# Q1. _a_ Define plane stress and plane strain conditions_ comment

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```					Q1.

(a) Define plane stress and plane strain conditions, comment on the difference
between fracture toughness and plane strain fracture toughness.
(10 Marks)

(b) A surface crack of 4 mm depth and 10 mm surface length is found in a thick
rectangular component. The component is subjected to variable stresses from zero to
200 MPa. The stress intensity factor K is known to vary along the crack front and is
given as:
1/ 4
1.12σ πa ⎛ 2      a2         ⎞
K=          ⎜ sin θ + 2 cos 2 θ ⎟
⎜                   ⎟
Φ    ⎝        c          ⎠

where          3π π a 2 , a is the depth and 2c is the surface crack length.
Φ=     +
8 8 c2

(i)         Find the location and magnitude of the highest stress intensity factor along
the crack front.                                                 (10 marks)

(ii)        If K IC = 80 MPa√m, find the minimum crack length at fast fracture.
(5 marks)

(iii)       Assuming the crack shape (aspect ratio, a/c) remains constant during crack
growth, Paris law constants for the material are C = 5.6×10-12 and m = 3,
find a conservative estimate of the life of the component to failure.
(5 marks)

10 mm

4 mm                            C
A       θ
σ                                                                            σ

Figure Q1
Q2.

A cylindrical steel pressure vessel of 5 m diameter and 40 mm wall thickness is to
operate at a working pressure of 8 MPa. The design assumes that failure will take
place by fast fracture from a crack which has extended gradually along the length of
the vessel by fatigue. To prevent fast fracture, the total number of loading cycles
from zero to full load and back to zero again must not exceed 2800 cycles. The
fracture toughness for the steel is 200 MPa. The fatigue crack growth may be
represented by

= C(ΔK )
da         4

dN

where C = 2×10-14 (MPa)-4m-1, da/dN is the crack growth per load cycle.

Find
(i) the critical crack length at fracture af;
(10 marks)

(ii) the initial allowable crack length ai if the total allowable number of cycles
is 2800;
(10 marks)
(iii) the minimum pressure to which the vessel must be tested before use to
guarantee against failure in under 2800 load cycles.
(10 marks)
Q3.

A cantilever of 1.2 m long with a rectangular cross section of 40 mm wide and 80 mm
deep is fully clamped at the right support. The yield stress of material is 280 MN/m2.
If a concentrated load P is applied at the left free end, determine:

(1)   the value of P to first cause the onset of yield;                    (9 marks)
(2)   the value of P to cause plastic penetration to a depth of 10 mm;     (13 marks)
(3)   the length of cantilever plastically stressed in case (2).           (8 marks)

bh 2
Mo =      σo
6

⎛ h 2 e2 ⎞
M 2 = b⎜ − ⎟σo
⎜ 4
⎝     3⎟ ⎠

Q4.

A two-bay rectangular plane frame is shown in Figure Q4. Determine the lowest
upper bound of the limit load P from all kinematically admissible velocity fields of
the frame.                                                              (30 marks)

4P                           10P

4m             4m            4m           4m
4P
2Mp                           2Mp

6m
Mp                           Mp                     Mp

Figure Q4
Q5.

A circular tube beam having a length 2L, a radius R and thickness t is fully clamped at
both ends and any axial movement at the supports is restricted. When subjected to a
static force at the middle point, the beam is deformed as shown in Figure Q5, where
W is the maximum displacement, θ and λ are the rotating angle and elongation of the
beam at the supports and the middle point, respectively.

M          ⎛π N ⎞
1).   Prove the yield relation        = cos⎜⎜ 2 N ⎟ , where Mo and No are the fully
⎟
Mo         ⎝    o ⎠

plastic bending moment and membrane force of the cross-section, respectively;
(12 marks)
& W
λ
2).   using the geometric relations, prove =                  & dλ and θ = dθ ;
, where λ =         &
&
θ    2                dt        dt
(8 marks)
⎛π N ⎞
3).   Using the normality rule of plasticity, prove W = 2R sin⎜  ⎜2 N ⎟ ⎟
⎝    o ⎠

for the case of N < No and W < 2R.                                     (10 marks)

L                     L
θ                                            θ
λ                                                       λ
W

λ    λ

Figure Q5

```
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