Plate Bending
not so long plate
previously have shown: M := −D⋅
2
d
w . this was for single axis bending.
2
dx
this relationship holds for the partial derivative in the respective direction fo both x and y;
assumptions:
plane cross section remains plane
ry
rx
dx
small deflections wmax < 3/4 t
dy
stress < yield
t
z
dz
w = w(x, y)
1
Rx
=−
2
d
2
1
w(x,y)
Ry
dx
=−
2
d
dy
2
w(x,y)
we are making no statement with respect to εy (or any other ε) as we did in long plate.
ε x :=
ε x :=
σx
E
1
Rx
−
⋅zz
ε x := −z⋅
ν ⋅σ
σy
E
2
d
2
ε y :=
w(x,y)
dx
ε y :=
ε x + ν ⋅ ε y collect , σ x , E →
σy
E
−
1
Ry
⋅zz
ν ⋅σ
σ x
x
ν ⋅ε y →
E
=>
σ x :=
y
dy
2
w(x,y)
E
1− ν
1
2
d
ε y := −z
(
)
⋅ ε x + ν ⋅εε y
2
notes_23_plate_bending.mcd
substituting
into
ε y := −z
σx →

1− ν 
E
2
2
d
dy
y
⋅  −z⋅
2
ε x := −z⋅
w( x, y)
2
d
2
E
σ x :=
w( x, y)
dxx
1− ν
2
(
)
⋅ ε x + ν ⋅εε y
d d
d d
 or ...
w( x, y)
w( x, y) − ν⋅z⋅
dxdx
dy dy

σ x( z) := −z
 d2

2
d
w( x, y) + ν⋅
w( x, y)
2
2  2
dy

1 − ν  dx
similarly:
σ y := −z
 d2

2
d
w( x, y) + ν⋅
w( x, y)
2
2  2
dx

1 − ν  dyy
E
E
⋅
⋅
as in bending (applied in each of x and y direction): see figure Hughes 9.3 (below)
the lower case m denotes moment per unit length
note that the designation is changed: the subscript on the m refers to the direction of axial stress
t
t
⌠2

mx :=  σ x( z)⋅z dz
− t
⌡
⌠
2
my :=  σ y⋅z dz
− t
⌡
2
mx →
1
12
3
⋅t ⋅E⋅
2
d d
d d
w( x, y) + ν⋅
w( x, y)
dxdx
dy dy
−1 + ν
with
D :=
E⋅ t
(
 d2
w(x,y) +
 dx2

12
3
⋅t ⋅E⋅
d d
d d
w( x, y) + ν⋅
w( x, y)
dy dy
dxdx
−1 + ν
ν
)
2
 d2

w(x,y)
 dyy 2



my := −D⋅
 d2

2
d
w(x,y) + ν
w(x,y)
2
 dy2
dx
x


in the general plate case there may exist a twisting moment mxy
we need to derive a similar expresssion for this moment
from basic shear strain relationships: γ :=
2
2
3
12⋅ 1 − ν
mx := −D⋅
my →
2
1
d
d
d
d
u; τ := G⋅γγ and τ := G⋅ v +
u
v +
y
dx
dy
dx
dyy
notes_23_plate_bending.mcd
x,u
dx
dw/dx
dy
u=-z*dw/dx
dv/dx
z
y,v
du/dy
from the geometry of the slope of w in each direction (x and y): u := −z⋅
=>
τ := −2⋅G⋅z⋅
d d
w ( x, y)
xdy
dx
d
w(x,y)
dxx
and v := −z⋅
d
w(x,y)
dy
y
and the twisting moment per unit length is determined by:
t
⌠ 2

mxy := − τ⋅z dz

⌡− t
mxy →
1 3 d d
⋅t ⋅G⋅
w( x, y)
6
dy dx
minus sign comes from sense of mxy +
and τxy(+)*z(+) see figure Hughes 9.3
below
2
using: G:=
E
2⋅ ( 1 + ν )
3
=> mxy :=
Et
d d

 w(x,y) ⋅ (
dx
x dy
12⋅
1 + ν)

(1 − ν )
Et
d d

⋅
 w(x,y) ⋅ (
dxx dy
 12⋅ 1 + ν ) ( 1 − ν )
3
multiply and divide by (1-ν) =>
mxy :=
d d

 w(x,y) ⋅
dxx dy

Et
(
3
12⋅ 1 − ν
)
2
mxy :=
⋅ ( 1 − ν )=>
d  d

mxy := D⋅ ( 1 − ν )   w(x,y) 
dxx dy

due to the complimentary nature of shear stress τ xy := τ yx
and due to the sign convention for + moment
mxy := −m
myx
d  d

=> myx := −D⋅ ( 1 − ν )   w(x,y) 
dxx dy

3
notes_23_plate_bending.mcd
now for equilibrium of the dx, dy, dz segment:
see Hughes figure 9.3 forces and moments in a plate (below)
t
qx
qy
my
mx
mxy
my+dmy/dy*dy
y
myx
mx+dmx/dx*dx
mxy+dmxy/dx*dx
myx+dmyx/dy*dy
x
qx+dqx/dx*dx
qy+dqy/dx*dx
z
Hughes figure 9.3 forces and moments in a plate
q in this context is the shear force per unit length on each face:
qx * the distance dy is the force on the x face etc...
equilibrium of vertical forces =>
q +
 x

p is the lateral (+z direction) load per unit area - not shown on figure
 d q  ⋅dx ⋅dy − q ⋅dy +  q +
 x

x
 y
dx 


 d q  ⋅dy ⋅dx − q ⋅dx + p⋅
dx⋅dy = 0
 y

y
dy  
=>
d
d
qx + qy + p = 0
dx
dy
moments wrt the x axis =>

m +  d m  ⋅dx ⋅dy − m ⋅dy + m ⋅dx −
 xy  xy

xy
y

dx  
 m +  d m  ⋅dy ⋅dx + q ⋅dx⋅dy = 0
 y 
y

y

dy  
= 0
and similarly: taking moments wrt the y-axis =>
d
d
myx + mx − qx = 0
dy
dx
d
 d
or ... since
mxy := −m
myx − mxy + mx − qx = 0
dy
 dx
4
notes_23_plate_bending.mcd
=>
d
d
mxy − my + qy
dx
dy
substituting the moment relations
 d m  + d m and
xy
x
dy
 dxx
qx := −
into the shear equation =>
2
d
2
dx
mx − 2⋅
d m  + d m
xy
y
dx  dyy
qy := −
 d  d  d
 d 
d  d
− mxy + mx + − mxy + my + p = 0
dx  dy
 dx  dy  dx  dy 
2
d d
d
myx +
m + p=0
2 y
dxdy
dy
substituting the relations fro mx, my and mxy above =>
  2
d
−D⋅
w(x,y) +
2
2

dx 
d
x

2
d
ν
 2
 
  
d
d d
d  d
  d2 −D⋅  d2 w(x,y) + ν⋅ d2 w(x,y) 
w(x,y)
− 2⋅  D⋅ ( 1 − ν )   w(x,y)   +
2
 dy2


dx dy
dx dy
  dy2   dy2
dx



= -p
4
2 2
4
p
the terms with ν cancel and the result is: d w(x,y) + 2⋅ d d w(x,y) + d w(x,y) =
4
2
2
4
D
dx
dx dy
dy
5
notes_23_plate_bending.mcd