Strain resolving method of composite plane plates
Ion FUIOREA
*Corresponding author
Institute for Theoretical and Experimental Analysis of Aeronautical Structures
STRAERO, Bucharest, Romania
B-dul Iuliu Maniu 220, Bucharest, 061126, CP 76/175
ifuiorea@straero.ro
Abstract: The paper deals with the extension of isotropic plates problem to the case of composite plates. In order to perform it, the Kirchhoff-Love hypotheses were “softened” by some additional ones. Considering the constitutive laws for composite materials the stress functions were eliminated by using Cauchy equations. As a result a partial derivative equation in displacements was obtained. Finally the boundary condition formulation was extended for the case of complex composite plates.
Key Words: composite materials, isotropic plates problem
1. INTRODUCTION
This paper aims at finding an approachable form in displacements of equations governing the mechanical response of composite plates. Unlike the solutions proposed by S. G. Lekhnitskij [1,2], utilizing the facilities offered by several simplifying assumptions that will be presented below, intermediate equations requiring an integration (usually numerical one) can be eliminated.
Eventually, a partial differential equation of 4th order (the analytical solution in displacements of the anisotropic problem of the composite plate) – which is a generalization for the anisotropic problem of the Sophie-Germain equation, can be determined.
Thus, an analytical alternative of the numerical solutions with stress solving that are proposed by most works dealing with the composite plates mechanics, is approached.
The starting point in this case is the Kirchhoff-Love hypothesis that allows a considerable simplification of the equations of the anisotropic elasticity theory, which in their turn are
“
weakened” by new hypotheses as will be shown below.In addition, exploiting favorably the assumptions made, analytical expressions are proposed for all types of boundary conditions.
2. WORKING HYPOTESIS
1. During deformation, the normal vectors to the central plane of the plate remain straight and normal to the median deformed surface.
This hypothesis implies that
0
xz yz (1)
From constitutive equation of orthotropic composites considering principal anisotropic directions, it results unequivocally.
0
xz yz (1’)“Weakening” of this hypothesis consists in interpreting previous relation (1’) as an indicator for the fact that unitary efforts xz,yz are very small with respect to the normal ones, but their gradient could be appreciable, namely
0
z
xz (1’’)
2. During deformation of the laminate its thickness is constant, which equates with the condition
0
z (2)
3. The small strain hypothesis is also considered thus resulting that the points located in median plane of the plate remain on the same vertical after deformation.
0 2 2 2
2 2
0 2 2 2
2 2
0
2
2
y x
w y
w x
w
z
y x
w y
w x
w
z
x v y u
y v x u
xy y x
(3)
3. THE PLATE PROBLEM SOLVED IN DISPLACEMENTS
The constitutive equation can be written with respect to the main directions of anisotropy in the form
12 2 1
66 22 12
12 11
12 2 1
0 0
0 0
Q Q Q
Q Q
(4)
where
12 66 21
12 2 22
21 12
2 12 12
21 12 1
11 ,
1 ,
1 ,
1 Q G
E Q
E Q
E
Q
(5)
In relation to any coordinate system equation (4) becomes
XY Y X
xy y x
Q Q Q
Q Q Q
Q Q Q
2 66 26 16
26 22 12
16 12 11
(6)
66 12 22 11 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 4 4 2 2 2 2 2 2 2 2 4 4 2 2 2 2 4 4 26 16 66 12 22 11 2 2 2 4 4 2 4 2 Q Q Q Q mn n m mn n m n m mn n m mn n m mn mn n m n m n m n m n m n m n m n m n m n m n m m n n m n m n m Q Q Q Q Q Q (7)with m = cos and n = sin ; is the angle made by the fibers direction (the principal anisotropic direction) and Ox axis direction
.
Replacing (3) in (6) we obtain the constitutive equation in relation with the strain function of medium plane; it results
0 2 2 2 2 2 66 26 16 26 22 12 16 12 11
2
y x w y w x w Q Q Q Q Q Q Q Q Q z xy y x (8)
If the mass forces are neglected in equilibrium equations, we can write
0 0 0 z y x z y x z y x z yz xz yz y xy xz xy x (9)
Because the ‘hard’ hypothesis of Kirchhoff does not involve the corresponding nullity of the unitary efforts, it can be ‘weakened’ assuming that these slides are small compared to the gradient of plate deformation– as in (10)
0 yz y w z v (10)
NOTE. “Weakening” Kirchhoff's first assumption is legitimate in order to ensure the balance of the plate element under transverse loads
To estimate the transverse efforts in plate the equilibrium relations (9) are considered. From the first relation in system (9) one deduces
y x z xy x zx (11)
y x
w Q y
w Q x
w Q y
y x
w Q y
w Q x
w Q x z z
zx
2
33 2
2
23 2 2
13
2
13 2
2
12 2 2
11
2 2
(12)
or
3 3
23 2
3
13 2
3
33 12
3 3
11 2 3
y w Q y x
w Q y
x w Q Q
x w Q z z
zx
(13)
If we note
3323 2
3
13 2
3
33 12
3 3
11 2 3
y w Q y x
w Q y
x w Q
Q x
w Q A
(14)
and integrating the (13) equation with respect to z variable it is obtained
x y f A zzx ,
2 1
2
(15)
Where f1(x,y) represents an arbitrary function which will be determined from outline
conditions. Thus from the assumption that on the plate surfaces (at
2 h
z ) there are no forces in plane(zx 0) one deduces
x y h Af
8 ,
2
1 (16)
namely in the end
A h z
zx
8 2
2 2
(17)
Where h is the plate thickness.
