INDIAN INSTITUTE OF TECHNOLOGY TIRUPATI
CE201L Strength of Materials
QUESTION SET 2
B Tech.
Semester III
1. A compression bar having a square cross section of width ‘b’ must support a load of 36 kN.
The bar is constructed from two pieces of material that are connected by a glued joint along
plane ‘PQ’, which is at an angle a 40° to the vertical. The material is a structural plastic for
which the allowable stresses in compression and shear are 8 MPa and 4 MPa, respectively.
Also, the allowable stresses in the glued joint are 5 MPa in compression and 3.5 MPa in shear.
Determine the minimum width ‘b’ of the bar.
2. During a tension test of a mild-steel specimen, the extensometer shows an elongation of 0.03
mm with a gauge length of 50 mm. Assume that the steel is stressed below the proportional
limit and that the modulus of elasticity, E = 200 GPa.
(a) What is the maximum normal stress σmax in the specimen?
(b) What is the maximum shear stress τmax ?
(c) Draw a stress element oriented at an angle of 35° to the axis of the bar and show all stresses
acting on the faces of this element.
3. A prismatic bar having cross-sectional area of 1200 mm2 is compressed by an axial load of 90
kN as shown in figure 1.
(a) Determine the stresses acting on an inclined section pq cut through the bar at an angle 25°.
(b) Determine the complete state of stress for the angle 25° and show the stresses on a properly
oriented stress element.
Figure 1
Figure 2
4. A plastic bar of diameter 32 mm is compressed in a testing device by a force P = 190 N applied
as shown in the figure 2.
(a) Determine the normal and shear stresses acting on all faces of stress elements oriented at
(i) an angle 0°, (ii) an angle 22.5° and (iii) an angle 45°. In each case, show the stresses on a
sketch of a properly oriented element. What are σmax and τmax ?
1
(b) Find σmax and τmax in the plastic bar if a re-centering spring of stiffness ‘k’ is inserted into
the testing device, as shown in the figure. The spring stiffness is 1/6 of the axial stiffness of
the plastic bar.
5. A steel bar with diameter of 12 mm is subjected to a tensile load of 9.5 kN.
(a) What is the maximum normal stress σmax in the bar?
(b) What is the maximum shear stress τmax ?
(c) Draw a stress element oriented at 45° to the axis of the bar and show all stresses acting on
the faces of this element.
6. A copper bar of rectangular cross section (breadth = 18 mm and height = 40 mm) is held snugly
between rigid supports as shown in figure 3. The allowable stresses on the inclined plane ‘pq’
at midspan, for which θ = 55°, are specified as 60 MPa in compression and 30 MPa in shear.
(a) What is the maximum permissible temperature rise, ∆T if the allowable stresses on plane
pq are not to be exceeded? (Assume α = 17x10−6 /°C and E = 120 GPa.)
(b) If the temperature increases by the maximum permissible amount, what are the stresses on
plane pq?
(c) If the temperature rises ∆T = 28°C, how far to the right of end A (distance 𝛽L, expressed
as a fraction of length L) can load P = 15 kN be applied without exceeding allowable stress
values in the bar? Assume that σa = 75 MPa and τa = 35 MPa.2
Figure 3
Figure 4
7. A prismatic bar is subjected to an axial force that produces a tensile stress σθ = 65 MPa and a
shear stress τθ = 23 MPa on a certain inclined plane as shown in figure 4. Determine the
stresses acting on all faces of a stress element oriented at θ = 30° and show the stresses on a
sketch of the element.
8. The normal stress on plane ‘pq’ of a prismatic bar in tension as shown in figure 5, is found to
be 52 MPa. On plane ‘rs’, which makes an angle 𝛽 = 30° with plane ‘pq’, the stress is found
to be 17 MPa. Determine the maximum normal stress and maximum shear stress in the bar.
9. A tension member is to be constructed of two pieces of plastic glued along plane ‘pq’ as shown
in figure 6. For purposes of cutting and gluing, the angle θ, must be between 25° and 45°. The
allowable stresses on the glued joint in tension and shear are 5 MPa and 3 MPa, respectively.
