B. L. Sharma
Assignment 2
Due on 14 Aug 2014, 11am
Remarks: This assignment concerns differential and integral calculus as well as forces, moments, and Cauchy
stress. Total: 120 marks.
2 , e
3 be the standard basis for E. Consider a vector valued
Home wo
B. L. Sharma
Practice Problem 1.1. Let
1 , e
2 , e
3 ,
e
form a right handed orthonormal basis for E. Let x E be represented by its components x1 , x2 , x3 along the
three axes. Consider the vector valued function v : E E,
S
T
x2
X
[v(x)]e = [v(x1 , x2 ,
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 17
29 Sep 2014
Practice Problem 17.1 Given the stress function : (x1 , x2 ) = dF3 x1 x22 (3d 2x2 ). Determine the stress
componen
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 4
Practice Problem 4.1 The state of stress throughout a body B, with respect to the standard basis e for E,
is given by matrix
S
Assignment 10 Solution
Solution 10.1
(ii)
(i) = (,ii ) = (,ii ),j = (,j ),ii =
.
k = (lkm ijk uj,il ) e
m ,
( u) = (ijk uj,i ) e
m = (li mj lj mi ) uj,il e
m ,
= (klm ijk uj,il ) e
m um,ll e
m = ( u) u.
=ul,lm e
(iii) ( u) = (ijk uj,i )
ek = ijk uj,l lk
B. L. Sharma
ME321: Advanced Mechanics of Solids: 2013
Assignment 1
Due on 7 Aug 2014, 11am
Remarks: This assignment concerns linear spaces and linear transformations. Inner-product is not needed
to solve some problems and hence only linearity of space is
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 9
Practice Problem 9.1 A small strain deformation is specified by the displacement
u1 px1 , x2 , x3 q 4x1 x2 ` 3x3
u2 px1 , x2 ,
ME321: Advanced Mechanics of Solids
2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 3
Problem 1. There is a disk of mass m and radius R as shown in Fig. 1. Also provided are two
orthonormal bases e , f R3 as shown
Assignment 9 Solution
c < x2
. Assume body forces are zero and only
)=
0 r
< x1
. Assume body forces are zero and only
ace example, consider the case of a concentrated force system acting at the origin, as shown in
This example is commonly called the Flam
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 12
Practice Problem 12.1 A simply connected body B has total strain under some mechanical forces and
change in temperature T 30 C
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 18
9 Oct 2014
Practice Problem 18.1 Consider the axisymmetric problem of an annular disk with a fixed inner radius
and loaded wit
Assignment 6 Due on 11 Sep 2014, 11am
Remarks: This assignment concerns Hookes relation, Superposition, thermal effects, and definition of plane
stress/strain. Consider the standard basis e for E for all problems in this assignment. The superscript e has
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 15
22 Sep 2014
Practice Problem 15.1 Verify the following relations for the case of plane strain with constant body forces:
2
2
Assignment 3 Solution
FIGURE P3.9
Figure 3.2: (Left): Plane for Problem
1. (Right): Cylinder of radius r and length L for Problem 2.
Cylinder of radius r and length L.
Home work Solution 3.1.
The equation of the plane ABC is 3x1 + 6x2 + 2x3 = 12, and the
Assignment 4 Due on 28 Aug 2014, 11am
Remarks: This assignment concerns Deformation, Strain, Small strain. Consider the standard basis e for E
e
for all problems in this assignment. The superscript e has been ignored, for example, xi = xi , i = 1, 2, 3.
T
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 25
8 Nov 2014
Practice Problem 25.1 A force doublet is commonly defined by two equal but opposite forces acting in
an infinite me
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 26
10 Nov 2014
Practice Problem 26.1 Consider a cylindrical bar of unstreched length L with cross-section area A. It has
been str
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 10
Practice Problem 10.1 (a) Prove that if the given displacement u and satisfy the equation
ij 1cfw_2pui,j ` uj,i q, i, j 1, 2,
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 6
Practice Problem 6.1 Suppose e is the standard basis in E. The components of the stress tensor at P are
given in MPa with respe
Assignment 5 Due on 4 Sep 2014, 11am
Remarks: This assignment concerns Small strain, compatibility, Hookes relation. Consider the standard basis
e
e for E for all problems in this assignment. The superscript e has been ignored, for example, xi xi , i 1, 2
Assignment 2 Solution
Home work Solution 2.1.
(i) The resultant force due to surface force t on
t(x)ds =
is given by
t(x)ds +
S
D
t(x)ds.
e
e
i on . Since e is the only basis in context the
Using the basis e, t(x) = ti (x)
ei for given x = xi e
superscrip
Assignment 11 Solution
Figure 11.1: Problem 1. (Left) Cylindrical coordinate system using (r, , z). (Right) Spherical coordinate system using (r, , ).
Solution 11.1 Consider an infinite body with a spherical cavity of radius a in it. Suppose there is a un
Assignment 4 Solution
Home work Solution 3.1.
2 ` 11
(a) The displacement vector at (3, 1, -2) is 10
e1 ` e
e3 .
2 2
The initial position vector of point is 3
e1 ` e
e3 .
Adding both, the position vector of the point after displacement is
13
e1 ` 2
e2 ` 9
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 8
Practice Problem 8.1 In certain deformation with the small strain, the strain is given as
S
T
1
3 2
W
1 2 X
[] = U 3
V.
2 2 6
C
Assignment 9: Posted on 16 Oct 2014: Due on 30 Oct 2014, 11am
Total: 120 marks.
Home work 9.1 (15 marks). Show that the stress function
(r, ) =
0 r 2
(sin2 log r + sin cos sin2 )
gives the solution to the problem of an elastic half space loaded by a unifo
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 24
3 Nov 2014
Practice Problem 24.1 For the case of zero body forces, the Galerkin vector is biharmonic. However, it
was pointed
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 5
Practice Problem 5.1 Suppose e is the standard basis for E. Ignore the superscript e as there is only one
basis involved. Suppo
Assignment 3 Due on 21 Aug 2014, 11am
Remarks: This assignment concerns Cauchy stress and local equilibrium. Assume that e is the standard basis
e
for E for all problems in this assignment. The superscript e has been ignored, for example, xi = xi , i = 1,
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 16
25 Sep 2014
e3
!
R
e2
e1
Figure 16.1: Problem 1.
Practice Problem 16.1 Consider the disk is rotating about its axial perpendic
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 7
Practice Problem 7.1 Consider a body and suppose that an arbitrary point is described by coordinates
(x1 , x2 , x3 ) in fixed b
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 11
Practice Problem 11.1 (a) Derive Hookes relation from a quadratic strain energy function W .
(b) Write the stiffness matrix C
Assignment 8 Solution
Solution 8.1 The displacement components in the Cartesian Coordinates and Polar Coordinates are related
as
u1 (x1 , x2 ) = ur (r, ) cos u (r, ) sin ,
u2 (x1 , x2 ) = ur (r, ) sin + u (r, ) cos .
(8.1)
(8.2)
The strain components in C
B. L. Sharma
ME321: Advanced Mechanics of Solids: 2013
Home work Solution 1.1.
Checking Linear Independence of Vectors: Let C1 , C2 , C3 be real numbers. Since b is a basis for
R3 , its elements are linearly independent vectors. So we need trivial solutio
ME321: Advanced Mechanics of Solids: 2014-15 1st Semester
Practice Problems prepared by Teaching Assistants, based on lecture 13
Practice Problem 13.1 Consider a hanging bar fixed at upper end as shown in Fig. 13.1 and suppose that
an arbitrary point is d