- Email: [email protected]
Buckling of orthotropic plates due to biaxial in-plane loads taking rotational restraints into account
BUCKLING OF ORTHOTROPIC PLATES DUE TO BIAXIAL IN-PLANE LOADS TAKING ROTATIONAL RESTRAINTS INTO ACCOUNT GEORGE S. JOHNSTON
Advanced Structures Department, D/72-26, Z-329, Lockheed Georgia Company, A Division of Lockheed Aircra]t Corporation, Marietta, Georgia 30063 ( USA)t SUMMARY Stability analysis based on the solution of the differential equation Jot long, orthotropic plates requires little computer time and capacity. Hence, it is usefuljor routine calculations and synthesis. Full biaxial loads and finite rotational restraints are taken into account. Results in the jorm o['buckling coefficients are presentedJbr sample cases.
INTRODUCTION
While not extending the state of the art of composite plate buckling as exemplified by reference 1, the present analysis is the missing link for the stability analysis of panels composed of flat orthotropic plates under biaxial loads. Reference 2 gives the rotational stiffness of compressed orthotropic plate elements with one edge supported or both edges supported. The combination of these and the present analysis gives allowables for the 'conventional plate buckling mode' (i.e. the buckling mode with nodes at each of the l~late junctions) taking rotational restraints into account. This applies to a variety of panel sections for use in wing covers and fuselage panels. Even the presence of the normal stress on the long edges, such as the hoop stress in longitudinally stiffened fuselage panels can be taken into account. Application includes metals and composite laminates that may be considered to be thin and orthotropic Note that the material axis and the plate axis are assumed coincident. Finite aspect ratio plates may be approximately analysed using the long plate boundary conditions because the wave form is periodic. The error can be minimised by using half wave lengths found by dividing the plate length by integers. This gives the accurate buckling load for rectangular plates with simply supported ends for loading conditions not involving shear. For pure shear loading on an t Present address: 6390 BridgewoodValleyRoad, Atlanta, Georgia 30328, USA. 435 Fibre Science and Technology 0015-0568/79/0012-0435/$02.25--© Applied Science Publishers Ltd, England, 1979 Printed in Great Britain
436
GEORGE S. JOHNSTON
isotropic plate with an aspect ratio of 3, the buckling stress using long plate theory is 7 ~oconservative for all edges simply supported. The effect of the rotational restraint on the long edges may be several times larger, and currently the long plate theory is a practical necessity for taking finite rotational restraints into account. Otherwise the edge rotations would not be sinusoidal and the analysis cannot be separated into plate elements. Note, the results supplied in the form of buckling coefficients are for the optimum wave length. These results are conservative for anything but long plates unless the optimum wave fits the boundary.
DISCUSSION
The present theory is a generalisation of the work of Stowell. 3 He considers a long, isotropic plate loaded in pure shear with equal, finite rotational restraints on the long sides. Stowell cites Southwell and Skan 4 as having obtained results for simply supported and clamped edge conditions for pure shear loading. Stowell and Schwartz 5 consider longitudinal compression and shear. Batdorf and Houbolt 6 substituted transverse (in-plane) compression for longitudinal compression. Here, longitudinal compression, transverse compression and shear loadings are considered for orthotropic material. Finite rotational restraint is included.
DERIVATION
Plate dimensions, loads and coordinate system are shown in Fig. l The differential equation may be written DI 1~74 w~ + 2(D12 +2D66 ) /~4w
D22 ~x 4
Dzz
+--+~%'
~2xp2y
cV*
~-~=2~2 w
kx bz Sx~' T(,2 ('}214,
~2 (~2 W
+ ks. Yb ~y~T+ 2k~ b 2 ~x ~ y
-- 0
(1)
where the buckling coefficients are defined in terms of the stiffness in the long transverse direction, D22:
k x = Nxb2/(jz2D22 ) !
k~,
N~.bX/(rcZD22)?
ks
Nx~.b2/(rcZDzz) )
(2)
The buckle wave form is assumed to be w = exp (inx/2 + imy/b)
(3)
437
BUCKLING OF ORTHOTROPIC PLATES
Y
N
Y
~
N y
Nx
,.,
l
,,¢
N
ttttttttYtttt Fig. 1.
tttttttt
N
tt
Y Loads, dimensions, coordinate system.
