shear force and bending moment diagrams | shear force and bending moment
shear force and bending moment diagrams | shear force and bending moment
shear force and bending moment
Shear and bending moment diagrams are analytical tools used in conjunction with structural analysis to help perform structural design by determining the value of shear force and bending moment at a given point of a structural element such as a beam.
These diagrams can be used to easily determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure. Another application of shear and moment diagrams is that the deflection of a beam can be easily determined using either the moment area method or the conjugate beam method.
The algebraic sum of the vertical forces at any section of a beam to the right or left of the section is known as shear force. It is briefly written as S.F. The algebraic sum of the moments of all the forces acting to the right or left of the section is known as bending moment.
It is written as B.M. In this chapter, the shear force and bending moment diagrams for different types of beams (i.e., cantilevers, simply supported, fixed, overhanging etc.) for different types of loads (i.e., point load, uniformly distributed loads, varying loads etc.) acing on the beams, will be considered
SHEAR FORCES AND BENDING MOMENT DIAGRAMS
Shear force
A shear force (SF) is defined as the algebraic sum of all the vertical forces, either to the left or to the right-hand side of the cut section.
Bending Moment
A bending moment (BM) is defined as the algebraic sum of the moments of all the forces either to the left or to the right of a section.
Sign convention of SHEAR FORCE
We can follow any one easy sign convention, i.e., to the right side of a section, the external force acting in the upward direction is treated as negative (remember this convention as RUN —» Right side of a section Upward force is Negative).
It is automatic that a downward force acting to the right side of a section be treated as positive. The sign convention is shown in Fig. The signs become just reversed when we consider the left side of the section.
Sign convention of Bending Moment
For the Bending moment Sign convention of SF The internal resistive moment at the section that would make the beam sag (means to sink down, droop) is treated to be positive. A sagged beam will bend such that it exhibits concave curvature at the top and convex curvature at the bottom. Positive bending moment is shown in Fig. 5.7 (A shear force that tends to rotate the beam in a clockwise direction is positive and vice versa)
The internal resistive moment that would hog the beam is treated as negative. A hogged beam will show convex curvature at the top and concave curvature at the bottom.
General Guidelines for SFD and BMD
1) The load, shear and bending moment diagrams should be constructed one below the other, in that order, all with the same horizontal scale.
2) The dimension on the beam need not be scaled but should be relative and proportionate (a 3 m span length should not look more than 5 m lengths!).
3) Ordinates (i.e., BM and SF values) need not be plotted to scale but should be relative. Curvature may need to be exaggerated for clarity.
4) Principal ordinates (BM and SF values at salient points) should be labeled on both SFD and BMD.
5) A clear distinction must be made on all straight lines as to whether the line is horizontal or has a positive or negative slope.
6) The entire diagram may be shaded or hatched for clarity if desired
SOLVED EXAMPLE
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