Plate Bending Introduction
see: bending with z load sheet for
derivations
review general beam, simply supported, clamped long plate
long plate, boundary conditions (end restrained) not so long plate
simply supported beam:
w
q⋅b
b
Q( x) :=
2
M 
⋅x − q⋅ x⋅
b
→
 2
σ x :=
t
b
x
q⋅bb
a
− q⋅ x
2
⌠
1
1
2
 Q( ξ ) dξ → ⋅q⋅b⋅x − ⋅q⋅x
⌡
2
2
0
M ( x) :=
t
q
x
M ( x)
I
1
8
z
x
1
d
M ( x) → ⋅q⋅b − q⋅ x
2
dx
2
1
M max :=
2
⋅q⋅b
8
= 0 @ x = b/2
2
⋅q⋅bb
σ x_max :=
maximum when z = t/2, m(x) = Mmax
⋅z
1
2
⋅q⋅b
M max t
⋅
I
2
+ tension other side of load
2
3
b
8
1 3
t
⋅t ⋅aa
σ x_max :=
⋅
σ x_max → ⋅q⋅
- compression on load side
I
2
2
4
12
t ⋅a
___________________________________________________________________________________________
I :=
clamped beam:
need to use delection
x
4
w
d
q
t a
w to solve
4
dx
 x2
M ( x) := −q⋅ 
result:

Given
d
M ( x) = 0
dx
2
−
bb⋅ x
2
+
2
b
b
12 
Find( x) →
1
2
z
⋅b
M 
b
 2
→
σ x_max :=
1
12
2
⋅q⋅b
I
⋅
t
2
24
2
⋅ q⋅b
M ( 0) →
M max = M ( x = 0, x = b)
1
1
σ x_max →
1
2
2
⋅q⋅
b
2
t ⋅a
−1
12
2
⋅ q⋅b
M ( b) →
−1
12
- compression other side of load
+ tension on load side
notes_22_plate_bending_intro.mcd
2
⋅ q⋅b
t
long plate:
treating unit length (away from end effects)
t
b
x
a
a := 1
w(x)
y
section at a
simply supported
free to pull in
z
t
q
x
w
a
t
da
t
b
z
strain in y constrained by adjacent plate
anticlastic curvature Rν = 1/ν ∗ R
ε x :=
σx
E
−
ν ⋅σ y
E
ε y :=
(1 − ν )⋅σσ x
σy
E
2
substituting => ε x:=
rearranging =>
σ x :=
or ... ε x:=
E
E⋅ε x
σx
E'
2
ν ⋅σ x
E
where E' = E/(1-ν^2)
ε x := −
z
=>
R
E
σ x := −
1− ν
E⋅tt
D :=
2
3
2
d
M := −D⋅
2
12⋅ ( 1 − ν )
8
I
⋅
t
2
⋅
d
2
M = My
w
dxx
w
σ x_max :=
3
4
1
2
⋅q⋅
b
t
σ x_max :=
2
2
Hughes 9.1.7 is of the form:
12⋅ ( 1 − ν )
2
2
clamped: (figure not shown)
2
⋅q⋅b
3
dx
x
moment relationships are the same:
simply supported:
1
2
M := −
E⋅ t
σ x_max := k⋅ q⋅
b
t
2
12
2
⋅q⋅b
I
⋅
t
2
σ x_max :=
k = 0.75 simply supported
k = 0.5 clamped
N.B. stress is + & - from bending
notes_22_plate_bending_intro.mcd
1
2
2
⋅q⋅
b
t
2
⋅
z
2 R
t
⌠2
2

E⋅ z
d
w⋅z dz
M := −
⋅
2 dx2

1− ν
− t
⌡
2
2
=
as in bending of beam:
t
⌠2

M :=  σ x⋅z dz

⌡− t
σ x_max :=
E
( 1−ν 2)
1− ν
define:
σy
=0
E
σx
or ... ε x:=
E
ν ⋅σ
σx
−