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To develop a more intuitive feeling for the force distribution in a structure, it is useful to try to determine correctly whether a member is in tension or compression by a careful qualitative inspection of each joint's equilibrium. Consider joint A in Figure 4-6(e)-(g), where it can be seen qualitatively that the directions assumed are intrinsically reasonable. The reaction is known to act upward. For equilibrium in the vertical direction to obtain, there must be a force acting downward.Only force of the two unknown member forces has a component in the vertical direction and would be capable of providing the downward force necessary to balance the upward reaction.Member AB is horizontal and thus has no component in the vertical direction.Force must therefore act in the downward direction shown. Thus, member AE must be in a state of compression. If force acts in the direction shown,it is evident that it has a horizontal component acting toward the left. For the joint to be in horizontal equilibrium, there must be some other force with a horizontal component acting to the right. The reaction acts only vertically and so does not enter into consideration. Force must therefore act to the right if equilibrium in the horizontal direction is to obtain. Member AB is hence in a state of tension. Looking next at joint E and noting that member AE is in compression, it is evident that member EB must be in tension to provide a downward component necessary to balance the upward one of the force in member AE. Member ED is horizontal and thus can contribute no component in the vertical direction. If forces and act in the directions shown, both have a component in the horizontal direction acting to the right. The force in member ED must therefore act to the left to balance the sum of horizontal components of the forces in the two diagonals.Member ED is thus in compression. Since the truss is symmetrical,the state of stress in the remaining members can be found by inspection (member DC must be in compression, BC in tension, and DB in tension). Note that joint B is also apparently in a state of equilibrium.Thus, the state of stress in all the members can be qualitatively determined without resort to calculation. This process is not possible with all trusses, but success is frequent enough to make the attempt worthwhile. This qualitative approach clearly does not yield numerical magnitudes of bar forces. These can be found only by formally writing the equations of equilibrium and solving for the unknown forces. The mathematical process, however, is conceptually similar to the qualitative one described before. Both are based on the principle that any element of a structure must be in equilibrium. To solve for numerical magnitudes, each joint is considered in turn. In the following, the reference axes used are vertical and horizontal.
◆ EXAMPLE
Determine all the member forces in the truss shown in Figure 4-6
Solution: Draw joint equilibrium diagrams for whole structure and then consider the equilibrium of each joint in turn. See Figure 4-7.
FIGURE 4-7
Joint A
Equilibrium in the vertical direction:
Member AE is in compression as assumed, since the sign is positive. See Figure 4-8.
Equilibrium in the horizontal direction:
Member AB is in tension as assumed,since the sign is positive.The next step is to proceed to an adjacent joint.Joint B involves three unknown forces for which only two independent equations are available for solution.Joint E,however,involves only two unknown forces.So go to this joint next.
Joint E
Equilibrium in the vertical direction:
Since is known, can be solved for: (tension)
Since these are the only two forces at this joint having components in the vertical direction, the components must be equal. Since the member angles are the same, the member forces must also be the same. Thus, the force in member EB could have been found by inspection. See Figure 4-8 (b).