Statically determinate yersus statically

Indeterminate structures

General. — In the analysis of a large group of engineering struc­tures the equations of static equilibrium (for coplanar structures F_x = 0, F_y = 0, and M=0) are sufficient to enable us to solve for unknown reactions, shears, and bending moments, and for unknown forces in various members in structures. The frame shown in Fig. 24 and the beam shown in Fig. 25 represent two such structures. The for­ces in the members of the frame, the reactions, shears, and bending moments at various points in the beam, may be completely determined by the foregoing equations.

Structural types that lend themselves to this method of analysis are designated by the term "statically determinate structures"

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W 16. per Ft.

Fig. 24. Fig. 25.

In other types of engineering structures the condition of static "equilibrium do not provide sufficient equations to solve for all the unknowns. If, for example, an additional bar e were introduced in Fig. 24, or if the beam in Fig. 25 were built into a wall instead of being hinged at the left support, one additional unknown would be introduced in each case. (In the former a new force would appear in bar e, and in the latter an additional bending moment would occur at the left end of the beam. Since we need as many equations as we have unknowns, the introduction of an additional unknown would call for an additional equation. If the equations of static equilibrium were just sufficient for the analysis of problems in the first case, they would obviously not be adequate for the analysis of those given in the second case. Structures of this latter type are l. -ignited by the term "stati­cally indeterminate structures".

This distinction between the two types of strictures is essentially a mathematical one. It explains and justifies the terminology by which the two main types of engineering structures are differentiated with regard to stress and strain analysis. Nevertheless, it is important also to realise clearly the physical distinction, which exists between the two types.

Physical distinction between statically determinate and statically indeterminate structures

In the analysis of structures such as those of Fig. 24 and Fig. 25 we assume: first, that the structure and its individual members shall be strong enough to resist whatever forces may be brought to bear upon them; second, that it shall not deform to the extent of disturbing materially the geometric relation between its various parts.

In Fig. 24 we find that every bar has to carry a definite force, but also that all the bars are necessary to the proper functioning of the structure. In Fig. 25 the left support must supply a horizontal as well as a vertical reaction. If it were changed into a sliding support, such as the right one, the structure would not be stable. If in Fig. 25 the right support were removed, the structure would likewise become unstable.

As a characteristic physical feature of statically determinate struc­tures it may be said, then, that every member, every support, every part, has a definite function to perform.

If we introduce (Fig. 24) an extra member e, it may be said that Fig. 24 contains more members than are absolutely necessary for pur­poses of equilibrium. Not only bar e, but also any other bar, may be regarded as superfluous. In fact, Fig. 24 may be looked upon as rep­resenting two trusses, ACB and ACD, both functioning to the same end, namely, that of carrying the load Q as shown in Fig. 24. The main difficulty in the analysis of Fig. 24 with the additional bar e lies in determining the value of that portion of Q which is carried by the truss ACB and the value of the portion of the load Q carried by the truss ACD. In Fig. 25 function of the right reaction is to hold up the right end of the beam. If, however, the beam is built into a wall at the left end, the beam might conceivably function as a canti­lever beam without aid of a reaction at the right end. It follows, then, that the bending moment supplied by wall and the right reaction func­tion to the same end.

A characteristic physical feature of statically indeterminate struc­tures, therefore, is that two or more members, two or more supports, two or more parts, function to one and the same purpose.

In determining the reactions, forces in members, etc., of statically determinate structures, the sizes of various members and the elastic behavior of the structure are immaterial, provided that the structure is strong enough to carry the superimposed loads, and provided that the deformations are not sufficient to affect materially the geometric rela­tions of its various parts.

This is not true in the analysis of statically indeterminate struc­tures. In Fig. 24 for example, if we assume all bars to be of equal size, bars a, e, c, and d to be of steel and bar b to be of rubber, it would seem that the truss ACD will have to carry the major part of the load. This is true simply because bar b is not capable of transfer­ring any large portion of the load to the truss ACD. Similarly, in Fig. 25 (with the left end built into a wall) the magnitude of the right reaction depends very largely on whether or not the reaction itself will yield, or whether or not the left support is completely rigid. If the right support (Fig. 25) should settle slightly when the loads are placed upon the beam, less reaction would result than there would be if the support permitted no such settling. Since the elastic properties of various members so materially influence the magnitude of the stress­es and forces set up in statically indeterminate structures, it would seem that we must take into account the elastic behavior of these structures if we are to make an analysis of the stresses and strains involved therein. We have stated before that, for the analysis of stat­ically indeterminate structures, the equations of static equilibrium are sufficient. Others involving the elastic behavior of structures must supplement these equations, for reason just given.

Deflections and displacements of engineering structures are rarely important for their own sake. As a key to the analysis of statically indeterminate structures, however, they are of the greatest significance.

In Fig. 25, for example, we may have little interest in the extent to which the beam may sag in the middle. However, if the displace­ment of the right end of the beam can be expressed in terms of the load on the beam and of its right reaction, if we know in advance that this displacement is a certain amount, zero for instance, such an expression will produce the needed additional equation. The expression "statically indeterminate"' does not mean that the equations of static equilibrium do not apply. These equations are always essential for a complete analysis. It means that the equations of static equilibrium must be supplemented.

Temperature Stresses. — In a statically determinate structure chan­ges in the temperature of its parts, or of the entire structure, do not affect the forces in the members. The structure readily deforms to allow for such changes in length as are produced by temperature changes. In a statically determinate truss, for example, one of the supports moves slightly in or out with the seasonal changes in temperature. In a statically indeterminate structure, however, the situation is very different. In a two-hinged arch, for example, the fact that the supports are unable to move laterally with respect to each other constitutes the reason for the statically indeterminate nature of the structure. Temper­ature changes may produce very high stresses, and it is important that we be able to compute them.