Q.8.5 Two thick walled, closed ended cylinders of the same dimensions are subjected
to internal and external pressure, respectively. The outer diameter of each is twice the
inner diameter. What is the ratio of the pressures for the following case?(a) The
Q.3.8 The thin cantilever shown in figure is subjected to uniform shearing stress 0
along its upper surface (y = +h) while surfaces y = -h and x = L are free of stress.
Determine whether the Airy stress function.
=
x y 2 x. y3 L. y2 L. y3
1
.0 (x.y - 2+
+
Q.3.32
A thin walled circular cylinder vessel of diameter d and wall thickness t is subjected
to internal pressure p .Given a small circular hole in the vessel wall, show that the maximum
5. p.d
p.d
and a =
, respectively.
tangential and axial stresses at
Question1.3:
A square prismatic bar of 1300-sq.mm cross-sectional area is composed of
two pieces of wood glued together along the x plane, which makes an angle with the
axial direction. The normal and the shearing stresses acting simultaneously on the joi
Quiz #3
November 16, 2001
Name _
x
P
c
L
[40 points] 1. A simply supported beam of length L, second moment of area I, Youngs
modulus E and Poissons ratio ?, carries a load P at a distance c away from its left end.
Obtain the beam deflection at the point w
QUIZ#1
APPLIED ELASTICITY
SEPTEMBER 21, 2001
q
b
x
P
2h
y
1-10. (30 points) The above cantilever beam, fixed at x=L, is loaded by a uniformly
distributed load q (units of q are of force/length) over the length L and a load P
(units of force) as shown. Ans
SAMPLE QUIZZES - ELASTICITY
-1
Sample Quiz Questions
1. Derive the following strain-displacement relationship similar to what we did in class
xy = / x + u / y
2. A thin rectangular plate a=20 mm and b=12 mm is acted upon by a stress distribution
resultin
Principle of Stationery Potential Energy
Among all admissible displaced configurations of a conservative system those that satisfy
equations of equilibrium make the potential energy stationery with respect to small
admissible variations of displacement. I
Principle of Stationery Potential Energy
Among all admissible displaced configurations of a conservative system those that satisfy
equations of equilibrium make the potential energy stationery with respect to small
admissible variations of displacement. I
Principle of Stationery Potential Energy
Among all admissible displaced configurations of a conservative system those that satisfy
equations of equilibrium make the potential energy stationery with respect to small
admissible variations of displacement. I
Problem: Find the stress state in a plate with a circular hole of radius a.
The plate is subjected to a simple tensile stress of 0 .
y
x
0
Solution:
The boundary conditions for the problem are
On the edge of the hole at r = a,
rr (a , ) = 0,
0 2
r (a ,
Fall 2006
EML 6653
APPLIED ELASTICITY
Class Room:
Instructor:
Telephone:
Office Hours:
Web Site:
ENG 003
Time:
3-6 Fridays
Autar K. Kaw, Professor
Office:
ENC 2215
813-974-5626
E-mail:
[email protected]
11:00-11:50AM TTh or drop by or call 813-974-5626 or e
Cartesian Co-ordinate System Equations
Equilibrium Equations
x xy zx
+
+
+X =0
x
y
z
y xy yz
+
+
+Y =0
y
x
z
z zx yz
+
+
+Z=0
z
x
y
2
1
x
Equations of Compatibility
StrainDisplacement Equations
u
v
w
,y = ,z =
x
y
z
u v
u w
v w
=
+ , zx =
+
+
, yz =
y
COMPOUND PRESSURE VESSEL
A cylinder for a high-pressure compressor having an internal diameter of 25 mm is to be made from a steel
grade (E=207 GPa) for which the maximum allowable hoop stress (tensile or compressive) is 400 MPa.
There is no limit on the
Question.10.25 A simply supported beam carries a load P at a distance c away from its
left end. Obtain the beam deflection at the point where P is applied .Use
the Rayleigh-Ritz method. Assume a deflection curve of the form,
v = a.x(L-x),
where a is to be
Q.8.30 Consider a steel rotating disk of hyperbolic cross-section with a = 0.125m, b =
0.625 m, t i = 0.125m, t o = 0.0625m. Determine the maximum tangential force that can
occur at the outer surface in newtons per meter of circumference if the maximum st
Question.4.8: A simply supported nonmetallic beam of 0.25 m height, 0.1 m width, and
1.5 m span is subjected to a uniform loading of 6KN/m. Determine the factor of safety for
this loading according to (a)the maximum distortion energy theory, and (b)the ma
Question 2.39:
1
, r = 150 mm) is subjected to hydrostatic
3
pressure p so that its volume is reduced by 0.5 %. Determine(a) the pressure p, and(b) the
4
strain energy U in the sphere.( Note: Volume of a sphere V = r 3 )
3
Solution:
A solid bronze sphere
Q.2.24
The following state of strain has been measured at a point on the surface
of a crane hook.
The readings are a = 1000 , b = 250 , c = 200 , a = 150 , b = 300 , c = 750 .
Determine strain components x , y , and .
xy
Solution.
We first write the formu
Question 2.9: A thin rectangular plate a=20 mm and b=12 mm is acted upon by a stress distribution resulting in the uniform strains, x = 300 , y = 500 , xy = 200 . Determine the changes in length of diagonals QB and AC. Solution: Y A b B
Q a
C
Since the di
Question
At a point in a loaded member, the stresses relative to an x,y,z coordinate system are
60 20 10
given by 20 40 5 MPa. Calculate the magnitude and direction of maximum
10 5 30
principal stress.
Solution
x = 60MPa, y = -40 MPa, z =30 MPa; xy= 20
Question 1.8:For what body forces will the following stress field describe a state of
equilibrium?
x = - 2.x2 +3.y2 -5.z
y = - 2.y2
z = 3.x + y + 3.z-5
xy = z + 4.x.y -7
yz = 0
zx = - 3.x + y +1
Solution.
We apply differential equations of equilbrium,
x +
1
Example of Solving a Two Dimensional Elasticity
Problem Using Airy Stress Function
t
x
2h
y
Starting from
=
d1 x 4 + d 2 x 3 y + d 3 x 2 y 2 + d 4 xy 3 + d 5 y 4 + c1 x 3 + c 2 x 2 y + c 3 xy 2 + c 4 y 3
+ b1 x 2 + b2 xy + b3 y 2 + a1 x + a 2 y + a3
fin