MIT - 16.20
Fall, 2002
16.20 HANDOUT #4
Fall, 2002
General (Shell) Beam Theory
REVIEW OF SIMPLE BEAM THEORY
[Force/Area]
width = b
cut beam
Axial Force:
Shear Force:
Bending Moment:
ε xx
ε xx
F =
∫
σ xx b dz
h
2
h
−
2
S = − ∫ σ xz b dz
h
2
h
−
2
M = − ∫ zσ xx b dz
d 2w
∂u
= −z 2 =
dx
∂x
σ
d 2w
= xx
M = EI 2
E
dx
dS
= p
dx
dM
= S
dx
Paul A. Lagace © 2002
h
2
h
−
2
σ xx
Mz
= −
I
σ xz
SQ
= −
Ib
I =
Q =
h
2
h
−
2
∫
∫
h
2
z
z 2 b dz
z b dz
Handout 4-1
MIT - 16.20
Fall, 2002
BEHAVIOR OF GENERAL (INCLUDING UNSYMMETRIC CROSSSECTION) BEAMS
•
Resultants:
F =
∫∫ σ
xx
dA
My = − ∫∫ σ xx z dA
Sy = − ∫∫ σ xy dA
Mz = − ∫∫ σ xx y dA
Sz = − ∫∫ σ xz dA
•
Displacement:
u ( x , y, z ) = u0 − y
dv
dw
− z
dx
dx
bending
about
z-axis
•
bending
about
y-axis
Strain:
ε xx
 d 2v 
 d 2w 
d u0
∂u
=
=
+ y− 2  + z− 2 
 dx 
 dx 
dx
∂x
= f1 + f2 y + f3 z
•
Strain:
σ xx = E ( f1 + f2 y + f3 z ) − E α ∆T
Paul A. Lagace © 2002
Handout 4-2
MIT - 16.20
Fall, 2002
MODULUS - WEIGHTED SECTION PROPERTIES
modulus of material
dA* =
E
dA
E1
reference modulus
*
= y * A*
∫∫ y dA
*
*
*
∫∫ z dA = z A
∫∫ dA = A
∫∫ y dA = I
∫∫ z dA = I
∫∫ y z dA = I
*
*
2
*
*
z
2
*
*
y
*
*
yz
top location of beam
also:
*
zT
∫z
•
z
E
bzdz = Q
*
E1
Thermal Resultants:
FT =
∫∫ E α ∆T dA
MyT = − ∫∫ E α ∆Tz dA
MzT = − ∫∫ E α ∆Ty dA
Paul A. Lagace © 2002
Handout 4-3
MIT - 16.20
•
Fall, 2002
Selective Reinforcement:
E
E1
E
b
•
Mechanical + Thermal Resultants
General Equations:
F TOT
d u0
f1 =
=
E1 A*
dx
(F
+ F T ) = F TOT
− ( Mz + MzT ) = − MzTOT
(
)
− My + MyT = − MyTOT
− I y* MzTOT + I yz* MyTOT
f2 =
(
E1 I y* Iz* − I yz*
2
)
− Iz* MyTOT + I yz* MzTOT
f3 =
(
E1 I y* Iz* − I yz*
σ xx
2
)
E  F TOT
=
−

E1  A*

d 2w
= − 2
dx
[I
*
y
]
MzTOT − I yz* MyTOT y
I y* Iz* − I yz*
−
Paul A. Lagace © 2002
d 2v
= − 2
dx
[I
*
z
2
]
MyTOT − I yz* MzTOT z
I y* Iz* − I yz*
2

− E1 α ∆T 

Handout 4-4
MIT - 16.20
Fall, 2002
COMMON SECTIONS
(See Handout #4A)
SHEARING AND TORSION (AND BENDING) OF SHELL BEAMS
Loadings
•
axial load: F
•
bending moments: My, Mz
•
shear forces: Sy, Sz
•
torque (torsional moment): T
Single Cell “Box Beam” (symmetric)
•
•
•
Location of modulus-weighted centroid
*
y =
•
use modulus-weighted
centroid of flanges
skin carries only shear loads
(no bending/axial loads)
*
i
∑A y
∑A
i
*
i
;
z
*
=
*
i i
*
i
∑A z
∑A
Section moments of Inertia
I y* =
2
∑ Ai* zi* ; Iz* =
2
∑ Ai* yi* ;
I yz* =
*
i
∑A
yi* zi*
sum over number of flanges (i = 1,…n)
•
Stresses in flanges
σ xx
Paul A. Lagace © 2002

