A preliminary approach to composite beam design using FEM analysis

Composite Structures 16 (1990) 259-275

A Preliminary Approach to Composite Beam Design Using FEM Analysis AndreaCorvi Dipartimentodi Meccanica,Universitadi Ancona,Via delleBrecceBianche, 60131 Ancona,Italy ABSTRACT A PCprogram is presented for the preliminary design of composite beams based on composite mechanics and the finite element method (FEM). This type of element is derived using Timoshenko beam theory which allows accounting for shear effects on beam deflection. Expressions of shear coefficients have been considered for a rectangular laminated beam and for a series of thin-walled sections (with reference being made in the latter case to the transformed section method). Application to an automotive leaf spring has demonstrated the feasibility of using the program as a tool in establishing the initial design considerations and in developing preliminary designs.

1 INTRODUCTION Over the last few years, the relationship between people and materials has undergone a complete reversal. Up to very recently, human creations were conditioned by the availability of materials, which meant that very often the possibilities of satisfying design requisites were seriously limited. This has changed: the new materials can now be adapted to people's needs instead of vice versa. In this sense, today's materials may be viewed as part and parcel of the object itself and as a 'supplier' of performance levels. The opportunity of designing materials tailored to specific requisites has, however, complicated the designer's selection process. This means that not only does the number of materials available continue to grow, but also that their manufacturing and component fabrication processes have an increasingly strong impact on material performance. The end result is multiplication of materials which can thus be more and more specialized to meet pre-established objectives. 259 Composite Structures 0263-8223/90/S03.50 © 1990 ElsevierSciencePublishers Ltd, England.Printedin Great Britain

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A. Corvi

The designer who works with composites must cope with the greater complexity of the materials stemming from their heterogeneity and the synergy effect exerted on the behavior of the matrix and reinforcement components. Also, he must deal with innovative geometries which are feasible, if not necessary, for rational use of the composite materials, as well as new processes and technologies which condition the properties of the materials themselves. From the structural design stage, the designer must make decisions from among increasingly numerous combinations of design parameters. He can then operate through already commercialized optimum design programs that take into account design requirements through objective functions which must be optimized in relation to the existing constraints on design parameters for combinations of design parameters.~ Decisions can also be made using preliminary design programs specialized according to well-defined categories of problems. The programs allow the designer to rapidly select a limited number of solutions which can then be further investigated using a more sophisticated structural analysis program.

2 A PROGRAM FOR COMPOSITE BEAM DESIGN A program for the structural analysis and design of composite beam components was developed in view of the foregoing considerations. The program structure is sketched in Fig. 1. The inputs can be roughly divided into two categories: (a) a category covering the material through its fiber and matrix component properties and laminate stacking sequence, and (b) a category covering the 2-D mesh beam geometry, applied loads, and the constraints imposed on the structure. Through FEM analysis, node displacements and rotations, as well as deformation and stress states, are evaluated in order to allow determination of the structure's collapse loads and locate failures. If the mechanical properties of the layers constituting the laminates are not known, the reinforcing fiber and matrix characteristics may be introduced in the program, together with their mutual volume percentages. Through lamina micromechanics equations (i.e. the rule of mixtures), a routine calculates the approximate values of the elastic and strength properties. On the basis of these values and the laminate stacking sequence, a second routine uses the classical lamination theory to evaluate the elastic properties of the laminate. 2 The beam element model for the structural analysis routine is selected with special care. In general, the deflection of a beam subjected to a

Composite beam design using FEM analysis

261

MATERIAL PROPERTIES giber-rosin) vol. % J /

I tAM'NA'RO"'T'ES; ( STACK,NGSmUENC.1 2-D MESH, LOADS ] CONSTRAINTS

LAMINATE PROPERTIES

J

FEM ANALYSIS 1 NODAL DISPLACEMENT~ ROTATIONS e oJ

1 Q COLLAPSE ANALYSISi Fig. 1.

Program structure.

