An effective curved composite beam finite element based on the hybrid-mixed formulation

Pergamon

0045-7949(94)EO259-5

Compuars d Sauctures Vol. 53, No. I, pp. 43-52. 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Gmt Britain. All rights mmvcd 004%7949194 s7.00 + 0.00

AN EFFECTIVE CURVED COMPOSITE BEAM FINITE ELEMENT BASED ON THE HYBRI D-MIXED FORMULATION H. R. Dorf~ and H. R. Busby7 Ohio State University, Department of Mechanical Engineering, 206 W 18th Ave. Columbus, OH 43210-1107, U.S.A. (Received 7 June 1993) Abstract-A laminated curved-beam finite element with six displacement degrees of freedom and three stress parameters is derived and evaluated. Both thermal and hygrothermal effects are included. The element is based on the Hellinger-Reissner principle and the hybrid-mixed formulation. The Timoshenko beam theory and classical lamination theory are employed in the finite element description. Within an element linear displacement interpolation is used; the generalized stresses are interpolated by either stress functions based on the equilibrium equations (P,) or constant stress approximation (PJ The beam element stiffness is obtained explicitly and numerical results show very good displacement prediction compared to analytical solutions. Generalized stresses are predicted accurately at the mid-point of the finite element only for constant stress interpolation. The P,-type element yields more accurate displacement and stress prediction.

1. INTRODUmION

one is the description of the geometry with a curvilinear coordinate system. For shallow arches a simplified theory may be used [lo, 111. In recent years the use of isoparametric degenerated threedimensional elements has gained more attention [12, 131,motivated by the systematic approach of the derivation. The literature on curved beams and corresponding finite element codes is extensive [lo, 11, 14-261 and several curved composite beams have been developed [27-351. Different variational principles are employed in the derivation of these elements. The most popular variational principle is the principle of minimum potential energy and the corresponding element models are usually denoted as displacement models. However, the variational principle can be generalized so that the variation of the functional with respect to the three field variables (displacement, stress, strain) recovers the field equations: the kinematic relationship, the constitutive law and the conservation law (linear momentum) [36]. This generalized principle is called the Hu-Washizu principle; all three field variables are approximated independently. By satisfying the kinematic and constitutive law a priori in the finite element formulation the displacement-based method is obtained. Since it is uneconomical to approximate all three field variables independently and sometimes undesirable to model only the displacement field, the Hellinger-Reissner principle is commonly used; it is derived from the general Hu-Washizu principle. The Hellinger-Reissner principle Type 1 satisfies a priori the constitutive law, thus displacement and stresses

Lightweight structures in aerospace applications have to be tailored to a maximum strength-to-weight ratio. This requires orthotropic or anisotropic material properties. Frequently, fiber reinforced matrix com-

posites are used, primarily because of their superior properties and the ease with which they can be optimized for a specific application. For these materials the thermal and hygrothermal properties are also much more pronounced than those of conventional materials [ 11. By laminating these materials in a suitable configuration, a number of desired structural as well as thermal characteristics can be designed. The beam is the simplest and most common structural element in aerospace, civil and mechanical engineering. Therefore its accurate analytical description has been of great interest. The classical analysis is based on the Bernoulli-Euler theory, assuming that planes normal to the neutral axis remain plane and normal to this axis during bending. Timoshenko [2] extended the theory lifting the assumption of normality and so allowing for shear deformation, an important effect in deep beams. The theory requires a shear correction factor to correct the strain energy of deformation. Higher order theories have been proposed to model the cross-sectional warping and to remove the shear correction factor [3-81. There are several approaches to the description of a curved-beam finite element [9]. The most popular t To whom correspondence should be addressed. 43

