1.050 Engineering Mechanics I
Lecture 27 Introduction: Energy bounds in linear elasticity
1
1.050 – Content overview
I. Dimensional analysis
1.
2.
On monsters, mice and mushrooms
Similarity relations: Important engineering tools
II. Stresses and strength
3.
4.
Stresses and equilibrium
Strength models (how to design structures, foundations..
against mechanical failure)
Lectures 1-3
Sept.
Lectures 4-15
Sept./Oct.
III. Deformation and strain
5.
6.
How strain gages work?
How to measure deformation in a 3D structure/material?
Lectures 16-19
Oct.
IV. Elasticity
7.
8.
Elasticity model – link stresses and deformation
Variational methods in elasticity
Lectures 20-31
Oct./Nov.
V. How things fail – and how to avoid it
9.
10.
11.
Elastic instabilities
Plasticity (permanent deformation)
Fracture mechanics
Lectures 32-37
Dec.
2
1.050 – Content overview
I. Dimensional analysis
II. Stresses and strength
III. Deformation and strain
IV. Elasticity
Lecture 20:
Lecture 21:
Lecture 22:
Lecture 23:
Lecture 24:
Lecture 25:
Lecture 26:
Lecture 27:
Lecture 28:
cont’d
…
Introduction to elasticity (thermodynamics)
Generalization to 3D continuum elasticity
Special case: isotropic elasticity
Applications and examples
Beam elasticity
Applications and examples (beam elasticity)
… cont’d and closure
Introduction: Energy bounds in linear elasticity (1D system)
Introduction: Energy bounds in linear elasticity (1D system),
V. How things fail – and how to avoid it
Lectures 32..37
3
Convexity of a function
f (x)
∂f
| (b − a) ≤ f (b) − f (a)
∂x x=a
secant
f (x)
tangent
a
b
x
4
Example system: 1D truss structure
We will use this example to illustrate all key concepts
5
Total external work
v r d vd r
W =ξ ⋅F +ξ ⋅R
d
Work done by
prescribed
forces
Displacements
unknown
Work done by
prescribed
displacements,
force unknown
6
Total internal work
N i
State equations
∂ψ i
Ni =
∂δ i
Complementary free energy
ψ
*
i
∂ψ
δi =
∂N i
*
i
ψi
Free energy
∑δ N
i
i
δ i
i
=ψ (N i ) + ψ i (δ i )
*
i
7
Combining it…
!
r
r
v
v
W d = ξ ⋅ F d + ξ d ⋅ R =ψ + ψ *
(
)
vd r !
v r
d
− ψ − ξ ⋅ R =ψ − ξ ⋅ F
*
Complementary
energy
=: ε com
Solution to elasticity problem
Complementary
energy
=: ε pot
− ε com = ε pot
8
Quiz II – Monday Nov. 19 • Focus on material presented in lectures 16-26
• Preparation: Problem sets, old quizzes, lecture
material
• Deformation and strain, isotropic elasticity, beam
deformation (beam bending and beam
stretching), forensic beam elasticity, sketch
solution of beam problems, concept of
superposition (frame structures)
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