Mechanics of Solids
ESO 202
Instructor: Dr. P. Chakraborty
Department of Aerospace Engineering
Office Location: 210/F NWTF Building
Office Phone: 0512-679-7951
email: cpritam@iitk.ac.in
4th January 2023
Outline
▪ Motivation & Objectives
▪ Topics
▪ Other Details
Motivation & Objective
Definition
▪ Deals with forces and motion of solid bodies
▪ In scope of this course:
Deformable solid bodies in equilibrium
▪ Equilibrium:
Body is in rest or constant velocity (zero
acceleration) under the action of forces
▪ Deformable:
Relative motion between points on the body
Deformable Body
2
2
1
1
Membrane before application
of force (pulling)
Membrane after application
of force (pulling)
A
B
A
B
Engineering Applications
▪ Deformable bodies in equilibrium
Cruising aircraft
Lift on wings
Wing bending & twisting
Engineering Applications
Steady rotation of shaft
Transmission of
torque
Ignoring
centrifugal force
Equal & opposite
torsion & twist
Engineering Applications
Transmission tower
Truss members
- light weight
- carry axial load
Pulled by power cables
Engineering Applications
Cables - axial load members
Bridge
Beams - bending members
Columns
Transverse loads
- Carry compressive load
- Design against buckling
Analysis of Engineering Applications
▪ Analysis required to design/modify
applications to meet requirements of load
without failure, deflection limits (avoid
interference), weight and cost savings
▪ Apply principles of mechanics to analyse
deforming structures in equilibrium under
the action of forces
Analysis Steps
▪ Select system of interest
▪ Idealization and simplification of the real system that
exhibit phenomena under consideration
▪ Apply principle of mechanics to the idealized model
Shear members
Actual Aircraft Wing
Axial members
Idealized cross-section for analysis
Course Objectives
▪ Simplified representation of structural members such
as truss, beams, pipes, shafts, columns, etc.
▪ Derivation of algebraic and differential equations
involving force, displacement and force-deformation
relations to represent the deformed state of structural
members in equilibrium
▪ Solving these equation to find deformation, forces,
etc. of the structural members
Topics
Topics
▪ Force; moments; idealization of joints/supports (pin,
fixed, etc.); friction; free body diagram; equilibrium
conditions
▪ Important relations to analyse deformable bodies:
Equilibrium conditions; deformation & compatibility;
force-deformation relations. Examples: Spring, bar,
truss, thin walled pressure vessels
▪ Definition of stress; equilibrium of a differential
element; relation between plane normal & stress
components; plane stress; Mohr’s circle
Topics
▪ Definition of strain; infinitesimal strain-displacement
relation; strain transformation; plane strain; Mohr’s
circle; strain rosette
▪ Tensile test; stress vs strain responses; uniaxial elastoplastic curves of different materials; linear elastic
constitutive relation in 3D; effect of thermal strain;
equations of elasticity; thick-walled pressurized
cylinder
▪ Torsion of circular shaft: Twist and strain; stress in
circular shaft with linear elastic constitutive model;
equilibrium condition; hollow thin-walled shafts
Topics
▪ Idealization of beams (slender member subjected to
transverse loads); shear force & bending moment
diagrams; shear force & bending moment differential
relation
▪ Stresses due to bending: Stress resultant force and
moment; curvature-strain relation; stress-curvature
for linear elastic beam; stress in symmetrical beams;
shear stress due to bending
▪ Deflection of beams: Moment curvature relation; load
deflection relation; superposition. (Must not use
singularity functions)
Topics
▪ Energy: Strain energy in elastic body; examples –
axial, torsion, bending; complementary energy;
Castigliano’s theorem.
▪ Stability: Stable & unstable equilibrium; elastic
instability of columns (elastic buckling).
Other Details
Course Material
• An Introduction to Mechanics of Solids (3rd Edition),
S. H. Crandall, N. C. Dahl, T. J. Lardner, M. S.
Sivakumar (text book).
• Engineering Mechanics of Solids, E. P. Popov
• Mechanics of Materials, Gere, Timoshenko
• Slides used during class will be shared on mookit
Break-up of Marks
5 Quiz (Announced)
Duration: 1 hour each
25%
Biometric attendance in class & tutorial
5%
Mid-of-semester exam
Duration: 2 hours
30%
End-of-semester exam
Duration: 3 hours
40%
Home Problems (submission not required)
0%
Other Policies
▪ Biometric attendance will be taken during classes and tutorials.
