Department of Chemical Engineering

Strength of Materials for Chemical Engineers (0935381)

Chapter 3

Shearing Force and Bending Moment Diagram

Beam: is a structural member subjected to a system of external forces at right angles to

axis.

Types of Beams

1- Cantilever beam: fixed or built-in at one end while it’s other end is free.

2- Freely or simply supported beam: the ends of a beam are made to freely rest on

supports.

3- Built-in or fixed beam: the beam is fixed at both ends.

4- Continuous beam: a beam which is provided with more than two supports.

5- Overhanging beam: a beam which has part of the loaded beam extends outside the

supports.

Statically Determinate Beams

Cantilever, simply supported, overhanging beams are statically determinate beams as the

reactions of these beams at their supports can be determined by the use of equations of

static equilibrium and the reactions are independent of the deformation of the beam.

There are two unknowns only.

Statically Indeterminate Beams

Fixed and continuous beams are statically indeterminate beams as the reactions at

supports cannot be determined by the use of equations of static equilibrium. There are

more than two unknown.

Types of Loads:

1- Concentrated load assumed to act at a point and immediately introduce an

oversimplification since all practical loading system must be applied over a finite

area.

2- Distributed load are assumed to act over part, or all, of the beam and in most cases

are assumed to be equally or uniformly distributed.

a- Uniformly distributed.

b- Uniformly varying load.

Shearing Force (S.F.)

Shearing force at the section is defined as the algebraic sum of the forces taken on one

side of the section.

Bending Moment (B.M)

Bending moment is defined as the algebraic sum of the moments of the forces about the

section, taken on either sides of the section.

1. S.F.and B.M. Diagrams for Beams Carrying Concentrated Loads Only:

• If the S.F. is zero the bending moment will remain constant.

• If the S.F. is positive the slope of the B.M. curve is positive.

• If the S.F. is negative the slope of the B.M. curve is negative.

• The difference in B.M. between any two points equals the area under the S.F.

curve for the same points.

• Between concentrated loads, there is no change in shear and the shear force curve

plots as a straight horizontal line.

• At each concentrated load or reaction, the value of the shear force changes

abruptly by an amount equal to the load or reaction force.

• The maximum bending moment occurs at a point where the shear curve crosses

its zero axis.

2. S.F.and B.M. Diagrams for Beams Carrying Distributed Loads Only:

3. S.F. and B.M. Diagrams for Beams Carrying Combined Concentrated and

Uniformly Distributed Loads:

Points of Contraflexure

It is a point where the curvature of the beam changes sign and occurs at a point where the

B.M. is zero (other than the ends).

In order to find the exact location of the contraflexure point you have to solve and find

the zeros of the bending moment equation applied in the interval where the curve crosses

the zero line.

For the above example find the zeros of the second order bending moment equation in the

third interval.

4. S.F. and B.M. Diagrams for Beams Carrying Couple or Moment:

At each couple or moment, the value of the bending moment changes abruptly by an

amount equal to the couple or moment.

Relationship between Shear Force and Bending Moment

• The maximum or minimum B.M. occurs where

0..== FS

dx

dM

•

Thus where S.F. is zero and crosses the zero axis B.M. is maximum or minimum.

•

If S.F. is zero then

constant00 =⇒=⇒=

∫∫

MdxdM

dx

dM

•

Since

..FS

dx

dM

=

then where the S.F. is positive the slope of the B.M. diagram is

positive, and where the S.F. is negative the slope of the B.M. diagram is also

negative.

•

The area of the S.F. diagram between any two points, from basic calculus

∫∫∫

=⇒=⇒= dxFSMdxFSdMFS

dx

dM

......

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