[American Institute of Aeronautics and Astronautics 37th Structure, Structural Dynamics and...

AIM-95-1 368-CP

ANALYSIS OF A N INTERNALLY PRESSIJRTZED ORTIIOCONALLY STIFFENED CYLINDRICAL SHELL WITH AN ASYMMETR.ICAL SECTION R.ING

Nnvrnn R.astogi* nnd Eric. l3. Johnson t Department of Aerospace and Ocean Enginrering

Virginia Polytechnic Institute and State lJnivorsity Blackshiirg,

ABSTRACT

The linear elastic response is determined for an intern ally pressur ized , long circular cy lin ti r ical s hell stiffened on the inside by a regular arrangement of idcntical stringers arid idmtical rings. Periodicit,y of this configuration permits the analysis of c7 unit cell coiisisting of a portion of Ilie shell wall centered over one stringer-ring joint. 'The stiffeners arf- modeled as discrrtc. beams. The stringer is assumed to have a symmetrical cross section and the ring a n asyrn- nirt,rir oprn st.ct,ion. Displacements are assumctl as truncated Fourier Series plus simplr terms in the axial coordinate to account for the closed-end pressure vcssel effect. Equilibrium is imposed by virtual work. Point-wise displacement continuity between the sliell and stiffeners is achieved by Lagrange multipliers wliicll represent the interacting line load distributions between the stiffeners and the inside shell wall. A composite material crown pane! typical of a large transport fuselage is used as numerical example. Structural models include effects of transverse shear deformations and warping deformati -m of the ring's cross section due to torsion. The interacting loads associated with the asymmetric response, and the outc-of-plane bending moment and torque in the ring, are very sensitive to these structural modeling effects.

INTRODUCTION

The design of stiffener-to-skin joints is cited by Jackson, et ai.' as one of the major technology issues in utilizing graphite-epoxy composites in the fuselage of a large transport aircraft. Stiffen- ers can be attached to the skin by either fasteners, cecuring, adhesive bonding, or sonie combi- nation of these methods. Where fasteners are required in a graphite-epoxy structure, aluminium fasteners cannot he iised because of galvanic corro- sion to the metal. More expensive fasteners, like

~~ ~

* Chaduate Research Assistant, Student Member

i Professor, Senior member AIAA. Member ASME iIIAA

This paper is declared a work of the U.S. Goveril- inelit and is not subject, to copyright protection in tlie Unitrd States.

Virginia 24061

tit aniiirii, arc wquired to avoid corrcsion. klence tc, rc~lurc manufncturing cost,s, mechanical fasten- rrs can be ctliminated in favor of hondcd joints. As a n example, a graphite-rpoxy crowii p;inPI fnr thr fuselagc of a 1 ; q c transport airrraft, was rc- cently fabricated without fasteners by cc-curing t hc stringers and co-hnnding the rings, ur frames. to tIiv skin {IIcewicz, et a\.'; liwansoii, ct d."). Also, the curved grapJiit+epoxy fiis4age frames were inaiiufact,ured I,y resin transfer rnolrliiig into two-clitncnsional braided prefornis of nct, strucl ural shapc (see Jackson')). (.'lcarly, the stlrength of the bond line is a critical issue for lhese primary fuselage struct,ures made from advanced composite mn- terials. The purpose of this paper is to analyze the load transfer in bonded stiffener-to-skin joints under cabin pressurization. Internal pressure is an important load to consider because it tends to cause peel stresses in the bond line, and peel stresses are par- ticularly debilitating in adhesive joints.

An idealized structural model of tlie fuselage is analyzed. This configuration is a long circular cylindrical shell stiffened on the inside by a regular arrangement of identical stringers and identical rings (frames). Periodicity of this configuration permits the analysis of a portion of the shell wall centered over a generic stringer-ring joint; Le., deformation of a structural unit cell determines the deformation of the entire stiffened shell. See Fig. 1. The stringer is assumed to have a symmetrical cross section, and the franie is assumed to have an asymmetrical open section. Asymmetrical open section frames are com- monly u.,ed as transverse stiffeners in the fuselage structiire. The stiffeners are modeled as discrete beams perfectly honded to the inside shell wall, so that the interacting loads between the stiffeners and shell wall are line load intensities. 'L'hcse line load intensities represent resiiil ants of the tractions inte- grated across the width of tlie attachment flanges of tahe stiffeners.

Mathematical formulations for the linear elastic response presented in this paper include the ef- f i x t of transverse shear deformations and the effect

1734

of wnrpinp of thc riiig's cross wcti t) t i dllt. to torsion. rhese effects arc iriiportnitt \ v l i i > t i I l ie ring 1 1 ~

nsylIll1iet ric;ll ('row SCTI ion, Iwc;lrisc* I l it> lobs 0 1 sytiinietry in t lie probletn resttlts i t i t orston of' I he ring, as I W I I as out-of-plane hc?rttlirig, atid ib conroiiii- tant rotation of' the $tit at th(1 stifbiicr itittwtr- tioil about the circiitnfrrc*ntinI :uis. 'This strinam ring-shell joint is modcletl i t 1 ; L I I idcdizrd iiianner: the stiffeners are iiiatbetnatically perltlit tetl to 1 ) ~ s through one another withsut contact but do intjrr- act indirectly through their mutual coiitact wi th the sliell a t the joint. Rest,rairrt of cross-srct ioilal isarp- ing, as occurs here in the ring due to contact, w i t h the shell, is an important contributor to the normal stresses in thin-walled open section hars. ;IS was demonstrated by HOP'. Based on transverse shear deformation and cross-sectional warping of the ring. four structitral models can be defined. The simplest model uses ~ioii-transverse-shear-deformnhle theory. or classical theory, and neglects warping dutb to torsion. 'rhe most complex model iltclitcles bot,li en'ects. Intermediate complexity moclels occurs for inclusion of one effect without the other.

For symmetric sectioli stiffeners, the response of the unit cell is symmetric about the stringer axis and the ring asis, and there is no rotation of stringer- ring-shell joint. For this bi-symmetric case, Wang and Hsu' presented results for the linear response, and Johnson and Rastogi7 presented results for thc geometrically nonlinear response.

MATHEMATICAL MODEL

The radius of the middle surface of the unde- formed cylindrical shell is denoted by R, and the thickness of the shell is denoted by 1 as shown in Fig. 1 . Axial coordinate z and the circun~ferential angle 8 are lines of curvature on the middle surface, and the thicknes.; coordinate is denoted by t, with -1/2 5 2 5 t / 2 . The origin of the surface coordi- nates is centered over the stiffeners intersection so that - 1 5 .e 5 1 and -0 5 0 5 0 , where 21 is the axial length, arid 2 R 0 is [lie circumferential arc length of the repeating unit.

On the h x k of thta syiniiietry about the r-axk for thc unit, only the interacting line load components tsngcnt a n d ttortiial to the stringer are included in thc analyses. The shell-stringer interacting force componwit,s per unit length along the contact lines are denoted by AEs(z) for the component tarlgr-nt to tlic stringer, and A 2 s ( . r ) for the

For all the strltrtrlrill mociels, the linear ~ J : i s - tic response of t he rrpea1,ing unit , to intmnal prrs siire i.5 oht,:iined hy utilizing R i t z ~net,Iiotl ;itid I l i t .

principle of' vir t ual work applietl srp;tr;it~ly t o t,11(. shell, stringer, and ring. 'rhc virtual work functionals are augmented by Lagrange niultipliers to en- force kinematic constraints between the st,ructiiral components of the repeating unit.. The Lagrange multipliers represent the interacting line loads between the stiffeners and the shell. Displacentents are separately assumed for the shell, stringer. and the ring.