Similarly, for the second component of the transverse effort we can successively deduce the following:
From the second relation of system (9) it can be deducted
y x z
y xy zy
(18)
or, using the relations from (8)
y x
w Q y
w Q x
w Q y
y x
w Q y
w Q x
w Q x z z
zy
2
23 2
2
22 2 2
12
2
33 2
2
23 2 2
13
2 2
or
3 3
22 2 3
23 2
3
33 12
3 3
13 2 3
y w Q y x
w Q y
x w Q Q
x w Q z z
zy
(20)
If denoting
3322 2 3
23 2
3
33 12
3 3
13 2 3
y w Q y x
w Q
y x
w Q
Q x
w Q B
(21)
And the (20) equation is integrated with respect to variable z it is obtained
x y f B zzy ,
2 2
2
(22)
Where f2(x,y) represents an arbitrary function which will be determined from outline conditions.
Thus from the assumption that on the plate surfaces (at
2 h
z ) there are no in plane forces in the plane(zx 0) one deduces
x y h Bf
8 ,
2
2 (23)
and finally
B h z
zy
8 2
2 2
(24)
In order to obtain an equation to be solved trough strain, in the third equation of (9) system, we determine
y x z
yz xz
z (25)
Or, using relations from (17) and (24) we write
B h z y A h z x z
z
8 2 8
2
2 2 2
2
(26)
or
4 4
22 3 4
23 2
2 4
33 12
3 4
13 4
4
11
2 2
4 2
2 4
2 8
y w Q y x
w Q y
x w Q Q
y x
w Q x
w Q
z h z
z
(27)
44 22 3 423 2
2 4
33 12
3 4
13 4
4
11 4 2 2 4
y w Q y x
w Q y
x w Q
Q y
x w Q x
w Q C
(28)
and integrating the (27) equation with respect to z variable it results
x y f C z z hz ,
6
8 3
3 2
(29)
Where f3(x,y) represents an arbitrary function which will be determined from outline conditions.
Thus from the assumption that on inner plate surface (at
2 h
z ) there are no normal
forces to the plane (z 0) and also on upper surface (at
2 h
z ) the p pressure is acting, it is successively obtained
x y h Cf
24 ,
3
3 (30)
or
C z z h h
z
6 8
24
3 2
3
(31)
and, for
2 h
z we consider z p.
I.e.,
p C h
12 3
(32)
Or, explicitly, in the final form
3 4 4 22 3 4 23
2 2
4 33 12
3 4 13 4
4 11
12 4
2 2
4
h p y
w Q y x
w Q
y x
w Q
Q y x
w Q x
w Q
(33)
The (33) equation represents the displacement solution of composite plane plate problem transversally loaded which can be numerically integrated.
As already stated it is a generalization of Sophie Germain equation for the composite plane plate.
4. BOUNDARY CONDITIONS
4.1 Generalities
To achieve an optimization code of the plate composite structure in relation to its dynamic response to interaction with the fluid field, it is interesting to define each possible case of boundary conditions, especially in strain for the plate defined by the equation (31).
x
y
x
y
z
x
xz
xy
2
h
2
h
Fig. 1 Geometry of the plate with unitar length and width
Thus, as shown in picture it can be noted
2
2 1
h
h y
x z dz
M – bending moment (34)
2
2 1
h
h x
y z dz
M – bending moment (35)
2
2 1
h
h xy
xy z dz
M – torque moment (36)
2
2 1
h
h xy
x dz
T – shear force on the face parallel to Oy (37)
2
2 1
h
h yz
y dz
4.2. Fixed edge
For the fixed edge the boundary conditions will block the movement of the points situated on the respective side along the vertical direction (Oz) as well as prevent the rotation of the plaque surface around its axis.
