reactionary force in both the vertical and horizontal direction. The right pin is a roller then only exerts a reactionary force in the vertical direction.
We also created individual free-body force diagrams for each joint and derived accompanying equilibrium equations. Next, we compared various crossbeam angles and lengths to finalize an economic design that would comfortably endure the load. By altering the interior angle denoted as Φ, the crossbeam length as well as the end angles changed. We employed this methodology to analyze our bridge with Φ values of 40°, 45°, and 50°.
After evaluating each member of the system, we had 26 equations and 26 unknowns. A system of equations of this magnitude requires a mathematical program to solve. Using the coefficients of these equations to make a matrix as well as defining a solution matrix, we solved for these values in Microsoft Excel. Through the use of Excel, we easily adjusted the θ and Φ values to find the angulation that can support the load effectively. After employing the equilibrium equations and solving for the forces in each member, we performed a stress analysis to find normal, shear, and bearing stress in each pin.
We found that the greater the Φ value, the lesser the bearing stress. However, when Φ = 40° the bearing stress is still minimal compared to the yield stress. Therefore, all of these designs can tolerate the 100 lb. load, however, we see that Φ = 50° has the lowest normal stress values, followed by Φ = 46° and ending with Φ = 40°. Our team decided that we could pick our design based on the most cost-effective design, considering that they all can withstand the normal stresses. After choosing the cheapest option, we did an FEA analysis in SolidWorks to confirm the results of our stress analysis.
Force Analysis
There are many variations of truss bridges, but we chose a Howe truss to base our design on. This type of bridge consists of vertical members in tension and diagonal members in compression. After extensive research of the Howe Truss, we began our calculations. With cost optimization in mind, we decided a six-panel bridge would be sufficient to bear the weight of 100 pounds. Using the Method of Joints, we constructed a global free-body force diagram to model the weight distribution.
FEA Analysis
Using the weldments feature in SolidWorks, a truss bridge can be created from a sketch and truss analysis studies within the program can calculate the normal stress in each member and pin. The sketch was converted into a 3-dimensional bridge by turning each line into a beam using the structural member feature. Each member’s stress value matches up with the values we expected from our hand calculations. The Finite Element Analysis gave us the normal stress in each member of the bridge. Once we validated our normal stress calculations, we knew that our shear stress and bearing stress were accurate as well. The FEA showed that the bridge was strong enough to support the 100 pounds of force, and the minimum factor of safety is 8.5. The simulation was run on a single part that resembled the bridge using weldments and structural members. However, our team created another assembly in SolidWorks to show how the bridge would actually be constructed using nuts, bolts, and washers to attach the various pieces.
The SolidWorks model used to run the FEA
The resulting FEA for one particular member of the bridge showing stress in Mega Pascals.
Final Report
Our full analysis was detailed in a technical report with all equations, diagrams, stress tables, and FEA outputs. A cost analysis was also performed by estimating the costs of all wood panels, nuts, bolts, and washers when varying the angle of the beams, Φ. The cheapest design was at Φ = 40°, resulting in an approximately $12 bridge. Our research indicated that this Howe Truss design is a reliable, durable, and relatively cheap choice for the 100-pound load. The design report received an A+ grade.
The final CAD model of the Howe Truss Bridge, complete with nuts, bolts, and washers