E  F TOT
=
 * − E1 f 2 y − E1 f 3 z − E1α∆T 
E1  A

Handout 4-5
MIT - 16.20
Fall, 2002
•
Shear flow:
(assume shear stress is
uniform through thickness)
q = σ xs t
shear
flow
shear stress
thickness
qout − qin = −
— At any flange (node):
dP
dx
for symmetric section (Iyz = 0) with Mz = 0:
qout − qin =
Qy Sz
Iy
where: Qy = Aizi* (moment of area about y)
•
Divide into 2 problems:
General
Pure Shear
Pure Torque
=
T ′ − Sz d = T
1. “Pure Shear” Problem
d = distance from Sz original line of action to shear center
use:
Æ
Joint equilibrium:
Æ
Torque equivalence:
Æ
No-Twist condition:
qout − qin =
∑T
internal
∫
Qy Sz
Iy
= Tapplied
q
ds = 0
t
2. “Pure Torsion” Problem
use:
Æ Joint Equilibrium: qout − qin = 0
Æ Torque equivalence:
∑T
internal
= Tapplied
3. Sum q’s from two problems
Paul A. Lagace © 2002
Handout 4-6
MIT - 16.20
Fall, 2002
UNSYMMETRICAL SHELL BEAMS (UNHEATED)
(− I M
y
z
+ I yz M y ) y + ( − I z M y + I yz Mz )z
•
σ xx =
•
q
•
Same procedure as previous case
out
I y I z − I yz2
− q
in
=
(I S
y y
− I yz Sz )Qz + ( I z Sz − I yz Sy )Qy
I y I z − I yz2
THICK SKIN SHELLS
•
Idealize by breaking up thick skin into a finite number of bending areas
Thick skin
Idealization
Bending
area
•
Skin here carries
only shear stress
Use previous techniques
DEFLECTIONS
v = vB
w = wB
deflection of shear center
+
+
vs
ws
due to
due to
bending
shearing
α = twist/rotation (due to torsion)
about shear center
I y Mz
d 2 vB
=
dx 2
E I y Iz − I y2z
(
Paul A. Lagace © 2002
)
−
I yz My
(
E I y Iz − I y2z
)
Handout 4-7
MIT - 16.20
Fall, 2002
Iz M y
d 2 wB
=
dx 2
E I y Iz − I y2z
(
)
−
I yz Mz
(
E I y Iz − I y2z
)
Sy
d vs
Sz
= −
−
dx
G Ayy
G Ayz
Sy
d ws
Sz
= −
−
dx
G Azz
G Ayz
dα
T
=
dz
GJ
SECTION PROPERTIES
•
•
Iy, Iz, Iyz as previously defined:
* 2
Iz =
∑A y
Iy =
∑A z
I yz =
∑ A yz
* 2
*
1
Ayy =
∫
•
(q )
y
t
•
(qz )2 ds
∫
∫
2A
q
ds
t
J =
where:
( qz
= shear flow due to unit Sz
t
1
q y qz
ds
t
Ayz =
Paul A. Lagace © 2002
ds
1
Azz =
∫
•
(
where: qy = shear flow due to unit Sy
2
where: q = shear flow due to unit T
Handout 4-8
MIT - 16.20
Fall, 2002
MULTI-CELL SHELL BEAMS
•
σxx equation the same
•
qout - qin problem the same
• Additional “No-Twist” Condition -- must be there for each cell as well
as overall beam (for “Pure Shear” case)
•
“Equal Twist” Condition (for “Pure Torsion” case) -- each cell must
twist the same amount.  dα the same
 dx

OPEN SECTION SHELL BEAMS
•
All qi determined from joint equilibrium
qout − qin = −
dP
dx
•
σxx equation the same
•
Use Membrane Analogy for “Pure Torsion” problem
τ =
Paul A. Lagace © 2002
2T
x
J
Handout 4-9