This document contains 15 problems related to determining stresses in beams undergoing bending and shearing. The problems involve calculating stresses in beams with various cross-sectional shapes under different loading conditions. The beams are made of materials like steel, wood, and brass. Parameters like moment of inertia, shear force, beam dimensions, and material properties are provided to calculate stresses.

1. Mechanics of Solids (NME-301) Assignment Bending and Shearing Stress in Beams
Yatin Kumar Singh Page 1
(1) Determine the maximum normal strain produced in a
steel wire of diameter d = 1/16 in. when it is bent around a
cylindrical drum of radius R = 24 in.
(2) A copper wire having diameter d = 3 mm is bent into a circle
and held with the ends just touching. If the maximum permissible
strain in the copper is = 0.0024, what is the shortest length L
of wire that can be used?
(3) A cantilever beam AB is loaded by a couple M0 at its free end.
The length of the beam is L= 1.5 m and the longitudinal normal
strain at the top surface is 0.001. The distance from the top surface
of the beam to the neutral surface is 75 mm. Calculate the radius of
curvature ρ, the curvature k, and the vertical deflection δ at the
end of the beam.
(4) A thin strip of steel of length L = 20 in. and thickness t = 0.2 in. is
bent by couples M0. The deflection δ at the midpoint of the strip
(measured from a line joining its end points) is found to be 0.25 in.
Determine the longitudinal normal strain at the top surface of the
strip.
(5) A bar of rectangular cross section is loaded and supported as
shown in the figure. The distance between supports is L = 1.2 m
and the height of the bar is h = 100 mm. The deflection δ at the
midpoint is measured as 3.6 mm. What is the maximum normal
strain at the top and bottom of the bar?
(6) A thin, high-strength steel rule (E = 30 × 10
6
psi) having
thickness t = 0.15 in. and length L= 40 in. is bent by couples M0 into
a circular arc subtending a central angle α = 45° (see figure). (a)
What is the maximum bending stress σmax in the rule? (b) Does the
stress increase or decrease if the central angle is increased?
(7) A simply supported wood beam AB with span length L = 3.5 m
carries a uniform load of intensity q = 6.4 kN/m (see figure).
Calculate the maximum bending stress σmax due to the load q if the
beam has a rectangular cross section with width b = 140 mm and
height h = 240 mm.
(8) A railroad tie (or sleeper) is subjected to two rail loads, each of
magnitude P = 175 kN, acting as shown in the figure. The reaction q
of the ballast is assumed to be uniformly distributed over the
length of the tie, which has cross-sectional dimensions b = 300 mm
and h = 250 mm. Calculate the maximum bending stress σmax in the
tie due to the loads P, assuming the distance L = 1500 mm and the
overhang length a = 500 mm.
(9) Determine the maximum tensile stress σt (due to pure bending
by positive bending moments M) for beams having cross sections
as follows (see figure): (a) a semicircle of diameter d, and (b) an
isosceles trapezoid with bases b1 = b and b2 = 4b/3, and altitude h.
(10) Determine the maximum bending stress σmax (due to pure
bending by a moment M) for a beam having a cross section in the
form of a circular core (see figure). The circle has diameter d and
the angle β = 60°.

