Consider an individual member from a pin-jointed plane-frame, e.g. member AB shown in Figure 3.8 with reference to a particular X-Y co-ordinate system.
If AB is a member of length LAB having a tensile force in it of TAB, then the components of this force in the X and Y directions are TAB Cos0 and TAB Sin0 respectively.
If the co-ordinates of A and B are (XA, YA) and (XB, YB), then the component of TAB in the x-direction is given by :
Figure 3.8 where
and is known as the tension coefficient of the bar. Similarly, the component of TAb in the y-direction is given by:
If at joint A in the frame there are a number of bars, i.e. AB, AC ... AN, and external loads XA and YA acting in the X and Y directions, then since the joint is in equilibrium the sum of the components of the external and internal forces must equal zero in each of those directions.
Expressing these conditions in terms of the components of each of the forces then gives:
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