# LECTURE #14 : 3.11 MECHANICS OF MATERIALS F03

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18 Ιουλ 2012 (πριν από 5 χρόνια και 10 μήνες)

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LECTURE #14 :
3.11 MECHANICS OF
MATERIALS F03
INSTRUCTOR : Professor Christine Ortiz
OFFICE : 13-4022 PHONE : 452-3084
WWW : http://web.mit.edu/cortiz/www
• Review : Beam Bending 3 : Normal Stresses
and Strains
• Transformations of Stress and Strain IBeam Theory 3 : Normal Stresses and Strains

y
z
ρ
Mo
N.A.

x
compression
Mo
p
m
e
y
f
dx
tension
n q
My
o
Flexure formula : σ =−
y
x
I
σ (max)c
σ =0
x
y
where :σ = normal stress in x-direction
x
M = internal bending moment
σ (y)
o
x
x
Mo
NA
y = vertical distance from NA axis
(see Gere Chapter 12, Appendix D, p 321)
I = moment of inertia of cross-sectional area
σ (max)T
x
Mx () y
[]
o max
max
(+ moment)
σ =−
()
x
max
I
h
for rectangular beams : y =
max
2
Mx () y
[]
My
o max
o max
εε =− , =−
()
xx
max
EI EI
EI = "flexural modulus"Beam Theory (Cont'd) : Shear Stresses and Strains
y
x
τ τ τ τ (y)
xy
NA
z
Derivation in Gere Section 5.8 :
2
Vh
2
τ y
(rectangular cross section)=− −
xy 
24 I

where : τ = shear stress
xy
V = shear force
h = height of cross sectional area
y= distance from NA
3V
τ (rectangular cross section) =−
xy max
2A
A = cross sectional area2-D (Plane) Stress State
y
στxxy 0

σ σ
σ σ
y

τσ yx y 0


000
τ τ
τ τ

xy
σ σ
σ σ
x
O
x

z
y
σ σ
σ σ
y
τ τ
τ τ
yx
τ τ τ τ
xy
σ σ
σ σ
O
x
x
•Given A State of Plane Stress, What Is The
Equivalent Stress State On An Element Rotated
By An Arbitrary Angle, θ θ + CCW
θ θ
y
O
x

y’
y
x’
x

Oy
y’
x’
x

Oy
y
y’
τ A /cosθ
x’y’ o
θ x’
σ A /cosθ
x’ o
θ
σ A
x o

x
O
τ A
xy o
τ A tanθ
yx o
σ A tanθ
y oy y
(1) t σσ x '=+τanθ+τtanθ
x yx x'' y
y’
ττ
xy x'' y
τ A /cosθ
(2) σσ x '=+ −
x’y’ o
y
θ x’
tanθθ tan
σ A /cosθ
x’ o
θ
σ A
x o

x
O
τ A
xy o
τ A tanθ
yx o
σ A tanθ
y oVariation of Stresses With θ θ θ θ
σσxy+−() σxσycos(2θ)
στ x'=+ + xysin(2θ )
22
σσxy+−() σxσycos(2θ)
στ y'=− − xy(sin(2θ )
22
() σσxy − sin(2θ)
ττ x'y' =− + xy cos(2θ )
2
STRESS TRANSFORMATION EQUATIONSVariation of Stresses With θ θ θ θ
For σ σ =0.2σ σ , τ τ =0.8σ σ
σ σ σ σ τ τ σ σ
Y x xy x
σσxy+−() σσx ycos(2θ)
στ x'=+ + xy sin(2θ )
22
σσxy+−() σxyσcos(2θ)
στ y'=− − xy(sin(2θ )
22
() σσxy − sin(2θ)
ττ x'y'=− + xy cos(2θ )
2Principal Stresses and Angles
σσxy+−() σσx ycos(2θ)
στ x'=+ + xy sin(2θ )
22
σσxy+−() σxyσcos(2θ)
στ y'=− − xy(sin(2θ )
22
() σσxy − sin(2θ)
ττ x'y'=− + xy cos(2θ )
2Principal Stresses and Angles
TAN(2θ )=2τ /(σ -σ )
p xy x y
10
0
-10

P
0 100 200 300
TAN(2θ )
P