Shear and Moment in Beams

Shear and Moment in Beams

Introduction Basic problem in strength of materials is to determine the relations between the stresses and deformations caused by loads applied to any structure.In axial or torsional loadings, we had little trouble in applying the stress and deformation relations because in the majority of the cases the loading either remains constant over the entire structure or is distributed in definite amounts to the component parts.The study of bending loads, however, is complicated by the fact that the loading effects vary from section to section of the beam. These loading effects take the form of a shearing force and a bending moment, sometimes referred to as shear and moment.Beams are classified as:Simply supported or freely supported beamCantilever beamOverhanging beamBuilt-in-beamPropped BeamRestrained BeamContinuous beam

Methods of supporting beamsSimple beam-is supported by a hinged reaction at one end and a roller support at the other, but is not otherwise restrained.

Cantilever beam-is supported at one end only, with a suitable restraint to prevent rotation of that end.

Overhanging beam-is supported by a hinge and a roller reaction, with either or both ends extending beyond the supports.

Statically determinate beamsSimple beamCantilever beamOverhanging beamStatically indeterminate beamsPropped beamFixed or restrained beamContinuous beamPropped beam

Fixed or Restrained beam

Continuous Beam

Types of LoadsConcentrated load- is one acts over so small a distance that can be assumed to act at a point.Distributed load-acts over a considerable length of the beamUniformly distributed- rectangular loadUniformly varying-triangular loadTrapezoidal load- combination of uniformly distributed and uniformly varying loadNonuniformBending Moment and Shear ForceShear Force- is defined as the algebraic summation of the unbalanced vertical force to the right or left of the section.Bending Moment- is defined as the algebraic summation of the moments of the forces, to the right or left of the section.Shear and Moment

Shear, VThe resisting shear VR, setup by the fibers in any section is always equal but oppositely directed to the shearing force V.In computing V, upward acting forces or loads are considered as positiveBending Moment, MBending moment is defined as the summation of moments about the centroidal axis of any selected section of all loads acting either to the left or to the right side of the section.Sign of Bending MomentUpward acting external forces cause positive bending moments with respect to any downward forces cause negative bending moments.

Shear and Moment

Shear and Moment DiagramsAre merely the graphical visualization of the shear and moment equations plotted on V-axis and M-axis, usually located below the loading diagram.The discontinuities in the shear diagram are joined by vertical lines drawn up or down to represent the abrupt changes in shear caused respectively by upward or downward concentrated load.Shear and Moment DiagramThe shear and moment diagram can be calculated numerically at any particular section. To know how these values vary, along the length of the beam, can be shown by plotting the shear or the moment as ordinates and the position of cross section of the beam as the abscissa. This is very useful, as they give the clear picture of the distribution of shear and moment along the beam.Illustrative ProblemWrite shear and moment equations for the cantilever beam carrying the uniformly varying load and concentrated load shown. Also, sketch the shear and moment diagram.Figure

Interpretation of Vertical Shear and Bending MomentThe resultant effect of the forces at one side of the exploratory section reduces to a single force and a couple that are respectively the vertical shear and the bending moment at that section.Relations among Load, Shear, and MomentIf there is concentrated load at a section of the beam, then the shear suddenly changes. But the bending moment remains unchanged.If there is no load between two points then the shear force does not change. But bending moment changes.If there is a Uniformly distributed load, between the two points, then the shear force changes linearly. But the bending moment changes according to the parabolic law.If there is uniformly varying load between two points, then the shear changes according to the parabolic law. But the bending moment changes according to cubic lawRelations among Load, Shear, and Momentequation4-1

equation4-2

equation4-3

equation4-4

Relations among Load, Shear, and Moment

Properties of Shear and MomentThe tangent drawn at any point of the moment diagram which makes angle with the x axis represents the shear at that section.The area of the shear force diagram between any two section is equal to the deflection between the moments between the section.The tangent drawn at any point of shear diagram which makes angle with x axis is equal to intensity of loading over that section.The slope of moment abruptly changes at the point of application of the concentrated load.If we consider any two section of a loaded beam then the total load getting between the considered any two section is equal to the difference of value of shear between the two considered section.Sign conventionsFor vertical forces:Upward = positiveDownward = negative

For moments:Clockwise = negativeCounter clockwise = positiveProcedures for constructing Shear and Moment DiagramsCompute the reactions.Compute the values of shear at the change of load points, using either V = (Fy)L or V =(area)load.Procedures for constructing Shear and Moment DiagramsSketch the shear diagram, determining the shape from equation 4-3; that is the intensity of the load ordinate equals the slope at the corresponding ordinate of the shear diagram Locate the points of zero shear

Procedures for constructing Shear and Moment DiagramsCompute values of bending moment at the change of load points and at points of zero shear, using either M = (M)L =(M)R or M = (area)shear, whichever is more convenient.Procedures for constructing Shear and Moment DiagramsSketch the moment diagram through the ordinates of the bending moments computed in step 5. the shape of the diagram is determined from equation 4-4; that is, the intensity of the shear ordinate equals the slope at the corresponding ordinate of the moment diagram.Illustrative ProblemsSketch shear and moment diagrams for the beam shown, computing the values at all change of loading points and the maximum shear and maximum moment.

Moving LoadsMoving LoadsThe value of left reaction is

The bending moment under P2 is then

Moving LoadsTo compute the value of x that will give maximum M2, we set the derivative of M2 with respect to x equal to zero:

From which

Moving LoadsThe bending moment under a particular load is a maximum when the center of the beam is midway between that load and the resultant of all loads then on the span.

The maximum shearing force occurs at, and is equal to, the maximum reaction. The maximum reaction is the reaction to which the resultant load is nearest.Illustrative ProblemA truck and trailer combination having the axle loads shown, rolls across the simple supported span of 12 m. Compute the maximum bending moment and the maximum shearing force.