(a) Determine the angle θ so that the bar will carry the largest load P. (Assume that the strength
of the glued joint controls the design.)
(b) Determine the maximum allowable load if the cross-sectional area of the bar is 225 mm2.
2
Figure 5
Figure 6
10. A load P = 1060 kN is applied to the granite block shown in figure 7. Determine the resulting
maximum value of (a) the normal stress, (b) the shearing stress. Specify the orientation of that
plane on which each of these maximum values occurs.
11. A centric load P is applied to the granite block shown in figure 7. Knowing that the resulting
maximum value of the shearing stress in the block is 18 MPa, determine (a) the magnitude of
P, (b) the orientation of the surface on which the maximum shearing stress occurs, (c) the
normal stress exerted on that surface, (d) the maximum value of the normal stress in the block.
12. Two wooden members of uniform rectangular cross section are joined by the simple glued
scarf splice shown in figure 8.
(a) Knowing that P = 11 kN, determine the normal and shearing stresses in the glued splice.
(b) Knowing that the maximum allowable shearing stress in the glued splice is 620 kPa,
determine, (i) the largest load P that can be safely applied, (ii) the corresponding tensile stress
in the splice.
Figure 7
Figure 9
Figure 8
Figure 10a
3
Figure 10b
13. The steel block as shown figure 9, is subjected to a uniform pressure on all its faces. Knowing
that the change in length of edge AB is −1.2x10−3 inches, determine
(a) The change in length of the other two edges.
(b) The pressure p applied to the faces of the block.
(c) The change in volume ∆V of the steel block when it is subjected to the hydrostatic pressure
p = 180 MPa. Assume E = 200 GPa and Poisson’s ratio, 𝜐 = 0.29. (1 inch = 25.4 mm).
14. A 60 mm cube is made from layers of graphite epoxy with fibers aligned in the x direction.
The cube is subjected to a compressive load of 140 kN in the x direction. The properties of the
composite material are Ex = 155 GPa, Ey = 12.1 GPa, Ez = 12.1 GPa, 𝜐𝑥𝑦 = 0.248, 𝜐𝑥𝑧 = 0.248
and 𝜐𝑦𝑧 = 0.458. Determine the changes in the cube dimensions, knowing that (a) the cube is
free to expand in the y and z directions (Figure 10a), (b) the cube is free to expand in the z
direction, but is restrained from expanding in the y direction by two fixed frictionless plates
(Figure 10b).
15. A prismatic bar of 500 mm2 cross sectional area is axially loaded with a tensile force of 50 kN.
Determine all the stresses acting on an element which makes 30o inclination with the vertical
plane.
16. The principal tensile stresses at a point across two perpendicular planes are 100 MPa and 50
MPa. Find the normal and tangential stresses and the resultant stress and its obliquity on a
plane at 20o with the major principal plane.
17. Direct tensile stresses of 120 MPa and 70 MPa act on a body on mutually perpendicular planes.
What is the magnitude of shearing stress that can be applied so that the major principal stress
at the point does not exceed 135 MPa? Determine the value of minor principal stress and the
maximum shear stress.
18. The magnitude of normal stress on two mutually perpendicular planes, at a point in an elastic
body are 60 MPa (compressive) and 80 MPa (tensile) respectively. Find the magnitudes of
shearing stresses on these planes if the magnitude of one of the principal stresses is 100 MPa
(tensile). Also find the magnitude of the other principal stress at this point. Sketch the Mohr
circle and mark the relevant data.
19. Two planes AB and BC which are at right angles are acted upon by tensile stress of 140 N/mm2
and a compressive stress of 70 N/mm2 respectively and by stress 35 N/mm2. Determine the
principal stresses and principal planes. Also find the maximum shear stress and planes on
which they act. Sketch the Mohr circle and mark the relevant data.
