Design and Optimization of a Balsa Wood Truss Bridge - PDF

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Design and Optimization of a Balsa Wood Truss Bridge A Truss bridge is a bridge composed of connected elements (typically straight) which may be stressed from tension, compression or sometimes both in response to dynamic loads. Truss bridges are one of the oldest types of modern bridges. A truss bridge is economical to construct owing to its efficient use of materials. The nature of a truss allows for analysis of a structure using a few assumptions and the application of statics. For the purpose of analysis, trusses are assumed to be pin jointed where the straight components meet. This assumption means that members of the truss (chords, verticals and diagonals) will only act in tension or compression. A more complex analysis is required where rigid joints impose significant bending loads upon the elements. Parts of a Truss bridge In the bridge illustrated, vertical members are in tension, lower horizontal members in tension, shear and bending, outer diagonal and top members are in compression, while the inner diagonals are in tension. The central vertical member stabilizes the upper compression member, preventing it from buckling. If the top member is sufficiently stiff then this vertical element may be eliminated. If the lower chord (a horizontal member of a truss) is sufficiently resistant to bending and shear, the outer vertical elements may be eliminated, but with additional strength added to other members in compensation. The ability to distribute forces in various ways has led to a large variety of truss bridge types. Some types may be more advantageous when wood is employed for compression elements or other types may be easier to erect in particular site conditions, or when the balance between labor, machinery and material costs have certain favorable proportions. Truss types commonly used in bridges are: 1. Allan truss 2. Bailey bridge 3. Baltimore truss 4. Bollman truss 5. Bowstring arch truss (Tied arch bridge) 6. Brown truss 7. Brunel truss 8. Burr arch truss 9. Cantilevered truss 10. Fink truss 11. Howe truss 12. Kingpost truss 13. Lattice truss (Town’s lattice truss) 14. Lenticular truss 15. Long truss 16. Parker (Camelback) truss 17. Pegram truss 18. Pennsylvania (Petit) truss 19. Post truss 20. Pratt truss 21. Queenpost truss 22. Truss arch 23. Waddell truss 24. Warren (non-polar) truss 25. Whipple Pratt truss 26. Vierendeel truss Among the above types each has its own advantages and they are mostly chosen depending on type of application. For our design process we have chosen a Truss design which is close to Warren truss because of its simplicity in design. It consists of longitudinal members joined only by angled cross members, forming alternately inverted equilateral triangle-shaped spaces along its length, ensuring the no individual strut, beam or tie is subject to bending or torsional straining forces, but only to tension or compression. Loads on the diagonals alternate between compression and tension (approaching the center), with no vertical elements, while elements near the center must support both tension and compression in response to live loads. This configuration combines strength with economy of materials and therefore can be relatively light. Warren Truss The aim of the project is to use a ProM structural model to design a balsa wood truss bridge according to the given specifications and then optimize the bridge design for its weight. The bridge is to span 15 inch length and be no more than 6 inches high and 6 inches wide. It should be designed to support a downward load of 100 lbf placed at the top center of the structure and the goal is have a minimum weight bridge that will support the load with a factor of safety of two for both tension/compression failure and buckling. The initial design is that of a Warren truss as discussed above. And to make sure that the structure does not fail we start off with a conservative design. We choose all the elements of the structure to have maximum area of cross section and use high density balsa wood. In this way we have a design that supports the given load without failing. The initial design looks as shown below: Initial Design The analysis for beam tensile/compressive stresses and buckling shows that the structure is a safe design to start with. But we would want to optimize the weight of the structure. So we would want to eliminate those elements which are redundant or taking very little load. From the static analysis we can plot graphs for beam tensile/compressive stresses. The graph looks as shown below: Beam tensile/compressive stresses From the above figure we could clearly see that some of the members don’t or take very little stress. So we start optimizing the weight of the structure by eliminating those members which are supposed to be redundant still making sure that we satisfy the constraints. Our initial design has 44 members of maximum cross section area and high density balsa wood. After the elimination of redundant members our structure has 35 elements. And it looks as shown in the