Attendance will be checked on a periodic basis. Any student
with < 50% attendance may be deregistered
▪
No make-up quizzes (announced & surprise) & mid-of
semester examination
▪
Relative grading
▪
Details of students found using unfair means during exams or
quizzes will be reported to competent authority and
appropriate action will be taken
Mechanics of Solids
ESO 202
Instructor: Dr. P. Chakraborty
Department of Aerospace Engineering
4th January 2024
Force
▪ Force is a directed interaction between bodies
▪ It is a simplified representation of complex physical
interactions that may exist between bodies
▪ Mathematically, force can be represented as a vector
▪ Classification:
▪ Contact forces – between bodies in physical contact
Force transmitted via
contact between hand
and shaft of cart
Pulling of a cart
Force transmitted via
contact between
roller and cage
Ball bearing
Force
▪ Classification:
▪ Non-contact forces – between physically separated
bodies
Gravitational force
Magnetic force
Force
▪ Newton’s laws of motion (relate force & motion)
▪ 1st Law: Effect of force on motion of a body
▪ 2nd Law: Mathematical relation between force and motion
▪ 3rd Law: Relation of forces between interacting bodies
▪ Unit of force obtainable from 2nd law
▪ SI unit – Newton (N)
▪ A newton is defined as the force which when acts on a body of
1 kg generates an acceleration of 1 m/s2
▪ 1 N = 1 kg-m/s2
Force
▪ The net effect of a system of forces acting at a point can be
expressed in terms of the resultant force
F1
F2
𝐹 = 𝐹1 + 𝐹2 + 𝐹3 + 𝐹4 + 𝐹5
F3
Y
X
Z
Resultant Force:
F5
5
𝐹 = ෍ 𝐹𝑖
F4
𝑖=1
F2
Can also be obtained from
vector diagram
F1
F3
F
F4
F5
Force
Components of a force in an
orthogonal axis-system
𝐹1 = 𝐹1𝑥 𝑖Ƹ + 𝐹1𝑦 𝑗Ƹ + 𝐹1𝑧 𝑘෠
Components of the resultant force
5
𝐹𝑥 = ෍ 𝐹𝑖𝑥
𝑖=1
5
𝐹𝑦 = ෍ 𝐹𝑖𝑦
𝑖=1
5
𝐹𝑧 = ෍ 𝐹𝑖𝑧
𝑖=1
Moment of a Force
▪ Force 𝐹 is applied at point P
▪ O is a fixed point in space
▪ 𝑟 is the position vector of P with respect to O
▪ Moment of 𝐹 about O is defined as
𝑀 =𝑟×𝐹
Moment of a Force
▪ Moment is a vector with a magnitude and direction
▪ Construct a plane consisting of vectors r and F
▪ Fcos is along OP and doesn’t contribute to
moment (definition of moment)
▪ Magnitude of moment
𝑀 = 𝐹𝑟𝑠𝑖𝑛𝜙
▪ Direction of moment determined by right hand screw rule
Moment of a Force
▪ Direction is in that of the thumb
of the right hand - when the
fingers curl in the direction that
the force F tends to turn about O
Direction of
moment
Moment of a Force
▪ Components of moment vector in an orthogonal
coordinate system
𝐹𝑥 𝑖Ƹ + 𝐹𝑦 𝑗Ƹ
𝑀 =𝑟×𝐹
= 𝑟𝑥 𝑖Ƹ + 𝑟𝑦 𝑗Ƹ × 𝐹𝑥 𝑖Ƹ + 𝐹𝑦 𝑗Ƹ
= 𝑟𝑥 𝐹𝑦 − 𝑟𝑦 𝐹𝑥 𝑘෠
𝑟𝑥 𝑖Ƹ + 𝑟𝑦 𝑗Ƹ
Moment of Forces
▪ Moment about point O due to multiple forces (fixed vector)
F1
F2
r1
r5
F5
r2
O
r4
F3
r3
F4
𝑀 = 𝑟1 × 𝐹1 + 𝑟2 × 𝐹2 + 𝑟3 × 𝐹3 +𝑟4 × 𝐹4 +𝑟5 × 𝐹5
5
= ෍ 𝑟𝑖 × 𝐹𝑖
𝑖=1
Couple
▪ A couple is a system of 2
parallel non-collinear
forces having equal
magnitudes and opposite
directions
M
𝑟1 × 𝐹1
r1
𝑟2 × 𝐹2
r2
F2
a
F1
𝑀 = 𝑟1 × 𝐹1 + 𝑟2 × 𝐹2 = 𝑟2 × 𝐹1 + 𝐹2 + 𝑎 × 𝐹1 = 𝑎 × 𝐹1
▪ The moment due to couple is independent of location
of point O