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4-3-3 Equilibrium of Joints
Equilibrium of Points Within Trusses. The fact that any portion of any structure must be in a state of equilibrium forms the basis for all truss-analysis techniques directed toward finding forces in truss members.In analyzing a truss by the classical method of joints, the truss is considered to be composed of a series of members and joints. Member forces are found by considering the equilibrium of the various joints which are idealized as points.Each of these joints or points must be in a state of equilibrium.
Figure 4-6(e) illustrates a typical truss that has been decomposed into a set of inpidual linear elements and a set of idealized joints. Free-body diagrams for all the inpidual members and
joints are shown.
By looking at the joints themselves. it can be seen that the system of forces acting on a joint is defined by the bars attached to it and by any external loads that might occur at the location of the joint. As shown in Figure 4-6(e). the forces on a joint are equal and opposite to those on the connecting members. Each joint must be in a state of equilibrium. The system of forces applied to the joint all act through the same point. From an analytical perspective, we are therefore interested in the equilibrium of a point. This requires consideration of translational equilibrium only.Rotational equilibrium is not a concern. since all forces act through a common point and thus produce no rotational effects. This is the key to analysis of trusses by the method of joints.For planar structures, two independent equations of statics exist for a concurrent force system(ƩFₓ = 0 and ƩFy = 0). Thus, two unknown forces can be found by application of these equations to the complete system of forces represented in the free-body diagram of a joint. If a joint with a maximum of two unknown forces is considered first,it is possible to calculate these forces. The starting point for an analysis of the forces in a truss is often at a support where the reaction has been determined by considering the rigid-body equilibrium of the whole structure.Once all the forces acting on the initial joint have been found (and thus also the forces in the bars attached to the joint), it is possible to proceed to another joint. Since the bar forces previously found can now be treated as known forces. it is convenient to next consider an adjacent joint.The next illustrated example will clarify the process.
A few words should be said about the arrow convention used. The arrows illustrate graphically the nature and direction of the forces developed on an element. Thus,the arrows shown on member DE in Figure 4-6(e) are used to indicate that forces causing the element to be in a state of compression are developed as a consequence of the loads on the larger structure. Note that the arrows seemingly subject the member to a compressive force. Conversely, the arrows shown on member BC in Figure 4-6(e) are used to indicate that a state of tension exists in the element. With respect to the joints, these arrows are shown to be equal and opposite. Thus, the action of member DE (in a state of compression) on joint E apparently is one of pushing against the joint. In actuality,a reaction is developed. In an analogous way, the action of member BC (in a state of tension) is seemingly to pull on joint C. It is quite useful to visualize a joint as being in a state of equilibrium when the "pushes" and "pulls" of the members framing into the joint balance each other.
FIGURE 4-6 Typical free-body diagrams for elemental truss pieces. These diagrams are based on the fundamental principle that any structure, or any portion of any structure, must be in a state of equilibrium.The free-body diagrams in (e), (f), and (g) are used for solution of bar forces by the method of joints.
Analysis of bar forces in a truss by the method of joints using hand-calculation techniques is generally straightforward for trusses with few members. For the truss shown in Figure 4-6, the first step is preferably to draw a set of free-body diagrams of the type shown in Figure 4-6(e). Alternatively, simplified diagrams of only the forces on the joints themselves may be drawn [see Figure 4-6(f)]. The equations of translational equilibrium (Ʃ Fₓ = 0 and Ʃ Fy = 0) are then applied in turn to each joint. In drawing the free-body diagrams and writing equilibrium equations,it is necessary to assume that an unknown bar force is either in a state of tension or compression.The state of stress assumed can be arbitrary. Whether or not the force is in the state of stress assumed will be evident from the algebraic sign of the force that is found after equilibrium calculations have been made. A positive sign means that the initial assumption was correct, while a negative sign means the converse.