This document contains 8 assignment sheets related to mechanical engineering concepts including: 1. Free body diagrams and reactions at supports 2. Internal reaction diagrams for beams 3. Axially loaded bars including stresses and deflections 4. Bending of bars including stresses, deflections, and internal reaction diagrams 5. Torsion of bars including shear stresses and angles of twist 6. Thin walled pressure containers including stress components and allowable pressures 7. Stress transformation including Mohr's circle and principal stresses 8. A problem involving stresses in a thin walled steel pressure container The assignments cover a range of load cases and ask students to calculate stresses, deflections, reactions and other mechanical properties.

1. Page 1 of 2
Cairo University MDP 124 – Mechanical Engineering
Faculty of Engineering 1st
Year Bio-Medical Engineering
Bio-Medical Engineering Dept. 2012-2013
Assignment Sheet # 1 – Free-Body Diagram
1. Draw the free-body diagram and find the reactions at the supports for the shown problems.
Consider two-force members conditions, joint equilibrium and use component free-body
diagrams to find the reactions as needed.
360 kN 360 kN 360 kN
4 m
3 m 3 m 3 m 3 m
(c)
(b)(a)
(d)
20 kN
P = 4 kNP P
16 kN
(e) (f)

2. Page 2 of 2
2. A child weighing 40 kg is seated on a chair. The
legs of the chair are 0.4 m apart, and the back of the
chair is 1.2 m high. Assuming that the frictional
forces at the front legs are large enough to prevent
slipping, what is the maximum horizontal force F
one could exert on the top of the back of the chair
without lifting the back legs?
3. Consider a person standing on a uniform,
horizontal beam that is resting on frictionless knife-
edge and roller supports. Let A and B be where the
supports contact the beam, C be the center of
gravity of the beam and D be the point on the beam
directly under the center of the gravity of the
person. Assume that the length of the beam is 5 m,
the distance between A and D is 3 m, the weight of
the beam is 900 N, and the mass of the person is 60
kg. Calculate the reactions on the beam at A and B.
4. A modified dolly is used to lift a 0.2 m diameter
log of mass 36 kg. Knowing that θ= 45º and that the
force exerted at C by the worker is perpendicular to
the handle of the dolly, determine the force exerted
by the arms of the worker.
5. The shown platform is used to measure the
weight and center of gravity position of a man’s
body. The weight of the man is determined while he
is standing at the middle of the platform. The C.G.
location is obtained through a dynamometer reading
while the man is laying horizontally with his head
resting on the shown head rest. If the two readings
of the dynamometer are wt1 and wt2 and the man’s
height is H, work out the expressions for the man’s
weight and the location of his body C.G. assuming
that all contacts are frictionless.
6. For a man having 172 cm height, the two
dynamometer readings are 420 N and 490 N,
determine the weight of the man and the location of
his C.G. referred to his feet bottom surface.

3. Page 1 of 1
Cairo University MDP 124 – Mechanical Engineering
Faculty of Engineering 1st
Year Bio-Medical Engineering
Bio-Medical Engineering Dept. 2012-2013
Assignment Sheet # 2 – Internal Reactions Diagrams
1. Draw the internal reaction diagrams for the beams shown below:
(a) (b)
(c) (d)
2. Draw the internal reaction diagrams for the beams shown below:
(P = 150 N, wn = 2.5N/mm, Mb = 1000 N.mm, L = 1000 mm, x = 0.25)

4. Page 1 of 2
Cairo University MDP 124 – Mechanical Engineering
Faculty of Engineering 1st
Year Bio-Medical Engineering
Bio-Medical Engineering Dept. 2012-2013
Assignment Sheet # 3 – Axially Loaded Bars
1. Rod ABC is made of aluminum for which
E = 70 GPa.
(a) If P = 6 kN and Q = 42 kN, determine
the deflection of points A and B.
(b) If P = 4 kN, determine the value of Q so
that the deflection at A is zero and determine
the corresponding deflection of point B.
2. A 250 mm bar of 15 x 30 mm
rectangular cross section consists of
two aluminum layers, 5 mm thick,
brazed to a center brass layer of the
same thickness. If it is subjected to
centric forces of magnitude P = 30
kN, and knowing that Ea = 70 GPa
and Eb = 105 GPa, determine the
normal stress in the aluminum and
brass layers.
3. A rod consisting of two cylindrical portions
AB and BC is restrained at both ends. Portion
AB is made of steel (Es = 200 GPa, s = 11.7
x 10-6
/o
C) and portion BC is made of brass
(Eb = 105 GPa, b = 20.9 x 10-6
/o
C).
Knowing that the rod is initially unstressed,
determine the compressive force induced in
ABC by a temperature rise of 50o
C and the
corresponding deflection of point B.
4. At room temperature (20o
C) a 0.5 mm gap
exists between the ends of the rods shown.
Determine the normal stress and the change in
length in the aluminum rod at a later time
when the temperature has reached 140o
C.

