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Chapter (2)

ANALYSIS OF INDETERMINATE STRUCTRES

1.1 Statically Indeterminate Structures A structure of any type is classified as statically indeterminate when the number of unknown reaction or internal forces exceeds the number of equilibrium equations available for its analysis. Most of structures are statically indeterminate. This indeterminacy may arise as a result of added support or members, or by the general form of the structure. For example, reinforced concrete buildings are almost always statically indeterminate since the columns and beams are poured as continuous members through the joints and over supports. Although the analysis of a statically indeterminate structure is more involved than that of one that is statically determinate, there are usually several very important reasons for choosing this type of structures for design. Most important reasons for a given loading the maximum stress and deflection of an indeterminate structure are generally smaller than those of its statically determinate counterpart. For example, the shown fixed beam in Fig.(l.l-a), the maximum moment of Mmax= PL/8, whereas the same beam when simply supported Fig.(l.l-b), will

Fig.(l.l-a)

Fig.(l.l-b),

be subjected to twice the moment, that is, Mmax = Pl/4 . As a result, the fixed support beam has one fourth the deflection and one half the stress. At its center of the one that is simplysupported. Another important reason for selecting a statically indeterminate structures is that it has a tendency to redistribute its load to its redundant supports in cases where faulty design or overloading occurs. In these cases, the structure maintains its stability and collapse is prevented. This is particularly important when sudden lateral loads, such as wind or earthquake, are imposed on the structure. To illustrate, consider again the fixed beam loaded at its center, Fig.(l-a), As load P is increased, the beam's matenal at the fixations and at the center of the beam begins to yield and forms a localized "Plastic Hinge", which causes the beam to deflect. Although the deflection becomes

large, the fixations will develop horizontal force and moment reactions that will be hold the beam and thus prevent it from totally collapsing. In the case of the simply supported beam, an excessive load P will cause the "Plastic Hinge" to form only at the center of the beam, and due to the large vertical deflection, the supports will not develop ;the horizontal force and moment reactions that may be necessary to prevent total collapse. Although statically indeterminate structures can support a loading with thinner members and with increased stability compared to their determinate counterparts, there are cases when this advantages may instead become disadvantages. The cost savings in matenal must be compared with the added cost necessary to fabricate the structure, since often time it becomes more costly to construct the supports and joints of an indeterminate structure compared to one that is determinate. More important, though, because statically lndeterrmante structures have redundant support reactions, one has to be very careful to prevent differential settlement of the supports, smce this effect will introduce internal stress in the structure. For example, if the fixed support at one end of fixed beam Fig.(l-a) were to settle, additional moment would be occur. On the other hand, if the beam was simply supported, then any settlement of its end would not cause the beam to deform, and therefore no stress or bending moment would be developed in the beam. In

general, then, any deformation, such as that caused by relative support displacement, or changes in member lengths caused by temperature or fabrication errors, will introduce additional stresses in the structure, which must be acknowledged when designing indeterminate structures. 1-2 Method of Analyses When analyzing any indeterminate structures, it is necessary to satisfy equilibrium, compatibility, and forcedisplacement requirements for the structure. Equilibrium is satisfied when the reactive forces hold the structure at rest, and compatibility is satisfied when the various segments of the structure fit together without intentional breaks or overlaps. The force -displacement requirement depend upon the way the material responds, which assumed linear-elastic response, hi general there are two different ways to satisfy these requirements when analysing a statically indeterminate structure, the force or displacement method. 1.2.1 Flexibility or Force Method The force method was originally developed by J.G. Maxwell and refined by Otto Mohr and Muller-Breslau. This method was one of the first available for the analysis of statically indeterminate structures. The force method consists of writing equations that satisfy the compatibility and forces displacement

requirements for the structure and involve redundant forces as the unknowns. The coefficients of these unknowns are called flexibility coefficients. Since compatibility forms the basis for this method, it has sometimes been referred to as the compatibility method or the method of consistent deformations. Once the redundant forces have been determined, the structure are determined by satisfying the equilibrium requirements for the structure. The fundamental principles involved in applying this method are easy to understand and develop. 1.2.2 Stiffness or Displacement Method The displacement method of analysis is based on first writing force displacement relations for the members and then satisfying the equilibrium requirements for the structures. In this case the unknowns in the equations are displacements and their coefficients compatibility are and called stiffness coefficients. equations. Once A the displacements are obtained the forces are determmed from the force-displacement matrix formulation of this method will given. 1.2.3 Comparison between the Flexibility and Stiffness Methods 1.2.3.1 Flexibility = deformation per unit force.

Cm/t

The deflection (d) of the shown

Spring Fig. (1-2) is; d = .P

1.2.3.2. Stiffness: Stiffness = force per unit deformation =K (t/cm or t.m/rad.)

For the shown spring in Fig. (2), the force required to produce a displacement d is; P=K.d The two methods can be compared as shown in the ollowing table.

Flexibility Method

- Unknown redundants X - Flexibility matrix 8

Stiffness Method

Unknown displacements d (,) Stiffness matrix K

- Displacements due to X =

.X

Force due d = K . d Force due to loads =Ro Equilibrium Ro + K.d = 0 Solve for d

- Displacements due to load =

o

- For Compatibility o + X = 0 - Solve for X Indeterminate released calculated. and deformation These Constraints the

are Additional restrains are added to of these restraints

resulting fix all degrees of freedom and the redundant calculated . The restrains are then and restore equilibrium. The

discontinuities values

actions are then replaced to removed to allow deformations restore the continuity.

resulting

equilibrium

equations

are solved to get determined.

displacements

and subsequently the forces are

The force or flexibility method of analyses discussed in the previous course of second year civil;. For example the methods of consistent deformation, three moment equation, and column analogy methods are considered a flexibility or force method of analysis whereas the slope deflection, moment distribution, and stiffness methods are a displacement or stiffness method of analysis.

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