structure problems

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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 kN P P

16 kN (e) (f)

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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.

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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)

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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

/oC) and portion BC is made of brass

(Eb = 105 GPa, b = 20.9 x 10-6

/oC).

Knowing that the rod is initially unstressed,

determine the compressive force induced in

ABC by a temperature rise of 50oC and the

corresponding deflection of point B.

4. At room temperature (20oC) 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 140oC.

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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.

Cairo University MDP 124 – Mechanical EngineeringFaculty of Engineering 1st Year Bio-Medical EngineeringBio-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 )

1000

mm

1000 mm

C

D

m

m

500

B

P

A

Sec. m-m

40

200

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




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




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.

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.

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 1116

b

b

MT

dM

,

2

3max 116

b

b

MT

dM

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 total length 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)

m

m Strain Gauges

T