Structural Analysis IDEFLECTION OF BEAMS Daniel Rumbi Teruna

School of Civil EngineeringUniversity of North Sumatera

DeflectionsIntroductionDeflection of beams and frames is the deviation of the configuration of beams and frames from their un-displaced state to the displaced state, measured from the neutral axis of a beam or a frame member. It is the cumulative effect of deformation of the infinitesimal elements of a beam or frame member. As shown in the figure below, an infinitesimal element of width dx can be subjected to all three actions, thrust, T, shear, V, and moment, M. Each of these actions has a different effect on the deformation of the element

DeflectionsEffect of thrust, shear, and moment on the deformation of an element.The effect of axial deformation on a member is axial elongation or shortening, which iscalculated in the same way as a truss members. The effect of shear deformation is thedistortion of the shape of an element that results in the transverse deflections of a member. The effect of flexural deformation is the bending of the element that results in transverse deflection and axial shortening. These effects are illustrated in the followingfigure.

DeflectionsEffects of axial, shear, and flexural deformations on a member.While both the axial and flexural deformations result in axial elongation or shortening, the effect of flexural deformation on axial elongation is considered negligible for practical applications. Thus bending induced shortening, , will not create axial tension in the figure below, even when the axial displacement is constrained by two hinges as in the right part of the figure, because the axial shortening is too small to be of any significance. As a result, axial and transverse deflections can be considered separately and independently.

DeflectionsBending induced shortening is negligible.We shall be concerned with only transverse deflection henceforth. The shear deformation effect on transverse deflection, however, is also negligible if the length-to depth ratio of a member is greater than ten, as a rule of thumb. Consequently, the onlyeffect to be included in the analysis of beam and frame deflection is that of the flexuraldeformation caused by bending moments. As such, there is no need to distinguish frames from beams. We shall now introduce the applicable theory for the transverse deflection of beams.

DeflectionsLinear Flexural Beam theoryClassical Beam TheoryClassical beam theory is based on the following assumptions:(1) Shear deformation effect is negligible.(2) Transverse deflection is small (

DeflectionsBeam element deformation and the resulting curvature .A small element s of a beam is shown in Figure 1 and is assumed to be bent in the shape of an arc of a circle of radius R. The slope of the elastic curveat one end of the element is . The change in slope of the elastic curve over the length of the element is , and the curvature, or rate of change of slope, over the element is:

DeflectionsThe slope of the beam is positive as shown and for small displacements is given by:In the limit:and:and:

DeflectionsThe slope of the elastic curve is given by the expression:The deflection of the elastic curve is given by the expression:Where:(1)(2)

DeflectionsExample 1. The beam shown has a constant EI and a length L, find the rotation anddeflection formulas.Integrating this expression with respect to x gives:C1=0 constant of integration

DeflectionsHence, the slope of the elastic curve is given by the expression:Integrating this expression with respect to x giveswhere: C2=constant of integration=0Hence, the deflection of the elastic curve is given by the expression:At x = l the deflection of the cantilever is:

DeflectionsExample 2. The beam shown has a constant EI and a length L, find the maximum rotation and deflection

DeflectionsExample 3. Find the maximum deflection of simple supported beam as shown below

DeflectionsConjugate Beam Method. In drawing the shear and moment diagrams, the basicequations we rely on are Eq. 3 and Eq. 4, which are reproduced below in equivalentforms(3)(4)Clearly the operations in Eq. 3 and Eq. 4 are parallel to those in Eq. 1 and Eq. 2. Ifwe defineas elastic load in parallel to q as the real load,We can now define a conjugate beam, on which an elastic load of magnitude M/EI is applied. We can draw the shear and moment diagrams of this conjugate beam and the results are actually the rotation and deflection diagrams of the original beam. Before we can do that, however, we have to find out what kind of support or connection conditions we need to specify for the conjugate beam.

DeflectionsAt a fixed end of the original beam, the rotation and deflection should be zero and theshear and moment are not. At the same location of the conjugate beam, to preserve theparallel, the shear and moment should be zero. But, that is the condition of a free end.Thus the conjugate bean should have a free end at where the original beam has a fixedend. The other conditions are derived in a similar way.

DeflectionsConversion from a real beam to a conjuagte beam.We can now summarize the process of constructing of the conjugate beam and drawing the rotation and deflection diagrams:

Deflections(1) Construct a conjugate beam of the same dimension as the original beam.(2) Replace the support sand connections in the original beam with another set of supports and connections on the conjugate beam according to the above table. i.e. fixed becomes free , free becomes fixed. etc. Place the M/EI diagram of the original beam onto the conjugate beam as a distributed load, turning positive moment into upward load. Draw the shear diagram of the conjugate beam, positive shear indicates counterclockwise rotation of the original beam. (5) Draw the moment diagram of the conjugate beam, positive moment indicates upwarddeflection.Example 2. The beam shown has a constant EI and a length L, draw the rotation anddeflection diagrams.A cantilever beam load by a moment at the tip.

DeflectionsSolution(1) Draw the moment diagram of the original beam.Moment diagram.(2) Construct the conjugate beam and apply the elastic load.Conjagte beam and elastic load.(3) Analyze the conjugate beam to find all reactions.Conjagte beam, elastic load and reactions.

DeflectionsExample 3. Find the rotation and deflection at the tip of the loaded beam shown. EI is constant. Find the tip rotation and deflection.Solution. The solution is presented in a series of diagrams below.ReactionsMoment diagram

DeflectionsAt the right end (tip of the real beam):

DeflectionsExample 4. Draw the rotation and deflection diagrams of the loaded beam shown. EIis constant.

Deflections

DeflectionsSolution process for rotation and deflection diagrams.

DeflectionsExample 5. Find the rotation and deflection of the loaded beam shown. EIis constant. Momen diagramConjugate beam

DeflectionsSpecial case,

Deflections

Deflections

****