TRANSVERSE SHEAR DEFORMATIOB ---- FORMULATIONS

A coiisiste:it first order t ratisverse shear deformation theory is devc.lqped to model the shell. Based on the assumption that the shell thickness I IS relatively srriall and hence, does not change during loading, thr displacements ai, an arbitrary material poitit in tho shell are approximatrd by

where u(x, e). ~ ( r , e) and w ( x , e) are the displace- nieiits of the points of the reference surface, and ~ & - ( L , O ) and $e( . r . ,O) iue the rotat,ions of the nor- nial to t h ~ reference siirfacc m shown i n Fig. : j ( i L )

1735

Using Eqs. ( 1 ) t o (3) arid assuming small rlisplace- nient gradients, the three-tliniensional engiiiecring strains are

In Eqs. (4) to (6), the two-dimensional, or shell, strain measures, which are independent of the 2-

coordinate, are defined by

The transverse shear strains ex, and eez given in Eqs. (6) were obtained through differentiation of Eqs. (1) to (3) with respect to z. However, Eqs. (1) to (3 ) are approximate in the z-coordinate, so that differentiating with respect to z cannot cap- ture the distribution of the transverse shear strains through the thickness of the shell. Since the material is assumed rigid in the z-direction ( e z z = 0). the distribution of the transverse shear strains, and consequently the distribution of the transverse shear stresses, does not influence the shell behavior. I t is the integral of the transverse shear stresses through the thickness, or transverse shear resultants. that influences shell behavior. Thus, Eqs. (6) should be viewed as average kalues of the transverse shear strains, or as the transverse shear strains evaluated a t the reference surface ( 2 = 0).

If we set the (average) transverse shear strains in Eq. (6 ) t o zero, then the rotations of the normal are s 11:

a .7: Q 3 . - (13)

so that

(15) l i ~ n c r , the thickness clistrihution of the shear strain reduces to

which coincides with the results of Novozliilov’s8 classical shell t,heory.

It is evident from Eq. ( 5 ) that three shell strain measures are needed to represent the in-plane shear strain distribution through the t#hickness in t,he transverse shear deformation shell theory. Whereas, only two shell strain measures are required in classical shell theory to represent the shearing strain distribution through the thickness (refer t o Eq. (16)). Also it. can be shown that under rigid body rotation of the shell, the nine shell strain measures, given by Eqs. (7) through (12) vanish. (For Novozhilov’s classical shell theory, six shell strain measures given by Eqs. (7-9) and (15) vanish under rigid body rotations).

The physical stress resultants and stress couples for the shell in terms of stress components are given, in tlhe usual way, by

A generalized 9 x 1 st,ress vector for thc . shell is defined by

1736

in which and $ll.g are the mathematical quan- tities conjugate to the triodi'ied twistifig memares ri5e and kxor respectively, and are defined in terms of the physical stress couples by

I I 2 A I J @ = ,(M,e + M e x ) i l x s = z ( M X e -

(19) The nine elements of the stress vector in Eq. (18) and the relations of Eq. (19) determine all the stress resultants and stress couples listed in Eq. (17) except for shear resultant N,.s. T h e shear stress resultant N,e is determined from moment equilibrium about the normal for an element of the shell. This secalled sixth equilibrium equation is

The generalized strain vector for the shell is

, €08 , Yxe, ~ x x , tceo %Set G e t 7 x 2 ~ e z l T cshci i =

(21) This strain vector is conjugate to the stress vector in the sense tha t the interiial virtual work for the shell is given by

bw:zt1l = JL 6 c h e 1 I z s h e l l dS (22)

where S denotes the area of the reference surface and dS = dzRd0 . This expression for the internal virtual work can be derived from three-dimensional elasticity theory by using Eqs. (4) to (6) for the thickness distribiitiom of the strains and the defini- tions of the resultants given by Eqs. (17) and (19).

Consistent with the transverse shear deformation theory, the linear elastic constitutive law for a laminated composite shell wall is given by

where

[C] =

' = [C]

it1 whicli stiffncsses A i , , B,, and 1' , Appendis.

givcw i n

The statement of virtual work is 6w;",l, = 6w;rt + bwy (25)

where the external virtual work for a cylindrical shell under constant internal pressure, including an axial load due t o the closed-end effect, is written as

- Q [ 6 ~ ( 1 , 0 ) - 6 ~ ( - 1 , 0 ) ] (27)

The axial force Q in Eq. (27) is an additional La- grange multiplier that accounts for axial load shar- ing between the stringer and shell.

S trinaer

Stringer displacements us( E ) and w,(x), and the rotation of the normal &s3(z) are shown in Fig. 3(b). Based on transverse shear deformation theory, the virtual work expression for the stringer is

/ [ N X , ~ C ~ S + M e s 6 6 e s + V 2 ~ 6 7 , ~ l d z J

- 1

1

1737

III which ,VZs is the axial furce 111 ilie atriugei, .\lv,, is the beiidiiig iiiuriient. \:, i5 the traiisvtme shear force. cx, is the rioriiial strain of the ct.litroida1 liiv.. ihe product t h ' d . . is tlie portioii uf tlie axial normal straiii due to txiidiiig, 3d3 is tlic traiisverse shear strain, and e, is the radial distance from th 1 stringer centroid to the contact line along the shell inside surface. The strain-displacement relations and Hooke's law for the stringer are

Ring

Ring displacements are denoted Itr ( e ) , Ur (8) , and w,. (S) , and the rotations are denoted by $rr(8), q5er(0), and pzr(0) as shown in Fig. 3(c). The structural model is based on transverse shear deformation theory sild includes cross-sectional warping due to to,sion. The extension of classical thin-walled, open section curved bar theory to laminated com- pgsite materials was developed by Woodson, John- son, and Haftkag. However, Woodson et al.9 did not consider transverse shear deformations.

The statement of virtual work is

+ Azr(e)[64zr(e) - u1hrr(e)]} ( 1 + z) ~o de 0

(31) in which Nor is the circuniferential idrce, Adxr is the in-plane bending moment, Adzr is the out-of- plane bending moment, M,r is the bimoment, TJr is the St. Venant's torque, I',r is transverse shear force in the 2-direction, Vzr is transverse shear force in the r-direction. is the circuinferential normal

% t r d l l l oftlle C'C11trI)l 1 ' 1 , ' h . r 15 tllr 1 1 1 ~ ~ J ~ ~ I l ~ 1 8 t : l < l -

ing rotatioil grarlit>iit. h ,. IS t hc oiit-of-planr twnd iiig rotation gradient. r, 15 i hv tbv i s i rat( -,=, IS traiibversr shear strain io 0 2 plane. ? z r is transverse shear strain ia 2 - 2 plane, e,. 13 the dislaocr froni the ring reference arc to the contact Iinfb alon: the shell inside surface, and Ro is the radius <Jf ririg reference arc. Parameters J O and w1 are the constant coefficients in the contour warping function, a(.) = uo t zw1, for the attachment flange of the ring. The rotations and strain-displacement relations are

(32) in wliich the over-dot deiiotes tin ordinary deriva- tive wi th respect to 8 . It is assumed that the shear forces are decoupled trorn extension, bending, and lorsional deformations of the ring. Thus, the material law for the ring is

Yzr

where

[C,] = -ESz -EIzx EI,, EI,, - / H a ]

E A ESx -ES, -ES, EH ESZ EIZz - E I z X -EI,Z E H , I EH EH, -EH, - E H , GJ

-ES, -EIWx EI,, EI,, - EH,

and

The stiffness in the submatrix [C,] were evaluated froin a computer code developed by Woodson, John- son, and Haftka9

For structural models in which the effect of %varping of the ring cross section is excluded the contribution of the bimoment, M w r , to the ring virtual work in Eq. (31) is neglected. The fourth row and column of the stiffness matrix, Eq. (33), are ignored Also, the contour warping function U ( Z ) is taken as zero.

