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Before solving mathematically for the bar forces in the example, by formal application of the equations of statics, it is useful to try to determine by qualitative inspection the senses of the member forces. Study equilibrium requirements in the subassembly shown to the left in Figure 4-12, for example. It can be seen that the force in member BD must act upward as shown to supply the vertical component necessary for balancing the difference between the upward reaction of 0.5P and the downward force of P acting on the subassembly (a net of 0.5P acting downward).Thus, the member must be in a state of tension. Since the subassembly to the left must be in rotational equilibrium also, the moment produced by the external forces must be exactly balanced by the moment developed by the internal forces (see Figure 4-12). By summing moments about point B, it can be seen that the sense of the force in member DE must be in the direction shown if moment equilibrium is to occur about point B. Remember that the sum of the moments produced by all forces must be zero about any point. Hence, member DE must be in a state of compression. The forces shown on the left subassembly are equal and opposite on the right subassembly. Since the right subassembly must also be in translatory and rotational equilibrium, the sense of the force in member BC can be found by summing moments about point D. For moment equilibrium to obtain about this point, force Fbc must act in the direction shown and so be in a state of tension. Thus, the states of stress in the unknown bar forces can be qualitatively determined. The mathematical process for determining the numerical magnitudes of the forces is conceptually similar to the process just described.
FIGURE4-12 Free-body diagrams for solution of forces in member , and by the method of sections
◆ ◆ EXAMPLE
Determine the forces in members DE, BD, and BC of the truss shown in Figure 4-10.
Solution:
Left Subassembly
Translatory equilibrium in the vertical direction:
Member BD is in tension, as assumed, since the sign is positive. We could next try ∑ =0, but this would involve two unknown forces( and ), and the equation could not be solved directly. Try to find an equation involving only a single unknown force by considering moment equilibrium about a point. By selecting point B, one unknown force, , acts through the moment center and consequently falls out of the moment equation (since its moment arm is zero), leaving an equation involving only the remaining unknown force and known external forces.
Moment equilibrium about point B:
The member is in compression, as assumed. The step where the moment developed by the internal forces is shown equal to that produced by the external forces is not actually necessary. It does, however, represent a fundamentally important way of looking at structural behavior and is important for design purposes. This way of looking at structures will be discussed further in Section 4-3-8. Now that two of the forces acting on the subassembly are known, the remaining unknown force can readily be found by application of the last unused equation of statics : ∑ =0.
Translatory equilibrium in the horizontal direction:
Member BC is in tension, as assumed. All unknown forces acting on the left subassembly have now been found. These forces act in an equal and opposite way on the right subassembly. As is obvious, it should also be in equilibrium. It is good to check this.
Right Subassembly
Moment equilibrium about point D:
Translatory equilibrium in the vertical direction:
Translatory equilibrium in the horizontal direction:
The right-hand subassembly is in a state of translational and rotational equilibrium.
The approach just illustrated can be used to find member forces in planar trusses when no more than three unknowns are involved, since there are only three independent equations of statics available for analyzing the equilibrium of planar rigid bodies. Care must be taken in isolating subassemblies so that only three unknowns are present. The decomposition shown in Figure4-6(d), for example, is perfectly valid, and each subassembly is indeed in a state of translatory and rotational equilibrium. However, the decomposition is not useful from an analytical viewpoint, since there are more unknown forces acting on each segment than could be solved for by using the basic equations of statics.