The Force Method of Analysis: Beams - 3 | Structural Analysis - Civil Engineering (CE) PDF Download

Instructional Objectives

After reading this chapter the student will be able to

1. Calculate additional stresses developed in statically indeterminate structures due to support settlements.
2. Analyse continuous beams which are supported on yielding supports.
3. Sketch the deflected shape of the member.
4. Draw banding moment and shear force diagrams for indeterminate beams undergoing support settlements.

Introduction

In the last lesson, the force method of analysis of statically indeterminate beams subjected to external loads was discussed. It is however, assumed in the analysis that the supports are unyielding and the temperature remains constant. In the design of indeterminate structure, it is required to make necessary provision for future unequal vertical settlement of supports or probable rotation of supports. It may be observed here that, in case of determinate structures no stresses are developed due to settlement of supports. The whole structure displaces as a rigid body (see Fig. 9.1). Hence, construction of determinate structures is easier than indeterminate structures.

The statically determinate structure changes their shape due to support settlement and this would in turn induce reactions and stresses in the system. Since, there is no external force system acting on the structures, these forces form a balanced force system by themselves and the structure would be in equilibrium. The effect of temperature changes, support settlement can also be easily included in the force method of analysis. In this lesson few problems, concerning the effect of support settlement are solved to illustrate the procedure.

Support Displacements

Consider a two span continuous beam, which is statically indeterminate to second degree, as shown in Fig. 9.2. Assume the flexural rigidity of this beam to be constant throughout. In this example, the support B is assumed to have settled by an amount Δ_b as shown in the figure.

This problem was solved in the last lesson, when there was no support settlement (vide section 8.2). In section 8.2, choosing reaction a B and C as the redundant, the total deflection of the primary structure due to applied external loading and redundant R₁ and R₂ is written as,

                     (9.1a)

              (9.1b)

wherein,  R₁ and R₂ and are the redundants at B and C respectively, and , and are the deflections of the primary structure at (Δ_L)₁, and (Δ_L)₂, are the deflections of the primary structure at B and C due to applied loading. In the present case, the support B settles by an amount Δ_b in the direction of the redundant R₁. This support movement can be readily incorporated in the force method of analysis. From the physics of the problem the total deflection at the support may be equal to the given amount of support movement. Hence, the compatibility condition may be written as,

Δ1 = -Δ_b                  (9.2a)

Δ₂ = 0                (9.2b)

It must be noted that, the support settlement Δ_b must be negative as it is displaces downwards. It is assumed that deflections and reactions are positive in the upward direction. The equation (9.1a) and (9.1b) may be written in compact form as,

                     (9.3a)

                  (9.3b)

Solving the above algebraic equations, one could evaluate redundants R₁ and R₂ due to external loading and support settlement.

Temperature Stresses

Internal stresses are also developed in the statically indeterminate structure if the free movement of the joint is prevented.

For example, consider a cantilever beam AB as shown in Fig. 9.3. Now, if the temperature of the member is increased uniformly throughout its length, then the length of the member is increased by an amount

Δ_T = α L T                           (9.4)

In which, Δ_T is the change in the length of the member due to temperature change, α is the coefficient of thermal expansion of the material and T is the change in temperature. The elongation (the change in the length of the member) and increase in temperature are taken as positive. However if the end B is restrained to move as shown in Fig 9.4, then the beam deformation is prevented. This would develop an internal axial force and reactions in the indeterminate structure.

Next consider a cantilever beam AB , subjected to a different temperature, T₁ at the top and T₂ at the bottom as shown in Fig. 9.5(a) and (b). If the top temperature T₁ is higher than the bottom beam surface temperature T₂, then the beam will deform as shown by dotted lines. Consider a small element dx at a distance x from A. The deformation of this small element is shown in Fig. 9.5c. Due to rise in temperature   on the top surface, the top surface elongates by

                            (9.5a)

Similarly due to rise in temperature T₂, the bottom fibers elongate by

                          (9.5b)

As the cross section of the member remains plane, the relative angle of rotation dθ between two cross sections at a distance dx is given by

                     (9.6)

where, d is the depth of beam. If the end B is fixed as in Fig. 9.4, then the differential change in temperature would develop support bending moment and reactions.

The effect of temperature can also be included in the force method of analysis quite easily. This is done as follows. Calculate the deflection corresponding to redundant actions separately due to applied loading, due to rise in temperature (either uniform or differential change in temperature) and redundant forces. The deflection in the primary structure due to temperature changes is denoted by (Δ_T)_i which denotes the deflection corresponding to i^th redundant due to temperature change in the determinate structure. Now the compatibility equation for statically indeterminate structure of order two can be written as

     (9.7)

wherein, {Δ_L} is the vector of displacements in the primary structure corresponding to redundant reactions due to external loads; {Δ_T} is the displacements in the primary structure corresponding to redundant reactions and due to temperature changes and {Δ} is the matrix of support displacements corresponding to redundant actions. Equation (9.7) can be solved to obtain the unknown redundants.