5. Page 2 of 2
5. The 4-mm-diameter cable BC is made of a steel
with E = 200 GPa. Knowing that the maximum
stress in the cable must not exceed 190 MPa and
that the elongation of the cable must not exceed 6
mm, find the maximum load P that can be applied
as shown.
6. The steel frame (E = 200 GPa) shown has a
diagonal brace BD with a cross-sectional area of
1920 mm2
. Determine the largest allowable load P
if the change in length of member BD is not to
exceed 1.6 mm.
7. Link BD is made of brass (E = 105 GPa) and has a
cross-sectional area of 240 mm2
. Link CE is made
of aluminum (E = 72 GPa) and has a cross-
sectional area of 300 mm2
. Knowing that they
support rigid member ABC, determine the
maximum force P that can be applied vertically at
point A if the deflection of A is not to exceed 0.35
mm.
8. Rod AB is made of steel for which the yield
strength is Y = 450 MPa and E = 200 GPa. Rod
BC is made of an aluminum alloy for which Y =
280 MPa and E = 73 GPa. Determine the
maximum strain energy that can be acquired by the
composite rod ABC without causing any
permanent deformations.

6. Cairo University MDP 124 – Mechanical Engineering
Faculty of Engineering 1st
Year Bio-Medical Engineering
Bio-Medical Engineering Dept. 2012-2013
Assignment Sheet # 4 - Bending of Bars
1. For the shown sections, determine the
location of the centriodal line and hence
give expressions to calculate Izz , Iyy for each
section. For each section two types of strips
are used in construction. Strip(1) is having
width (W) and thickness (t1) while strip(2) is
of a height (H) and thickness (t2). Give
numerical values for these sections if
W=100 mm, t1=10 mm, H= 200 mm, and
t2=5 mm.
Strip(2)
Strip(1)
y
z
2. A wooden shelf 1000 mm long is used to
support a set of books as shown. The shelf
width is 200 mm while its thickness is 15
mm. The weight of the books is assumed to
be distributed force with intensity of 3
N/mm. Obtain the location of the critical
section and hence determine the maximum
and minimum stress for the proposed two
arrangements. 600 mm
1000 mm
Arrangement (2)
200 mm
Arrangement (1)
Simple support
3. ABC is an elastic beam which is kept in
position through the shown circular tie bar
BD. If a load P of 20 kN is applied at point
A obtain the value and direction of reactions
at supports C and D. Also get the stress and
strains in the tie rod BD. Draw the internal
reaction diagrams for the beam ABC and the
max. and min. stresses at the critical
section. Draw a sketch for the stress
distribution at this section. Obtain the
displacement of point A in direction of the
applied force P.
(H=200 mm, W= 40 mm, d = 20 mm, E =
210 GPa,  = 0.30 )
1000mm
1000 mm
C
D
m
m
500
B
P
A
Sec. m-m

40
200

7. 4. The shown bar is subjected to a force F=
125 kN where the bar section is made of one
of the shown assemblies. Determine the
maximum and minimum stresses acting on
each of these sections. Locate the neutral
line for each section and draw the stress
distribution. For section assemblies, strip (1)
has a width of W=100 mm and a thickness
of t1=10 mm and for strip (2), the height is
H= 200 mm, and thickness is t2=5 mm.
Obtain the displacement of the applied force
if material elastic constant is E= 75 GPa.
FF
L
Strip(2)
Strip(1)
F
F
Force Action

8. Cairo University MDP 124 – Mechanical Engineering
Faculty of Engineering 1st
Year Bio-Medical Engineering
Bio-Medical Engineering Dept. 2012-2013
Assignment Sheet # 5 - Torsion of Bars
1- A 25 mm diameter shaft runs at 1200 rpm and transmits 30 kW. Calculate the
maximum shear stress and give the angle of twist in degrees if its length is 3 meters.
If a hole of 18 mm is dilled along the shaft length, what will be the maximum stress
and what will be the angle of twist for the hollow shaft if it is transmitting the same
power at the same speed. (G= 85.5 GPa.).
2- A hollow steel shaft is 30 mm outside diameter and 20 mm inside diameter. Calculate
the maximum transmitted power if the shaft is rotated at 1500 rpm and its material
can stand maximum shear stress of 70 MPa. If this shaft is replaced by another solid
one, determine its outer diameter if it is used to transmit the same power at the same
speed and is made of the same material. Compare between the resulting angles of
twist for the two shafts if each is of 750 mm length. (E= 210 GPa.,  = 0.30).
3- A solid shaft is used to transmit 7.5 kW at 2000 rpm. If the shaft is not to twist more
than 1o
on a length of 12 times its diameter and its maximum shear stress is not to
exceed 45 MPa., Determine the minimum shaft diameter. (G = 73 GPa).
4- Two steel shafts are assembled as shown. The hollow shaft is of 50 mm outer
diameter and 35 mm inner diameter. The solid shaft is 35 mm and is welded to the
inner surface of the hollow one. Obtain the maximum allowed power transmitted by
this assembly if the maximum shear stress does not exceed 60 MPa., and both are
rotated at 2500 rpm. What will be the total angle of twist for this compound shaft.
(G= 81 GPa., L= 300 mm.).
300 mm 300 mm
80 mm Solid Shaft

