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°.

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/mm2 and maximum shearing

stress is 1N/mm2, 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 × 104

mm4 . Calculate the shear stress at the neutral

axis and at the junction of the web and the flange.

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 × 104. 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/mm2 and in brass = 75

N/mm2 . 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/mm2. What is the maximum stress in the steel in each

case? Take Es = 2 × 105 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/mm2 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 × 105 N/mm

2 and for wood = 1 × 10

4

N/mm2.