transverse load depends both on the bending moment and on shearinduced warping. If the beam is made of an isotropic material and is sufficiently thin, the shear-induced deflection component can be negligible with respect to its bending deflection counterpart. This is due to the fact that the shear's contribution to the deflection is proportional to the ratio a2/L 2, in which a indicates beam thickness and L indicates beam length. 3,4

In the case of an anisotropic beam whose material has a longitudinal modulus of elasticity to shear modulus ratio exceeding that of an isotropic material by more than an order of magnitude, the shearinduced deflection component may no longer be ignored. This is evident in the following expression valid for a circular-section beam made of a transverse isotropic material 5 Wshear

- -

Wbending

Exx a

2

= 1.125 - - L---~ - 0-75 Vy~ G zx

where E = is the longitudinal modulus, Gzx is the shear modulus, and vrz is the Poisson ratio. For many cases of composite materials (E=/G=) can range from 20 to 50, meaning that the type of element used in the structural analysis must also account for shear effects.6-9

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A. Corvi

The constant-section beam element that bends in-plane can be analyzed by means of alternative formulas. The principles underlying them are that the flexural displacements are slight with respect to the size of the beam section and that the stresses perpendicular to the neutral axis are negligible. The best known relationships are given by the Euler-BernouUi beam theory, which hypothesizes that the sections normal to the beam axis remain flat during deformation of the loaded beam so that shear effects are totally ignored. Conversely, the Saint Venant theory assumes that the axial stresses are linearly distributed over the beam. As this hypothesis is not verifiable for composite laminates in which layers can differ as to type and orientation, the Saint Venant approach is not applicable.4-6 The theory developed by Timoshenko is better suited to our purposes. It is bsaed upon the hypothesis that prior to deformation the sections normal to the neutral axis remain in-plane, although they are not necessarily normal to the neutral axis once deformation has taken place. This approach highlights the contribution of shear-induced warping to deformation, which is considered linear through the beam section and thus represents an approximate scheme of real behavior (Fig. 2).6-9 According to the Timoshenko theory, the total rotation (d w/dx) of the section normal to the neutral axis is, under load action, the sum of

Fig. 2.

Beam deformation according to Timoshenko theory.

b,

,4_2Y L

z I

Fig. 3. Beamgeometry.

x

Composite beam design using FEM analysis

263

rotations due to bending (7)) and to shear sliding (0) dw

dx

7)+ 0

(1)

Referring to Figs 2 and 3, the equations for a homogeneous beam material are

My= EIy d~o dx Q =kGAlp where My is the bending moment, E is the elastic modulus of the beam material, Iy is the modulus of inertia of the beam section, Q is the shear force, k is the shear coefficient accounting for the cross-section warpings, G is the shear modulus and A is the beam section area. When load and constraints are known, the above equations can be solved and the solution expressed in general form as

w(x)=f,(x) + f2(x) Ely kAG The result for a tip-loaded cantilever of length L with force P is ft = P ( 3Lx2 -x3) f2 = P x

The tip-section beam deflection under load thus equals

w(L ) =

PL 3 PL +-3EIy kAG

The first term in this equation represents the deflection component due solely to bending moment, whereas the second term represents the deflection component due solely to shear action. In calculating shear coefficient k, it is necessary to account for beam section geometry and material. In the case of a composite beam, as the material is macroscopically heterogeneous, the effects of different layups have to be considered in evaluating k. In the case of a laminated rectangular beam, some authors have developed expressions for k to

A. Corvi

264

Fig. 4.

Laminated rectangular beam.

account for the elastic properties of the laminate's single layers and their locations in the laminate. For symmetric laminates, we get (Fig. 4) ~~-~3

k

hE

4q= 1

h -l)

1

h,L1)

with Xp =(011)pOll q-(012)p/~12 where the subscripts q and p refer to a generic lamina and Qij represents the terms of its stiffness matrix in relation to a reference system oriented at an angle a with respect to the principal orthotropic directions. The/)~j obtainable from 1)ij= ( - 1)i+JDij/(D,1D,2 - 022) constitute the terms of the inverse of the stiffness matrix accounting for the bending moment that relate (for symmetric laminates) moments M i to rotations Pi of the laminate

