Modified Moment-Area Method for Deep Cantilever Beams

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otation. Hence, the strain energy due to shearing force from the initial position to the final rotation β = V / λ is y U shear -w(x) L = L ∫ 0 2 λβ dx (1) yshear ybending Fig. 1 Deep cantilevered beam and its elastic curve with positive sign convention. Comparing Eq. 1 with the well-known strain energy due to shearing force from L 2 V classical theory, U shear = ∫κ dx , we have 2GA 0 λ κ GA = (2) Now, let us consider Fig. 1, the total deflection of the beam can be written as y = ybending + yshear (3) Differentiating Eq. 3 twice, we have 2 2 2 d y d ybending d yshear = + 2 2 2 dx dx dx 2 d ybending dβ = + dx dx (4) According to the Bernoulli-Euler beam theory and the previously mentioned sign 2 d ybending conventions, we have 2 dx M = and dV / dx = −w( x) . Also, if the factor λ is EI independent of x for a given portion of the beam, then 2 d y dθ M d( V / λ) = = + 2 dx dx EI dx (6) x
M w dθ = dx − dx (7) EI λ This equation states that the change d θ in the slope of the tangent on either side of an incremental element dx correspond to infinitesimal area under the M / EI diagram and the w / λ diagram, respectively. Integrating from point A on the beam to point B , where B / A B B M θ B / A = ∫ dx − EI ∫ A A w dx λ θ is the angle of the tangent at B measured with respect to the tangent at A and has unit in radian. The positive sign indicates that the angle is in clockwise direction. For a small displacement, the vertical deviation dt of the tangent on each side of the incremental element dx can be determined by using the circular arch formula dt = xdθ . Then, the deviation of the tangent at point A to point B can be found as B B M w t B / A = ∫ xdx − xdx EI ∫ (9) λ A A where positive sign indicates the upward vertical deviation. It should be noted that the Eq. 9 must be modified in the case of concentrated load due to dV / dx = 0 . Since we have d ( Vx) / dx = [ V + x( dV / dx)] = V , then B M V t B / A = ∫ xdx − xdx EI ∫ (10) λ A A Example: Determine the deflection and rotation at the end B of the deep cantilevered concrete beam as shown in Fig. 2 by using the modified moment area method. The beam has a rectangular cross-section with a constant width, b . Fig. 2 Deep cantilevered beam having rectangular cross-section with a constant width. in Fig. 3. y w(x) A B L/3 L/3 L/3 L B V w M 1. Draw the diagram, diagram, and diagram due to the loads as shown λ λ EI P h h/5 h/5 x (8)
Page 1: Modified Moment-Area Method for Dee
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Modified Moment-Area Method for Deep Cantilever Beams

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