17 38

CLASSICAL FORMULATIONS

Shell

The shell is modeled with Sanders'" theoty tor Define a generalized strain vrL'tor in thin shells.

terms of the shell strain measures by

(34)

The first five strain measures of the shell rcftwnrc surface in Eq. (34) are related to the displacoiiients by Eqs. (7-9), and the sixth strain nieasure, t i r e r is given by

where the rotation about the iiorinal, q5:, is

(35)

and the rotations & and Cps of the normal are given by Eqs. (13) and (14).

Define a generalized stress vector in terms of the stress resultants and couples of Sanders' theory by

such that the internal virtual work is still given by Eq. (22) except Eqs. (34) and (37) are used. Quan- tities Nie and M:@ are the modified shear resultant. and twisting moment resultant in the Sanders theory. Hooke's law for a laminated composite shell wall is

in which rhe 3 x 3 sub-matrices A , B and D are given by classical lamination theory (Jones"). The external virtual work expressions for the classical shell theory are still given by Eqs. (26) and (27), but the rotations in Eqs. (27) are given by Eqs. (13) and (14) .

Stringer

The stringer is modeled with Euler-Bernoulli beam theory thereby neglecting the transverse shear

strain Heiict., rqiiaciiig -,:* i n 1:q (29) to ~ e i o results i n the following cspression for rpb lS .

(31)

It may be iioted that neglecting the transverse shcnr strain would also ~iiodify the virtual work statenlent given by Eq. (%), and tile third equatioii iri tlic Hooke's law, Eq. (30), is neglected

Ring

The ring is niodeled with the classical ~ l i i n - walled, opeii section curved bar theory developed by Woodson, Johnson, and Haftka9. For classical formulations, the transverse shear strains are iir-

glected. Hence, equating I c r and iZr in Eq. (32) to zero results in the following expressioils for thc rotatioiis $,,, and $2r:

It may be noted that neglecting the transverse shear strains would also modify the virtual work statement given by Eq. (31), and the submatrix [Cz] in the Hooke's law, Eq. (33), is neglected.

DISPLACEMENT CONTINUITY

In order to maintain continuous deformation between the inside surface of the shell and stiffeners along their lines of contact, the following displacement continuity constraints are impr,sed:

Along the shell - stringer interface (i.e., - 1 5 2 5 I, e = o),

(44) (45)

17 39

The veriational form of t,hese constraints are

1

0

(49) J [bXrrgrr + b h r g e r + 6Xzrg.zr + 6AerGer

+ bAZrGzr] (Ro + e,) dB = 0 -0

The constraint that the elongation of the shell at 0 = 0 and the elongation of the stringer are the same is

FOURIER SERIES APPROXIMATIONS

The periodic portions of the displacements and rotations are represented by truncated Fourier Se- ries having fundamental periods in the stringer and ring spacing. The non-periodic portions of the displacements due to axial stretching are represented by simple terms in r. The Fourier Series reflect symmetry about the x-axis for the repeating unit. For the shell, displacements of the middle surface (see Fig. 3a) are represented as

M N

m = l n = l

M N

m=O n=l

M N

m = l n=l

M N

( 5 3 )

and rotations of the normal are M N

m = l n=O hf N

m = l n=l

M N (54)

m=O n=l

M N

+ C d s 2 m t i ~ i n ( c r m t ) s i n ( ~ n e ) m=l n=l

(55) in which a m = 7 and pn = where m and n are non-negative integers. Note that some terms in the truncated Fourier Series of Eqs. (51-55) have been omitted. The coefficients of the omitted terms are ~ 2 0 0 , U2m0, ~ 0 n t WZmot 4zzoor 4zzmOr and 4 z 2 0 n i in which rn = 1 , 2 , ..., A4 and n = 1 , 2 , ..., N . The ra- tionale for their omission is discussed in the following sub-section. The displacements of the centroidal line of the stringer (see Fig. 3b) are

rn=l

m= 1

(57) and the rotation of the normal of the stringer about the 8-axis is

M M

(58 ) where the coeffcients us20 , W,ZO and 4 0 ~ 2 0 are omit- tea. Coefficient qo in the axial displacement field of the shell and q1 in the axial displacement field of the stringer represent elongations of each respective element caused by either an axial mechanical load or due to closed-end pressure vessel effects. The displacements of the reference circle of the ring (see Fig. 3c) are

N

1740

N loads were determiped from rigid body equilibriuni conditions for the iing and stringer, and from displacement continuity conditions between the shell and the stiffeners. The external virtual work for the stringer and ring must vanish for any possible rigid body motions of these elements. For the stringer these rigid body motions are spatially uniform x- direction and z-direction displacements. (A rigid body rotation of the stringer in the r-z plane is not considered since this motion would violate longitudinal periodicity of the repeating units.) Vanishing of the external virtual work for an arbitrary rigid body displacement of the stringer in the axial direction leads to the E-direction equilibrium equation

N

n=O

and rotations are

- I where the coeffcients up0 and der0 are omitted. The distributions of the interacting loads, or Lagrange multipliers, are taken as

similarly, the equilibrium equation for a rigid bod,, displacement of the stringer in the 2-direction is

M 1 I Azs(z) dx = 0

- I

(73)

If the ring is considered in its entirety, that is, as made up of an integer number of repeating units around its circumference, the rigid body motions that lead to non-trivial equilibrium conditions are a displacement in the z-direction and a rotation about an axis through the centroid of the ring parallel to the E-direction. The z-direction equilibrium equa-

N

n = l

N

(74)

and the moment equilibrium equation about the r- axis is

(75)

where the coefficients X x 2 s ~ , A z ~ s ~ , ArrO, and Aero are omitted.

Equilibrium Eqs. (72) to (74) imply that coefficients

AX820 = 0 Terms Omitted in the Fourier Series

Terms omitted in the truncated Fourier Series for the displacements, rotations, and interacting

in the Fourier Series for the interacting loads, and these conditions have been taken into account in

1741

Eqs. (65) to (67). The sine series for Xer given in Eq. (68) satisfies the equilibrium condition given in Eq. (75).

Consider the variational form of the constraints, Eqs. (48) and {49), for the spatially uniform conipo- nents of the virtual interacting loads. These equations are

(77)

(78) n = O

+ er4sro)] b ~ r t o = o (79)

Since these equations are satisfied on the basis that 6A,,20 = 0, SAzS2o = 0 and 6X,,o = 0, consistent with Eq. (76), the bracketed terms in Eqs. (77) to (79) do not necessarily vanish. The implica- tion that these bracketed terms in Eqs. (77) to (79) do not vanish is that displacement continuity conditions are not satisfied pointwise. Pointwise continuity can be achieved by taking each Fourier coefficient appearing in the bracketed terms of Eqs. (77) to (79) to be individually zero. Fourier Series given in Eqs. (51), (54), (56), (58), (59), and (63) reflect this choice. Moreover, Fourier coefficients 21200, ~ ~ 2 0 , and uro represent rigid body displacement in the axial direction for the shell, stringer, and ring, respectively, and setting them to zero can be justified on the basis that rigid body displacement does not contribute to the deformation of the structural elements. Since Fourier coefficient w,20

represents rigid body displacement of the stringer in the z-direction, it would seem that it should be set to zero as well. However, to maintain continuity between the stringer and the shell in the z-direction, we impose the condition

to determine 2 0 ~ 2 0 after obtaining the solution for the displacement components that deform the shell;

i.e., Fourier coefficients LVlOn, n = 1, ..,, N , are taken to be non-zero independent degrees of freedom since the stringer coefficient w,?O is not a part of the solution vector.

