Shear Force and Bending Moments

Beam 

Normally beams stand for horizontal structural members which transmit loads horizontally along their length to the supports. where the loads are generally transformed into vertical forces.

The objective of beams is to withstand vertical loads, shear forces and bending moments, couples that lie in a plane containing the longitudinal section of the bar. 

According to determent, a beam may be determinate or indeterminate.

Statically Determinate Beams 

Statically determinate beams are those beams in which the reactions of the supports may be determined by the use of the three static equilibrium equations . 

Statically Indeterminate Beams 

If the number of reactions exerted upon a beam exceeds the number of equations in static equilibrium, the beam is said to be statically indeterminate. In order to solve the reactions of the beam, the static equations must be supplemented by equations based upon the elastic deformations of the beam.

The degree of indeterminacy is taken as the difference between the umber of reactions to the number of equations in static equilibrium that can be applied.

Types of Supports 

The supports of the beam may consists of three types

a) Simply support or Roller Support 

b) Hinge Support or Pinned Support 

c) Fixed Support or Built in support 

Types of Loading 

Loads applied to the beam may consist of 

a) concentrated load (load applied at a point), 

b)uniform distributed load (u.d.l.), 

c) uniformly varying load, 

d) an applied couple or moment.

Internal Forces in Beams

When a beam or frame is subjected to transverse loadings, the three possible internal forces that are developed are the normal or axial force, the shearing force, and the bending moment.

Consider a cantilever beam as shown in above figure to predict the the magnitudes of these forces. 

Take a section at point k and divide the beam in to two parts as shown in figure. These two parts are in equilibrium by introducing the internal forces.

The forces and moments on either side of the section must be equal in order to counteract each other and maintain a state of equilibrium. so the same internal forces and moments will result from summing of the forces or moments, regardless of which side of the section is selected.

Normal or Axial Force (N)

The normal force at any section of a structure is defined as the algebraic sum of the axial forces acting on either side of the section.

Tensile forces are considered as positive and compressive forces are considered as negative. 

Shear Force (V)

The internal vertical resistance is called Shear Force.It is nothing but the unbalanced vertical force at the section. 

The Shear Force at a section may be defined as "the algebraic sum of the total vertical forces on either side of the cross section".

The unit of shear force is : N,or kN 

Sign convention for S.F

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If a left portion of a section is considered ,upward forces will be positive and right side portion is considered down ward forces will be positive.

If a left portion of a section is considered downward forces will be negative and right side portion is considered upward forces will be negative.

Moment of Resistance (M)

The internal resistance against bending moment is called Moment of Resistance, which is always equal to bending moment.

Bending moment 

The moment which bends the beam is called Bending moment.

The Bending moment at a section through a structural element may be defined as "the algebraic sum of the moments about that section of all external forces acting to one side of that section". 

The unit of bending moment is : N-mm, or kN-m

Sign convention for B.M.

It is more common to use the sign convention that a clockwise bending moment to the left of the point,and anticlockwise on right side are under considered as positive. a set of these positive moment will cause "sagging" 

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Anti clockwise bending moment to the left of the point, and clockwise on right side are under considered as negative. a set of these negative moments will cause  "hogging". 

Note:

1.Bending moment at a section is +ve if it is sagging and -ve if it is hogging. 

2.Shear force at a section ,such that the portion to the right of the section slides up wards with respect to the left of the section are +ve and vice versa.

Shear Force and Bending moment diagrams

A Shear Force diagram (SFD) is one which shows the variation of the shear force along the length of the beam. 

A bending moment diagram (BMD) is one which shows the variation of the bending moment along the length of the beam. 

Point of contraflexure

A point of zero bending moment within a beam is called as point of contraflexure.

This is the point of transition from hogging to sagging or vice versa.

Note: 

1. Bending Moment is maximum at the point where the Shear Force is zero or where it changes direction from +ve to -ve or vice versa.

2. Bending moment varies linearly over unloaded sections, and parabolic over uniformly loaded sections. 

3.The point of contra-flexure or point of inflexion is the point where the Bending moment changes its direction from +ve to -ve or vice versa. Where the magnitude of B.M. is zero.

4.Maximum shear force usually occur at the support or at the point under concentrated load.

Procedure for drawing of S.F and B.M diagrams:

1.Draw the free-body diagram of the structure.

Check the stability and determinacy of the structure. If the structure is stable and determinate, proceed to the next step of the analysis.

2.Determine the unknown reactions by applying the conditions of equilibrium. (For cantilevered beams we need not find reactions by considering the free-end of the structure as the initial starting point of the analysis)

3. Pass an imaginary section perpendicular to the neutral axis of the structure at the point where the internal forces are to be determined. The passed section divides the structure into two parts. Consider either part of the structure for the computation of the desired internal forces.

4.For shearing force and bending moment computation, first write the functional expression for these internal forces for the segment where the section lies, with respect to the distance x from the origin.

5.Compute the principal values of the shearing force and the bending moment at the segment where the section lies.

6.Draw the shearing force, and bending moment diagram for the structure, noting the sign conventions.

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To draw SFD or BMD the negative values are plotted to scale below a horizontal reference line and positive values are plotted above the line.