Substituting eqn. (3) into eqn. (1) yields
- -
022 \/]-/
-
~*
7~
kx -- 0
(4)
This equation is no more difficult to solve than eqn. A-6 of reference 3 for isotropic plates loaded in pure shear. The derivation there is complete except for the explanation of the relationship between the coefficients of the terms of the polynomial and the roots; reference 7 contains this explanation. The stability criterion is derived by imposing the boundary conditions on the general solution of the differential equation. The general sol~ation includes all four values of 'm' in the ),-function. The following four boundary conditions are used: w = O,
y = b/2
w =
y
82w {?),z ?2w 33'2
-
O,
=
-
e [w b ~3" e i~w b 3)"
b/2
y=
b/2
t .
y
=
-
h/2
(5)
438
GEORGE S. JOHNSTON
where e is the non-dimensional rotational restraint given by -
Ob
(6)
D22
and 0 is the m o m e n t per unit length per radian. The buckle form of equilibrium becomes possible when the determinate for the coefficients for the four exp (imy/b) terms becomes zero. Expanding the determinate yields the stability criterion. Since the preceding part of the derivation parallels reference 3 and the boundary conditions are the same, the stability criteria are identical: 2ctfl(4? 2 -- ~82 )(cosh 2acos 2/3 -- cos 47) -- [4y2(f12 -- 0~2) -- (f12 + (X2)2
_
(4?,2 _/32 + c~2)~_ sinh 2c~sin 2/3 + e[c~(4?2 + ~2 +/32) cosh 2c~sin 2/3 +/3(472 _ c~2 _/32) sinh 2~ cos 2/3 - 4c~/37sin 4?] = 0
(7)
This form was obtained by substituting functions of ~, [3 and 7 for the m's. The new values of c~,/3 and ? are + 2c,]. ,2 + c 2 + c~/4 + ?': + ~c t
(8)
/3 = x/~,/4?" + 2c,72 + c 2 + c~/4 - 72 - ~c,
(9)
4?2(474 + 2c~y 2 + c 2 + c~/4) = c~
(10)
= V / x / 4~?
where
c,
=
c2C3 =
4
L
k,,
(1 l)
1z4[D" (b~4-(b)ek~,] 16L =2
(12)
D22
-
8 ~fi./ s
(13)
ALGORITHM
Starting with known values of 2/b, the ratio of the halt-wave length to the plate width, the stiffness ratios for the orthotropic plate, the rotational restraint, and the buckling coefficients (which represent applied loads), the stability condition of the plate may be determined as follows: 1. 2.
Calculate c l, c 2 and c 3 using eqns. (11), (12) and (13). Find the root of eqn. (10) between zero and x~/c32/16.
BUCKLING OF ORTHOTROPIC PLATES
3. 4.
439
Calculate • and fl from eqns. (8) and (9). (Note, if fl is imaginary use trigometric to hyperbolic formulas in Step 4.) Evaluate the stability criterion, eqn. 5; if the left hand side is positive, the plate is stable. (Note, false positive values result if allowable loading of clamped plates are exceeded too far.)
This algorithm was developed by Spencer a for use with reference 5.
BUCKLING COEFFICIENTS
The algorithm was applied in an iterative scheme to find the critical buckling coefficients for the optimum wave length shown on Figs. 2-7. The buckling stresses are calculated
a x = k~n2DE2/(bZt)) ar
krnZD22/(b2t)?
z
k~n2D2z/(bZt).)
(14)
Sample materials are: (a) isotropic, D~I/D:E= 1, 2(D1:+2D66)/Daa=2 , (b) +45 ° graphite epoxy (interspersed) D~/D22 = 1, 2(DIa + 2D6o)/D22 = 5'1, (c) ( 0 ° / + 4 5 ° / 0 ° / - 4 5 °) graphite epoxy, D~ ~/D22 = 3.4, 2(D~2 + 2D~,6)/D2z = 4-5. Only simply supported and clamped boundary conditions are shown. Owing to the
- I .00 0.00
2
0.80
o
-'15.00
v
-I0.00
-5.00
0.00
5.00
BUCKLING COEFFICIENT K X Fig. 2.