Finally, consider the constraint equation associated with 6Aor0, the spatially uniform component of the interacting moment intensity, which was omitted in the series given by Eq. (70). Derived from Eq. (49), this constraint equation is

M

[ c a m W 2 r n 0 + dsro16~sro = 0 (81) m = l

We equated the constant component of the twist, c#JerO, to zero from the considerations associated with Eq. (79). Consequently, a non-zero value of the constant component of the interacting moment intensity, Aero # 0, would not contribute to the equilibrium of the ring, since Aero and 4oro are conjugate variables in the external work for the ring (refer to Eq. (31)). Since 4erO = 0, it is neces- sary that Aero = 0 to achieve consistent conditions for the torsional and out-of-plane bending equilibrium of the ring. With 6Asro = 0 in Eq. (81), the bracketed term does not necessarily vanish, and as a result pointwise rotational continuity betwen the shell and the ring is not assured. Pointwise rotational continuity is achieved if we take the coefficients wzm0 = 0, rn = 1, ..., M , as is done in the Fourier Series for the normal displacement of the shell given by Eq. (53).

DISCRETE EQUATIONS AND THEIR SOLUTION

Transverse Shear Deformation Model

The discrete displacement vector for the shell is the (lOMN + 3M + 3N + 2) x 1 vector

T

1742

where m = 1, ..., M The ( 6 M + 1) x 1 discrete displacement vector for the stringer and ( 6 N + 1) x 1 vector for the ring are

Ust r = [q i , ~ ~ 1 1 , ~ 3 2 1 , w s i i , wsal , desi 1 des2i ...(

U , i M , U d 2 M , w s l M , w s 2 M i ~ B d 1 M , ~ B s ? M ] T

(85) U r i n g = [WrO, url) urlr wr1, berl- 4m-1, d r r l t U r N 3

(86)

T VrN t WrN 7 derN, d i r N 7 dzrN]

in which the term w,o for the stringer has been omitted as discussed in reference to Eq. (80). The 4M x 1 discrete interacting loads vector for the shell- stringer interface and ( 5 N + 1) x 1 vector for the shell-ring interface are

The discrete displacement vector for the shell is the ( 6 M N + 2M + 2M + 2) x 1 vector

in which subvectors are

um = [Ulmol w1rn01 Ulrnlj ~ m 1 , u1rnll V Z r n l l w l m l ,

w2ml , ..., UlmN, UZrnN, ~ l m N ~ V Z r n N r ~ l m N 1

WZmN] T

(911 where rn = 1, . . . l A4

The (4M + 1) x 1 discrete displacement vector for the stringer and (4N+ 1 ) x 1 vector for the ring are

(92)

(93)

The 4M x 1 discrete interacting loads vect,or for the shell-stringer interface and ( 5 N + 1) x 1 vector for the shell-ring interface are the same as for t h e sliear deformation model and are given by Eqs. ( 8 7 ) and (88), respectively.

The approximations in Eqs. (51) through (64) for the displacements and Eqs. (65) through ( 7 1 ) for the interacting loads are substituted into the virtual work functionals for each structural element, and also substituted into the variational form of displacement continuity constraints. Then integra- tion over the space is performed. (The test space of virtual displacements and the virtual interacting loads is the same space used for the approximations in Eqs. (51-71).) This process results in a lOMN + 13M + 14N + 6 system of equations for the transverse shear deformation model and 6 M N + 10M + 11N + 6 system of equations for the classical model, governing the displacements and the interacting loads, The governing equations are of the form

= { 1) where the system matrix [L ] is given as

[Ll =

(94)

in which sub-matrices K l l , I i22 and K33 are the stiffness matrices for the shell, stringer, and ring, respectively. The sub-matrices Bij i, j = 1 ,2 ,3 , in Eq. (94) are determined from the external virtual work terms involving the interacting loads, and the constraint Eqs. (48) to (50). The vector on the right-hand-side of Eq. (94) is the force vector, determined from the external virtual work terms involving pressure. The constrailit equations correspond to the last three rows of the partitioned matrix in Eq. (94). Equation (94) is first solved for the displacements in terms of interacting loads, then this solution is substituted into the constraint equations

1743

to determine the interacting loads. Thus, the total solution is obtained.

NUMERICAL EXAMPLE

Data for the numerical example are represen- tative of a large transport fuselage str,ictixre. The shell radius R = 122.0 in., frame spacing 21 = 22 in., and stringer spacing 2 R 0 = 15.0 in The shell wall is a 13-ply [f45,90,0, &60,90, &60,0,90, f451, laminate of graphite-epoxy AS4/938 tow prepreg with total thickness of 0.0962 in. The ply thickness is 0.0074 in., and the lamina material properties are assumed to be E1 = 19.21 x 1061b/in.', E:! = 1.36 x 1O6lb/in.', GI' = G13 = G23 = 0.72 x 1061b/in.2, and v12 = 0.32. For the transverse shear deformation model, the shell wall stiffness sub-matrices of Eq. (23) are computed using these ply data and the exirtessions for the stiffness elements given in the Appendix. The numerical results are

0.5774 0.2619 i ] x lo6 lblin.

0 0.2889

I 3.893 0 0.1847 0.1847 B = [ 0 -5.043 -0.2213 0.2213 lb

0 -0.443 -1.1351 1.1351

D =

-474.9 256.1 45.1 0 256.1 615.2 54.0 - 0 . 4 7 ~

10-5

10-5

10-5 10-5 10-5

45.1 54.0 277.0 - 0 . 7 5 ~

0 - 0 . 4 7 ~ - 0 . 7 5 ~ 0 . 7 5 ~

lb in.

and elements of the transverse shear stiffness matrix in Fq. (24) are

A44 = A55 = 0.69264 x lo5 lblin., A45 = 0

The bending and stretching-bending coupling sub- matrices for classical lamination theory, Eq. (38), are given by [ 474.9 256.1 ]

D = 256.11 615.2 lb in. B = 0 0 0 277.0

The extensional stiffness sub-matrix A is the same for classical theory and the transverse shear deformation theory.

'Cross sections of the stiffeners and their dimen- sions are shown in Fig. 4. The stringer is a 12- ply [*45, 0 2 , 90, f 15 ,90 , 0 2 , &45], laminated hat- section of graphite-epoxy AS4/938 tow prepreg tape with a total thickness of 0.0888 in. The stiffenesses in Hooke's law for the stringer in Eq. (30) are given by

( E A ) s = 0.6675 x 1071b, (GA)s = 0.843 x 1061b

( E I ) s = 0.2141 x lo7 lb in.2

We assume a 2-D braided frame consisting of 0' and 90" tows. The wall thickness is 0.141 inches, tnd the elastic modulii are assumed to be E1 = 7.7G x 1O6lb/in.', E2 = 8.02 x 1061b/in.', GI' = G13 = G23 = 1.99 x 1O6/b/in.', and VI:! = 0.187. Usirig the ring material properties and the cross sectional data, the stiffeness matrix for the ring in Eq. (33) is computed, and the non-zero numerical results are given by

EA = 0.91 x 1071b, EI,, = 3.92 x 1071b in.',

EI,, = 0.187 x 1071b in.', EI,, = 0.299 x lo7& in.',

EI,, = -1.3 x 1071b ~ I Z . ~ , G J = 0.135 x 1051b in.',

EI,, = 1.7 x 1071b in.4, EI,, = -0.187 x 1061b in.3,

GA,e = GA,e = 0.2396 x lo7 Ib

The contour warping function of the top flange of the ring in contact with the inside surface of the shell is

where

wo = 1.0132 ina2 and w1 = 2.7535 in.

