FAKULTI KEJURUTERAAN MEKANIKALUNIVERSITI TEKNIKAL MALAYSIA MELAKA

TUTORIAL 2 (Solution)SOLID MECHANICS 2 (BMCS 2333)

SEMESTER 2 SESSI 2005/2006Mohr’s Circle, Strain Transformation and Thin & Thick Cylinder

1. σx = -15 MPa, σy= 35 MPa and τx y= 60. By using Mohr’s circle,a) find principal

stresses and their orientation planes, and

(75MPa, -55MPa, -33.7, 56.3)b) find maximum

shear stress and its orientation plane. (65MPa, 11.3)

Solution

,

X=(-15,-60) , Y=(35,60)

a) ,

,

, b)

,

2. A 45 strain rosette (see figure) mounted on the surface of an automobile frame that is being tested gives the following readings: gauge ,

; gauge , ; gauge , . Determine the principal

strains and maximum shear strains.

(33210-6, -18210-6, 51510-6)

Solution, ,

45

45

x

y

O

C

B

A

(MPa)

(MPa)

X

Y

avg 12

max

0

22 21

2s

3. A cylindrical tank containing compressed air has wall thickness and inside radius (see figure). The stresses in the wall of the tank acting on an inclined element have the values shown in the figure. What is the air pressure in the tank?

(6.6MPa)

Solution

,

,

…….(1)

………(2)

Solve equation (1) and (2),

124MPa28MPa82MPa

4. A cylinder of hydraulic jack has diameter 150mm operated with internal pressure 13.8Mpa. Determine thickness of the cylinder wall if maximum tensile stress that can be afford by the cylinder material is 41.4MPa.Also sketch the distribution of hoop stress and radial stress along the cylinder wall and determine maximum shear stress of the cylinder.

(21.97mm,27.6MPa)SolutionBoundary condition,At , At , At , where R is internal radius

Therefore, ……………...(1)

……………..(2)

……………….(3)

Solve the 3 equations,, ,

Wall thickness,

At , ,

At , ,

Maximum shear stress,

5. A thick cylinder has inside diameter of 200mm and outside diameter 300mm was pressurized internally 60Mpa and externally 30Mpa. Definea) hoop stress

and radial stress on internal and external surface of the cylinder, and

b) hoop stress and radial stress radius r=125mm, and

(28.56Mpa, -40.56Mpa)c) maximum and

minimum shear stresses.(54MPa,24MPa)

SolutionBoundary condition,At , At ,

Therefore, ……….(1)

……….(2)

Solve the 2 equations,,

a) Hoop stress,

Radial stress,

b) At

c) At ,

,

At ,

,

Therefore,

r (mm) 75 53

41.4

27.6

0

-13.8

(MPa)

h

r

6. A compound thick cylinder has a bore diameter of 100 mm, a common diameter of 200 mm and outside diameter of 300 mm. The outer tube is shrunk on to the inner tube, and the radial stress at common surface owing to shrinking is 10 MN/m2. If this compound cylinder is subjected to internal pressure of 150 MN/m2, determine:a) hoop and radial stresses, andb) sketch the stresses distribution.

SolutionStep 1: (Inner cylinder only, )

, , , Boundary condition, at ,

at ,

Therefore, ……………….(1)

…………….(2)

Solve the two equations,,

Hoop stress,

Radial stress,

Step 2: (outer cylinder only, ), , ,

Boundary condition, at , at ,

Therefore, ………….(1)

…………….(2)

Solve the two equations,,

Hoop stress,

Radial stress,

Step 3: (Both cylinders, ), , ,

Boundary condition, at , at ,

Therefore, ………….(1)

…………….(2)

Solve the two equations,,

Hoop stress,

Radial stress,

(MPa) (MPa)(m) 0.05 0.10 0.15 0.05 0.10 0.15

Step 1

0 -10 - -26.7 -16.7 -

Step 2

- -10 0 - 26 16

Step 3

-150 -2.6 0 1200 390 240

S1+S3

-150 -12.6 - 1173 373 -

S2+S3

- -12.6 0 - 416 256

S1+S3: For S2+S3: For

r (mm)

0.05

416

1173

373

0

-12.6

(MPa)

h

r

256

-150

0.10 0.15