2. Mechanics of Solids (NME-301) Assignment Bending and Shearing Stress in Beams
Yatin Kumar Singh Page 2
(11) Determine the maximum tensile stress σt and maximum
compressive stress σc due to the load P acting on the simple beam
AB (see figure). Data are as follows: P = 5.4 kN, L = 3.0 m, d = 1.2 m,
b = 75 mm, t = 25 mm, h =100 mm, and h1 = 75 mm.
(12) A cantilever beam AB, loaded by a uniform load and a
concentrated load (see figure), is constructed of a channel
section. Find the maximum tensile stress σt and maximum
compressive stress σc if the cross section has the dimensions
indicated and the moment of inertia about the z axis (the neutral
axis) is I = 2.81 in.4 (Note: The uniform load represents the
weight of the beam.)
(13) A cantilever beam AB of triangular cross section has length L =
0.8 m, width b = 80 mm, and height h = 120 mm (see figure). The
beam is made of brass weighing 85 kN/m
3
. (a) Determine the
maximum tensile stress σt and maximum compressive stress σc due
to the beam’s own weight. (b) If the width b is doubled, what
happens to the stresses? (c) If the height h is doubled, what
happens to the stresses?
(14) A beam ABC with an overhang from B to C supports a uniform
load of 160 lb/ft throughout its length (see figure). The beam is a
channel section with dimensions as shown in the figure. The
moment of inertia about the z axis (the neutral axis) equals 5.14
in.
4
Calculate the maximum tensile stress σt and maximum
compressive stress σc due to the uniform load.
(15) A beam of T-section is supported and loaded as shown in the
figure. The cross section has width b = 2 1/2 in., height h = 3 in.,
and thickness t = 1/2 in. Determine the maximum tensile and
compressive stresses in the beam.
(16) A cantilever beam AB with a rectangular cross section has a
longitudinal hole drilled throughout its length (see figure). The
beam supports a load P = 600 N. The cross section is 25 mm wide
and 50 mm high, and the hole has a diameter of 10 mm. Find the
bending stresses at the top of the beam, at the top of the hole, and
at the bottom of the beam.
Shear Stresses in Beams
(1) A timber beam of rectangle section is simply supported at the
ends and carries a point load at the centre of the beam. The
maximum bending stress is 12N/mm
2
and maximum shearing
stress is 1N/mm
2
, find the ratio of the span to the depth.
(2) An I-section beam 350mm × 150mm has a web thickness of
10mm and a flange thickness of 20mm. If the shear force acting on
the section is 40kN, find the maximum shear stress developed in
the I-section. Sketch the shear stress distribution across the
section. Also calculate the total shear force carried by the web.
(3) The shear force acting on a section of a beam is 50 kN. The
section of the beam is of T-shaped of dimensions 100mm × 100mm
as shown in fig. The moment of inertia about the horizontal neutral
axis is 314.221 × 10
4
mm
4
. Calculate the shear stress at the neutral
axis and at the junction of the web and the flange.

3. Mechanics of Solids (NME-301) Assignment Bending and Shearing Stress in Beams
Yatin Kumar Singh Page 3
(4) The shear force acting on a beam at an I-section with unequal
flanges is 50 kN. The moment of inertia of the section about N.A. is
2.849 × 10
4
. Calculate the shear stress at the N.A. and also draw the
shear stress distribution over the depth of the section.
(5) A beam of square section is used as a beam with one diagonal
horizontal. The beam is subjected to a shear force F, at a section.
Find the maximum shear in the cross section of the beam and draw
the shear distribution diagram for the section.
(6) Fig shows a section which is subjected to a shear force of 100
kN. Determine the shear streses at A,B,C and D. Sketch the shear
stress distribution also.
(7) Two rectangular plates one of steel and the other of brass each
40 mm wide and 10 mm deep are placed together to form a beam
40 mm wide and 20mm deep, on two supports 1 m apart, the brass
plate being on top of the steel plate. Determine the maximum load,
which can be applied at the centre of the beam, if the plates are:
(i) separate and can bend independently,
(ii) firmly secured throughout their length.
Maximum allowable stress in steel = 112.5 N/mm
2
and in brass = 75
N/mm
2
. Take Es = 2 × 10
5
N/mm
2
; Eb = 8 × 10
4
N/mm
2
(8) A timber beam 100 mm wide and 200 mm deep is to be
reinforced by bolting on two steel flitches each 150 mm by 12.5
mm in section. Calculate the moment of resistance in the following
cases: (i) flitches attached symmetrically at the top and bottom; (ii)
flitches attached symmetrically at the sides. Allowable stress in
timber is 6 N/mm
2
. What is the maximum stress in the steel in each
case? Take Es = 2 × 10
5
N/mm
2
; Et = 1 × 10
4
N/mm
2
(9) A flitched beam consists of a wooden joist 10 cm wide and 20
cm deep strengthened by two steel plates 10 mm thick and 20 cm
as shown in figure. If the maximum stress in the wooden joist is 7
N/mm
2
find the corresponding maximum stress attained in steel.
Find also the moment of resistance of the composite section. Take
Young’s Modulus for steel = 2 × 10
5
N/mm
2
and for wood = 1 × 10
4
N/mm
2
.