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
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
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 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 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
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
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
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
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
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 18
9 Oct 2014
Practice Problem 18.1 Consider the axisymmetric problem of an annular disk with a fixed inner radius
and loaded wit
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
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 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
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 ,
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 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 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
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,
Heat Exchangers
What is a heat exchanger?
Heat Exchangers (HX) are devices where heat is
transferred between two fluids at different
temperatures without any mixing of fluids
Classification of HX
Hundreds of types of heat exchangers depending on designs,
Natural Convection:
Correlations and slides
General Considerations (cont)
Pertinent Dimensionless Parameters
Grashof Number:
g Ts T L3
GrL
2
Buoyancy Force
Viscous Force
L characteristic length of surface
coefficient of thermal expansion
1
T p
Perf
Internal Flow:
Forced Convection
Entrance Conditions
Entrance Conditions
Must distinguish between entrance and fully developed regions.
Hydrodynamic Effects: Assume laminar flow with uniform velocity profile at
inlet of a circular tube.
Velocity bounda
Transient Conduction:
Spatial Effects
Heat Transfer, Autumn 2016
IIT Kharagpur
Plane Wall
Solution to the Heat Equation for a Plane Wall with
Symmetrical Convection Conditions
If the lumped capacitance approximation can not be made, consideration must
be
Conduction: Theory of Extended
Surfaces
Why extended surface?
h, T
q
q hA(Ts T )
Increasing h
Increasing A
2
Fins as extended surfaces
A fin is a thin component or appendage attached to a
larger body or structure
Hot
surface
In the context of heat transfe
Condensation Heat
Transfer
Condensation on a Vertical Surface
Heat transfer to a surface occurs by condensation when the
surface temperature is less than the saturation temperature of an
adjoining vapor.
Film Condensation
o Entire surface is covered by t
CORRELATIONS FOR CONVECTIVE HEAT TRANSFER
I. CORRELATIONS FOR FORECD CONVECTION
1. Forced convection from flat plate
Flow regime
Laminar, local
Range of application
Tw const , Re x 5 105 ,
Correlation
1/3
Nu x 0.322 Re1/2
x Pr
0.6 Pr 50
Nu x
Tw const , R