Structural Analysis – II TABLE OF CONTENT UNIT TOPIC Unit – 1 Unit – 2 Unit – 3 Unit – 4 Unit – 5 Unit – 6 Unit – 7 Unit – 8 Rolling load and influence lines Slope deflection method Moment distribution method Sway analysis Kanis methods Flexibility matrix method of analysis Stiffness matrix method of analysis Basic principles of dynamics PAGE NO 4 14 31 43 57 67 77 86
Structural Analysis – II UNIT - 1 ROLLING LOAD AND INFLUENCE LINES 1 Introduction: Variable Loadings So far in this course we have been dealing with structural systems subjected to a specific set of loads. However, it is not necessary that a structure is subjected to a single set of loads all of the time. For example, the single-lane bridge deck in Figure1 may be subjected to one set of a loading at one point of time (Figure1a) and the same structure may be subjected to another set of loading at a different point of time. It depends on the number of vehicles, position of vehicles and weight of vehicles. The variation of load in a structure results in variation in the response of the structure. For example, the internal forces change causing a variation in stresses that are generated in the structure. This becomes a critical consideration from design perspective, because a structure is designed primarily on the basis of the intensity and location of maximum stresses in the structure. Similarly, the location and magnitude of maximum deflection (which are also critical parameters for design) also become variables in case of variable loading. Thus, multiple sets of loading require multiple sets of analysis in order to obtain the critical response parameters. Figure 1 Loading condition on a bridge deck at different points of time Influence lines offer a quick and easy way of performing multiple analyses for a single structure. Response parameters such as shear force or bending moment at a point or reaction at a support for several load sets can be easily computed using influence lines.
Structural Analysis – II For example, we can construct influence lines for (shear force at B ) or (bending moment at) or (vertical reaction at support D ) and each one will help us calculate the corresponding response parameter for different sets of loading on the beam AD (Figure 2). Figure 2 Different response parameters for beam AD An influence line is a diagram which presents the variation of a certain response parameter due to the variation of the position of a unit concentrated load along the length of the structural member. Let us consider that a unit downward concentrated force is moving from point A to point B of the beam shown in Figure 3a. We can assume it to be a wheel of unit weight moving along the length of the beam. The magnitude of the vertical support reaction at A will change depending on the location of this unit downward force. The influence line for (Figure3b) gives us the value of for different locations of the moving unit load. From the ordinate of the influence line at C, we can say that when the unit load is at point C . Figure 3b Influence line of for beam AB
Structural Analysis – II Thus, an influence line can be defined as a curve, the ordinate to which at any abscissa gives the value of a particular response function due to a unit downward load acting at the point in the structure corresponding to the abscissa. The next section discusses how to construct influence lines using methods of equilibrium. 2 Construction of Influence Lines using Equilibrium Methods The most basic method of obtaining influence line for a specific response parameter is to solve the static equilibrium equations for various locations of the unit load. The general procedure for constructing an influence line is described below. 1. Define the positive direction of the response parameter under consideration through a free body diagram of the whole system. 2..For a particular location of the unit load, solve for the equilibrium of the whole system and if required, as in the case of an internal force, also for a part of the member to obtain the response parameter for that location of the unit load.This gives the ordinate of the influence line at that particular location of the load. 3. Repeat this process for as many locations of the unit load as required to determine the shape of the influence line for the whole length of the member. It is often helpful if we can consider a generic location (or several locations) x of the unit load. 4. Joining ordinates for different locations of the unit load throughout the length of the member, we get the influence line for that particular response parameter. The following three examples show how to construct influence lines for a support reaction, a shear force and a bending moment for the simply supported beam AB . Example 1 Draw the influence line for (vertical reaction at A ) of beam AB in Fig.1 Solution: Free body diagram of AB :