ME 223: Solid Mechanics and Strength of Materials
Autumn 2023, Indian Institute of Technology, Bombay
Prof. R. Ganesh
Tutorial/Homework 1
16 August 2023
Problem 1
Two equal cylinders, each weighing 900 N are placed in a box as shown in fig. 1. Neglecting friction between the
cylinders and the box, estimate the reactions at A, B, C.
Figure 1
Problem 2
A rigid rod with negligible weight and small transverse dimensions carries a load W whose position is adjustable.
The rod rests on a small roller at A and bears against the vertical wall at B, as shown in fig. 2. Determine the
distance x for any given value of θ such that the rod will be in equilibrium. Assume that friction is negligible.
Figure 2
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Problem 3
A wooden diving board is hinged at one end and supported 1.5 m from this end by a spring with a constant of 35
kN/m (as shown in fig. 3). How much will the spring deflect if a person weighing 600 N stands at the end of the
board? Does the assumed rigidity (or lack thereof) of the wooden board play a role in the deflection of the spring?
Figure 3
Problem 4
The stiff member AB is horizontal before the load of 20 kN is applied at A. The steel bars ED, BD and BC are
fastened with pins at their ends. Find:
(a) The force in the bar BD.
(b) The horizontal and vertical movement of the point A.
Note: Ignore the values given in fig. 4, and use the following values for your calculation: LAE = 4 m, LEB = 2
m, LBC = 2 m, AED = 1.5 × 10−3 m2 , ABC = 7 × 10−4 m2 , AED = 1.5 × 10−3 m2 . Young’s Modulus of Steel,
E = 2 × 1011 Pa.
Figure 4
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Problem 5
A very stiff horizontal member is supported by two vertical steel rods of different cross-sectional area and length
(as shown in fig. 5). If a vertical load of 120 kN is applied to the horizontal beam at point B, estimate the vertical
deflection of the point B.
Figure 5
Problem 6
A Rigid bar ABC, shown in fig. 6 supported with a pin at A and an elastic steel rod (of diameter 25 mm) at C, is
subjected to a uniform load q on span AC and a point load at B. Calculate the change in length of the elastic rod
assuming E = 200 GPa.
Figure 6
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Problem 7
A bar consisting of three prismatic segments A1 , A2 and A3 is subjected to point loads along its length at B, C, E,
as shown in fig. 7. Calculate the displacements of points B, D, and E, using the following numerical data: A1 =
6000 mm2 , A2 = 5000 mm2 , and A3 = 4000 mm2 . The bar is made of steel with E = 200 GPa. Ignore the weight
of the bar. Draw the Axial Force diagram for this composite bar.
Figure 7
Problem 8
A rectangular bar of length L has a slot in the middle half of its length (see fig. 8). The bar has width b, thickness
t, and modulus of elasticity E. The slot has width b/4.
(a) Obtain a formula for the elongation δ of the bar due to the axial loads P .
(b) Calculate the elongation of the bar if the axial stress in the middle region is 160 MPa. Assume L = 1500 mm,
and E = 210 GPa.
(c) If the total elongation of the bar is limited to δmax = 0.475 mm, what is the maximum length of the slotted
region? Assume that the axial stress in the middle region remains at 160 MPa.
Figure 8
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Problem 9
A wood pile, driven into the earth, supports a load P entirely by friction along its sides (see fig. 9(a)). The friction
force f per unit length of the pile is assumed to be uniformly distributed over the surface of the pile. The pile has
a length L, cross-sectional area A and modulus of elasticity E.
(a) Derive a formula for the shortening δ of the pile in terms of P, L, E and A.
(b) Draw a diagram showing how the compressive stress σc varies throughout the length of the pile.
(c) Repeat parts (a) and (b) if skin friction f varies linearly with depth (see fig. 9(b)).
(d) Draw the Axial Force Diagram and Axial Displacement Diagram for both cases.
Figure 9
Problem 10
The non-prismatic cantilever circular bar shown in fig. 10 has an internal cylindrical hole of diameter d/2 from 0
to x, so that the net area of the cross section for the segment 1 is (3/4)A. Load P is applied at x, and load P/2 is
applied at x = L. Assume that E is constant.
Figure 10
(a) Find reaction force R1 .
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(b) Find internal axial forces Ni in segments 1 and 2.
(c) Find x required to obtain axial displacement at joint 3 of δ3 = P L/AE.
(d) In part (c), what is the displacement at joint 2, δ2 ?
(e) If P acts at x = 2L/3 and P/2 at joint 3 is replaced by βP , find β so that δ3 = P L/EA.
(f) Draw the axial force diagrams (AFD: N (x), 0 ≤ x ≤ L) and axial displacement diagrams (ADD: δ(x), 0 ≤
x ≤ L) using results from parts (b) through (d).
Problem 11
A post AB supporting equipment in a laboratory is tapered uniformly throughout its height H (see fig. 11). The
cross sections of the post are square, with dimensions b × b at the top and 1.5b × 1.5b at the base. Derive a formula
for the shortening δ of the post due to the compressive load P acting at the top. (Assume that the angle of taper
is small and disregard the weight of the post itself.)
Figure 11
Problem 12
Consider the pin-connected framework loaded as shown in fig. 12. Find the axial force in each bar. The two outer
bars are identical with cross-sectional areas A0 ; the inner bar has a cross-sectional area A. All bars have the same
modulus of elasticity E.
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Figure 12
Credit Author Statement
This document was prepared predominantly by Siddhant Naik, along with the help of fellow TAs (listed in Alphabetical Order): Arunav Choudhury, Ayush Saxena, Mritunjay Hiremath, and Sorab Khan. If there are any
typos/questions, please follow up with the Instructor (R. Ganesh) regarding the same. The figures included in this
document have been adapted from the citations listed below, and are meant ONLY for reference.
References
[1] J.M. Gere and B.J. Goodno. Mechanics of Materials, Enhanced Edition. Cengage Learning, 2021.
[2] S.H. Crandall, N.C. Dahl, M.S. Sivakumar, and T.J. Lardner. An Introduction to the Mechanics of Solids: (in
SI Units). McGraw-Hill, 2012.
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