Problem 3 Pratt truss bridge. A truss bridge has a load-bearing superstructure composed of triangular elements. The Pratt truss appears to be one of the oldest truss designs still in use. In the analysis of trusses in general we assume that all loads are applied at joints only. If a weight of a member (or link) is significant then half of this weight may be considered to be applied to the joint at each end of the member. It is assumed that every member is in pure compression or pure tension. Shear, bending moments, etc are all considered to be negligible. The forces acting upon the bridge shown in Figure 3.1 can be decomposed into forces acting in the x-y plane and forces acting in the x-z plane (i.e. the forces in the lateral bracing members). Show how this observation permits a very simple analysis of the structure whereby the cross-bracing members can be simply ignored and where the truss to be analysed is a planar truss, i.e. a truss in which all members lie in one plane. A planar truss is statically determinate if the number of joints j and the number of members m satisfy the relation 2j-m+3. Show that the planar truss which adequately models the bridge of Figure 3.1 is statically determinate. The analysis of a statically determinate truss is relatively straightforward entailing only a balance of forces at each joint. On each side of the bridge of Figure 3.1 each horizontal member and each vertical member is 4 m in length. All of the heavier members of the bridge are assumed to carbon steel S275 bars having a square cross-section of side 120 mm. The lighter cross-bracing members have negligible weight in comparison. The bridge deck accommodates just a single lane of traffic. It is 4 m wide and made of reinforced concrete (of thickness 0.15 m) with an asphalt overlay (of thickness 0.05 m) supported by transverse floor panels also of reinforced concrete. The floor panels are square, of side 4 m and have a thickness of 0.025 m. You should have no difficulty consulting either websites or texts to find the density of carbon steel S275, reinforced concrete and asphalt A 2-axle lorry has a wheelbase of 4 m. The front axle load is 10 kN and the rear axle load is 36 kN Given the speed limit on the bridge the movement of the lorry at this limit has the dynamical effect of increasing the axle loads by 45% For a given position of the lorry on the bridge determine all of the forces in the various members Write a MATLAB function M-file to calculate the forces in all of the members (excluding the cross- bracing members) for a given position of the lorry. Write MATLAB code to create a graphical representation of the bridge. Let the colour of a member be chosen to reflect the force in it, with blue indicating a member in which the force is very low and red a member in which the force is very high and colour smoothly transitioning from blue to red as forces increase. Employing a number of patch graphics objects (see laboratory MS0.2 and module notes SIM2) and employing the animation facilities within MATLAB (see module notes SIM2) create either a MATLAB animation of the movement of the lorry across the bridge, with changing member colours indicating the changing forces within the various members Using DSM or otherwise find the deflections in the members as the lorry crosses the bridge and identify the most critical member and the most critical time. Figure 3.1: 2-axle lorry crossing a Pratt truss bridge. Problem 3 Pratt truss bridge. A truss bridge has a load-bearing superstructure composed of triangular elements. The Pratt truss appears to be one of the oldest truss designs still in use. In the analysis of trusses in general we assume that all loads are applied at joints only. If a weight of a member (or link) is significant then half of this weight may be considered to be applied to the joint at each end of the member. It is assumed that every member is in pure compression or pure tension. Shear, bending moments, etc are all considered to be negligible. The forces acting upon the bridge shown in Figure 3.1 can be decomposed into forces acting in the x-y plane and forces acting in the x-z plane (i.e. the forces in the lateral bracing members). Show how this observation permits a very simple analysis of the structure whereby the cross-bracing members can be simply ignored and where the truss to be analysed is a planar truss, i.e. a truss in which all members lie in one plane. A planar truss is statically determinate if the number of joints j and the number of members m satisfy the relation 2j-m+3. Show that the planar truss which adequately models the bridge of Figure 3.1 is statically determinate. The analysis of a statically determinate truss is relatively straightforward entailing only a balance of forces at each joint. On each side of the bridge of Figure 3.1 each horizontal member and each vertical member is 4 m in length. All of the heavier members of the bridge are assumed to carbon steel S275 bars having a square cross-section of side 120 mm. The lighter cross-bracing members have negligible weight in comparison. The bridge deck accommodates just a single lane of traffic. It is 4 m wide and made of reinforced concrete (of thickness 0.15 m) with an asphalt overlay (of thickness 0.05 m) supported by transverse floor panels also of reinforced concrete. The floor panels are square, of side 4 m and have a thickness of 0.025 m. You should have no difficulty consulting either websites or texts to find the density of carbon steel S275, reinforced concrete and asphalt A 2-axle lorry has a wheelbase of 4 m. The front axle load is 10 kN and the rear axle load is 36 kN Given the speed limit on the bridge the movement of the lorry at this limit has the dynamical effect of increasing the axle loads by 45% For a given position of the lorry on the bridge determine all of the forces in the various members Write a MATLAB function M-file to calculate the forces in all of the members (excluding the cross- bracing members) for a given position of the lorry. Write MATLAB code to create a graphical representation of the bridge. Let the colour of a member be chosen to reflect the force in it, with blue indicating a member in which the force is very low and red a member in which the force is very high and colour smoothly transitioning from blue to red as forces increase. Employing a number of patch graphics objects (see laboratory MS0.2 and module notes SIM2) and employing the animation facilities within MATLAB (see module notes SIM2) create either a MATLAB animation of the movement of the lorry across the bridge, with changing member colours indicating the changing forces within the various members Using DSM or otherwise find the deflections in the members as the lorry crosses the bridge and identify the most critical member and the most critical time. Figure 3.1: 2-axle lorry crossing a Pratt truss bridge