bending of beam

7/24/2019 Bending of Beam

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2.1

Chapter 2

BENDING OF

BEAMS

The following sections discuss singularity functions and sign

convention commonly used in bending of beams

2.1 SINGULARITY FUNCTIONS

Singularity Functions can be used to describe the load

intensity w(x), shear force Fxy and bending moment Mxz

distributions along a beam. These functions are as shown

below:

x a

x a

x a

x a

n

n

0

where n 0 (n = 0, 1, 2 ....)

Integration of a singularity function takes the form:

x a dx x a

n

nn

1

1

Examples for the use of beams:

Supporting floors of buildings, bridges

Automotive axles, leaf springs

Airplane wings, brackets.



2.2

10)(

Nma xw xw

Examples of load intensity distributions using singularity

functions are:

(i) Uniformly distributed load acting downward (negative).

(ii) Linearly-increasingly load intensity (load actingdownward is negative).

11)(

Nma xm xw

m(a/2)

a/2



2.3

(iii) Parabolically-distributed load (load acting upward is

positive)

12

)(

Nmc xk xw

x

c

3c

4kc2 Nm-1

w

3c



2.4

There are two special cases of singularity function:

(i) The concentrated moment or unit doublet function:

Note that x a

2 has no physical meaning, this is

just to enable us to do certain calculations later on.

(ii) The concentrated load or unit impulse function:

11)(

Nma x P xw

These two functions integrate according to thefollowing:

11

0

2

0

Nma x M dxa x M

N a x P dxa x P 01

12

0)(

Nma x M xw



2.5

2.2 Sign Convention

Notation:

F : Fxy (in y direction)

M : Mxz (about z)

Left plane of cut beam

Fxy

y

MxzMx

x

PP

Fxy

Positive plane Negative planez

(direction of normal to the plane on which F acts)

(direction of normal to the plane on which M acts)

Cut section

y

Right plane of cut beam

z



2.6

Sign convention for bending moment:

+ve moment, smiling

-ve moment, frowning

)int( curveensityload under area

wdx F

)( curve force shear under area

Fdx M

Shear force (F) and bending moment (M) relationships inbending of beams are given by:

bending of beam

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