1.050 Engineering Mechanics
II. Stresses and Strength
Application in Structural Mechanics
Program 8th Lecture
1-050 CONTENT
I.
II.
Dimensional Analysis:
Stresses & Strength
2.
Stresses and Equilibrium
1.
2.
3.
3.
III.
Strength Models
Deformation and Strain
4.
IV.
How Strain Gages work?
Elasticity
5.
6.
V.
Discrete Model
Continuum Model
Beam Model
TODAY:
1. Scales of Structural mechanics:
Section vs. Beam structure
2. Link between stresses and
forces and moments
3. Beam Equilibrium Conditions
4. Example
Elastic Model
Variational Methods in
Elasticity
How Things Fail? And How
to avoid it.
Goal: Construct a Force-Moment Beam Model
Appreciate the link between Continuum
Model and Beam Model
Three Scale Approach
• Beam Scale defined
by beam length
• Cross-section scale
(height, width)
(h,b ) << l
• Continuum scale
O (dΩ1/ 3 ) << (h,b ) << l
From the Continuum Scale to the Cross Section Scale
• Continuum Quantity:
– Stress vector
z, ez
T (n = e x ) = σ ⋅ e x
y, e y
• Section Quantities:
|
x, ex
h
|
dΩ
n dS
– Forces
F S = ∫ σ ⋅ e x dS
S
b
– Moments
M S = ∫ x × (σ ⋅ e x )dS
S
From the Cross Section Scale to the Beam Length Scale
• Differential Force
equilibrium
ez ⎛
dM z ⎞
dx ⎟ ez
⎜Mz +
dx
⎝
⎠
ey
dM y ⎞
⎛⎛
⎟e y
⎜⎜ M y + dx dx ⎟⎟e
⎠
⎝
⎝
− M x ex
f ext
− Fx ex
− exx
− dx −
− Fy e yy
− Fz ez
− M yey
− M z e
ez
− eey
− ez
ext
dF S
+ f =0
dx
ex
dM x ⎞
⎛
dx ⎟ex
⎜Mx +
dx
⎝
⎝
⎠
• Differential Moment
equilibrium
dM S
+e x × F S = 0
dx
Formulation of a Beam Boundary Value Problem
• Example
z
y
f
ext
= − ρgS e z
R
• Force and Moment
Boundary Conditions
• Sum of all forces and
Moments along x is
zero
x
• Differential
Equilibrium of
– Section forces
– Section moments