ENGINEERING NOTES SFD AND BMD

 SFD AND BMD NOTES (MECHANICS)

Shear Force and Bending Moment Diagrams

INTRODUCTION

 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. 

TYPES OF BEAMS

 The following are the important types of beams: 1. Cantilever beam, 2. Simply supported beam, 3. Overhanging beam, 4. Fixed beams, and 5. Continuous beam. 

TYPES OF LOADS 

Concentrated Load; This type of load acts relatively on a smaller area. For example, the force exerted by a chair or a table leg on the supporting floor or load exerted by a beam on a supporting column are both considered to be concentrated

Uniformly Distributed Load (UDL) As the name itself implies, uniformly distributed load is spread over a large area. Its magnitude is designated by its intensity (N/m or kN/m). The water pressure on the bottom slab of a water tank is an example of such a loading. If a floor slab is supported by beams, Mechanics of Materials 18MEI33 the load of the slab on the beams is certainly uniformly distributed.

Uniformly Varying Load (UVL) This type of load will be uniformly varying from zero intensity at one end to the designated intensity at the other end. A triangular block of brickwork practically imposes such a loading on a beam. The water pressure distribution on the walls of a water tank could be another example. Here again, equivalent concentrated load (equal to area of the loading triangle) is to be used while dealing with this load.

Concentrated Moment If for some purpose, a beam is to accommodate a load on a bracket mounted on it, what gets transmitted on the beam is a concentrated moment

REACTIONS AT SUPPORTS OF BEAMS

 A beam is a structural member used to support loads applied at various points along its length. Beams are generally long, straight and prismatic (i.e. of the same cross-sectional area throughout the length of the beam). 

Types of Supports:

 Beams are supported on roller, hinged or fixed supports.

 Simple Support: If one end of the beam rests in a fixed support, the support is known as simple support. The reaction of the simple support is always perpendicular to the surface of support. The beam is free to slide and rotate at the simple support. 

 Roller Support: Here one end of the beam is supported on a roller. The only reaction of the roller support is normal to the surface on which the roller rolls without friction. Examples of roller supports are wheels of a motorcycle, or a handcart, or an over-head crane, or of a car, etc.

 Hinged Support: At the hinged support the beam does not move either along or normal to its axis. The beam, however, may rotate at the hinged support. The total support reaction is R and its.

Fixed Support:  Types of supports and reactions At the fixed support, the beam is not free to rotate or slide along the length of the beam or in the direction normal to the beam. Therefore, there are three reaction components, viz., vertical reaction component (V), horizontal reaction component (H) and the moment (M)

SHEAR FORCES AND BENDING MOMENT DIAMGRAMS

 Definition of Shear force and bending moment 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 section.

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 SF and BM For Shear force:

 We shall remember one easy sign convention, i.e., to the right side of a section, external force acting in 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 signs become just reversed when we consider left side of section.

For Bending moment:

 The internal resistive moment at the section that would make the beam to sag (means to sink down, droop) is treated to be positive. A sagged beam will bend such that it exhibits concave curvature at top and convex curvature at bottom.

SFD and BMD definitions 

It is clear from foregone discussions that at a section taken on a loaded beam, two internal forces can be visualized, namely, the bending moment and the shear force. It is also understood that the magnitude of bending moment and shear force varies at different cross sections over the beam. The diagram depicting variation of bending moment and shear force over the beam is called bending moment diagram [BMD] and shear force diagram [SFD]. Such graphic representation is useful in determining where the maximum shearing force and bending moment occur, and we need this information to calculate the maximum shear stress and the maximum bending stress in a beam. The moment diagram can also be used to predict the qualitative shape of the deflected axis of a beam.

 General Guidelines on Construction of SFD and BMD 

Before we go on to solving problems, several standard procedures (or practices) in relation with construction of shear force and bending moment diagrams need to be noted.

 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 length!).

 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.

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.

Variation of shear force and bending moment diagrams ; Load UDL UVL Shear Force Constant Linear Parabolic Bending Moment Linear parabolic Cubic