figure below: Intermediate design of the structure Beam tensile/compressive stresses From the above tensile/compressive stresses we could clearly see that the weight of the structure can be further reduced. However, we do not want to reduce the elements any further. But we could still reduce the weight of the beam by changing the area of cross section and density of remaining members. We do this process by taking the elements which are experiencing minimum stresses in the structure and iteratively reduce the weight still keeping in mind the failure criterion constraints. This iterative optimization process is carried out until our stresses reach the maximum admissible stresses. However during this process there is a chance that the structure may fail due to buckling. So we have to make sure the buckling load factors are greater than or equal to two. And the maximum Tensile and compressive beam stresses have to be monitored for every iteration such that they do not cross the max allowed stresses ie ------------------------------------. We continue this process until we achieve a right balance between the weight of the structure, the maximum beam stresses, and the buckling load factors for the 1st few modes. In the iterations shown below, by the 16th iteration we find that the maximum beam stresses do not increase much but the buckling load factors reduce drastically with any changes in the design. We compensate for this buckling by adding cross beams on the top surface of the design, and reducing the density and cross-sections of the other beams to compensate for the additional weight. We continue doing this until we get a BLF close to 2, at which point we stop the iteration process. Notations used for the analysis:  The materials used are defined as ‘H’ for heavy balsa and ‘L’ for light weight versions.  Cross sections predefined are o 1 – 0.0625 in square o 2 – 0.125 in square o 3 – 0.25 in square o 4 – 0.375 in square o 5 – 0.5 in square The steps taken in the iterative process: 1. Initial design and analysis 2. 6 Vertical elements removed 3. Removed 3 horizontal beams at the base 4. Density change for the horizontal beam(7) from H to L 5. Density change for all cross connecting elements(1,2,5,6,25,26,29,30) 6. Cross section change from 5 to 3 for yellow elements (3,4,27,28) 7. Cross section change for horizontal cross connecting elements (42,41,40,22,21,20,18) from 5 to 3 8. Increase density for 7 to avoid buckling 9. Changed the cross section back to 4 for (3,4,27,28) 10. Changed the CS of (35,13,17,38,) to 4 11. Changed the CS of (39,10,7,31) to 4 12. Changed the CS of (8,9,23,24) to 4 13. Changed the CS of (37,36,16,14) to 4 14. Changed the CS of (8,9,23,24) to 2 15. Changed the CS of (8,9,23,24) to 3 16. Additional cross links at the top with cross section 2 and L balsa 17. CS change for (14,16,36,37,13,35,38,17) changed to 3 18. CS changed to 4 for (6,5,29,30,1,2,25,26)/ and material changed to L for (3,4,27,28) 19. CS changed to 3 for (28,27,3,4,1,2,25,26,29,30,5,6) 20. CS changed to for the rest of the elements except the cross beams at the top 21. (40,41,42) material changed to H 22. (3,4,27,28) material changed to H 23. Final design The results are tabulated in the table as shown below: Table # of iterations No of elements Mass(Slugs) Mass(grams) Beam Tensile Beam Compression BLF mode 1 BLF mode 2 BLF mode 3 1 44 3.24E-02 4.54E+02 104.3 -102.4 3.43E+01 6.32E+01 6.98E+01 2 38 2.72E-02 3.81E+02 103.7 -106.2 2.92E+01 5.55E+01 6.12E+01 3 35 2.46E-02 3.44E+02 103.7 -106.2 2.82E+01 5.23E+01 5.23E+01 4 35 2.39E-02 3.34E+02 103.7 -106.2 2.80E+01 4.88E+01 4.89E+01 5 35 2.08E-02 2.92E+02 103.7 -106.2 2.00E+01 3.94E+01 4.49E+01 6 35 1.94E-02 2.71E+02 104.3 -102.4 1.93E+01 3.67E+01 3.92E+01 7 35 1.88E-02 2.64E+02 108.5 -107.2 1.88E+01 -2.18E+01 -2.18E+01 8 35 1.66E-02 2.32E+02 108.5 -107.2 8.05E+00 -2.12E+01 -2.17E+01 9 35 1.67E-02 2.34E+02 175.7 -107.2 8.07E+00 2.44E+01 -2.83E+01 10 35 1.72E-02 2.41E+02 105.2 -107.2 8.13E+00 2.53E+01 3.65E+01 11 35 1.56E-02 2.18E+02 184.9 -106.9 7.92E+00 2.41E+01 -2.72E+01 12 35 1.39E-02 1.95E+02 185.2 -189.1 4.69E+00 1.42E+01 2.24E+01 13 35 1.26E-02 1.77E+02 187.1 -188.7 3.98E+00 1.19E+01 1.56E+01 14 35 1.10E-02 1.54E+02 187.1 -190.4 3.68E+00 1.01E+01 1.45E+01 15 35 9.55E-03 1.34E+02 189 -127 1.09E+00 1.09E+00 1.12E+00 16 35 1.01E-02 1.41E+02 188.1 -326 2.47E+00 6.90E+00 8.75E+00 17 39 7.88E-03 1.10E+02 420.3 -430.8 4.22E+00 4.28E+00 4.83E+00 18 39 6.84E-03 9.58E+01 426.5 -430 4.01E+00 4.14E+00 4.47E+00 19 39 6.62E-03 9.26E+01 427.5 -430.5 3.97E+00 4.14E+00 4.35E+00 20 39 6.16E-03 8.63E+01 429.4 -430.3 3.67E+00 3.98E+00 4.22E+00 21 39 4.99E-03 6.98E+01 429.7 -432.1 1.63E+00 3.62E+00 4.17E+00 22 39 5.49E-03 7.69E+01 429.8 -432.5 2.02E+00 3.92E+00 4.51E+00 The final design as idealized in Pro E: The beam tensile/compressive stresses in the final design are shown in the figure below: Screen shot of final design and analysis showing the mass: Beam tensile/compressive stresses in final design Screenshot of the Buckling mode analysis results table with BLFs for the 1st three modes: Buckling mode1 in final design Buckling mode 2 in the final design