[Mi] = [ Dii[ [Pj] [pj] = IOij I [M,] The deflection expression thus assumes the form

PL 3 PL w(L ) 3Exxi,+ k ' A G ~ where Exx is the material longitudinal modulus, Iy is the beam section moment of inertia, and k' is the modified shear coefficient. The laminate shear modulus Gzx could be obtained through experimental testing, but this requires characterizing all the laminates that could possibly be considered in the course of design. This would be a huge effort and as such conflict with the objectives of the program, i.e. to provide a first approach design tool. Hence, in the place of Gzx, we can substitute the shear modulus of the same matrix used in composite Gin, since the

Composite beam design using FEM analysis

265

matrix acts as an interfiber and interlamina connector, thereby notably conditioning the laminate's shear behavior. In the case of a thin-walled beam whose panels are obtained from laminates with different stacking sequences, the deflection expression, having established constraint and load conditions, is

PL 3 w(L)=3e'xxfy

PL k'AE'xx

where E'xx is the longitudinal modulus of the material transformed section, I'y is the moment of inertia of the transformed beam section, and k' is the modified shear coefficient. 7,14 I'y is calculated as follows: (1) select a reference value for Egg (generally for one of the laminates constituting the beam panels), (2) modify the panel thicknesses in order to obtain a beam section with the same total axial stiffness, and (3) compute the moment of inertia from the transformed sections. Reference 8 gives the general expression for k' and the expressions calculated for three sections: the rectangular box beam, the I-beam, and the T-beam (Table 1 and Fig. 5). The element model can then be applied to the beam obtained through section transformation. In the case of the Timoshenko beam shown in Fig. 2, the displacement w(x) and rotation of section ~p(x) can be considered independent variables that are functions only of x

w(x) =al + a2x = a3 + a4x

The shape functions which correlate the displacements on the element to the node displacements [ U] are easily obtained

1 -x/L

0

x/L

0

0

1 -x/L

0

x/L

IUl =

(a)

I

b

(b) l

J

a Y 2G2V2 E1GlVl~

,y

tl Z

a

b

(c)

t2

--y

E1 Girl\ I

a

b

't

G1V ... E2G2v t2

~y

tl Z

Z

Fig. 5. Thin-walled beam sections analyzed in the program with shear coefficient expressions: (a) rectangular box beam; (b) I-beam; (c) T-beam.

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266

TABLE 1

Modified Shear Coefficient for the Thin-Walled Beam Sections Analyzed in the Program s Rectangular box beam k'=20(2+3m)2/I ~ (60m2n2+602mn2)+~ (180m3+3002m2+14422m+2423)

+ vl( - 30 m2n 2 _ 50 2mn 2) + v2(30 m 2 + 6 2m - 422)]

b bt~ E~ m=--, 2- " a at2 E~

where n = - ,

1-beam

k '= 20(2 + 3 m)2 / [ - ~ (60 m2 n2 + 60 2mn2) + G2 E~ ( 1 8 0 m 3 + 3002m2 + 1 4 4 2 2 m + 2423)

+ vl(60 m2n 2 + 40 2mn 2) + v2(30 m 2 + 6 2m - 4 22)]

b 2bt~ E, m= ,2= a at~ E~

where n = - ,

T-beam k ' = 102(2 + 4m)2(2 + m ) 2 / [ ~

( 1 2 2 6 + 12025m+48024m2+84023m 3

+ 6 6 0 2 2 m 4 + 1 9 2 2 m -s) + ~El (30mn

22(2 + m ) 3 ) + v l ( - 2 2 5 - 5 2 4 m -

1523m 2

- 20 22m 3 + 40 2m 4 + 48 m 5) + v2(20 24ran 2 - 15 24m + 80 23mZn 2 - 75 23m 2 + 12022m3n 2 - 60 22m 3 + 80 2m4n 2 + 20 mSn2)]

b a

bt~ E~ 2 = ~" at2 Et

where n = - , m = - - ,

Composite beam design using FEM analysis

267

The deformation state of the Timoshenko beam as derived from eqn (1) is the sum of the bending and shear warping components

