CLASSICAL STRUCTURAL ANALYSIS OF STATICALLY INDETERMINATE STRUCTURES

Since indeterminate structures have more support reactions and/or members than required for static stability, the equilibrium equations alone are not sufficient for determining the reactions and internal forces of such structures, and must be supplemented by additional relationships based on the geometry of deformation of structures. These additional relationships, which are termed the compatibility conditions, ensure that the continuity of the displacements is maintained throughout the structure and that the structure’s various parts fit together. For example, at a rigid joint the deflections and rotations of all the members meeting at the joint must be the same. Thus the analysis of an indeterminate structure involves, in addition to the dimensions and arrangement of members of the structure, its cross-sectional and material properties (such as cross-sectional areas, moments of inertia, moduli of elasticity, etc.), which in turn, depend on the internal forces of the structure .

SHEAR AND MOMENT DIAGRAMS OF BEAMS USING MOMENT

DISTRIBUTION METHOD

The method of analyzing beams and frames using moment distribution was developed by Hardy Cross, in 1930. At the time this method was first published it attracted immediate attention, and it has been recognized as one of the most notable advances in structural analysis during the twentieth century.
Moment distribution is based on the principle of successively locking and unlocking the joints of a structure in order to allow the moments at the joints to be distributed and balanced.
Procedures for determining the end moments of beam spans using moment distribution:

1. Distribution Factors and Fixed End MomentsThe joints on the beam should be identified and the stiffness factors for each span at the joints should be calculated. Using these values, the distribution factors can be determined from DF = K/∑ K. Remember that DF = 0 for a fixed end and DF = 1 for an end pin or roller support. The fixed-end moments for each loaded span are determined using the table given. Positive FEMs act clockwise on the span and negative FEMs act counterclockwise.
2. Moment Distribution ProcessAssume that all joints at which the moments in the connecting spans must be determined are initially locked. Then,a.) Determine the moment that is needed to put each joint in equilibrium.b.) Release or “unlock” the joints and distribute the counterbalancing moments into the connecting span at each joint.c.) Carry these moments in each span over to its other end by multiplying each moment by the carry-over factor + 1/2.

SHEAR AND MOMENT DIAGRAMS OF BEAMS USING THREE MOMENT EQUATION

The three moment equation was developed by French engineer Clapeyron in 1857. The principle is applied on a continuous beam at three points of supports which is based on the conjugate beam method. From the figure, the roller supports at A, B and C on the real beam becomes hinges at points A, B, and C on the conjugate beam. The A1/EI1 and A2/EI2 represents the total area under their respective M/EI diagrams and “a” and “b” represents the centroids of the areas from A and C respectively. Since the slope of the real beam is continuous over the support at B, then V1 + V2 = -(V3 + V4) for the conjugate beam.

When the supports are on the same level:

Considering settlements in any supports:

SHEAR AND MOMENT DIAGRAMS OF BEAMS USING METHOD OF

CONSISTENT DEFORMATION

If the free body diagram were drawn, there would be four unknown support reactions; and since three equilibrium equations are available for solution, the beam is indeterminate to the first degree. Consequently, one additional equation is necessary for solution. By choosing one of the support reactions as “redundant” and temporarily removing its effect on the beam so that the beam then becomes statically determinate and stable. This beam is referred to as the primary structure. When the restraining action of the rocker at B is removed, the load P will cause B to be displaced downward by an amount ∆B as shown in the figure. By superposition, however, the unknown reaction at B, (BY), causes the beam at B to be displaced ∆′BB upward. The first letter in this double-subscript notation refers to the point (B) where the deflection is specified, and the second letter refers to the point (B) where the unknown reaction acts.

A force method of analysis consists of writing equations that satisfy compatibility and the force-displacement requirements, which then gives a direct solution for the redundant reactions. Once obtained, the remaining reactions are found from the equilibrium equations.

Maxwell’s Theorem of Reciprocal Displacements; Betti’s Law

The theorem of reciprocal displacements is stated as follows: The displacement of a point B on a structure due to a unit load acting at point A is equal to the displacement of point A when the unit load is acting at point B, that is, fBA = fAB .

This theorem can be easily demonstrated by using the method of virtual work. For example, consider the beam shown below. When a real unit load acts at A, assume that the internal moments in the beam are represented by mA. To determine the flexibility coefficient at B, that is, fBA , a virtual unit load is placed at B, and the internal moments mB are computed.

