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, (X_A, Y_A, M_A, X_B, and Y_B). The number of static equilibrium equations, e, equals 3, (F_x = 0, F_y = 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, e_c. 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 e_c = 1. The degree of indeterminacy, n, is calculated as:

n = r - (e + e_c)
  = 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,

Figure 4(a)(i) - Moment diagram of primary structure	Figure 4(a)(ii) - Moment diagram by superposition

Figure 4(b)(i) - Moment diagram with MA = 1 ft-k	Figure 4(b)(ii) - Deflected shape with MA = 1 ft-k

Figure 4(c)(i) - Moment diagram with XB = 1 k	Figure 4(c)(ii) - Deflected shape with XB = 1 k

The above relationships yield the flexibility coefficients f_aa, f_ab, f_ba, and f_bb. The flexibility coefficient f_ij 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:

  (1)

  (2)

• solve deformation equations

Solve Equations (1) and (2) in the previous step to obtain the unknown redundants M_A and X_B. Notice that if the answers of M_A and X_B 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., X_A, Y_A, and Y_B of the indeterminate structure by imposing the calculated values of M_A and X_B in the correct directions and utilizing the three equilibrium equations, (F_x = 0, F_y = 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