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Equilibrium in the horizontal direction:(compression)
is merely the sum of the horizontal components in the two diagonals. Next proceed to an adjacent joint (either B or D).
Joint B
Equilibrium in the vertical direction:
Since is known, can be solved for:
(tension)
A symmetry argument would yield the same result, that is, must be similar to since the truss is geometrically symmetrical and is symmetrically loaded. Otherwise, there is no a priori reason to believe the forces to be the same. See Figure 4-8 (c).
Equilibrium in the horizontal direction:
Joint D
Equilibrium in the vertical direction:
Equilibrium in the horizontal direction:
All forces are known.The equation sums to zero.This is a good check on the accuracy of calculations.See Figure 4-8(d).
Joint C
Equilibrium in the vertical direction:
Again,this is a check,since all force are known.See Figure4-8(e)
Equilibrium in the horizontal direction:
Again,this is a check.All member forces are already known.
Graphic Statics. The truss in Figure 4-6 may be also analyzed by using graphic statics techniques based on joint equilibrium. It is best to first redefine members and loads in terms of adjacent spaces, as shown in Figure 4-7. Only spaces separated by loads are designated. The load P becomes ba, the reaction becomes ac, the force becomes cl, and so forth. For joint A, ac is laid off to scale. Proceeding clockwise around the joint, remaining forces are drawn. The directions of the unknown forces in cl and al are drawn through c and a, respectively. The location of point l is necessarily defined at the intersection of the lines of action of these two forces so that a closed polygon (necessary for equilibrium) is formed. Unknown forces in cl and la can be measured directly. With cl known, a similar process may be repeated at joint E and other joints. This process can be streamlined by first drawing a force polygon for the reactions and loading (simply a vertical line) and drawing the diagram for A. Since cl is common in the solution of both joints A and E, a composite diagram for A and E may next be drawn, as illustrated. Proceeding to joint B on the composite diagram, point 3 is located by extending lines of action of 23 and b3 from known points. Proceeding to joint D, the process is repeated to form the final diagram. With practice, diagrams of this type may be drawn directly without the intermediate steps shown in
Figure 4-9 and forces values measured directly. Determining the sense of the forces is difficult and best done through normal joint equilibrium considerations, as described previously.
Complex Trusses. Many trusses have geometries that render the analysis procedure more difficult. The following example involves solving simultaneous equations since inpidual equilibrium expressions cannot be solved directly. A way around this complexity is then presented.
◆ ◆ EXAMPLE
Determine the forces in members and in the truss shown in Figure 4-10
Solution: and are known. Each equation still involves two unknown forces ( and ). Solving the equations simultaneously yields
and are known. Each equation still involves two unknown forces ( and ). Solving the equations simultaneously yields