Simply supported isotropic plate.
I0.00
1.~ .00
440
GEORGE S. JOHNSTON
- 4 ,00
0.00
3.20
45.0o
4o.oo
!
-~.00
°0'.00
10.00
.00
BUCKLING COEFFICIENT
175 -00
KX
Fig. 3. Clamped isotropic plate.
KY
O
u~
- 1 .00 0.00
O
0.80 to
C
,
°
-15.00
r 7
I0.00
BUCKLING COEFFICIENT KX Fig. 4.
Simply supported _+45 ° G/E.
~S .00
441
BUCKLING OF ORTHOTROPIC PLATES
~Y -4.00
0.00
3.20
-15.00
I0 O0
5 O0
0 O0
5 O0
BUCKLING COEFFICIENT
I0.00
[email protected]
KX
Fig. 5. Clamped +45° G/E.
KY tO_
-i .00 0.00 0.80
tO"
-15.00
-~o.oo -~.oo %'.00 s.oo B U C K L I N G C O E F F I C I E N T KX
r
10.00
Fig. 6. Simply supported (00/+450/0o/-45 °) G/E.
7
15.00
442
GEORGE S. JOHNSTON
~Y -4.00
0.00
3.20
-'is.oo
~~
1o oo
s oo o oo s oo lo oo BUCKLING COEFFICIENT KX Fig. 7.
l~.oo
2o~oo
Clamped (00/+450/00/-45 °) G/E.
variation of rotational restraint with wave length in real panels, many more graphs would be required to properly consider finite rotational restraint. The practical way to do this analysis is with a computer. A comparison of the clamped and the simply supported buckling coefficients reveals the compelling reason for taking finite rotational restraints into account; the buckling stress is significantly higher for restrained plates; a condition which always exists when one plate in an assembly is unstable. Present results are in agreement with the results for isotropic plates from reference 9. This includes finite rotational restraint for pure shear and pure compression and the interaction curves for compression in either direction and shear as well as the values for all the single loading conditions. The present results are in agreement with the results for long, orthotropic plates collected in reference 10. This includes single loading conditions for simply supported and clamped edge conditions as well as biaxial compression and combined compression and shear for simply supported edges. REFERENCES
1. A. V. VISWANATHAN,M. TAMEKUNland L. L. BAKER,'Elastic Stability of Laminated, Flat and Curved, Long Rectangular Plates Subjectto Combined Inplane Loads,' NAS CR-2330,June 1974.
BUCKLING OF ORTHOTROPIC PLATES
443
2. P. P. BIJLAARD and G. P. FISHER, 'Interaction of Column and Local Buckling in Compression Members,' NACA TN 2640, March 1952. 3. E. 7. STOWELL, 'Critical Shear Stress of an Infinitely Long Flat Plate with Equal Elastic Restraints Against Rotation Along the Parallel Edges,' NACA ARR 3K12, November 1943. 4. R. V. SOUTHWELLand S. W. SKAN, On the stability under shearing forces of a flat elastic strip, Proc. Royal Society, Serial A, 105 (733) (May 1924) pp. 582-607. 5. E.Z. STOWELLand E. B. SCHWARTZ,'Critical Stress for an Infinitely Long Flat Plate with Elastically Restrained Edges Under Combined Shear and Direct Stress,' NACA ARR 3Kl 3, November 1943. 6. S. B. BATOORFand J. C. HOUaOLT, 'Critical Combinations of Shear and Transverse Direct Stress for an Infinitely Long Flat Plate with Edges Elastically Restrained Against Rotation,' NACA Report 847, 1946. 7. H. W. TURNBULL, Theory of Equations, Oliver and Boyd, Edinburgh and London (revised 1952), p. 66. 8. B. R. SPENCER, Private communication, October 31, 1967. 9. G. GERARD and H. BECKER, 'Handbook of Structural Stabdity, Part I--Buckling of Flat Plates,' NACA TN 3781, July 1957. 10. S. G. LEKHNITSKII(as translated by Tsai, S. W.) Anisotropic Plates, Gordon and Breach Science Publishers, New York, London and Paris, 1968, pp. 445-78.
