# Introduction To Structural Analysis Displacement And Force Methods Pdf

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## Introduction to Structural Analysis : Displacement and Force Methods

Displacement methods also known as stiffness or equilibrium method Each method involves the combination of a particular solution which is obtained by making the structure statically determinate, and a complementary solution in which the effects of each individual modification is assessed. In the force methods, the behaviour of the structure is considered in terms of unknown forces, while in the stiffness method, the behaviour of the structure is considered in terms of unknown displacements.

By implication, both analysis methods always involve reducing the structure to a basic system a determinate system. In the force method, the basic system involves the removal of redundant forces, while the stiffness method involves restraining the joints of the structure against displacement. Whenever we are using the force method, our basic system is dependent on the degree of static indeterminacy, which a system that must be statically determinate and stable. The choice of redundant constraints to remove is based on experience, geometrical configuration, stability of the structure, and ease of analysis.

In this step, we calculate the deflection corresponding to each redundant force separately due to applied loading and other redundant forces from force-displacement relations. Deflection due to redundant force cannot be obtained without knowing the magnitude of the Downloaded from www. Therefore, we apply a unit load in the direction of redundant force and determine the corresponding deflection. Since the principle of superposition is valid in elastic analysis, the deflections due to redundant force can be obtained by multiplying the unknown redundant with the deflection obtained from applying unit value of force.

Now, we calculate the total deflection due to the applied loading and the redundant force by applying the principle of superposition which must be compatible with the existing boundary condition. For more than one set of redundant forces, we construct a set of simultaneous equations with the redundant forces as unknowns and flexibility coefficients as coefficients of the equations. These flexibility coefficients are also called the influence coefficients.

The total number of equations equals the number of unknown redundant forces. The fundamental approach in the displacement method is the opposite of force method. First of all, we calculate the deformations at the ends of the members and then the internal forces in the members. Thus, the primary unknowns in the displacement method are the displacements.

Analysis of any statically indeterminate structure by the displacement method begins with determining the degree of kinematical indeterminacy. In general, the degrees of kinematical and static indeterminacy are not equal.

The primary system of the displacement method is obtained from the given one by introducing additional constraints to prevent rotation of all rigid joints and all independent displacements of various joints.

These introduced constraints are shown by the shaded squares and the double lines, respectively. Downloaded from www. The number of primary unknowns, n, for each structure equals to the degrees of its kinematical indeterminacy. Often times in the classroom, the lecturer often specify the method he wants the students to use, based on the scheme of work.

Apart from that, sometimes it is required that manual methods be used to verify some results produced by computer programs, or when you are verifying the results produced using a MATLAB code you have written.

The following are simple guides to choice of method. Check the degree of static and kinematic indeterminacy using equations 1 and 5 above. The one that has the less value is usually the most favourable when you are using your classroom non-programmable calculator.

However, when you have programs that can help you solve the matrices e. Check the loading configuration on the structure. The more complex the loading, the less advantageous force method becomes over the displacement method. The more complex the loading, the more complex the diagrams are expected to be, and of course the more number of sections you are expected to cut in order to generate moment equations for your integration.

When you are seeking to obtain the rotation at the nodes, the displacement method gains more advantage over the force method because the solution of the canonical equation gives the displacement of the structure.

However, if force method is used, it is imperative to obtain the final bending moment diagram of the structure, before proceeding. This is also the case when you are looking to obtain the side sway of the frame. Based on the nature of the analysis methods, it is faster to obtain support reactions of frames using force method since the solution of the canonical equations give the solution.

However in displacement method, support reactions can be obtained from 1 st principle after analysis of the bending moments. A variation of this is in the direct stiffness method, which is better handled using a standard computer program. It is not handy for manual calculations at all. The force method handles frames of non-symmetric configuration better than the displacement method when computing manually.

For instance, it is easier to handle frames that are made up of slanted members with force method than with displacement method since Downloaded from www.

When slanted or non- symmetric members are involved in displacement method, the side sway mechanism can get complicated, and errors can creep in. When writing structural analysis programs in computer, displacement method specifically direct stiffness method is beyond the competition.

For some structural configurations, it can be a quite daunting task. Even during manual analysis, it is common to see some students getting the basic system wrong. But the basic system in displacement method is easy to conceive. Step 2: Reduce the structure to a basic system As we realised from the first step, we will need to remove two redundant supports in order to make the structure statically determinate.

However, we must ensure that the selected system must be stable. If we remove the roller at supports E and F, we will obtain a good basic system. So in this example, we are removing the reactive forces at supports E and F and replacing them X1 and X2 which will be assigned unit values. The basic system is shown below. Step 3: Analysis of the various load cases X1, X2 and externally applied load The values X1 and X2 represent the redundant forces that have been removed from the system, and they will be assigned unit values in order for us to progress in our analysis.

We are going to treat each of them as independent load cases on the structure. Our main interest in this discuss will be on the moment diagrams that they produce. You are expected to be familiar with the analysis of statically determinate frames by now. Since we have known these values, the structure has become determinate, and we can analyse it by the law of statics, or progress using the force method. In this example, we are using the force method to obtain the final moment diagram, and not the shear and axial forces see further examples for shear and axial.

To start with, we consider the number of unknown nodal displacements that the structure can undergo. A little consideration will show that the structure is kinematically indeterminate to the 4th degree. These unknown displacements are the rotations at nodes C, E, G, and the lateral translations at F side sway. To make the structure kinematically determinate, we rigidly fix up nodes C labelled 1 , E labelled 2 , and F labelled 3 , and also put a constraint at nodes F labelled 4 to prevent lateral displacement.

## INTRODUCTION TO FORCE AND DISPLACEMENT TO FORCE AND DISPLACEMENT METHODS OF ... the unknown...

Superposition helps us solve these problems by breaking the member down as many times as necessary for each force acting on it. Once all the stresses or deflections for the point of interest are found, they can then be added all together to get a final answer. Distributed Load:. Uniform external forces that acts on the surface of a member over a specific length. External Load:. Are the forces acting on the surface of a member. These can include support reactions,applied forces, normal force etc.

Compatibility and material information are essential. Indeterminate Structures. ForceMethod Page 1. Page 2. Maxwell's Theorem of Reciprocal displacements;.

## Introduction to Structural Analysis: Displacement and Force Methods

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The method is based on transforming a given structure into a statically determinate primary system and calculating the magnitude of statically redundant forces required to restore the geometric boundary conditions of the original structure. The basic steps in the force method are as follows: a Determine the degree of static indeterminacy, n of the structure. This is accomplished by releasing external support conditions or by creating internal hinges. The system thus formed is called the basic determinate structure. Use suitable method given in Chapter 4 to calculate displacements at each of the released constraints in the basic determinate structure.

It is organized logically, presented concisely and written well. Classical analysis methods and modern matrix stiffness methods are of good balance. While the force method is important for students to. Read more

Faster previews. Personalized experience. Get started with a FREE account. Structural Analysis with the Finite Element Method.

It may be recalled that, in the case of indeterminate structures either the reactions.

Fundamentals of Structural Engineering pp Cite as. The previous chapter dealt with the force method, one of two procedures for analyzing statically indeterminate structures. In this chapter, we describe the second procedure, referred to as the displacement method.

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