H. R. Dorfi and H. R. Busby

44

are varied independently. In Type 2, the displacements and strains are chosen as independent variables, a less common approach. Elements using these principles are referred to as mixed models because of the mixed use of field variables. Hybrid-mixed models are frequently computationally more effective. In these formulations multifield descriptions of the solution are still used, but some field variables are eliminated on the element level. There are numerous possibilities on how to formulate hybrid elements. A very successful formulation by Pian [37], the hybrid stress method, assumes compatible displacements between elements and stress variations that satisfy the equations of equilibrium. The stress parameters are eliminated on the element level. Thus a displacementbased finite element program can be used, since the global nodal point unknowns are the displacements only. In recent years attention has focused on the mixed and hybrid formulations of curved-beam elements. The effects of shear and membrane locking encountered in the classical displacement-based formulation of curved structural members [lo, 11,20,21] can be avoided in these formulations and lower order approximations in the displacement field result in simpler and more effective element [23,31,32]. In this paper a linear two-dimensional curved composite beam element will be presented, based on the Timoshenko beam theory and the hybrid-mixed formulation. The composite element model is based on the classical lamination theory and includes thermal and hygrothermal effects. It is important to note that composite beams with general lamination sequence may deform transverse to the load pfane. But in many problems stacking sequences are used, which do not exhibit this behavior. This makes a twodimensional analysis of curved composite beams practically useful. 2. MATHEMATICAL

FORMULATION

The curved-beam element is based on the Hellinger-Reissner principle (Type 1). For a linear elastic material, small deformations and a domain with volume V and surfaces S,, S,, it is given by [38]

r-I=

I

(-f&8

V

isfied, thus only the stress and displacement fields in the above functional can be varied independently. The last term in eqn (1) vanishes for compatible displacement fields. In the following derivation this assumption will be satisfied. Figure 1 shows a curved element with two nodes and six displacement degrees of freedom. At the two nodal points three displacements (u, LJ,0) are specified. The three stress resultants (N. Q, M) are given in terms of stress parameters and these are eliminated prior to element assembly, thus the element has six global degrees of freedom. The curved beam is modeled in the X-J plane, using a polar coordinate system. The beam is circular with radius R, opening angle ‘p,,, local coordinate cp and has a symmetric cross-section with respect to the y-axis. The stacking sequence and ply orientation is given by hr, the distance from the central axis to the top surface of lamina k and its ply angle s(~. For a deep curved beam, normal and shear-strain components cXand yXyin a material fiber at distance y from the reference surface are given by f23] Yo

Co-yli

t,r = -,

1 -y/R

yyr= 1 _y,R’

where the generalized strains eo, y. and K are defined by the linear strain+5splacement relationship t = Bu as

For a single ply denoted by k the stress-strain relations for a two-dimensional beam analysis are [39]

+ =

Gk,f(~h:y

where Ez is the effective Young’s modulus, Ct the shear modulus, q{ the coefficient of thermal expansion, ~2 the hygrometric coefficient and AT, Am the changes in temperature and moisture. The function f(y) in eqn (4) represents a parabolic weighting

tare ---f%

-(an + aH’)‘E) dV

where e denotes the stress vector, S is the compliance matrix, u is the displacement vector, L are the strains, T tractions on the boundary, f body forces, uTh, aHy are thermal and hygrothermal loads and (-) are prescribed quantities, The linear strain~ispla~ment relationship c = Bu is assumed to be implicitly sat-

(4)

Fig. I. Curved laminated beam geometry.

45

A laminated curved-beam finite element = k,[ 1 - y/(/~/2)~] and includes a shear Factor to approximate the shear stress distribution. Et, Gt, v(: and ~2 are found from the properties in principal material directions and the angle ak between the fiber direction and the beam length axis [39] function a

I (h:- h:- M

1 Z=~+(--&-~)coszo,sin2n,i~

I

(5) Gk,= G,, cos2 ak + Gz3sin* ak

and the thermal and hygrothermal are given by

(6) hk

vi: = rj, cos2 uk + q2 sin* aL

NTh=

@k

2

v:AT

s

k=l

PI: = p, cos2 ak + p2 sin2 uk.

k=l

dy

(8)

MT&--

=,$Il_,‘:Yb’dy

(9)

M”Y = -

5 E;bk k=l

h&

M=-

a,y dA = - 2 sA

a>ybk dy,

(10)

s hr_1

k=l

where NL denotes the number of laminas and bk the width of the kth layer. Substitution of the kinematic relations (2) and the constitutive equations (4) into the equations for stress resultants yields after simplification [40] N =

I

s s

v:ATYdv

hk -I

hk

P:A~Y

dy.