Advanced Structures Department, D/72-26, Z-329, Lockheed Georgia Company, A Division of Lockheed Aircra]t Corporation, Marietta, Georgia 30063 ( USA)t SUMMARY Stability analysis based on the solution of the differential equation Jot long, orthotropic plates requires little computer time and capacity. Hence, it is usefuljor routine calculations and synthesis. Full biaxial loads and finite rotational restraints are taken into account. Results in the jorm o['buckling coefficients are presentedJbr sample cases.
INTRODUCTION
While not extending the state of the art of composite plate buckling as exemplified by reference 1, the present analysis is the missing link for the stability analysis of panels composed of flat orthotropic plates under biaxial loads. Reference 2 gives the rotational stiffness of compressed orthotropic plate elements with one edge supported or both edges supported. The combination of these and the present analysis gives allowables for the 'conventional plate buckling mode' (i.e. the buckling mode with nodes at each of the l~late junctions) taking rotational restraints into account. This applies to a variety of panel sections for use in wing covers and fuselage panels. Even the presence of the normal stress on the long edges, such as the hoop stress in longitudinally stiffened fuselage panels can be taken into account. Application includes metals and composite laminates that may be considered to be thin and orthotropic Note that the material axis and the plate axis are assumed coincident. Finite aspect ratio plates may be approximately analysed using the long plate boundary conditions because the wave form is periodic. The error can be minimised by using half wave lengths found by dividing the plate length by integers. This gives the accurate buckling load for rectangular plates with simply supported ends for loading conditions not involving shear. For pure shear loading on an t Present address: 6390 BridgewoodValleyRoad, Atlanta, Georgia 30328, USA. 435 Fibre Science and Technology 0015-0568/79/0012-0435/$02.25--© Applied Science Publishers Ltd, England, 1979 Printed in Great Britain
436
GEORGE S. JOHNSTON
isotropic plate with an aspect ratio of 3, the buckling stress using long plate theory is 7 ~oconservative for all edges simply supported. The effect of the rotational restraint on the long edges may be several times larger, and currently the long plate theory is a practical necessity for taking finite rotational restraints into account. Otherwise the edge rotations would not be sinusoidal and the analysis cannot be separated into plate elements. Note, the results supplied in the form of buckling coefficients are for the optimum wave length. These results are conservative for anything but long plates unless the optimum wave fits the boundary.
DISCUSSION
The present theory is a generalisation of the work of Stowell. 3 He considers a long, isotropic plate loaded in pure shear with equal, finite rotational restraints on the long sides. Stowell cites Southwell and Skan 4 as having obtained results for simply supported and clamped edge conditions for pure shear loading. Stowell and Schwartz 5 consider longitudinal compression and shear. Batdorf and Houbolt 6 substituted transverse (in-plane) compression for longitudinal compression. Here, longitudinal compression, transverse compression and shear loadings are considered for orthotropic material. Finite rotational restraint is included.
DERIVATION
Plate dimensions, loads and coordinate system are shown in Fig. l The differential equation may be written DI 1~74 w~ + 2(D12 +2D66 ) /~4w
D22 ~x 4
Dzz
+--+~%'
~2xp2y
cV*
~-~=2~2 w
kx bz Sx~' T(,2 ('}214,
~2 (~2 W
+ ks. Yb ~y~T+ 2k~ b 2 ~x ~ y
-- 0
(1)
where the buckling coefficients are defined in terms of the stiffness in the long transverse direction, D22:
k x = Nxb2/(jz2D22 ) !
k~,
N~.bX/(rcZD22)?
ks
Nx~.b2/(rcZDzz) )
(2)
The buckle wave form is assumed to be w = exp (inx/2 + imy/b)
(3)
437
BUCKLING OF ORTHOTROPIC PLATES
Y
N
Y
~
N y
Nx
,.,
l
,,¢
N
ttttttttYtttt Fig. 1.
tttttttt
N
tt
Y Loads, dimensions, coordinate system.
