Under ‘bars’ we consider the analysis of members subject to axial forces only. These members are having one dimension (length) considerably large compared to cross sectional dimensions. Tension bars and columns fall under this category. In case of pin connected frames (trusses), members can be assumed to have only axial forces. In case of pin connected frames (trusses), members can be assumed to have only axial forces.
The typical member considered for explaining the procedure is shown in Fig.11.1. In this problem we see cross section varies in 3 steps A1 , A2 and A3 . There are three point loads P1, P2 and P3. The surface forces are xs1, xs2, and xs3 and Xb is the body force. The surface forces may be due to frictional forces, viscous drag or surface shear. The body force is due to self-weight. The material of the bar is same throughout.
Step 1: Selecting suitable field variables and elements:
In all stress analysis problems, displacements are selected as field variables. In the tension bar or columns at any point there is only one component of displacement to be considered, i.e., the displacement in x direction. Since there is only one degree of freedom and it needs only Co continuity, we select bar element shown in Fig. 11.2. In this case there are only two nodes.
Step 2: Discritise the continua
In this problem there are geometric discontinuities at x = 200 mm, 500 mm and 650 mm. There is additional point of discontinuity at x = 350 mm, where concentrated load P1 is acting. Hence we discritise the continua as shown in Fig. 11.3 using four bar elements.
Hence nodal displacement vector is
In finite element analysis the nodes may be numbered in any fashion, but to keep the band width minimum we number the nodes continuously. In this problem there are five nodes and in all such problem there is definite relationship between number of nodes and number of element i.e. Number of node = Number of elements + 1.
For any element local node number is 1 and 2 only, but global coordinate numbers for each element are different. For example, local coordinate numbers 1 and 2 for element 3 refers to global numbering system 3 and 4 respectively. The relation between the local and global node number is called connectivity details. In this problem the connectivity detail is as shown in Fig. 11.4. From this Figure it can be seen that the connectivity detail can be easily generated also. Thus
For element (i),
Local node number 1 = i
Local node number 2 = i + 1
Figure 14.9 shows the uniformly distributed load acting on the typical element. Noting that the load acting is in y‘ -direction, the nodal force system in local coordinate system is