(17)

kk -I

While thermal and hygrothermal effects are included, it is assumed that the problems are uncoupled and the temperature and moisture distribution are thus given independently of the deformation state. Also the temperature variation of the elastic constants is assumed to be negligible. For the further analysis it is convenient to define the generalized stress vectors as

a,+&+ &?r60 1

-

B,+~D,+~F,

1 K -

_{;}

_LF Rh2 -

Bx+;D,+;Fx

xy

&=Hy={Z}.

(18)

With eqns (11-13) and (18) a generalized constitutive law can be defined as

-

4 R2h’H,.

1 1

1

Yo (12)

co

&fx

#TL{;}

NTh- NHY (11)

Axy+;Bxy+

Q=k,

M=

hk-

dy

hk

Epk

k-1

Q =J/,dA

I

area the

hk +”

hk

(7)

By integrating over the cross-sectional stress resultants can be found as

c

stress resultants

K - MTh-MHy,

(13)

u

=Cc -an-nHY,

(19)

where C denotes the stiffness matrix of the curved composite beam. The compliance matrix S is defined as S=C-‘. The Hellinger-Reissner functional for a curved beam can now be written in terms of generalized stresses and the displacement vector as

where the coefficients are defined as (-ja=S~+a%~)ds-

(14)

i

W

(20)

where s,, = Rq, is the length of the reference axis of the beam and W is the contribution of the external forces: the thermal and hygrothermal loads, the distributed load vector p = (p,, p,,~,~) in normal and

H. R. Dorfi and H. R. Busby

46

tangential direction and the nodal point loads. It is given by

where the stiffness matrix K is defined by

K = G%-‘G.

where NEN equals the number

of element nodes.

3. FINITE ELEMENT FORMULATION

In this section a curved composite beam finite element with two nodal points, six displacement degrees of freedom and three stress parameters will be developed. Following the usual notation in the literature the interpolation of the displacement vector u and the element stress vector u are written as u=Nq

a=P/J,

+ #IQ

- RTq,

(23)

where the matrices H and G are defined as H=

PrSPds s so

G=

Pr’RN ds s .%

[(oTh+ uHy)% + p7N ds $ R&,

P, = (25)

where R,, denotes applied nodal forces. So far the generalized stresses and the displacement vector have been treated independent and they will be modeled by independent interpolation functions. However, if we choose the stresses to be discontinuous along element boundaries but continuous within the element (C-i approximation), we can eliminate the stress parameters on element level, obtaining only nodal displacements as unknowns. Invoking the stationarity of the functional with respect to the stress parameter yields the necessary equation as fi = H-‘Gq.

(26)

With eqn (26) the functional (23) can be expressed in terms of the displacement vector only as I-I = fq%q - RTq,

There are several guidelines on how to select appropriate stress functions. For a review the reader is referred to Saleeb and Chang [23]. We will investigate both stress functions, which satisfy the homogeneous equilibrium conditions and constant stress interpolation within an element. The stress interpolation matrices are thus chosen as [40]

(24)

and the load vector R is found as

R’=

So far the general formulation of the finite element has been outlined. An important step is now the choice of appropriate interpolation functions. A twonode element will be considered herein and its stiffness matrix derived. Lagrange polynomials are chosen to interpolate the displacements. The degrees of freedom are tangential and normal displacement and rotation; the interpolation matrix N is thus given by

(22)

where N and P are the respective interpolation functions, q is the nodal displacement vector and /I the vector of stress parameters. Substitution of eqn (22) into (20) yields II = -f/VII/I

