University of Mumbai 2007-1st Sem M.E Mechanical Engineering Theory of Elasticity & Materials Behaviour - - Question Paper

(4 Hours)    [Total Marks : 100

N.B. ; (1) Ques:ion No. 1 is compulsory.

(2)    Solve any three questions from the remaining questions.

(3)    Assume suitable data if necessary.    _s - UaAju'oU

N'tTT     rT/P tW/    * Vi W $'*'*-

1.    (a) A component subjected to cyclic axial loading -$5 kN to 60 kN is to be designed from 13

a steel having S_u = 800 N/mm² and has E = 20 x 10⁴ N/mm^?. The component is hollow ci'cular cylinder with inside diameter equal to half the outer diameter. The related material data is as follows

b = Fatigue strength exponent    =    -0 071

k, = Fatigue strength correction factor    = 1*326

e'₍ = Fatigue ductility coefficient    =    105

<r'_f = Fatigue strength coefficient    =    1180 N/mm²

Factor of safely =1-5

Design cross-section of the component for 8 x 10^s cycles.

(b) A machine component is repeatedly subjected to following axial load    12

+12 kN, -10 kN. +20 kN, -7-0 kN, +10 kN, -18 kN repeated cycle. Determine the number of cycles of the above load spectrum (you may apply any method for counting cycles) and give the range of each cycle.

2.    (a) Discuss the 'Griffith Theory which gives the energy needed to create the new surfaces 10

lor crack propagation. Using this theory, establish the critical stress intensity property of the material.

Kl_c

Where c_c is the stress applied (or farfield stress field) and ajn is the critical crack length.

(b) An Bdge crack, detected on large plate, is of length 2mm under a constant amplitude 15 cyclic load having a_max = 300 MPa and c_inin = 180 MPa. If the plate is made of steel with following properties

Kl_c x 160MPa Vm , values of c and m* for Paris law equation are 6-8 x 10~¹² and 3*0 respectively.

Determine the life of the plate if maximum crack propagation length is to be restricted tc 25 mm. The geometric constant for tho crack may be taken as Y = 1.

3.    (a) A solid circular tension member, 1-5 meter long and 25 mm diameter is take up a 13

tensile load of 45 kN, for 10 years at 400C. The maximum extension of the member permitted is 3 mm. The material properties of the material, from which the tension member is to be made, are given below -

(i)    Modulus of elasticity (E) * 140 GPa

(ii)    Coefficient of thermal expansion (a) = 4-7 x 10~⁶ per degree centigrade.

(iii)    Material constant for power law (B) = 1 2 x 10^-22 at 400C

(iv)    Stress exponent (N) = 6-9 at 400*C e_c = B t o^N

Where ct,    - failure stress in MPa

e_c    - creep strain

t    - time in days

B and N - Constants for the power law

(v)    Room temperature (T) = 20C.

Determine the extension of the member and state if the member can survive for the duration or not.

(i)    Estimate the maximum temperature to which the turbine blades can be exposed.

(ii)    What would be the reduction in design life, if the turbine ran 10C hortter. LMP (Larson Miller Parameter) is 27500 at 150 MPa and constant 'C = 20.

The state of stress a? a point P is given by -2-2 O'

-2 -J2 0 , 0 0 -V2 ,

Wit!i resoect to the first coordinate system. A second co-ordinate system is introduced whose x\ y and z* axes are given by -

' 0 '		|W		'-1&
		V2	1	M2
,Vv'2,		I-V2J		{-n

What are the components of a_p (o_p In new coordinate system) ?

Determine the principal stresses and their associated directions at a point P where 12 the state of stress is defined by the rectangular components as -

Oj( ⁼ 1, Cy 1, o ⁼ 11 ⁼ 2, 1 and 1

How are measurements performed with a resistance strain gauge ?    3

What is the main advantage of semi-conductor strain gauge ?    2

Eigen values of ay are calculated from the characteristic equation det (Ojj - 8ya) = 0. Show that this can be written in the form

C³ - l-CT² +

Where l_ff, II_a and III are the First, Second and Third stress invariants. What are their values ?

(b) The displacement componets are given by -

i_x = K (x² + 2x), u_y = K(4x + 2y² + z), u₂ = 4 KZ².

Where K is small quantity. Calculate the linear strain at the point (2, 2, 3) in the direction n_x 0, n_y 1/V2 , n₂ -= 1./V2

By considering a small cubic element of a material, show that -

= In

How many temperature compensation be performed on resistance strain gauges ? How many a brittle coating be used for strain measurement ?

6. (a) If the stress tensor is in old coordinate system x, y_t z and is stress tensor after 10 rotation in new x\ y\ z* coordinate system, then prove that -

^Ti'j' ~~ ^aie^ajm'^Tem

(b)    At a point in a body the components of strain are = 0 01, ty = - 0 055, e* * 0 003

Y_xy =0 02, 7 =0 005, =0 004, Find principal strains.

(c)    What is strain sensitivity ?

(d)    What is meant by a strain-gauge rosette ? How it is used ?

(e)    What is strain sensitivity ?

(f)    Define gauge factor for strain gauges.

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Attachment: university-of-mumbai-2007-m-e-mechanical-engineering-51627-1.pdf

Earning:   Approval pending.