Substituting eqn. (3) into eqn. (1) yields
- -
022 \/]-/
-
~*
7~
kx -- 0
(4)
This equation is no more difficult to solve than eqn. A-6 of reference 3 for isotropic plates loaded in pure shear. The derivation there is complete except for the explanation of the relationship between the coefficients of the terms of the polynomial and the roots; reference 7 contains this explanation. The stability criterion is derived by imposing the boundary conditions on the general solution of the differential equation. The general sol~ation includes all four values of 'm' in the ),-function. The following four boundary conditions are used: w = O,
y = b/2
w =
y
82w {?),z ?2w 33'2
-
O,
=
-
e [w b ~3" e i~w b 3)"
b/2
y=
b/2
t .
y
=
-
h/2
(5)
438
GEORGE S. JOHNSTON
where e is the non-dimensional rotational restraint given by -
Ob
(6)
D22
and 0 is the m o m e n t per unit length per radian. The buckle form of equilibrium becomes possible when the determinate for the coefficients for the four exp (imy/b) terms becomes zero. Expanding the determinate yields the stability criterion. Since the preceding part of the derivation parallels reference 3 and the boundary conditions are the same, the stability criteria are identical: 2ctfl(4? 2 -- ~82 )(cosh 2acos 2/3 -- cos 47) -- [4y2(f12 -- 0~2) -- (f12 + (X2)2
_
(4?,2 _/32 + c~2)~_ sinh 2c~sin 2/3 + e[c~(4?2 + ~2 +/32) cosh 2c~sin 2/3 +/3(472 _ c~2 _/32) sinh 2~ cos 2/3 - 4c~/37sin 4?] = 0
(7)
This form was obtained by substituting functions of ~, [3 and 7 for the m's. The new values of c~,/3 and ? are + 2c,]. ,2 + c 2 + c~/4 + ?': + ~c t
(8)
/3 = x/~,/4?" + 2c,72 + c 2 + c~/4 - 72 - ~c,
(9)
4?2(474 + 2c~y 2 + c 2 + c~/4) = c~
(10)
= V / x / 4~?
where
c,
=
c2C3 =
4
L
k,,
(1 l)
1z4[D" (b~4-(b)ek~,] 16L =2
(12)
D22
-
8 ~fi./ s
(13)
ALGORITHM
Starting with known values of 2/b, the ratio of the halt-wave length to the plate width, the stiffness ratios for the orthotropic plate, the rotational restraint, and the buckling coefficients (which represent applied loads), the stability condition of the plate may be determined as follows: 1. 2.
Calculate c l, c 2 and c 3 using eqns. (11), (12) and (13). Find the root of eqn. (10) between zero and x~/c32/16.
BUCKLING OF ORTHOTROPIC PLATES
3. 4.
439
Calculate • and fl from eqns. (8) and (9). (Note, if fl is imaginary use trigometric to hyperbolic formulas in Step 4.) Evaluate the stability criterion, eqn. 5; if the left hand side is positive, the plate is stable. (Note, false positive values result if allowable loading of clamped plates are exceeded too far.)
This algorithm was developed by Spencer a for use with reference 5.
BUCKLING COEFFICIENTS
The algorithm was applied in an iterative scheme to find the critical buckling coefficients for the optimum wave length shown on Figs. 2-7. The buckling stresses are calculated
a x = k~n2DE2/(bZt)) ar
krnZD22/(b2t)?
z
k~n2D2z/(bZt).)
(14)
Sample materials are: (a) isotropic, D~I/D:E= 1, 2(D1:+2D66)/Daa=2 , (b) +45 ° graphite epoxy (interspersed) D~/D22 = 1, 2(DIa + 2D6o)/D22 = 5'1, (c) ( 0 ° / + 4 5 ° / 0 ° / - 4 5 °) graphite epoxy, D~ ~/D22 = 3.4, 2(D~2 + 2D~,6)/D2z = 4-5. Only simply supported and clamped boundary conditions are shown. Owing to the
- I .00 0.00
2
0.80
o
-'15.00
v
-I0.00
-5.00
0.00
5.00
BUCKLING COEFFICIENT K X Fig. 2.
