PH8151 Engineering Physics Chapter 1

34 CHAPTER 1. PROPERTIES OF MATTER

R- Radius of curvature of the neutral filament of the bar.
In the equilibrium position, moment of the external bending couple = internal bending
moment

Wa=

YIg

R

(1.37)

Since for a given loadW, the values ofa,Y andIg are constants,Ris also a constant
so that the beam is bent uniformly into an arc of a circle of radiusRas shown in Figure
1.30.

C D

E

F

O

l /2 l /2

R R

R

(R-y

)

y

EF = y, OF = OC = OD = R, OE = OE - EF = R - y, CE = DE = l /2

A B

Figure 1.30: Uniform bending - circular geometry of neutral filament

Now,CD=landyis the elevation of the midpoint E of the beam so thaty=EF.
From Pythagoras theorem we have,

CE^2 =OC^2 ≠ OE^2. Or

A

l

2

B 2

=R^2 ≠(R≠y)^2 =2Ry≠y^2

That is,

A

l

2

B 2

=2Ry≠y^2

Since the radius of curvatureRis usually very large compared to the elevationy, the term
y^2 is negligible compared to 2 Ry.

)

A

l

2

B 2

¥ 2 Ry

Hence
1
R

=

8 y

l^2

(1.38)

34