Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Stresses in Bending
DR PRAVIN SINGRU
INTRODUCTION
IN ORDER TO MAINTAIN EQUILIBRIUM, A SHEAR FORCE
,V & BENDING MOMENT, M HAVE TO ACT ON THE
CROSS SECTION.
GEOMETRY OF DEFORMATION OF A
SYMMETRICAL BEAM SUBJECTED TO PURE
BENDING
ASSUMPTIONS
STRAIGHT UNIFO
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter 2
Introduction to Mechanics
of deformable bodies
Analysis of deformable bodies
1. Identification of a system
2. Simplification of this system
3. To develop model which can be analyzed
Steps to analysis System
1. Study of forces and equilibrium req
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter: 1
Fundamental Principles of
Mechanics
Mechanics of Solids
Vikas Cahudhari
1
Mechanics
Study of Force & Motion
Mechanics of Solids
Vikas Cahudhari
2
Mechanics
Study of Force & Motion
Gross/Overall Motion
(Dynamics)
KINETICS
Mechanics of Solids
KIN
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter 5
Suitability of a structure or machine may
depend on the deformations in the structure as
well as the stresses induced under loading.
Sometimes structures must be designed to
accommodate or produce certain deformations
Certain structural elements
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
CHAPTER 4
STRESS AND STRAIN
Objectives
STRESS
To know state of stress at a point
To solve for plane stress condition
applications
STRAIN
To know state of stress at a point
To solve for plane stress condition
applications
STRESS
Lets considered that th
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter 3
Forces and Moments
Transmitted by Slender
Members
Contents:
Slender members
Determination of Forces and moments under
point loads
Sign conventions for shear force and Bending
moment
Shear force and Bending moment Diagram
Distributed Loading
Diff
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
AMME2301/AMME5301
Combined Loadings
Lecturer: Dr Li Chang
Room s503, Building J07
Tel: 93515572
email: [email protected]
State of Stresses Caused by Combined Loadings
Loading Types
Stresses Caused by Combined Loadings Principle of Superposition
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
AMME2301/AMME5301
Torsion
Lecturer: Dr Li Chang
Room s503, Building J07
Tel: 93515572
email: [email protected]
Torsion of Circular Section: Maximum Shear Stress
Engineer's Theory of Torsion ( ETT )
where
T = the internal torque at the analyzed
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Week 6 Discussion Handout
Strain compatibility conditions for a simply connected body
At every point in the current conguration C, which is a simply connected domain,
curl curl = 0.
(1)
In some Cartesian basis, the above vector equation looks like the fol
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Week 4 Discussion Handout
Denitions
Let f , v and A be a scalar, a vector and a second order tensor elds, respectively, dened on an open domain
U of V and continuously dierentiable on U . Let c V be xed. Then we have the following denitions. Last
denition
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Let V be the three dimensional Euclidean vector space and L be the set of all linear maps
from V to V . The set of real numbers is denoted by R. Let E be the three dimensional Euclidean
point space. It can be identied with V .
Derivatives of elds
By elds
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Week 2 Discussion Handout
V is the three dimensional Euclidean space, which is a three dimensional vector space equipped with the
Euclidean inner product x y = xi yi , where x, y V and z = zi ei is the representation of any z V the
in
standard orthonormal
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Let V be the three dimensional Euclidean vector space and L be the set of all linear maps
from V to V . The set of real numbers is denoted by R. The identity element in L is denoted
by I.
Two theorems for symmetric tensors According to the spectral theore
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
AMME2301/AMME5301
Energy Methods Castiglianos
Second Theorem
Lecturer: Dr Li Chang
Room s503, Building J07
Tel: 93515572
email: [email protected]
Energy Methods Strain Energy
y
Internal Work Strain Energy
Total Strain Energy inside deformable b
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
AMME2301/AMME5301
Axial Load
Lecturer: Dr Li Chang
Room s503, Building J07
Tel: 93515572
email: [email protected]
Stress & Strain: Hookes Law
= E
Axial Load Stress: Stress Concentration & Saint Venants Principle
a
F
F
a
Stressconcentration fa
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Quiz 3: ME 321 (Total 10 Marks)
Name: Roll Number:
Problem 1 (2): Investigate what problem is solved by the stress function ¢ = ——§§wy2(3d — 23/) applied to a region
included in y = 0, y = d, m = 0, on the side X positive.
*QL “‘4
gLvMJ.
Problem 2 (2+3)
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Chapter 14
THERMOELASTICITY
Most materials tend to expand iftheir temperature rises and, to a first approximation,
the expansion is proportional to the temperature change. If the expansion is unre
strained, all dimensions will expand equally 7 i.e. there
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
/ Quiz 5: ME 321 (Total 10 Marks)
Name: Roll Number:
Problem 1 (3): Prove that 15% : a’  (13, i = 1,.,3, where ('1; = a x d.;, d; = 9 x di, and Q = midi. [Hintz
b><c><g=(bg)ci[b'c)gl '
Ln’ = to a
1.33114? “rent, = illXi,1—J}xgli
W L._—.—._.a
ext
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Quiz 4: ME 321 (Total 10 Marks) «C  ﬁat 7*{3 . mi. :5)
Name: a Roll Number: ' "H
T :. K "
Problem 1 (5):  I4 In.
‘H‘ “’1.
‘r 69
f3 = :
th ‘13
I“. ._ 1.4.x A.
d ‘i’z
7" = "f
Find expression for the shear stresses in a thin walled tube of the sec
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Quiz 1 Makeup: ME 321 (Total 10 Marks) C S OLUT (0N)
Name: Roll Number:
Problem 1 (2): Show that if A  B 2 0 is true for every tensor B then tensor A = 0.
Mam 2
E Can/n bu. My. (3. (dims.
*3
WG Cam m {giLMH ML W CLIVE“. 13.: gg' ®3=J m
“J z"! ng
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Quiz 1: ME 321 (Total 10 Marks) ( S 0 L U T' W")
Name: Roll Number:
Let u, v, and W be arbitrary vectors.
Problem 1 (2): Show that det(u (8 v) = 0.
AJ(961)= [wager (Q®!)é;_ @Cg‘ilgzl (41%” M'm'h'm)
= [Cy£09 (2949 (2.9.921
61:.) (y. at) 91.5.1) C E 9 a1
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Q. 1. An axisymmetric composite cylinder is composed of a solid inner shaft, of radius a shear
modulus 1 , and an outer sleeve of outer radius b and shear modulus 2 . The shaft and sleeve
are ideally bonded at their interface and the composite cylinder is
Birla Institute of Technology & Science, Pilani  Hyderabad
Mechanics of solids
AMME 2301

Spring 2013
Problem 1.
Figure 1:
A curved beam, as shown in Fig. 1, is xed at one end and subjected to a radial force (per unit
thickness of the beam) of magnitude P at the free end ( = 0). The inner and outer radii are a and
b respectively. The Airy stress function