All the results presented are for an internal pressure p = 10 psi, and the Fourier Series are truncated at twenty-four terms in the x- arid &directions ( M = N = 24).

RESULTS AND DISCUSSION

Interacting Load Distributions

The distributions of the interacting load intensities between the stiffeners and the shell are shown in Figs. 5 through 11. The effects of transverse shear

1744

deformations and of warping of the ring’s cross section due to torsion on the magnitudes of the interacting line loads are summarized in Table 1. For the component Azs tangent to the stringer (Fig. 5 ) ) there are only small differences in the distributions as predicted by the four structural models. How- ever, the peak value of the component normal to the stringer, A,,, is reduced in the transverse shear deformation models with respect to its peak value in the classical models (Fig. 6 and Table 1).

The distributions of axial force intensity, A,,, between the ring and shell predicted by the classical and shear deformation models with warping are nearly the same (Fig. 7). However, the distributions of this force intensity predicted by the classical and shear deformation models without warping have significant differences. Thus, this interacting load intensity is more sensitive to the inclusion or exclusion of warping of the ring’s cross section into the strue- tural model. As shown in Fig. 8, the differences in the results for circumferential force intensity, A e r ,

between the ring and shell from the four models are small, except in the vicinity of the stiffener intersection where the effects of including the transverse shear deformation into the models are manifested. However, the differences in Aer occur over one wave length of the highest frequency i.e., AO/@ = 2/24. Differences occuring over the shortest wavelength may not be significant; more terms in the Fourier Series are required to verify this. The distributions of the normal force intensity, Atrl between the ring and shell predicted by the four models are essentially the same (Fig. 9). The distributions of the Circumferential moment component, Aer , predicted by the classical models have higher magnitudes as compared to shear deformation models (Fig. 10 and Table 1). Also note the change in sign of ne,. distributions in the vicinity of the joint as a result of inclusion of warping into the models. The classical theory predicts much larger magnitudes of normal moment component, A,,-, compared to the transverse shear deformation theory for the models in which warping is included (Fig. 11 and Table 1). However, the reverse is true for the structural models with warping excluded. Also, there is a change in sign in the distributions of A,, for classical models with and without warping effects.

The distribution of the normal component of the traction across the width of the attachment flasqe of the ring is represented by iine force intensity A t r

and line moment intensity A@,-. The values of A z r

are nearly the same in the classical and transverse shear deformation models (Fig. 9), but magnitudes of Ae, are substantially decreased in the transverse shear deformation models with respect to the classical models (Fig. 10). Thus, the asymmetry of the normal traction across the flange width of the ring is decreased in the transverse shear deformation models with respect to the classical models.

The distribution of the circumferential component of the traction across the width of the attachment flange of the ring is represented by line force intensity Aer and line moment intensity Alp . The values of ABr are nearly the same in the classical and transverse shear deformation models (Fig. 8). However, the magnitude of A,, is substantially in- tread in the transverse shear deformation model with respect to the classical model with warping excluded, and is substantially decreased in the transverse shear deformation model with respect to the classical model with warping included (Fig. 11). Thus, the asymmetry of the circumferential traction across the flange width of the ring is increased in the transverse shear deformation model with respect to the classical model without warping, and is decreased in the transverse shear deformation model with respect to the classical model with warping.

For the stiffened shell configuration with asymmetrical cross section ring, the inclusion of transverse shear deformation and warping of ring’s cross section into the analyses influences the distributions and magnitudes of interacting line load components A,,, A i r l Aer , ne,, and A z p . The distributions of interacting line load components A,, and A,, remain essentially the same. The cause of sensitivity to transverse shear deformations is two-fold: First, the tangential displacements of the shell along the contact lines are de-coupled from the out-of-plane rotations of the reference surface of the shell, and for the stiffeners the longitudinal displacements along the contact lines are de-coupled from the rotations of the longitudinal reference axes. Second, in the transverse shear deformation model, the torsional rotation of the ring at the shell-stringer-ring joint is de-coupled from the in-plane bending rotation of the stringer at the joint, thereby allowing for increased joint flexibility. In the classical model, the torsional rotation of the ring at the joint is constrained to be the same as the bending rotation of the stringer (see Fig. 12). The values of these joint rotations

1745

Table 1: Effect Of 'Ikansverse Shear And Warping On The Interacting Load Intensities And Their Distributions Along The Contact Lines'

~~

Peak Values of the Interacting Load Intensity - Comments on the

Warping' Warping Warping Warping number Neglected Included Neglected Included

CLTb Model CLT Model SD'pd Model SDT Model distribution, and figure Component

130. (9 xfl = - 0.22

Antisymmetric; Non- zero over entire stringer length; Fig. 5

131. @ 132. @ 127. @ Ib/in . dl = - 0.22 dl = - 0.22 x/l = - 0.21

ha lb/in.

767. @ x/l = - 0.02

613. @ di = 0.01

571. @ X A = O

571. (3 X n = O

Symmetric; Small magnitudes except near origin; Fig. 6

A X 1

Ib/in. 90.8 @ e m = o

74.6 @ e/@ = 0

10.7 43 e/o = o

80.7 (3 6/@ = 0

Symmetric; Small magnitudes except near origin; Fig. 7

Antisymmetric; Non- zero over entire ring length; Fig. 8

Symmetric; Small values except near origin; Fig. 9

her lb/in.

63.2 @ e/@ = i-0.04

55.2 @ 49.0 @ e/@ = k0.04 e m = 3 .21

A;, Ib/in.

- 886. @ e/o = 0

- 883. 43 e m = o

- 854. @ e/o = o

- 852. 0 e/o = o

4 1

in.-lb/in. - 198. @ e m = o

73.1 @ e/@ = 0

- 30.2 @ e m = o

22.1 @ e m = o

Symmetric; Nearly zero except near origin; Fig. 10

4 1 in.-lb/in.

3.21 @ - 3.95 @ - 1.55 @ Antisymmetric; Nsn- 7 e/@ = f0.20 zero over entire stringer

length; Fig. 11

a. Results for Fourier series truncated at 24 terms in the axial and circumferential directions. b. CLT is classical iamiliation theory. c. Out of plane warping of the ring's cross section due to torsion d. First order transverse shear deformation theory

010 = f

1746

for the four structural models are given in Table 2. Notice from Table 2 that the sense of the rotation changes if warping is included, and that the transverse shear deformation results in a torsional rotation of the ring that is about twice as much as the bending rotation of the stringer.