(28)

(271

cos rp -sin cp i -R(coscp - 1)

sinfp

0

costp -R sincp

0 I

(30)

In the following numerical evaluation we will refer to the two beam elements as being of P, or P, type de~nding on the stress inte~olation employed. Substitution of the displacement and stress interpolation matrices (29), (30) into eqns (24) and (28) yields the stiffness matrices K, and K, for the respective stress interpolation matrix. These matrices are listed in the Appendix. Since the integration can be performed explicitly, the stiffness are obtained in closed form. For a given tem~rature and moisture dist~bution and distributed loads the load vector is calculated based on eqns (16), (17) and (25). Since the stress parameters are eliminated prior to element assembly, it is straightforward to implement

A laminated curved-beam finite element

41

Table 1. Straight composite beam: material and geometric data Material

Geometry

Kevlar epoxy [39] E22=

5.5 GPa vu = 0.34

E,, = 76GPa G = 2.3 GPa

Stacking sequence: [90/45/-45/45/-45/O] beam length: 0.5 m 12mm beam height: 20 mm beam width:

the curved composite beam element into a conventional displacement based finite element program [41]. Based on the displacement finite element solution the generalized stresses are found from (26). For a laminated composite structure the local lamina stresses are of greater interest, since they determine the ply failure. To calculate these stresses from (2) and (4) the generalized strains have to be found. There are two possibilities on how to calculate the strains: based on the displacement approximation and the strain-displacement relationship (3) or the stress approximation and the generalized constitutive law (19). Since we approximated the stresses independently, it is preferable to calculate the local stresses based on the generalized stress approximation rather than the displacement approximation. 4. NUMERICAL

for further details. Two numerical examples are presented here. The results are compared to the corresponding theoretical solutions based on the Timoshenko beam theory and the classical lamination theory. Bending-extensional beam

coupling of a straight composite

A distributed triangular load is applied in the axial direction of a simply-supported straight beam with asymmetric ply stacking sequence (Fig. 2). Because of the asymmetry bending-extensional coupling occurs and the beam bends under pure axial load. The problem is analyzed with 2, 4, 8 and 16 elements modeling the beam structure. Material and geometric data are given in Table 1. Figure 3 compares the normalized transverse displacement prediction of the finite element solution to the theoretical deformation [40]. The numerical solution converges rapidly toward the exact solution.

RESULTS

In this section the numerical performance of the two-node curved comoosite beam element will be evaluated. Earlier works showed that beam elements based on the hybrid mixed formulation exhibit very good numerical properties [23,40]; in particular no shear and membrane locking was observed. These properties are also observed in the presented composite beam element; the reader is referred to ref. [40]

I :.- x

L

Fig. 2. Straight beam under triangular axial load.

Normeked Transverse Msplaeemeslt

-

0

0.2

4 Elements

OA

0.6

0.6

NormalkedBeam~npth Fig. 3. Transverse beam displacement due to bending-extensional CAS WI-0

coupling.

1

48

H. R. Dorfi and H. R. Busby

44 0

1

2

3

4

5

Binary Lagarlthm of Element Number

Fig. 4. Convergence of transverse displacement at midpoint. Both element types (P, and P2 stress interpolation) yield identical results. Similar convergence is observed for the axial displacement and beam rotation. Figure 4 shows the rate of convergence of a characteristic displacement, the transverse displacement at the mid-point, as a function of the element

size. The double logarithmic plot has a constant slope of 2; thus the rate of convergence is quadratic. For this particular problem the stress approximation is also identical for both types of stress interpolation, a piece-wise constant normal stress approximation is obtained. Figure 5 compares the finite element stress approximation of four and eight

0.6 - Normalized hruull Force 0.4 --

Fig. 5. Normal force approximation.

