Instructional Objectives:
The objectives of this lesson are as follows.
Introduction
In previous lessons, we have studied the development of influence lines for beams loaded with single point load, UDL and a series of loads. In similar fashion, one can construct the influence lines for the trusses. The moving loads are never carried directly on the main girder but are transmitted across cross girders to the joints of bottom chord. Following section will explain load transmission to the trusses followed by the influence lines for the truss reactions and influence lines for truss member forces.
Bridge Truss Floor System
A typical bridge floor system is shown in Figure 40.1. As shown in Figure, the loading on bridge deck is transferred to stringers. These stringers in turn transfer the load to floor beams and then to the joints along the bottom chord of the truss.
It should be noted that for any load position; the truss is always loaded at the joint.
Influence lines for truss support reaction
Influence line for truss reactions are of similar to that a simply supported beam. Let us assume that there is truss with overhang on both ends as shown in Figure 40.2. In this case, the loads to truss joints are applied through floor beams as discussed earlier. These influence lines are useful to find out the support, which will be critical in terms of maximum loading.
The influence lines for truss reactions at A and B are shown in Figure 40.3.
Influence lines for truss member forces
Influence lines for truss member force can be obtained very easily. Obtain the ordinate values of influence line for a member by loading each joint along the deck with a unit load and find member force. The member force can be found out using the method of joints or method of sections. The data is prepared in tabular form and plotted for a specific truss member force. The truss member carries axial loads. In the present discussion, tensile force nature is considered as positive and compressive force nature is considered as negative.
Numerical Examples
Example 1:
Construct the influence line for the force in member GB of the bridge truss shown in Figure 40.4.
Solution:
Tabulated Values:
In this case, successive joints L_{0}, L_{1}, L_{2}, L_{3}, and L_{4} are loaded with a unit load and the force F_{L2U3} in the member L_{2}U_{3} are using the method of sections. Figure 40.5 shows a case where the joint load is applied at L_{1} and force F_{L2U3 }is calculated.
Figure 40.5: Member Force F_{L2U3} Calculation using method of sections.
The computed values are given below.
Influence line: Let us plot the tabular data and connected points will give the influence line for member L_{2}U_{3}. The influence line is shown in Figure 40.6. The figure shows the behaviour of the member under moving load. Similarly other influence line diagrams can be generated for the other members to find the critical axial forces in the member.
Example 2:
Tabulate the influence line values for all the members of the bridge truss shown in Figure 40.7.
Solution:
Tabulate Values:
Here objective is to construct the influence line for all the members of the bridge truss, hence it is necessary to place a unit load at each lower joints and find the forces in the members. Typical cases where the unit load is applied at L_{1}, L_{2} and L_{3} are shown in Figures 40.8-10 and forces in the members are computed using method of joints and are tabulated below.
Member | Member force due to unit load at: | ||||||
L_{0} | L_{1} | L_{2} | L_{3} | L_{4} | L_{5} | L_{6} | |
L_{0}L_{1} | 0 | 0.8333 | 0.6667 | 0.5 | 0.3333 | 0.1678 | 0 |
L_{1}L_{2} | 0 | 0.8333 | 0.6667 | 0.5 | 0.3333 | 0.1678 | 0 |
L_{2}L_{3} | 0 | 0.6667 | 1.3333 | 1.0 | 0.6667 | 0.3336 | 0 |
L_{3}L_{4} | 0 | 0.3336 | 0.6667 | 1.0 | 1.3333 | 0.6667 | 0 |
L_{4}L_{5} | 0 | 0.1678 | 0.3333 | 0.5 | 0.6667 | 0.8333 | 0 |
L_{5}L_{6} | 0 | 0.1678 | 0.3333 | 0.5 | 0.6667 | 0.8333 | 0 |
U_{1}U_{2} | 0 | -0.6667 | -1.333 | -1.0 | -0.6667 | -0.333 | 0 |
U_{2}U_{3} | 0 | -0.50 | -1.000 | -1.5 | -1.0 | -0.50 | 0 |
U_{3}U_{4} | 0 | -0.50 | -1.000 | -1,5 | -1.0 | -0.50 | 0 |
U_{4}U_{5} | 0 | -0.333 | -0.6667 | -1.0 | -1.333 | -0.6667 | 0 |
L_{0}U_{1} | 0 | -1.1785 | -0.9428 | -0.7071 | -0.4714 | -0.2357 | 0 |
L_{1}U_{1} | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
L_{2}U_{1} | 0 | -0.2357 | 0.9428 | 0.7071 | 0.4714 | 0.2357 | 0 |
L_{2}U_{2} | 0 | 0.167 | 0.3333 | -0.50 | -0.3333 | -0.3333 | 0 |
L_{3}U_{2} | 0 | -0.2357 | -0.4714 | 0.7071 | 0.4714 | 0.2357 | 0 |
L_{3}U_{3} | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
L_{3}U_{4} | 0 | 0.2357 | 0.4714 | 0.7071 | -0.4714 | -0.2357 | 0 |
L_{4}U_{4} | 0 | -03333 | -0.3333 | -0.50 | 0.3333 | 0.167 | 0 |
L_{4}U_{5} | 0 | 0.2357 | 0.4714 | 0.7071 | 0.9428 | -0.2357 | 0 |
L_{5}U_{5} | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
L_{6}U_{5} | 0 | -0.2357 | -0.4714 | -0.7071 | -0.9428 | -1.1785 | 0 |
Influence lines:
Using the values obtained in the above given table, the influence line can be plotted very easily for truss members.