Considering the system of axis from figure 1 the following conditions can be written a) on the edge parallel to the Oy axis
0 0
x w w
(39)
b) on the edge parallel to the Ox axis
0 0
y w w
(40)
4.3. Simply supported edge
For simply supported edge the boundary conditions will block the movement of the points situated on the respective side along the vertical direction (Oz). On the other hand, the bending moment on this surface must be null because the simply support does not introduce reaction moment. Making reference to the system of axes in figure 1 the following conditions can be written:
a) on the edge parallel to the Ox axis
0 0
x
M w
(41)
The second relation from (41), taking (34) into account leads to the condition
0 1
2
2
h
h y
x z dz
M (42)
Namely, if relation (8) is used it can be written
0 2
2
2
2
23 2
2
22 2 2
21
h
h
dz z y x
w Q y
w Q x
w Q
z (43)
Or, noting
y x
w Q y
w Q x
w Q D
21 22 22 22 2 23 2 (44)
It results
0 0
12 0
3
3 3 2
2
D h
D z
D
h
h
So, finally, the relation (41) becomes
0 2
0
2
23 2
2
22 2 2
21
y x
w Q y
w Q x
w Q
w
(46)
b) on the edge parallel to the Oy axis Similarly for the other side is written
0 0
y
M w
(47)
The second relation from (47), taking (35) into account leads to the condition
0 1
2
2
h
h x
y z dz
M (48)
Namely, if relation (8) is used it can be written
0 2
2
2
2
13 2
2
12 2 2
11
h
h
dz z y x
w Q y
w Q x
w Q
z (49)
Or, by noting
y x
w Q y
w Q x
w Q D
11 22 12 22 2 13 2
1 (50)
it results
0 1 0
12 1 0
3 1
3 3 2
2
D h
D z
D
h
h (51)
So, finally, the relation (51) becomes
0 2
0
2
13 2
2
12 2 2
11
y x
w Q y
w Q x
w Q
w
(52)
4.4. Free edge
For the free edge the boundary conditions stem in the condition that all efforts in that section are null.
Making reference to the system of axes mentioned in figure 1 the following conditions can be written
a) on the edge parallel to the Ox axis
0 ,
0 ,
0
xy x
y M T
M (53)
force Tx and the components of the torque moment Mxy (as distributed moment) are steady,
from the last two equations from (53) it results the relation:
0
y M
Tx xy (54)
Meaning, from (53) the boundary conditions become
0 0
y M T
M
xy x
y
(55)
The first relation from (55) is equivalent with the second relation from (47) i.e.
0 2
2
13 2
2
12 2 2
11
y x
w Q y
w Q x
w
Q (56)
After processing the second relation from (55) it results
0 2
2 2
2
h
h xy h
h
xz zdz
y
dz
(57)
or,
0 2
8 2
2
2
2
33 2
2
23 2 2
13 2
2
2 2
h
h h
h
dz z y x
w Q y
w Q x
w Q z y dz A h z
(58)
where B is the notation from (21). If, denoting
y x
w Q y
w Q x
w Q E
13 22 23 22 2 33 2 (59)
Then (58) becomes
0 8
2
2
2 2 2
2
2 2
h
h h
h
dz z y dz A h z
(60)
or
0 0
12 12
3 3
y E A h
E y h
A (61)
And, replacing
2
3 2 02
33 2 2
23 2 2
13 3
3
23 2 3
13 2 3
33 12 3 3
11
y x
w Q y
w Q x
w Q y y
w Q y x
w Q y x
w Q Q x
w
Q (62)
0 2 4 4 0 2 3 3 23 2 3 33 12 2 3 13 3 3 11 2 13 2 2 12 2 2 11 y w Q y x w Q Q y x w Q x w Q y x w Q y w Q x w Q (63)b) on the edge parallel to the Oy axis
0
,
0
,
0
xy
yx
M
T
M
(64)Similar to the previous case it results the relation:
0 x M
Ty xy (65)
Or, from (64) the boundary conditions become
0 0 x M T M xy y x (66)
The first relation from (66) is equivalent with the second relation from (41) i.e.
0 2 2 23 2 2 22 2 2
12
y x w Q y w Q x w Q (67)
After processing the second relation from (66) it results
0 2 2 2 2
h h xy h h yz zdz x dz (68) or, 0 2 8 2 2 2 2 33 2 2 23 2 2 13 2 2 2 2
h h h h dz z y x w Q y w Q x w Q z x dz B h z (69)where A is the notation of (14). If the notation from (59) is considered, then (69) becomes
0 2 8 2 2 2 2 33 2 2 23 2 2 13 2 2 2 2
h h h h dz z y x w Q y w Q x w Q z x dz B h z (70) or 0 0 12 12 3 3 x E A h E x h B (71) And, replacing
2
3 2 02 33 2 2 23 2 2 13 3 3 22 2 3 23 2 3 33 12 3 3
13
Finally, after processing, the two boundary conditions are written
0 2
4 4
2
0 2
3 3
22 2
3
23 2
3
33 12 3 3
13
2
23 2
2
22 2 2
21
y w Q y x
w Q y x
w Q Q x
w Q
y x
w Q y
w Q x
w Q
(73)
5. CONCLUSIONS
In the case of composite plane plates subjected to transverse loads and if small displacements assumption is considered, the equation (33) is a relatively simple and easy solution to integrate, which eliminates the considerable calculus effort and additional errors introduced by classical numerical methods.
The boundary conditions are easily applicable and allow a simple interpretation of the phenomena which have generated the mathematical conditions.
By dropping the assumption of anisotropy the Sophie-Germain equation is re-found.
The proposed method requires a comparison between classical and numerical method in terms of solving time and accuracy of the calculus.
REFERENCES
[1] S. G. Lekhnitskij, Theory of Elasticity of an Anisotropic Body, Mir Publishers, 1981.
[2] S. G. Lekhnitskij, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, San Francisco, 1963. [3] V. Bia, V. Ille, M. Soare, Rezistenta Materialelor si Teoria Elasticitatii, Ed. Didactica si Pedagogica,
Bucuresti, 1983.