A

R 2F

2F

. . . . . . . . _.._.._

F

j

91...:

K

-oFig. 6. Composite

ring

under compressive point load.

beam elements to the theoretical normal force distribution. It is important to note that the stress prediction is most accurate at the midpoint of the finite elements. Composite ring under compressive load

A composite ring with a radius of 0.1 m and thickness of 20 mm (R/h = 5) is subjected to compressive point loads in radial direction (Fig. 6). Only a quarter of the geometry has to be modeled because of the double symmetry of the specimen and loading. Material and geometric data is given in Table 2. The problem is analyzed with 1, 2, 4, 8 and 16 elements modeling the beam structure and compared

Table 2. Curved

composite

to its analytical solution [42]. Both P, - and P,-type elements are used in the analysis (equilibrium and constant stress interpolation). Figure 7 shows the tangential displacement approximation of two and four beam elements for the P, and P2 element type. Clearly the P,-type element gives more accurate results. It is instructive to compare the rate of convergence of the two element types. Figure 8 shows the error of the transverse displacement prediction at the loading point as a function of the number of elements. Both methods converge essentially quadratic; however the elements based on the equilibrium stress interpolation (P, ) yield a more accurate deformation prediction for the same finite element discretization. The accurate prediction of the generalized stresses is of importance for the subsequent calculation of the ply stresses. Figure 9 shows the finite element approximation of the normal force. For the P, -element type the exact theoretical normal force distribution is recovered independent of the number of elements used in the discretization. This is not surprising, since the P, stress interpolation is based on the solution of the homogeneous equilibrium equations. The normal force distribution for constant stress approximation

beam: material

and geometric

Material graphite epoxy [39] E,, = 289 GPa G = 4.13 GPa

49

laminated curved-beam finite element

data

Geometry Ej2 = 6.06 GPa VIZ= 0.31

stacking sequence: radius: beam height: beam width: shear factor:

[90/45/45/45/45], 0.1 m 20 mm 20 mm 1.2

Notmalkd TmgenUsl 0.002

oisplacement

0.001

0

15

45

60

An&I*1 Fig. 7. Composite

ring: normalized

tangential

displacement.

75

90

H. R. Dorti and H. R. Busby

-3

,

, I

0

1

I 1

I I

, I

2

3

4

5

Binary Lqarlthm of Ekment Number

Fig. 8. Composite ring: convergence of transverse displacement at loading point.

-0.8

-1 0

16

30

4s

60

76

90

An&I01 Fig. 9. Composite ring: normal force approximation. gives good results at the midpoint of the finite element only. Very similar results are obtained for the shear force and bending moment dist~bution. 5. CONCLUSIONS

In this paper a curved composite beam finite element has been proposed and numerically evalu-

ated. The hybrid-mixed formulation yields a numeritally very effective element. Both equilibrium stress (P,) and constant stress (Pz) interpoIation are employed in the stiffness derivation and it is demonstrated that the P, element type yields better results for both displacement and stress approximation. The element also models precisely the bendingextensional coupling of asymmetric laminates.

