Mechanical Engineering Principles

BENDING OF BEAMS 107

The maximum stress occurs at the outermost fibre
of the beam’s cross-section fromNA,namelyatyˆ.

By inspection, yˆ=

0. 2

2

= 0 .1m

Hence σˆ=

Mˆyˆ

I

=

4500 N m× 0 .1m

4. 637 × 10 −^5 m^4

i.e.the maximum bending stress,

σˆ= 9. 70 × 106 N/m^2 = 9 .70 MPa

Now try the following exercise

Exercise 41 Further problems on the

bending of beams

1. A cantilever of solid circular cross-
   section is subjected to a concentrated
   load of 30 N at its free end, as
   shown in Figure 8.8. If the diameter of
   the cantilever is 10 mm, determine the
   maximum stress in the cantilever.
   [367 MPa]


2. If the cantilever of Figure 8.8 were
   replaced with a tube of the same external
   diameter, but of wall thickness 2 mm,
   what would be the maximum stress due
   to the load shown in Figure 8.8.
   [421 MPa]

1.2 m

30 N

Figure 8.8

3. A uniform section beam, simply sup-
   ported at its ends, is subjected to a cen-
   trally placed concentrated load of 5 kN.
   The beam’s length is 1 m and its cross-
   section is a solid circular one. If the
   maximum stress in the beam is limited
   to 30 MPa, determine the minimum per-
   missible diameter of the beam’s cross-
   section. [75 mm]


4. If the cross-section of the beam of
   Problem 3 were of rectangular shape,
   as shown in Figure 8.9, determine its

dimensions. Bending can be assumed to

take place about thexxaxis.

[0.172 m× 0 .086 m]

D x

D/2

x

D/2

Figure 8.9

5. If the cross-section of the beam of
   Problem 3 is a circular tube of external
   diameter d and internal diameter d/2,
   determine the value ofd. [0.166 m]


6. A cantilever of length 2 m, carries a
   uniformly distributed load of 30 N/m,
   as shown in Figure 8.10. Determine the
   maximum stress in the cantilever.
   [39.1 MPa]

30 N/m

(b) Cross-section

(solid)

(a) Contilever

2 m

25 mm

Figure 8.10

7. If the cantilever of Problem 6 were
   replaced by a uniform section beam,
   simply supported at its ends and carrying
   the same uniformly distributed load,
   determine the maximum stress in the
   beam. The cross-section of the beam may
   be assumed to be the same as that of
   Problem 6. [9.78 MPa]


8. If the load in Problem 7 were replaced
   by a single concentrated load of 120 N,
   placed at a distance of 0.75 m from the
   left support, what would be the maximum
   stress in the beam due to this concen-
   trated load. [36.7 MPa]


9. If the beam of Figure 8.10 were replaced
   by another beam of the same length, but
   which had a cross-section of tee form,