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### Consistent Deformations - Force Method

Introduction

The method of consistent deformations, or sometimes referred to as the force or flexibility method, is one of the several techniques available to analyze indeterminate structures. The following is the procedure that describes the concept of this method for analyzing externally indeterminate structures with single or double degrees of indeterminacy.

• determine the degree of indeterminacy

Determine the degree of indeterminacy of a given structure. This can be accomplished by calculating the number of unknown reactions, r, minus the number of static equilibrium equations, e. For example, considering the frame shown below (Fig.1), the number of unknown external reactions, r, equals 5, (XA, YA, MA, XB, and YB). The number of static equilibrium equations, e, equals 3, ( Fx = 0, Fy = 0 and M = 0). Therefore, the number of degree of indeterminacy, n, is calculated as:

n = r - e
= 5 - 3
= 2 Figure 1 - Indeterminate frame structure

In the frame shown in Fig. 2, there is another equation of statics that can be written at the hinge h. In other words, the fact that the moment at h = zero for either part of the structure on the right or the left side of the hinge h can be used. This equation is referred to as equation of condition, ec. In this case, the number of unknown reactions is r = 6, the number of equations of statics is e = 3, and the number of equations of condition ec = 1. The degree of indeterminacy, n, is calculated as:

n = r - (e + ec)
= 6 - (3 + 1)
= 2 Figure 2 - Indeterminate frame structure with hinge

• select redundants

Select a number of the support reactions equal to the degree of indeterminacy as redundants. The choice of the redundants will vary since any of the unknown reactions can be utilized as a redundant. In the example shown in Fig. 1, the X and Y reactions at Support B can be selected as redundants. Another alternative is to select the X reaction at B and the moment at A as redundants. The later choice was selected and utilized through the remainder of this procedure.

• remove restraints at the redundants

Remove the support reactions (restraints) corresponding to the selected redundants from the indeterminate structure to obtain a primary determinate structure, or sometimes referred to as a released structure. This determinate system must represent a stable and admissible system.

• sketch deflected shape

Sketch the deflected shape of the primary determinate structure under the applied loads, and label the deformations at the removed restraints, (see Fig. 3). Figure 3(a) - Primary structure Figure 3(b) - Primary structure deflected shape

• calculate deformations at redundants

Calculate the deformations corresponding to the redundants, i.e., the rotation at Support A, A0, and the translation, B0, at Support B. This can be accomplished as follows; using the virtual work method,

(a) Draw the moment diagram, M0, for the primary structure under the applied loads, (see Fig. 4(a)(i)). The method of superposition can also be utilized when drawing the M0 diagram, (see Fig. 4(a)(ii). This will simplify the integration needed to calculate the deformation A0 and B0.  Figure 4(a)(i) - Moment diagram of primary structure Figure 4(a)(ii) - Moment diagram by superposition

(b) Apply a unit load at the location of the first redundant. In this case, apply a unit moment, MA = 1 ft-k at Support A. Sketch the deflected shape, label the deformation at the removed restraints and draw the moment diagram of the primary structure when subjected to this load, see Fig. 4(b).  Figure 4(b)(i) - Moment diagram with MA = 1 ft-k Figure 4(b)(ii) - Deflected shape with MA = 1 ft-k

(c) Calculate the rotation, A0, at Support A using the following equation: (d) Apply a unit load at the location of the next redundant. i.e., apply a unit force, XB = 1 k at Support B. Sketch the deflected shape, label the deformation at the removed restraints and draw the moment diagram of the primary structure when subjected to this load, see Fig. 4(c).  Figure 4(c)(i) - Moment diagram with XB = 1 k Figure 4(c)(ii) - Deflected shape with XB = 1 k

(e) Calculate the translation, B0, at Support B using the following equation: (f) Calculate the deformations of the primary structure when subjected to the redundant MA, see Fig. 4(b), or the redundant XB, see Fig. 4(c). This is accomplished by using the following relationships:   The above relationships yield the flexibility coefficients faa, fab, fba, and fbb. The flexibility coefficient fij is defined as the deformation corresponding to the redundant i, due to a unit value of the redundant j.

• write consistent deformation equations

Write consistent deformation equations that correspond to each redundant. In this case:

(a) Rotation at Support A = 0 since Support A is a fixed support that prevents rotations. (1)

(b) Translation at Support B = 0 since Support B does not allow horizontal translation. (2)

 solve deformation equations Solve Equations (1) and (2) in the previous step to obtain the unknown redundants MA and XB. Notice that if the answers of MA and XB are positive, this means that the assumed directions of the applied force in Figures 4(b) and 4(c) are correct. determine support reactions Determine the remaining support reactions, i.e., XA, YA, and YB of the indeterminate structure by imposing the calculated values of MA and XB in the correct directions and utilizing the three equilibrium equations, ( Fx = 0, Fy = 0 and M = 0). draw resultant structural diagrams Once all reactions have been evaluated, the axial, shear, and moments diagrams can be drawn. With this information, an approximate deflected shape can also be sketched. The following examples illustrate the application of the consistent deformation method to analyze statically indeterminate structures.
 Examples Once statically indeterminate beam    - Vertical reaction at B as redundant Once statically indeterminate beam    - Moment reaction at A as redundant Once statically indeterminate frame    - Horizontal reaction at D as redundant Once statically indeterminate frame    - Vertical reaction at D as redundant Twice statically indeterminate frame    - Horizontal and vertical reactions at D as redundants Once statically indeterminate truss    - Horizontal reaction at E as redundant