Resultants at the Stiffener Intersection

The interacting line load intensities acting on the inside of the shell wall can be resolved into a resultant at the stiffener intersection (x = 0 = 0). In general this resultant consists of a force with components F,, Fe and F,, and a couple with moment components Cr, Ce and C,. These components are shown in their positive sense on the inside of the shell wall in Fig. 13. The components of the resultant force vector are defined by

(95)

1 2

Fe = 1 [A,,. Cos0 + A,, SinB] ( R - -) d0 (96)

F, = A,,dz+ j! [X,,CosO - XerSinB](R - - )de,

and the components of the moment resultant of the couple are

-0

t 2

- I - Q

(97)

t 2

C, = 1 [A,,.Sin0 - ( 1 - CosB)Aer] ( R - - ) 2 d0 -0

(98) I

0 (99)

t

It is found tha t substituting for interacting load approximations given by Eqs. (65-71) into Eqs. (95- loo), and performing the line integrals results in

+ A,,COSO] ( R - ? ) d e

components Fr = Fe = Cx = C', = 0. Thus, at the stiffener intersection, the only non-zero rcsul- tants are a radial force resultant, F,, and a circuinferential moment resultant Ce (refer Fig. 13). In Eq. (99) the circumferential moment component, Ce, consists of two line integrals; the first integral being the contribution of shell-stringer interacting loads, and the second representing the contributions of shell-ring interacting loads. The contribution to the radial force resultant F, in Eq. (97) comes only from the shell-ring interacting load intensitites since the resultant from the stringer vanishes by Eq. (73) .

The values of the radial force and circumferential moment resultants are computed using Eqs. (97) and (99) for the four structural models under consideration, and are given in Table 3. The differences predicted by the four structural models in the magnitudes of the radial force resultant F, are very small, and are within 0.4% of each other. There are substantial differences i n the magnitudes of circumferential COP. onent of the moment predicted by the four mode'.,. The values of Ce predicted by the models with -, tarping included are much larger than those predicted by the models without warping effects. The individual contributions of the stringer and ring t o Ce are also affected by the change in the model as shown in Table 3, It may be noted that Ce is more sensitive to the effect of warping than to transvcrse shear deformation.

Singular Behavior at the Joint

In Table 1 the comparison of peak values of the interacting liae load intensities is meant t,o convey the influences of transverse shear deformations and warping in the structural modeling. The peak values of components AIS , Axrr A,,, and Aer occur a t the joint, but these peak values do not exhibit conver- gence with an increasing number of terms retained in the Fourier Series. It appears that these components are singular at the joint. However, the resultants F, and Ce determined from these line load intensities are found to converge quite rapidly. See Johnson and Rastogi7 for further discussion of this point.

Stiffener Actions

The Jistributions of the force and moment resultants in the stiffeners are shown in Figs. 14 through 19. The stringer axial force and bending

1747

a b l e 2 Rotations About The Circumferential Axis At The Stiffener Intersection.

Rotations in radians ~~~

Classical Theory Transverse Shear Theory Description of the Rotation of the Structural Component

No Warping Warping No Warping Warping

Shell normal I$x (0,O) - 2.56 2.58 - 1.06 2.65

Ring twist (Der (0) - 2.56 2.58 -2.67 3.54

Stringer normal I$e, (0) - 2.56 2.33 -0.29 1.85

mble 3: Resultants At Stiffener Intersection.

Classical Theory Transverse Shear Theory

No Warping Warping No Warping Warping

Components of the Resultant

C, from stringer, - 1.1696 - 0.0921 - 0.2953 - 0.7797 lb-in.

Ce from ring, Ib-in. 1.627 5.645 1.363 6.0192

Ce total, lb-in. 0.457 5.5526 1.0676 5.2396

F,, lb. - 564.06 - 564.56 - 562.27 - 563.15

moment distributions (Fig. 14) are slightly asymmetric about the origin. The bending moment distributions in the stringer are more sensitive to the change in models as compared to the axial force distributions. The distribution of stringer shear force, V,,, is shown in Fig. 15, and it is asymmetric about the origin. Only small differences are predicted by the four structural models in the distribution of V,,.

The distributions of the circumferential force and in-plane bending moment in the ring are shown in Fig. 16. The differences in these distributions predicted by the four models are very small. The distributions of the in-plane shear force, Vzr, in the ring predicted by the four structural models have negligible differences, as shown in Fig. 17. The out- of-plane bending moment Mz, and torque T r in the ring are more sensitive to the change in models as shown Fig. 18. The distributions of the out-of-plane bending moment are symmetric about the origin, and their magnitudes predicted by the models with warping included are substantially larger as compared to the magnitudes predicted by the models without warping. The distributions of total torque, Tr (= T,, - &fur/&), are antisymmetric about the origin. As shown in Fig. 18, the torque has reduced magnitudes in the transverse shear deformation model compared to the classical model when warping is included. The torque predicted by the models without warping is St. Venant’s torque Tar , and this is negligible as shown in Fig. 18. The distributions of out-of-plane shear force, V,,., in the ring are shown in Fig. 19, and these distributions are antisymmetric about the origin. The magnitudes of VZr predicted by the transverse shear deformation model are larger compared to the classical model when warping is included. However the reverse is true for the Vz+ distributions without warping. The distributions for Me,, Vz,, Adz,, Tr and Vs, shown in Figs. 14, 15, 18 an3 19, respectively, indicate that these stiffener actions are sensitive to both transverse shear deformations and warping deformations.

Shell ResDonse

The distribution of the normal displacement along r-curve midway between the stringers ( 0 = -e), and along the &curve midway between the rings (z = -!), are shown in Fig. 20. As depicted in this figure, there is a negligible difference between the results from the transverse shear deformation model and classical model (warping is included in

both models). Also, there is negligible difference in the axial and circumferential normal strain distributions between the two models as shown in Figs. 21 and 22. Thus, the normal displacement and in-plane normal strains for the shell are not significantly influenced by the inclusion of either transverse shear deformations or warping effects into the structural models, in part because the shell is very thin for the example studied.

A Ring with Symmetric Cross Section

As a benchmark for comparing transverse shear deformation model with the classical model, analyses were performed for a ring with symmetrical cross section. In this case the changes made to the numerical example under discussion arf: to set the bending-coupling stiffeness EI,, , the out-of-plane bending to warping coupling stiffness EI,,, and the contour warping function parameter wo of the ring, all to zero. Consequently, the @-axis, as well as the z-axis, are axes of symmetry for the repeating unit in terms of geometry, load, and material properties. Symmetry about the 0-axis implies there is no out-of-plane bending and torsion of the ring; i.e., .,(e) = der(8) = d z r ( 8 ) = &rr(d) = &+(e) = h t r ( 0 ) = 0 for -0 5 6 5 0. Thus, for the symmetrical section stiffeners only the interacting line load components tangent and normal to the stiffeners are non-zero. Since the internal pressure loading is syrn- metric, warping of the ring’s cross section does not play any role in the analyses.

The distributions of the tangential interacting load intensity between the shell and ring are shown in Fig. 23. The differences in the results from the two models are small except in the vicinity of the stiffener intersection. The peak magnitude of Xer in the transverse shear deformation model is smaller than the peak value for Aer in the classical model (50.8 lb/in versus 64.5 lb/in,). However, this difference occurs over one wave length of the highest harmonic retained in the analysis, and may not be significant. The distributions of the tangential and normal interacting load intensities between the shell and stringer, and the normal load intensity between the shell and ring are not significantly different in the two models.

For a symmetrical cross section ring, in Eqs. (95) through (100) Fz = Fe = C, = Ce = C, = 0. The only non-zero component of the force resultaqt

1749

is the radial force F,. The values of F, computed from the classical and transverse shear deformation models are -563.72 lb. and -561.89 lb., respectively.

CONCLUDING REMARKS

The asymmetrical section ring complicated the analysis of the unit cell, since symmetry about the plane of the ring is lost. Out-of-plane bending and torsion of the ring occur as well as a rotation of the shell-stringer-ring joint about the circumferential axis. Inclusion of transverse shear deformations, and warping deformation of the ring’s cross section due to torsion, into the mathematical model significantly influenced several aspects of the response.

