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Member Forces: General This section considers classical analytical methods for determining the nature and distribution of forces in the members of a truss that has known geometrical characteristics and carries known loads. The principle underlying the analytical techniques to be developed is that any structure, or any elemental portion of any structure, must be in a state of equilibrium. This principle is the key to common truss analysis techniques. The first step in analyzing a truss of many members is to isolate an elemental portion of the structure and consider the system of forces acting on the element. If some of these forces are known, it is usually possible to calculate the others by use of the basic equations of statics, since it is known that the element must be in equilibrium. These equations are merely formal statements of the notion that any set of forces, including both those that are externally applied and those internally developed, must form a system whose net result is zero.36112
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The extent of the portion of the structure selected for study is not restricted. A whole segment consisting of several members and joints could be considered, or attention could be limited to a single joint or member.
(a)Deformed shape of a simple pin-connected structure without diagonals.The original distance between points A and C tends to be increased and that between B and D tends to be decreased.
(b)A cable placed between points A and C will have tension forces developed in it because it resists the tendency of the two points to separate.The cable will stabilize the structure and keep it from collapsing.
(c) A rigid element placed between A and C will serve the same function as a cable. Tension forces will be developed in the member.
(d) Placing a cable between points B and D is useless in keeping the structure from collapsing. The points move toward one another.A cable placed between these two points would simply buckle out of the way. The same thing would happen to the cable in (b) if the direction of the load were reversed.
(e) Placing a rigid element between points B and D can stabilize the structure. Compressive forces would be developed in the member.
(f),(g)In order to completely stabilize the structure with respect to loads from either direction, by using cables, it is necessary to use a crossed-cable system.Under a specific loading, one cable will operate effectively in stabilizing the structure, while the other simply buckles out of the way. Reversing the loading causes a reversal in which member operates effectively.
(h),(i)Crossed rigid elements can also be used,but a certain degree of redundancy is involved,since a single diagonal is capable of stabilizing the structure alone with respect to loads in either direction.
FIGURE 4-5 Diagonal bracing
To find the internal forces present at a particular location in a structure, it is necessary that the structure be decomposed at least at that point. Figure 4-6(b)-(e) illustrates free-body diagrams for typical elemental pieces of the truss shown in Figure 4-6(a).Each of the pieces shown must be in a state of equilibrium under the action of the force system present on the piece. The force system considered consists of not only any external loads applied to the piece, but also those which are internal to the structure as a whole. The latter are forces developed in members in response to the external loading on the whole truss, which are necessary to maintain the equilibrium of elemental portions of the truss. In considering the equilibrium of an element, it is often useful to think of these as applied forces. At points where the truss is decomposed, internal forces are shown to be equal in magnitude but opposite in sense in adjacent elements. This follows from the discussion in Chapter 2, where we considered Newton's third law: that the forces of action and reaction between elements in contact with each other have the same magnitude and line of action, but are of opposite sense. Since the members of a truss are pin-connected and their ends are free to rotate, only forces, and not moments, can be transmitted from one member to another at the point of connection. Thus, only vector forces are indicated in the free-body diagrams shown. Since only forces and not moments can be transmitted at the connections, the forces at either end of a member must act collinearly (assuming that the member itself is not subjected to external loads) if the member is indeed to be in a state of equilibrium. (See the discussion in Section 2-3-2 0n two-force members.) If the member itself is aligned with these forces, it follows that the member is only axially loaded and not subject to bending moments.