Question 2.6.1: Figure 2.6-1(a) shows a statically determinate truss acted o...
Figure 2.6-1(a) shows a statically determinate truss acted on by a force F applied at joint B. Using the method of joints, determine graphically the axial forces in all the members and the support reactions at A and D.
Since each joint is in equilibrium under the action of the member axial forces (and the applied forces and support reactions if any), then these forces acting on any individual joint must form a closed polygon. If, at any joint, only two force components are unknown then these can be determined by drawing a force polygon. An examination of Fig. 2.6-1 (a) shows that there are at least three unknowns at every joint.
However, since the entire truss is in equilibrium under the action of the applied force and the support reactions, the latter can be determined by a force triangle, noting that the direction of the reaction R_A must be such that the three forces F, R_A, and R_D intersect at one point. With R_A and R_D determined from a force triangle (Fig. 2.6-l(b)), a force triangle can be drawn for joint A (Fig. 2.6-l(c)), thereby determining the axial forces S_1 and S_5 in members 1 and 5 respectively. Figure 2.6-l(c) shows that S_1 acts towards joint A, that is member 1 is in compression; likewise, S_5 acts away from joint A, showing that member 5 is in tension. Arrow-heads are now added in members 1 and 5 to indicate that (a) member 1 is in compression, so that its axial force acts towards both joints A and B; (b) member 2 is in tension so that its axial force acts away from both joints A and E.
Now that S_1 is known, a force polygon can be constructed for joint B (Fig. 2.6-l(d)), determining S_2 and S_6. Arrow-heads are now added on to members 2 and 6 (Fig. 2.6-1 (a)) to indicate compression in both members.
In a similar way, force polygons are drawn successively for joints C and D (Figs. 2.6-l(e) and (f)). The force polygon for joint E (Fig. 2.6-l(g)) is drawn from the member forces acting on that joint which are now all known. The closing of this polygon is an indication that the previous force polygons have been constructed correctly.
In each of the force polygons in Figs. 2.6-l(b) to (f) inclusive, the unknown forces determined from the polygon have been shown by dotted lines.
Attention is drawn to two significant points: (a) the layout dimensions of the truss are immaterial as far as the member forces are concerned—such forces depend only on the shape of the truss; (b) the cross-sectional sizes of the members and their modulus of elasticity have no effect on member forces.