The sense of the rotations of the structural elements at the joint is changed with the inclusion of warping deformation in the ring, and the twist rotation of the ring at the joint increases by 40% with the inclusion of transverse shear deformation, as is shown in Table 2. That is, joint flexibility increases since element rotations at the joint are de-coupled by using transverse shear deformation models.

The interacting loads that are strongly affected by the structural modeling are those components associated with the asymmetric response. These are :he axial force intensity Azr , the normal moment intensity A,, , and the tangential moment intensity A@, between the ring and shell. At the joint the magnitude of Azr is increased by the inclusion of warping and reduced by the inclusion of transverse shear (Fig. 7). The normal moment intensity is a measure of the asymmetry in the distribution of the circumferential traction across the width of the ring’s attachment flange. The sense of Azr is changed when both transverse shear and warping are included, and its magnitudes are reduced by the transverse shear effect (Fig. 11). However, the magnitude of the resultant of the circumferential traction across the flange width of the ring, as measured by the line load intensity Aer, is only moderately affected by the changes in the structural mljdels (Fig. 8). The tangential moment intensity As, is a measure of the asymmetry in the distribution of the normal traction across the width of the ring’s attachment flange. At the joint, the sense of A@, is changed by the inclusion of warping deformation and additionally its magnitude is reduced by the inclusion of transverse shear deformation (Fig. 10). However, the magnitude of the resultant of the normal traction across the flange width, as measured by

the line load intensity A I , , is essentially unaffected by changes in the structural models (Fig. 9) .

The distributions of the normal actions between the shell and stiffeners ( A z s , A,,, and A e r ) all show significant magnitudes only in the vicinity of the joint, with much smaller magnitudes away from the joint. In fact, they all appear to exhibit singular behavior at the joint, but only finite magnitudes are represented by the truncated series approximations. The distributions of the actions tangent to the stiffeners (A,,, A o r , and Az,) , on the other hand, have small magnitudes in the vicinity of the joint and larger magnitudes away from the joint. These tangential actions do not exhibit singular behavior.

Inspite of the singular behavior of the line load intensities associated with the normal actions, the resultant of these distributions resolved at the joint converge relatively quickly with the number of terms retained in the series approximations. The resultant consists of a radial force F, and a moment Cs about the circumeferential axis. F, represents the action of the stiffeners to pull the shell radially in- ward against the pressure load, and Ce is primarily due to asymmetry in the actions between the ring and shell. Force F, is essentially unaffected by the structural modeling (Table 3), and its magnitude for the example studied is about 17% of the total pressure load carried by the unit cell. The remain- ing pressure load is carried by the shell itself. The moment Ce is very sensitive to the structural modeling, in particular to the effect of warping as shown in Table 3. This moment Ce vanishes for a completely symmetric problem.

The out-of-plane bending moment and torque in the ring are very sensitive to the structural modeling, as might be expected. The magnitudes of the both the out-of-plane bending moment and torque increase with the inclusion of warping and transverse shear into the mathematical model (Fig. 18). However, shell’s normal displacement and strains are insensitive to the changes in the structural models for the very thin shell (R / t =; 1268) used in the numerical example.

ACKNOWLEDGMENT

This research is supported by NASA Grant NAG-1-537, and Dr. James H. Starnes, Jr., NASA Langley Research Center is the technical monitor.

1750

REFERENCES

lJackson, A. C., Campion, M . C., and Pei, G., “Study of Utilization of Advanced Composites in Fuselage Structures of Large TPansports,” N A S A Con t ra ct o r Report 1 724 04, Contract N AS 1 - 1 74 15 , September, 1984.

211cewicz, L. B., Smith, P. J., and Horton, R. E. , “Advanced Composites Fuselage Technol- ogy,” Proceedzngs of the T h a d N A S A / D o D Ad- vanced Composztes Technology Conference, NASA CP-3178, 1992, pp. 97-156.

3Swanson, G. D., Ilcewicz, L. B., Walker, T . H., Graesser, D., Tuttle, M., and Zabinsky, Z . , ‘ ( L e cal Design Optimization for Composite Transport Fuselage Crown Panels,” Proceedzngs of the Second N A S A / D o D Advanced Composztes Technology Con- ference, NASA CP-3154, 1992, pp. 243-262.

4Jackson, A. C., “Development of Textile Com- posite Preforms for Aircraft Structures,” Proceed- ings of the A I A A / A S M E / A S C E / A H S / A S C 35ih Structures, Structural Dynamacs and Matenals Con- ference (Hilton Head, SC), Part 2, Washington, D.C., April 1994, pp. 1008-1012, (Paper 94-1430).

Hoff, N . J . , “The Applicablity of Saint Venant ’s Principle to Airplane Structures,,’ Journal of Aero- nauiaca! Sczence, Vol. 12, 1945, pp. 455-460.

‘Wang, J . T-S., and Hsu, T-H., “Discrete Anal- ysis of Orthogonally Stiffened Composite Cylindri- cal Shell,” A I A A Journal, Vol. 23, No. 11, 1985, pp. 1753-1761.

7Johnson, E. R., and Rastogi, N . , “Interact- ing Loads in an Orthogonally Stiffened Composite Cylindrical Shell,” Proceedzngs of the A I A A / A S M E / A S C E / A H S / A S C 35th Structures, Structural Dy- namics and Matenals Conference (Hilton Head SC), Part 5, AIAA, Washington, D.C., April 1994, pp. 2607-2620, (Paper 94-1646).

‘Novozhilov, V. V. , Thzn Shell Theory, Noord-

9 W ~ ~ d ~ ~ n , M . B., Johnson, E. R., and Haftka, R. T., “A Vlasov Theory for Laminated Composite Circular Beams with Thin-walled Open Sections,” Proceed angs of the A IA A / A S M E / A S C E / A HS/A SC 34th Strzlctures, Structural Dynamtcs and Materzals

hoff Ltd., The Netherlands, 1964, p. 23.

Conference (Ldolla , CA), Part 5, AIAA, Washing- ton, D.C., April 1993, pp. 2742-2760, (Paper 93- 1619).

l’Sanders, J . L . , “An Improved First Approx- imation Theory for Thin Shells,” N A S A Technical Report R-24, June, 1959.

”Jones, R. M., Mechanics of Composite Mate- rials, Scripta Book Co., Washington DC, 1975, pp. 51, 154, 155.

APPENDIX

Based on the transverse shear deformation theory the elements Ai j , Bij , and Dij of the stiffness matrix in Eq. (23), for the constitutive law for a laminated shell wall, are given by

1751

where Qij are the transformed reduced stiffnesses given in the text by Jones". The transverse shear stiffnesses in Eq. (24) are given by

A44 =/C44(1+ t i ) d z

2 -1 A55 = /Cs~ ( l+ t z) dr

where C44 =G~~COS'CY + G23Sin2cu Cas =(GI3 - G?3)CosaSina C55 =G~~COS'CY + G13Sin2CY

in which CY is the ply orientation angle.

e \

c

Fig. 1. stiffened cylindrical shell.

Repeating unit of an orthogonally

i"

Fig. 2. Interacting line load intensities shown in the positive sen- acting on the inside surface of the sheil.

1752

f OZ

Stringer Q = 1.4 in. e, = 0.8151 in.

Fig. 3. (b) stringer, and (c) ring

Displacements and rotations for (a) shell,

1 b J Ring h = 5.6 in. b = 1.4 in. e, = 2.854 in.

Fig. 4. Stiffener cross sections.

