Guiding.Rules.in.CBM

Guiding.Rules.in.CBM - Guiding Rules in the Conjugate Beam...

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1 Guiding Rules in the Conjugate Beam Method ( Excerpt from an IJEE Paper) Courtesy: Int. J. Engng. Ed. , Vol. 26, No. 6, pp. 1422-1427, 2010 icjong@uark.edu 1.1 Pedagogy of the conjugate beam method. The conjugate beam method is actually a natural extension of the moment-area theorems . It is an elegant, efficient, and powerful method published by Westergaard [1] some nine decades ago, although some considered Mohr (1868) and Breslau (1865) to have prior influences. Elementary presentation of this method did appear in early textbooks in mechanics of materials [2, 3]. For reasons unknown, this method is missing in most such current textbooks. The pedagogy of the conjugate beam method lies in teaching and applying the rules in this method [1, 11]. These rules are summarized as follows: Rule 1: The conjugate beam and the given beam are of the same length . Rule 2 : The load on the conjugate beam is the elastic weight , which is the bending moment M in the given beam divided by the flexural rigidity EI of the given beam. (This elastic weight is taken to act upward if the bending moment is positive to cause top fiber in compression in beam convention.) For each existing support condition of the given beam, there is a corresponding support condition for the conjugate beam. The correspondence is given by rules 3 through 7 as follows: Existing support condition in the given beam: Corresponding support condition in the conjugate beam: Rule 3: Fixed end Free end Rule 4: Free end Fixed end Rule 5: Simple support at the end Simple support at the end Rule 6: Simple support not at the end Unsupported hinge Rule 7: Unsupported hinge Simple support Rule 8 : The conjugate beam is in static equilibrium . Rule 9 : The slope of the given beam at any cross section is given by the “ shear force at that cross section of the conjugate beam. (This slope is positive, or counterclockwise, if the “ shear force ” is positive tending to rotate the beam element clockwise in beam convention.) Rule 10 : The deflection
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Guiding.Rules.in.CBM - Guiding Rules in the Conjugate Beam...

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