20. At a certain point in a strained material, the intensities of tensile stresses on two planes at right
angles to each other are 20 MPa and 10 MPa. They are accompanied by a shear stress of
magnitude 10 MPa. Find the location of principal planes and evaluate principal stresses using
graphical method and conventional equations.
21. An elemental cube is subjected to tensile stresses of 30 MPa and 10 MPa acting on two
mutually perpendicular planes and a shear stress of 10 MPa on these planes. Draw the Mohr’s
circle of stresses and determine the magnitudes and directions of principal stresses and the
greatest shear stress.
4
22. The stresses at a point in a bar are 200 MPa (tensile) and 100 MPa (compressive). Using
graphical and numerical calculations, determine the resultant stress in magnitude and direction
on a plane inclined at 60o to the axis of the major axis. Also determine the maximum intensity
of shear stress in the material at the point.
23. At a point in a loaded member, a state of plane stress exists, and the strains are 𝜀 x = 270x10-6,
𝜀 y = -90x10-6 and 𝛾xy = 360x10-6. If the elastic constants E, G and 𝜈 are 200 GPa, 80 GPa and
0.25 respectively. Determine the normal stress 𝜎x and 𝜎y and the shear stress 𝜏xy at the point.
24. Draw the Mohr’s circle for the principal stresses of 54.6 MPa, 0 MPa and -84.6 MPa. Find out
the maximum shear stress.
25. Two planes AB and BC which are at right angles carry shear stresses of intensity 26 N/mm²
while the plane AB carries a tensile stress of 60 N/mm² (tensile) as shown in figure 11. Find
the normal and shear stresses on plane AC inclined at 25° to the plane AB.
Figure 11
Figure 12
26. An element of material in plane strain undergoes the following strains: 𝜀 x = 340x10-6, 𝜀 y =
110x10-6 and 𝛾xy = 180x10-6. These strains are shown highly exaggerated in figure 12, which
shows the deformations of an element of unit dimensions. Since the edges of the element have
unit lengths, the changes in linear dimensions have the same magnitudes as the normal strains
𝜀 x and 𝜀 y. The shear strain 𝛾xy is the decrease in angle at the lower-left corner of the element.
Determine the following quantities:
(a) The strains for an element oriented at an angle 𝜃 = 30°
(b) The principal strains
(c) The maximum shear strains (Consider only the in-plane strains and show all results on
sketches of properly oriented elements.)
27. A thin rectangular plate in biaxial stress is subjected to stresses 𝜎x and 𝜎y, as shown in part (a)
of the figure 13. The width and height of the plate are b = 160 mm, h = 60 mm, 𝜀 x = 410x10-6
and 𝜀 y = -320x10-6. With reference to part (b) of the figure 13, which shows a two-dimensional
view of the plate, determine the following quantities:
(a) The increase Δd in the length of diagonal Od
(b) The change Δ𝜙 in the angle 𝜙 between diagonal Od and the x axis
(c) The change Δ𝜓 in the angle 𝜓 between diagonal Od and the y axis
5
Figure 13
28. In a material in a state of plane strain, it is known that the horizontal side of a 10 x 10 mm
square elongates by 4 𝜇m, while its vertical side remains unchanged, and that the angle at the
lower left corner increases by 0.4 x 10-3 rad. Determine (a) the principal axes and principal
strains, (b) the maximum shearing strain and the corresponding normal strain.
29. A plane-stress condition exists at a point on the surface of a loaded structure, where the stresses
have the magnitudes and directions shown on the stress element of figure 14. Determine the
stresses acting on an element that is oriented at a clockwise angle of 15° with respect to the
original element.
Figure 14
Figure 15
30. An element in plane stress is subjected to stresses 𝜎x = 150 MPa, 𝜎y = -210 MPa, 𝜏xy = -16 MPa
as shown in figure 15. The material is brass with modulus of elasticity E 100 GPa and Poisson’s
ratio 𝜈 = 0.34. Determine the following quantities:
(a) The strains for an element oriented at an angle 𝜃 = 50°
(b) The principal strains
(c) The maximum shear strains. Show the results on sketches of properly oriented elements.
6