9. Cairo University MDP 124 – Mechanical Engineering
Faculty of Engineering 1st
Year Bio-Medical Engineering
Bio-Medical Engineering Dept. 2012-2013
Assignment Sheet # 6 - Thin Walled Pressure Containers
1-The tank of the air compressor is
subjected to an internal pressure of 0.63
MPa. If the internal diameter of the tank
is 550 mm, and the wall thickness is 6
mm, determine the stress components
acting at point A.
2-A water tank is filled to the top as
shown. The tank has a wall thickness of
20 mm at the lower section and a
diameter of 10 m. Determine the hoop
stress at the base of the tank. (The mass
density of the water = 1000 Kg/m³.)
3-A scuba diver's tank is made of
ASTM-A242 steel. The ends are
hemispherical with wall thickness of 3
mm. and diameter of 0.3 m. based on a
factor of safety against yielding of 4.0;
determine the largest internal pressure
that the tank can hold. ( 350 MPa)
4-The open-ended polyvinyl chloride
pipe has an inner diameter of 100 mm
and thickness of 5 mm. if it carries
flowing water at 0.42 MPa pressure,
determine the state of stress in the walls
of the pipe. If the flow of water within
the pipe is stopped due to the closing of
a valve, determine the state of stress in
the walls of the pipe. Neglect the weight
of the water. Assume the supports only
exert vertical forces on the pipe.

10. 5-A spherical gas tank has an inner
radius of r=1.5 m. if it’s subjected to an
internal pressure of p=300 KPa,
determine its required thickness if the
maximum normal stress is not to exceed
12 MPa.
6-The thin walled cylinder can be
supported in of two ways as shown.
Determine the state of stress in the wall
of the cylinder for both cases if the
piston P causes the internal pressure to
be 0.3 Mpa. The wall thickness is 6 mm.
7-Air pressure in the cylinder is
increased by exerting forces P=2 KN on
the two pistons, each having a radius of
45 mm. if the cylinder has a wall
thickness of 2 mm, determine the state
of stress in the wall of the cylinder.
8-The cap on the cylindrical tank is
bolted to the tank along the flanges. The
tank has an inner diameter of 1.5 m and
a wall thickness of 18 mm. if the largest
normal stress is not to exceed 150 MPa,
determine the maximum pressure the
tank can sustain. Also, compute the
number of bolts required to attach the
cap to the tank if each bolt has a
diameter of 20 mm. the allowable stress
for the bolts is 180 MPa.

11. Cairo University MDP 124 – Mechanical Engineering
Faculty of Engineering 1st
Year Bio-Medical Engineering
Bio-Medical Engineering Dept. 2012-2013
Sheet 7 - Stress Transformation
1. For the following loads acting on a bar having 50 mm diameter and 1000 mm length , sketch
the stress element and draw Mohr’s circle for stress and determine the values and directions of
principal stresses and maximum shear stresses:
a) Axial load of 235 kN.
b) Axial load of 235 kN with pressure on the bar surface of 600 bar.
c) Bending moment of 1.2 kN.m and a torque of 1.0 kN.m.
d) All the above given loads.
2. A shaft having diameter of d subjected to bending moment Mb and torsional moment of T,
prove that the maximum normal stress and the maximum shear stress are given by the following
expressions:















2
3max 11
16
b
b
M
T
d
M

 ,
2
3max 1
16







b
b
M
T
d
M


Check these relations using values and results of problem (1-c)
3. A closed thin walled pressure container having mean diameter D, thickness t and length L is
subjected to internal pressure p. Derive the expressions for the maximum normal stress and
maximum shear stress.
4. The shown thin walled steel container is used
to keep gas under pressure of 30 bar. The
container’s diameter is 200 mm and its totallength
is 700 mm. If the allowable normal stress for the
used steel weld is 100 MPa., determine the
required wall thickness and give the values of the
stresses developed in the weld lines. If an inclined
weld line, which makes an angle of 15o
with the
cylinder’s axis, .replaces the axial one, what will
be the values of normal and tangential stresses
acting on this inclined weld line
Weld Lines
Spherical Head
5. Strain gauges are applied to the surface of a
circular bar having diameter of 50 mm subjected
to an axial force of 235 kN and torsional moment
of 1.2 kN.m. If the setting of the of the strain
gauges are at 45o
to the bar axis as shown.
Determine the strain readings of both gauges. (E =
210 GPa.,  =0.33)
mm
Strain Gauges
T