SHEAR AND MOMENT DIAGRAMS USING SLOPE DEFLECTION METHOD

The slope-deflection method was originally developed by Heinrich Manderla and Otto Mohr for the purpose of studying secondary stresses in trusses. Later, in 1915, G. A. Maney developed a refined version of this technique and applied it to the analysis of indeterminate beams and framed structures. It relates the unknown slopes and deflections to the applied load on a structure. In order to develop the general form of the slope-deflection equations, consider the typical span AB of continuous beam shown in the figure below, which is subjected to the arbitrary loading and has a constant EI. Relate the beam’s internal end moments MAB and MBA in terms of its three degrees of freedom, namely, its angular displacements θA and θB, and linear displacement ∆ which could be caused by a relative settlement between the supports. Take note that moments and angular displacements will be considered positive when they act clockwise on the span. The linear displacement is considered positive as shown, since this displacement causes the cord of the span and the span’s cord angle ψ to rotate clockwise.

The slope-deflection equations relate the unknown moments at each joint of a structural member to the unknown rotations that occur there. The following equation is applied twice to each member or span, considering each side as the “near” end and its counterpart as the far end.
Once the slope-deflection equations are written, they are substituted into the equations of moment equilibrium at each joint and then solved for the unknown displacements. If the structure (frame) has sidesway, then an unknown horizontal displacement at each floor level will occur, and the unknown column shears must be related to the moments at the joints, using both the force and moment equilibrium equations. Once the unknown displacements are obtained, the unknown reactions are found from the load-displacement relations

SHEAR AND MOMENT DIAGRAMS OF BEAMS USING MATRIX METHOD

The classical methods of structural analysis (also known as system approach of analysis), such as the method of consistent deformation, slope-deflection methods, etc., considering the behavior of the entire structure for developing equations necessary for analysis. Hence, these methods are suitable only for simple or small structures. For larger structures, the use of classical methods of analysis will be difficult and time-consuming. The analysis procedure can be concisely written using matrix notations and are suitable for computer programming.

There exist two ways by which a structure is analyzed using matrix methods, namely the flexibility method or the force method and the stiffness method or the displacement method.

FORCE METHOD/FLEXIBILITY METHOD

When computing the flexibility coefficients, fij (or aij), for the structure, it will be noticed that they depend only on the material and geometrical properties of the members and not on the loading of the primary structure. Hence these values, once determined, can be used to compute the reactions for any loading.
For a structure having n redundant reactions, Rn, we can write n compatibility equations, namely:

STIFFNESS METHOD/ DISPLACEMENT METHOD

The stiffness method is the preferred method for analyzing structures using a computer. It first requires identifying the number of structural elements and their nodes. The global coordinates for the entire structure are then established, and each member’s local coordinate system is located so that its origin is at a selected near end, such that the positive x' axis extends towards the far end.

In the stiffness method, first of all, the compatibility conditions are used to relate the end displacements of the elements with the nodal displacements of the structure. Then the force–displacement relations using stiffness coefficients are established between the end actions and displacements of the elements, and between the nodal forces and displacements of the Structure. Finally, from the equations of equilibrium using the FBD of the nodes, the unknown nodal displacements (active DOF) are determined. Using this, the support reactions, element forces, and element displacements are found out.
In the stiffness method, the displacements are treated as independent variables. Hence in this method, coordinates indicate displacements. Once the coordinates are chosen, then they also indicate the direction of corresponding forces.

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REFLECTION:

In this module, I learned that most of the real-life structures are statically indeterminate. Also, I have realized that statically indeterminate structures can be solved by using displacement method as if unknown displacements and rotations were chosen. From a system of equilibrium equations deformations can be calculated from which internal forces and reactions are calculated. The displacement method is superior to the force method when the number of unknown forces exceeds the number of unknown displacements and rotations.Thus the analysis of an indeterminate structure involves, in addition to the dimensions and arrangement of members of the structure, its cross-sectional and material properties (such as cross-sectional areas, moments of inertia, moduli of elasticity, etc.), which in turn, depend on the internal forces of the structure. The design of an indeterminate structure is, therefore, carried out in an iteratively, whereby the (relative) sizes of the structural members are initially assumed and used to analyze the structure, and the internal forces thus obtained are used to revise the member sizes. The iteration continues until the member sizes based on the results of an analysis are close to those assumed for that analysis. Although designing indeterminate structures seems cumbersome to do, great majority of structures being built today are statically indeterminate (e.g., modern reinforced concrete buildings, truss bridges, etc.).

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