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Module 4 : Deflection of Structures

Lecture 2 : Conjugate Beam Method

4.3	Conjugate Beam Method
	The conjugate beam method is an extremely versatile method for computation of deflections in beams. The relationships between the loading, shear, and bending moments are given by
		(4.7)
	where M is the bending moment; V is the shear; and w ( x ) is the intensity of distributed laod.
	Similarly, we have the following
		(4.8)
	A comparison of two set of equations indicates that if M / EI is the loading on an imaginary beam, the resulting shear and moment in the beam are the slope and displacement of the real beam, respectively. The imaginary beam is called as the “ conjugate beam ” and has the same length as the original beam.
	There are two major steps in the conjugate beam method. The first step is to set up an additional beam, called "conjugate beam,” and the second step is to determine the “ shearing forces ” and “ bending moments ” in the conjugate beam.
	The loading diagram showing the elastic loads acting on the conjugate beam is simply the bending-moment diagram of the actual beam divided by the flexural rigidity EI of the actual beam. This elastic load is downward if the bending moment is sagging.
	For each existing support condition of the actual beam, there is a corresponding support condition for the conjugate beam. Table 4.1 shows the corresponding conjugate beam of different types of actual beams. The actual beam as well as the conjugate beam are always in static equilibrium condition .
	The slope of (the centerline of) the actual beam at any cross-section is equal to the “ shearing force ” at the corresponding cross-section of the conjugate beam. This slope is positive or anti-clockwise if the “ shearing force ” is positive — to rotate the beam element anti-clockwise — in beam convention . The deflection of (the centerline of) the actual beam at any point is equal to the “ bending moment ” of the conjugate beam at the corresponding point. This deflection is downward if the “ bending moment ” is positive — to cause top fiber in compression — in beam convention . The positive shearing force and bending moment are shown below in Figure 4.7.