1.050 – Content overview
I. Dimensional analysis
1.
2.
1.050 Engineering Mechanics I
On monsters, mice and mushrooms
Similarity relations: Important engineering tools
Lectures 1-3
Sept.
II. Stresses and strength
3.
4.
Lecture 34 Stresses and equilibrium
Strength models (how to design structures,
foundations.. against mechanical failure)
Lectures 4-15
Sept./Oct.
III. Deformation and strain
5.
6.
How things fail – and how to avoid it
How strain gages work?
How to measure deformation in a 3D
structure/material?
Lectures 16-19
Oct.
IV. Elasticity
Additional notes – energy approach
7.
8.
Elasticity model – link stresses and deformation
Variational methods in elasticity
Lectures 20-32
Oct./Nov.
V. How things fail – and how to avoid it
9.
10.
11.
1
Elastic instabilities
Fracture mechanics
Plasticity (permanent deformation)
Lectures 33-37
Dec.
2
Selection of boundary conditions: Euler
buckling
1.050 – Content overview
I. Dimensional analysis
Clamped cantilever beam
II. Stresses and strength
e=2
P
III. Deformation and strain
IV. Elasticity
V. How things fail – and how to avoid it
Lecture 33 (Mon): Buckling (loss of convexity)
Lecture 34 (Wed): Fracture mechanics I
Lecture 35 (Fri – teaching evaluation): Fracture mechanics II Lecture 36 (Mon): Plastic yield
Lecture 37 (Wed): Wrap-up plastic yield and closure
P < Pcrit =
el
π 2 EI
(el )
2
“effective length”
Single supported beam
e =1
Double clamped
cantilever beam
e=
3
1
2
4
1
Example: Concrete column w/
circular cross-section
Equation provides critical buckling load: Force must be
below this load to avoid failure
P < Pcrit =
EI =
Eπd
64
Recall – lecture 33
Analyzed buckling instability (displacement convergence at
point where load is applied) using two approaches:
P
π EI
2
(2l )
2
• (i) iterative solution using conventional small deformation
beam theory (divergence of series)
I=5m
4
Circular cross-section
d = 0.2 m
• (ii) application of large-deformation beam theory
(nonexistence of solution since determinant of
coefficient matrix is zero - bifurcation point)
E = 25 GPa
Pcrit = 194 kN
• (iii) instability is equivalent to loss of convexity (energy
approach)
5
Express potential energy for large
deformation beam theory
6
Express potential energy for large
deformation beam theory
P
Goal: Show that elastic instability corresponds to a loss of convexity
Reminder:
ε pot = ε pot (ϑ y , ω y )
∆
Slight imperfection
New term
large-deformation beam theory
Governing equations:
dM y
dx
dM y
dX
= Qz
Conventional, small deformation beam theory
Moment generated by rotation of beam
= Qz + ω y Q x
Large deformation beam theory
Moment generated by rotation of beam
Note: Potential energy in large-deformation beam theory
depends also on rotation (because rotations create moments)
Calculation of potential energy
7
ε pot (ϑ y , ω y ) = ∫
l
r
1
(
EIϑ y2 + Fxω y2 )dX − W (ξ 0 ,ω y )
2
8
2
Express potential energy for large
deformation beam theory
Express potential energy for large
deformation beam theory
P
ε pot (ϑ y , ω y ) ≤ ε pot (ϑ y ', ω y ' )
As studied previously:
Assume K.A. displacement field:
Potential
energy at
solution
Potential energy at any
other K.A. displacement
field
Lower bound approach
f ( X / l)
Solution corresponds to absolute min of
potential energy for K.A. displacement
field
l
Necessary condition for minimum of potential energy:
⎛ ⎛ d2 f
= δ ∫ ⎜ EI ⎜⎜ 2
⎜ ⎝ dx
∂δ
l ⎝
∂ε pot
Now express potential energy as function of parameter δ :
1
2
⎛
⎛ d2 f
2
⎝ dx
ε pot = δ 2 ∫ ⎜ EI ⎜⎜
l
⎜
⎝
⎞
⎛ df ⎞
⎟⎟ − P ⎜ ⎟
⎝ dx ⎠
⎠
2
10
Express potential energy for large
deformation beam theory
r
1
(
EIϑ y2 + Fxω y2 )dX − W (ξ 0 ,ω y )
2
2
must satisfy BCs…:
9
Express potential energy for large
deformation beam theory
ε pot (ϑ y , ω y ) = ∫
Parameter
ε pot
Also, the expression of
⎞
⎟dX − P ∆ δ
⎟
l
⎠
∂ 2ε pot
Convexity:
ε pot
11
∂δ 2
>0
δ
2
⎞
⎛ df ⎞
⎟⎟ − P ⎜ ⎟
⎝ dx ⎠
⎠
2
⎞
!
⎟dX − P ∆ = 0
⎟
l
⎠
must be convex!
Loss of convexity:
Elastic instability!
∂ 2ε pot
ε pot
∂δ 2
≤0
12
δ
3
Express potential energy for large
deformation beam theory
Loss of convexity:
∂ 2ε pot
∂δ 2
∂ 2ε pot
Express potential energy for large
deformation beam theory
Notes:
≤ 0
∂δ 2
⎛ ⎛ d2 f
= ∫ ⎜ EI ⎜⎜ 2
⎜ ⎝ dx
l ⎝
2
⎞
⎛ df ⎞
⎟⎟ − P ⎜ ⎟
⎝ dx ⎠
⎠
2
• Critical load obtained from potential energy approach
yields upper bound of actual critical buckling load
⎞
⎟dX
⎟
⎠
• Iterations with first order displacement field leads to
identical series expansion as in the iterative approach
with small deformation beam theory
Instability occurs if…
P≥
⎛ ⎛ d2 f
∫l ⎜⎜ EI ⎜⎜⎝ dx 2
⎝
2
⎞
⎟⎟
⎠
2
⎞
⎟dX
⎟
⎠
=P
⎛ df ⎞
∫l ⎜⎝ dx ⎟⎠ dX
crit
Approximate solution:
=
Pcrit
13
EI
= 2.5 2
l
Actual solution:
>
Pcrit =
π 2 EI
4 l
2
EI
2
14 l
= 2.4674
Summary
Analyzed buckling instability (displacement convergence at
point where load is applied) using two approaches:
• (i) iterative solution using conventional small deformation
beam theory (divergence of series)
• (ii) application of large-deformation beam theory
(nonexistence of solution since determinant of
coefficient matrix is zero - bifurcation point)
Fracture mechanics
How brittle materials fail
“crack extension”
• (iii) instability is equivalent to loss of convexity (energy
approach)
15
16
4
Brittle failure – crack extension
Brittle fracture
Brittle fracture = typically uncontrolled response of a structure, often leading to
sudden malfunction of entire system
Images removed due to copyright restrictions.
Images removed due to copyright restrictions: Photograph of fault line,
World Trade Center towers, shattered wine glass,
X-ray of broken bone.
Snapshots show microscopic processes as a crack extends.
17
During crack propagation, elastic energy stored in the material is dissipated
18
by breaking atomic bonds
M.J. Buehler, H. Tang, et al., Phys. Rev. Letters, 2007
5