A laminated curved-beam finite element REFERENCES

1. M. M. Schwartz, Composite Materials Handbook. McGraw-Hill, New York (1984). 2. S. Timoshenko, Theory of Elasticity. McGraw-Hill, New York (1934). 3. N. G. Stephen and M. Levinson, A second order beam theorv. J. Sound Vibr. 67. 293-305 (1979). 4. M. L&inson, A new rectangular beam theory. J. Sound Vibr. 74, 81-87 (1981). 5. M. Levinson, Further results of a new beam theory. J. Sound Vibr. 77, MO-444 (1981). 6. Z. Rychter, On the accuracy of a beam theory. Mech. Res. Commun. 14, 99-105 (1987). 7. W. B. Bickford, A consistent higher-order beam theory. Devl. theor. appl. Mech. 11, 137 (1982). 8. T. Kant and B. S. Manjunath, Refined theories for composite and sandwich beams with Co finite elements. Comput. Struct. 33, 755-764 (1989). 9. D. G. Ashwell and R. H. Gallagher (eds), Finite Elements for Thin Shells and Curved Members. John Wiley, London (1976). 10. D. J. Dawe, Curved finite elements for the analysis of shallow and deep arches. Comput. Struct. 4, 559-580 (1974). I 1. D. J. Dawe, Numerical studies using circular arch finite elements. Comput. Struct. 4, 729-740 (1974). 12. K. J. Bathe. Finite Element Procedures in Engineering Analysis. Prentice-Hall, Englewood Cliffs, NJ-(1987).13. T. J. Hughes, The Finite Element Method. Prentice-Hall, Englewood Ciffs, NJ (1987). 14. H. P. Lee, Generalized stiffness matrix of a curved beam finite element. AIAA Jn17, 2043-2045 (1969). 15. P. C. Kohnke and W. C. Schnobrich, Analysis of eccentrically stiffened cylindrical shells. J. Struct. Mech. Div. Proc. ASCE ST7, 1493-1509 (1972). 16. G. H. Ferguson and R. D. Clark, A variable thickness, curved beam and shell stiffening element with shear deformations. Int. J. numer. Meth. Engng 14, 581-592 (1979). 17. A. K. Noor, Nonlinear finite element analysis of curved beams. Comp. Meth. appl. Mech. Engng 12, 289-307 (1977). 18. A. K. Noor, Mixed models and reduced/selective integration displacement models for nonlinear analysis of curved beams. Int. J. numer. Meth. Engng 17, 615-631 (1981). 19. H. R. Meek, An accurate polynomial displacement function for finite ring elements. Comput. Struct. 11, 265-269 (1980). 20. H. Stolarski and T. Belytschko, Membrane locking and reduced integration for curved elements. J. appl. Mech. 49, 172-176 (1982). 21. H. Stolarski and T. Belytschko, Shear and membrane locking in curved Co elements. Comput. Meth. appl. Mech. Enang 41. 279-296 (1983). 22. G. Prathap: The curved beam;deep arch/finite ring revisited. Int. J. numer. Meth. Engng 21, 3899407 (1985). 23. A. F. Saleeb and T. Y. Chang, On the hybrid-mixed formulation of Co curved beam elements. Comput. Meth. appl. Mech. Engng 60, 95-121 (1987).

51

24. K. Surana and S. Nguyen, Completely hierarchical two-dimensional curved beam element for dynamics. Comput. Struct. 40, 957-967 (1991). 25. R. Wen and B. Suhendro, Nonlinear curved-beam element for arch structure. J. Struct. Engng 117, 3496-3515 (1991). 26. J. Sandhu et al., 3-D, co-rotational, curved and twisted beam element. Comput. Struct. 35, 69-79 (1990). 27. A. Venkatesh and K. P. Rao, A laminated anisotropic curved beam and shell stiffening element. Comput. Struct. 15, 197-201 (1982). _ 28. W. C. Chao and J. N. Reddv. Analvsis of laminated composite shells using a degenerated 3-D element. Int. J. numer. Mech. Engng 20, 1991-2007 (1984). 29. S. Nagarajan, Finite element model for orthotropic, laminated beam analysis. Comput. Struct. 20, 443-449 (1985). 30. 0. Bachau and C.-H. Hong, Large displacement analysis of naturally curved and twisted composite beams. AIAA Jnl 25, 1469-1475 (1987). 31. D. Karamanlidis, A simple and efficient curved beam element for the linear and non-linear analysis of laminated composite structures. Comput. Struct. 29, 623-632 (1988). 32. A. Benedetti and A. Tralli, A new hybrid F.E. model for arbitrarily curved beam-I. Linear analysis. Comput. Struct. 33, 1437-1449 (1989). 33. F.-G. Yuan and R. Miller, A new finite element for laminated composite beams. Comput. Struct. 31, 737-745 (1989). 34. T. Ray, Analysis of stresses and deformations of a curved beam (arch) of orthotropic composites by finite element method. J. Inst. Engr (India), Part ME: mech. Engng Div. 70, 18-20 (1989). 35. F. Orth and K. Surana, Two-dimensional laminated composite curved beam element based on piecewise hierarchical p-version displacement approximation for linear static and geometrically non-linear behavior. Composite Mater. Technol. 1992, ASME PD45,189-204 (1992). 36. K. Washizu, Variational Method in Elasticity and Plasticity, 3rd edn. Pergamon Press, Oxford (1982). 37. T. H. H. Pian, Derivation of element stiffness matrices by assumed stress distributions. AIAA Jnl2, 1333-1336 (1964). 38. J. N. Reddy, Energy and Variational Methods in Applied Mechanics. John Wiley, London (1984). 39. J. R. Vinson and R. L. Sierakowski, The Behavior of Structures Composed of Composite Materials. Martinus Nijhoff, Dordrecht (1986). 40. H. Dorh, A laminated curved beam finite element based on the hybrid mixed formulation. M.Sc. thesis, Department of Mechanical Engineering, The Ohio State University (1988). 41. J. K. Lee, FEMP-finite element program. Department of Engineering Mechanics, The Ohio State University (1988). 42. H. Dorfi, Analytical solutions of curved composite structures. Internal Report, Department of Mechanical Engineering, The Ohio State University (1993).

