Previously, we considered the analysis of statically determinate structures. In this part ,we focus our attention on the analysis of statically indeterminate structures. As discussed previously, the support reactions and internal forces of statically determinate structures can be determined from the equations of equilibrium (including equations of condition, if any).
However, since indeterminate structures have more support reactions and/or members than required for static stability, the equilibrium equations alone are not sufficient for determining the reactions and internal forces of such structures, and must be supplemented by additional relationships based on the geometry of deformation of structures. These additional relationships, which are termed the compatibility conditions, ensure that the continuity of the displacements is maintained throughout the structure and that the structure’s various parts fit together. For example, at a rigid joint the deflections and rotations of all the members meeting at the joint must be the same. Thus the analysis of an indeterminate structure involves, in addition to the dimensions and arrangement of members of the structure, its cross-sectional and material properties (such as cross-sectional areas, moments of inertia, moduli of elasticity, etc.), which in turn, depend on the internal forces of the structure. The design of an indeterminate structure is, therefore, carried out in an iterative manner, whereby the (relative) sizes of the structural members are initially assumed and used to analyze the structure, and the internal forces thus obtained are used to revise the member sizes; if the revised member sizes are not close to those initially assumed, then the structure is reanalyzed using the latest member sizes.
The iteration continues until the member sizes based on the results of an analysis are close to those assumed for that analysis. Despite the foregoing difficulty in designing indeterminate structures, a great majority of structures being built today are statically indeterminate; for example, most modern reinforced concrete buildings are statically indeterminate. In this article, we discuss some of the important advantages and disadvantages of indeterminate structures as compared to determinate structures and introduce the fundamental concepts of the analysis of indeterminate structures.
Advantages and disadvantages of indeterminate structures
The advantages of statically indeterminate structures over determinate structures include the following:
1.Smaller Stresses : The maximum stresses in statically indeterminate structures are generally lower than those in comparable determinate structures. Consider, for example, the statically determinate and indeterminate beams shown in Fig. 11.1(a) and (b), respectively. The bending moment diagrams for the beams due to a uniformly distributed load, w, are also shown in the figure. (The procedures for analyzing indeterminate beams are considered in subsequent articles.) It can be seen from the figure that the maximum bending moment—and consequently the maximum bending stress—in the indeterminate beam is significantly lower than in the determinate beam.
2. Greater Stiffnesses : Statically indeterminate structures generally have higher stiffnesses (i.e., smaller deformations), than those of comparable determinate structures. From Fig. 11.1, we observe that the maximum deflection of the indeterminate beam is only one-fifth that of the determinate beam.
3. Redundancies : Statically indeterminate structures, if properly designed, have the capacity for redistributing loads when certain structural portions become overstressed or collapse in cases of overloads due to earthquakes, tornadoes, impact (e.g., gas explosions or vehicle impacts), and other such events. Indeterminate structures have more members and/or support reactions than required for static stability, so if a part (or member or support) of such a structure fails, the entire structure will not necessarily collapse, and the loads will be redistributed to the adjacent portions of the structure. Consider, for example, the statically determinate and indeterminate beams shown in Fig. 11.2(a) and (b), respectively. Suppose that the beams are supporting bridges over a waterway and that the middle pier, B, is destroyed when a barge accidentally rams into it. Because the statically determinate beam is supported by just the sufficient number of reactions required for static stability, the removal of support B will cause the entire structure to collapse, as shown in Fig. 11.2(a). However, the indeterminate beam (Fig. 11.2(b)) has one extra reaction in the vertical direction; therefore, the structure will not necessarily collapse and may remain stable, even after the support B has failed. Assuming that the beam has been designed to support dead loads only in case of such an accident, the bridge will be closed to traffic until pier B is repaired and then will be reopened.
The main disadvantages of statically indeterminate structures, over determinate structures, are the following:
1.Stresses Due to Support Settlements Support : settlements do not cause any stresses in determinate structures; they may, however, induce significant stresses in indeterminate structures, which should be taken into account when designing indeterminate structures. Consider the determinate and indeterminate beams shown in Fig. 11.3. It can be seen from Fig. 11.3(a) that when the support B of the determinate beam undergoes a small settlement DB, the portions AB and BC of the beam, which are connected together by an internal hinge at B, move as rigid bodies without bending—that is, they remain straight. Thus, no stresses develop in the determinate beam. However, when the continuous indeterminate beam of Fig. 11.3(b) is subjected to a similar support settlement, it bends, as shown in the figure; therefore, bending moments develop in the beam.
2. Stresses Due to Temperature Changes and Fabrication Errors Like support settlements, these effects do not cause stresses in determinate structures but may induce significant stresses in indeterminate ones. Consider the determinate and indeterminate beams shown in Fig. 11.4. It can be seen from Fig. 11.4(a) that when the determinate beam is subjected to a uniform temperature increase DT, it simply elongates, with the axial deformation given by d ¼ aðDTÞL (Eq. 7.24). No stresses develop in the determinate beam, since it is free to elongate. However, when the indeterminate beam of Fig. 11.4(b), which is restrained from deforming axially by the fixed supports, is subjected to a similar temperature change, DT, a compressive axial force, F ¼ dðAE=LÞ¼ aðDTÞAE, develops in the beam, as shown in the figure. The effects of fabrication errors are similar to those of temperature changes on determinate and indeterminate structures.
from deforming axially by the fixed supports, is subjected to a similar temperature change, DT, a compressive axial force, F ¼ dðAE=LÞ¼ aðDTÞAE, develops in the beam, as shown in the figure. The effects of fabrication errors are similar to those of temperature changes on determinate and indeterminate structures.
Regardless of whether a structure is statically determinate or indeterminate, its complete analysis requires the use of three types of relationships:
Member force-deformation relations.
The equilibrium equations relate the forces acting on the structure (or its parts), ensuring that the entire structure as well as its parts remain in equilibrium; the compatibility conditions relate the displacements of the structure so that its various parts fit together; and the member force deformation relations, which involve the material and cross-sectional properties (E;I, and A) of the members, provide the necessary link between the forces and displacements of the structure. In the analysis of statically determinate structures, the equations of equilibrium are first used to obtain the reactions and the internal forces of the structure; then the member force-deformation relations and the compatibility conditions are employed to determine the structure’s displacements.
In the analysis of statically indeterminate structures, the equilibrium equations alone are not sufficient for determining the reactions and internal forces. Therefore, it becomes necessary to solve the equilibrium equations in conjunction with the compatibility conditions of the structure to determine its response. Because the equilibrium equations contain the unknown forces, whereas the compatibility conditions involve displacements as the unknowns, the member force-deformation relations are utilized to express either the unknown forces in terms of the unknown displacements or vice versa. The resulting system of equations containing only one type of unknowns is then solved for the unknown forces or displacements, which are then substituted into the fundamental relationships to determine the remaining response characteristics of the structure.
Method of Analysis
- Claypeyron’s theorem of three moment
- Slope deflection method
- Moment distribution method
- Kani’s method