Simply supported isotropic plate.
I0.00
1.~ .00
440
GEORGE S. JOHNSTON
- 4 ,00
0.00
3.20
45.0o
4o.oo
!
-~.00
°0'.00
10.00
.00
BUCKLING COEFFICIENT
175 -00
KX
Fig. 3. Clamped isotropic plate.
KY
O
u~
- 1 .00 0.00
O
0.80 to
C
,
°
-15.00
r 7
I0.00
BUCKLING COEFFICIENT KX Fig. 4.
Simply supported _+45 ° G/E.
~S .00
441
BUCKLING OF ORTHOTROPIC PLATES
~Y -4.00
0.00
3.20
-15.00
I0 O0
5 O0
0 O0
5 O0
BUCKLING COEFFICIENT
I0.00
[email protected]
KX
Fig. 5. Clamped +45° G/E.
KY tO_
-i .00 0.00 0.80
tO"
-15.00
-~o.oo -~.oo %'.00 s.oo B U C K L I N G C O E F F I C I E N T KX
r
10.00
Fig. 6. Simply supported (00/+450/0o/-45 °) G/E.
7
15.00
442
GEORGE S. JOHNSTON
~Y -4.00
0.00
3.20
-'is.oo
~~
1o oo
s oo o oo s oo lo oo BUCKLING COEFFICIENT KX Fig. 7.
l~.oo
2o~oo
Clamped (00/+450/00/-45 °) G/E.
variation of rotational restraint with wave length in real panels, many more graphs would be required to properly consider finite rotational restraint. The practical way to do this analysis is with a computer. A comparison of the clamped and the simply supported buckling coefficients reveals the compelling reason for taking finite rotational restraints into account; the buckling stress is significantly higher for restrained plates; a condition which always exists when one plate in an assembly is unstable. Present results are in agreement with the results for isotropic plates from reference 9. This includes finite rotational restraint for pure shear and pure compression and the interaction curves for compression in either direction and shear as well as the values for all the single loading conditions. The present results are in agreement with the results for long, orthotropic plates collected in reference 10. This includes single loading conditions for simply supported and clamped edge conditions as well as biaxial compression and combined compression and shear for simply supported edges. REFERENCES
1. A. V. VISWANATHAN,M. TAMEKUNland L. L. BAKER,'Elastic Stability of Laminated, Flat and Curved, Long Rectangular Plates Subjectto Combined Inplane Loads,' NAS CR-2330,June 1974.
BUCKLING OF ORTHOTROPIC PLATES
443
2. P. P. BIJLAARD and G. P. FISHER, 'Interaction of Column and Local Buckling in Compression Members,' NACA TN 2640, March 1952. 3. E. 7. STOWELL, 'Critical Shear Stress of an Infinitely Long Flat Plate with Equal Elastic Restraints Against Rotation Along the Parallel Edges,' NACA ARR 3K12, November 1943. 4. R. V. SOUTHWELLand S. W. SKAN, On the stability under shearing forces of a flat elastic strip, Proc. Royal Society, Serial A, 105 (733) (May 1924) pp. 582-607. 5. E.Z. STOWELLand E. B. SCHWARTZ,'Critical Stress for an Infinitely Long Flat Plate with Elastically Restrained Edges Under Combined Shear and Direct Stress,' NACA ARR 3Kl 3, November 1943. 6. S. B. BATOORFand J. C. HOUaOLT, 'Critical Combinations of Shear and Transverse Direct Stress for an Infinitely Long Flat Plate with Edges Elastically Restrained Against Rotation,' NACA Report 847, 1946. 7. H. W. TURNBULL, Theory of Equations, Oliver and Boyd, Edinburgh and London (revised 1952), p. 66. 8. B. R. SPENCER, Private communication, October 31, 1967. 9. G. GERARD and H. BECKER, 'Handbook of Structural Stabdity, Part I--Buckling of Flat Plates,' NACA TN 3781, July 1957. 10. S. G. LEKHNITSKII(as translated by Tsai, S. W.) Anisotropic Plates, Gordon and Breach Science Publishers, New York, London and Paris, 1968, pp. 445-78.