APPENDIX: STIFFNESS MATRICES

Two different stiffness matrices are obtained: the stiffness matrix K, is calculated based on the stress interpolation matrix equilibrium equations. It is found from

P,, which satisfies the homogeneous

K, =G~H;‘G,,

(AlI

H. R. Dofi and H. R. Busby

52

where G, and H, are given by -1

0 0

-1

0

G, =

0 0

O-l

2R sin” 2 -sin 2 sin (pO cos ‘PO -R sin To 0 coscp,

1

0 (pO

i (-42)

(a, + 2R*a,) ? + 2a, R sin(cp,) +

(a4 + a, cos(cp,))sin2 9 0

0,

W. + a2 sin(cp,1

ppo a3 sin(2cp,) fz--4 2

H,=R

’

symm.

fA3)

Q5‘PO

with the abbreviations a,=S,,-2RS,,+S,,fR2S3,,

a, = S13 - RS,,

a,=S,,-2RS,3-S,,+R’S,,,

a,=S,,-S,,-R”S,,,

a,=S,,.

644)

The S, denote the components of the compliance matrix S = C-l. The matrices can be simplified for a straight beam (R --+a?, Rqq,-+ L). If we denote the length of the straight beam by L, G, and H, simplify to -1 G; =

0

0

-1 0

0

ii

0

1 0

0

0

0

-L

-1

1

0 0

-f;&/2 S22+ L2S3,13

S,, HF=L

I

WI I

s13

WI

-L&j2

S33 i

1 symm.

The symmetric stiffness matrix K, for the composite beam with constant stress interpolation matrix Pz is found as

c,, + C,,w’

c,,w - c*,o c,,tJ+

C,,

C,,

C,,w

-

C22RwZ

+ C,,Ro

C,, + C,,R’w’

K&

-c,,

-c,,w -C,,

+

c,,w

C*,w’

- C,,w - C,,Ro’

c,, + Cz~2

0

+

c,,w

C,,o?-

G2

C,,w

- C,>Ro

-c,,w

+ c,,w

C,,WZ+Czz

symm.

-C,, - C,,Ru’ -C,,w -C,,

+ C,,Rw + C,,R*w’

C,, - GRW2

-C,,w

- C2,Rw

C ,7 + C 22Rzio2

I

(A71

where we used the abbreviation o = (pot2 and the C,j denote the components of the beam stiffness matrix defined in eqn (19). For a straight beam K, simplifies to

‘G

0 G2

c,,

-c,,

C22Li2

0

c,, f C,, L%

-c,,

CL,

.

symm.

0 - e** -C,,

Lj2

0 c22

-cl3 C**L/2 -c,,

+ G, L2/4

CL3 -Cz2L/2 c,, + cz2 L ‘I4