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EXAM 1 MEEM 4150 Oct. 4th, 2011
1. (a) Show the non-zero stress components in the r, , and
spherical coordinate system on the A,B, and C faces of
the stress element shown.
(b) Show the direction of shear stress on thefour
relevantsurfaces at points A and B on the given stress cubes.
Circle the
correct component and sign..
(c) Assuming a positive shear force Vy in bending (a) sketch the
direction of the shear flow along the center-line on the thin
cross-sections shown. (b) At points A, B, C, and D, determine if
the stress component is xy orxz and if it is positive, neg
ative or zero. Circle the appropriate answers
(d, e) The stresses at a point were found to be rr= 100 MPa (T),
= 200 MPa (T) and r= 140 MPa . The
material has a modulus of elasticity of 80 GPa and Poissons
ratio of0.25. Determine rr
(d) assumingPlaneStress
(e) assumingPlaneStrain(f) The displacements u and v in the x
and y directions respectively were measured by Moire'
interferometry. Displacements
of 9 points on the body and are as given below. Determine the
shear strain xy at point 8.
A
C
B
r
x
y
z
rr 0= r 18 ks i= r 0ksi=
r 18 ks i= 10ks i C( )= 25ksi=
r 0= 25ksi= 20ks i T( )=
x
y
A
BTxz
A
B C
D
y
z
Pointu
(mm)v
(mm)Point
u(mm)
v(mm)
1 0.000 0.000 6 -0.080 0.240
2 -0.112 0.144 7 0.128 0.384
3 -0.128 0.256 8 -0.048 0.336
4 0.112 0.176 9 0.128 0.384
5 -0.032 0.224
x
y
1 2 3
h = 0.0005 mm
h h
h
h4 5 6
7 8 9
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2. The principal stresses for the given stress matrix are ; ;
and
. (a) Determine the three stress invariants (I1, I2, I3). (b)
Determine the principal angles associ-
ated with the third principal stress 25.58 ksi (C) (c) Determine
the stress xx. (d) The material yield stress is 60 ksi
in tension. Determine the factor of safety (K) at the point
using octahedral shear stress theory. (e) The critical stress
intensity factor for the material is , what is the critical
crack length for the material.
3. A positive shear force of Vy=10 kN and a bending moment ofMz
= 5 kN-m acts on the thin cross-sections shown
(not drawn to scale). The cross-section has a uniform thickness
of 10 mm and an area moment of inertia about the
z-axis (a) Determine the shear flow along center line for
segment BF and DF in terms of s1
and s2, respectively. (b) The principal stresses at point F on
the cross-section. Point F is just below the flange.
Report your answers with proper units.
Answers
1 (b) (xy)A = positive; (xz)B = positive; (c) (xy)A positive;
(xz)B zero; (xz)C positive; (xy)D positive; (d)
= 218.7 (e) = 1953.1 ; (f) xy =- 32 rads
2. ; ; ; ; ; ; ; ;
3. ; ;
1 28.10 ksi T( )= 2 7.48 ksi T( )=
3 25.58 ksi C( )=
22ksi in
xx 10 5
10 15 0
5 0 25
ks i
Stress Matrix
Izz 3.7518 106
( )mm4
=
y
z 100 mm
100 mm
25 mm
s1
s2
C
25 mm
D
B
FA
E
70.66 mm
I1 10 ksi= I2 700 ksi2
= I3 5377 ksi3
= xx 20 ksi (T)= x 83.38o
= y 88.37o
= z 173.2o
= K 1.27=
2a crit 0.39 in.=
q1 13.3s12
1883.4s1[ ] N/m= q2 782s2 N/m= 1 1.03 MPa (T)= 2 33.5 MPa (C)= 3
0=