1753

Claaalcal theory ___ Shear delorm. theory - Claaalcal thaory w/o warping _ - Shear delorm. theor9 w/o warping 1 26.27

ClSSSlClll theory - -- Shear deform. theory - Clasrlcal theory wlo warplng - .

140.1

I I 1 3 105.1 600.0

Shear deform. theory w/o warping

n 800.0 150.0 r

19.70

13.13

6.57 2-

0.00

$ -6.57 3

CI

-13.13

-19.70

100.0

50.0 400.0 70.1

IY n E

0.0 c4 C c p 200.0 Y

35.0 2 3 3 Y

m dX

-50.0 0.0 0.0

-100.0 I -200.0 C I

\I I -35.0 1 -150.01 ' ' ' ' I " ' ' I ' ' ' ' I '

I ' I -26.27 -400.0 1 -70.1

-1 -0.5 0 0.5 1

X/l

-1 -0.5 0 0.5 1 xll

Fig. 5 in axial direction.

Stringer-shell tangential force intensity Fig. 6 Stringer-shell normal force intensity.

Claaslcal theory -- Shear deform. theory - - -

. Classical theory wlo warping Shear deform. theory wlo warping

- Shear deform. theory . . CIassIcaI theory - _ _ .Shear deform. theory wlo warping -

- . - Clasaical theory wlo warping

- -

-60.0

80.0

60.0

40.0

14.01

9.34 75.0 13.1

8.8 4.67

n 50.0 E c P Y

- 20.0 P C c Y 0.0

n s 3 I

A

0.00 3 3 Y

L 25.0 dX

4.4 2 -20.0

-4.67 0.0 0.0

-40.0

-9.34 t -25.0 1 -4.4

J

'80.0 L L - -14.01 -1 -0.5 0 0.5 1

we -50.0 3 -8.8

-1 -0.5 0 0.5 1

ole Fig. 8 F!ln$shell tangential force intensity in circumferential d;rectiori. Fig. 7 Ring-shell tangential force intensity in axial direction.

1754

200.0

0.0

-200.0 h E = - -400.0 P . AN

Claaslcal lheory - Shear deform. lhaory - CIaeslcal lheory wlo warping - . Shear deform. theory wlo warping

35.0

0.0

-35.0

-70.1

-600.0 -

-000.0 -

-1000 .0 - ' " " " " " ' ' I ' " ' -175.1 -1 -0.5 0 0.5 1

om

Fig. 9 Ring-shell normal force intensity.

-105.1

-140.1

n c = c .I

5.0

2.5

0.0 L

4"

-2.5

-5.0

A z 3 -

Shear delorm. theory CIaaaIcaI lheory Shear deform. theory wlo warping CIaasIcaI theory wlo warping

- _-

_ _ 100.0 _ _

50.0 i 0.0 0.0 >

9 - z - -222.4 3

' I z 3 3 ' I - -444.0 'Y

C

2 -50.0

CI

& < -100.0

-150.0

i /

' I -667.2 i L 1

I -200.0 " " I ' ' " " " I " " -889.6

-1 -0.5 0 0.5 1

818

Fig. 10 Ring-shell tangential moment intensity in circumferential direction.

Shear deform. theory . . . . . . . . - Classical theory - - -Shear deform. theory w/o warping

Classical theory w/o warping - . \_ - . . . ... . . , ---. ..

-

-

' I'

. . \ - J ;" * . I - . . . *

, , 1 , , , , 1 , , , , 1 , , , ,

-1 -0.5 0 0.5 1 WO

22.24

14.83

7.41 n z

0.00 3 3 3

-7.41 2

-14.83

-22.24

Fig. 11 Ring-shell normal moment intensity.

1755

7

$,tO,O) = cp,,(O) = cP,r(0) $x (090) + $e8 (0) + $er (0)

(a) (b)

Fig. 12. Rotations at the joint in (a) classical theory, and (b) transverse shear theory.

Fig. 13. Components of the resultant of the interacting line load intensities resolved on the inside wall of the shell at the origin.

1756

300.0 r

.

Shear deform. theory

. Classical theory

. Shear deform. theory wlo warping LI

U Classical theory wlo warping

9 50.0

4000.0

n E 9 3000.0 Y

.- L

3" 5 2000.0 CI s Y

zg 1000.0

-1 .o

-

- . - Shear deform. theory

Classical theory

- --- Shear deform. theory wlo warping

- Classical theory w/o warping

-

-0.5 0.0 0.5 1.0

dl

Fig. 14 Stringer axial force and bending moment distribution.

\

M.. I o.om

-1000.0 \ -1 .o -0.5 0.0 0.5 1 .o

019

Fig. 16 Ring circumferential force and in-plane bending moment dirtribution.

. Claaalcal theory - Shear deform. theory - . Cla8alcal theory wlo warplng -_ . Shear deform. theory wlo warplng

2oo.o F 150.0

100.0

50.0 n e

0.0 Y

3 > -50.0

-100.0

-150.0

667.2

444.8

222.4 <

E 0.0 A

z Y

-222.4

-444.8

-667.2

-200.0 7 -889.6 -1 -0.5 0 0.5 1

xll Fig. 15 Strlnger trenaverse .ahear force dlstributlon.

150.0

- Shear deform. lheory

- - - Shear deform. theory wlo warplng - . Classical theory wlo warplna

. . Classlcal theory

667.2

100.0

50.0 CI s >* cc . 0.0

-50.0

-100.0

-150.0

444.8

222.4

c - 0.0 h

2 - -222.4

-444.8

-667.2 -1 -0.5 0 0.5 1

018

Fig. 17 Ring in-plane shear force distribution.

1757

Shear deform. theory . . . . . . . Classical theory

60.0 I . - Classical theory ~ I O warping - - . Shear deform. theory wlo warping - -

-

-80.0 I , , , , I , , , , I , , , ,

-1 .o -0.5 0.0 0.5 1 .o

6779.1

451 9.4

2259.7

0.0

-2259.7

-451 9.4

-6779.1

-9038.8

Fig. 18 Ring out-of-plane bending moment and torque distributions.

= >*

I Y

L

25.0

16.7

8.3

0.0

Classical theory

- - - - _

7-

Shear deform. theory ......... Classical theory - _ - Shear deform. theory wlo warping

wlo warping -

- - - ~ .

I

-

-

I ,

1 -25.0

-1 -0.5 0 0.5 8/63

111.2

74.1

37.1

0.0

-37.1

-74.1

-1 11.2

Fig. 19 Ring out-of-plane shear force distribution.

1758

Fig. 20 Distribution of shell's normal displacement.

/ Shear deform. theory

Classical theory

0.0020 __

E 0.0016

0.001 2

'ee

0.0008

inner

0.0025

I -- 0.0020


Classical theory

0.0010

€XX

0.0005

0.0000

inner L - L -0.0005

-0.0010

-0.0015 -1.0 -0.8 -0.6 -0.4 -0.2 0.0

xll

Fig. 21 Distributions of the axial normal strain on the inner outer shell sarfaces midway between the stringers (e P - 0).


Classical theory 14.01

9.34 60.0 f 40.0

i outer I

I

B -2

-20.0

-40.0

-60.0

I I I I ---I 0.0000 ' -1 .o -0.8 -0.6 -0.4 -0.2 0.0

818

-80.0 -1 -0.5 0 0 5 1

019

4.67 P z h

0.00 $ 3 Y

-4.67

-9.34

-14.01

Fig. 22 Distributions of the circumferential normal strain on the inner snd outer shell surfaces midway between the rings (x = - I). Fig. 23 Ring-shell tangential force intensity in circumferential

direction for a ring with symmetrical cross section.

1759

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