[e(x)] = d[~]

where d____~= - 7'1 + 7,2 dx L O=

--

Wl "[- W2

L

- - 7 , 1 + ~32

7,~

L

x

Matrix [B[, which joins up the deformations and node displacements [U],

[e]=IBI[U] can be written IBI =

0

-1/L

-1/L

-l+x/L

0

1/L

1/L x/L

Likewise, we can calculate the material stiffness matrix [ R[ joining up the stress and deformation states

[a]=lRl[e] This results in a 2 x 2 configuration in which there is a term related to bending effects and a term related to shear effects

IRI=

E"fl'y 0

k'

OE,

x

Stiffness matrix IKel can be obtained from the sum of two stiffness matrices, one for beam bending behavior [Kbl and one for shear behavior 1Ks[

IK~l---[Kbl+lKsl

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A. Corvi

where i

0

0

1

0

0 -1

0 0

IKbl = E'xxI'~ L

IK

L0

k'A E'x~,

f=-L

-

0 -1

0 1

L/2

- 1

L/2

L2/3

- L/2

L2/6

-L/2

1

-L/2

-L/2

L2/3

IL/2 L2/6

This element constitutes the basis of the program developed for PCs. Convergence tests of the solutions and their approximations were conducted, first on the structural behavior of isotropic beams of known behavior and then on composite beams whose data were obtainable from the literature. The results of several analyses were compared with those obtained on analogous structures using reliable FEM codes such as ABAQUS and NASTRAN. Overall, the program showed excellent reliability, especially for general structure behavior, with lesser approximation appearing only in the local responses in areas of marked discontinuity (load and constraint application points). These observations are confirmation that the program can be useful in making general structural evaluations among constructional solutions, which enormously aids the designer by reducing the range of solutions requiring verification. Should the need arise for a more sophisticated analysis of structural behavior, it is necessary to use more sophisticated codes that can only be run on more powerful computers by trained operators.

3 THE APPLICATION OF THE PROGRAM IN AUTOMOTIVE LEAF SPRING DESIGN As part of a joint design project undertaken with a major Italian automobile manufacturer, the program was applied in the preliminary design of a composite leaf spring for a commercial vehicle. Manufacturers are well aware of the importance of research in designing leaf springs that allow weight reductions resulting in enhanced performance and payload.

Composite beam design using FEM analysis

269

Composite leaf springs, for example, can be up to 50% lighter than their steel counterparts.~5-18 In designing a leaf spring, the designer must make decisions regarding the selection of materials (fiber and matrix) and the determination of optimal lay-ups. This requires a verification of a wide variety of different solutions for evaluating structural behavior and a description of the variations in component efficiency parameters as design parameters vary. To do this, one or more parameters capable of significantly describing leaf spring behavior must be found. In the analysis illustrated herein, a comparison parameter was leaf spring compliance. Another important characteristic of proper leaf spring behavior is the maximum energy that can be stored during load application and thereafter during deformation. The amount of elastic energy that can be stored by a leaf spring volume unit may be written as the common expression 1 o2 S ~ - m

2E

where OL is the stress originating when a maximum allowable load is applied to the leaf spring and E is the modulus of elasticity, both in the longitudinal direction. Generally speaking, referring to the maximum load applicable at the leaf spring F, we can also express the overall energy of deformation S =FU m

where Um represents the displacement of the leaf spring's centerline at the load application section. Evidently, design solutions should aim to produce leaf springs capable of storing the greatest amount of energy along with the greatest reduction in weight) 8-2° The reference leaf spring geometry is shown in Fig. 6. In the design analysis, all dimensions, including laminate thickness, were kept fixed. Of the loads acting on the leaf spring illustrated in Fig. 7, the vertical and transverse ones were considered in this analysis. The results are presented in dimensionless form.

Fig. 6.

Leaf spring geometry.

270

A. Corvi

S

57

Fig. 7. Loads acting on the leaf spring.

A preliminary investigation was c o n d u c t e d to evaluate leaf spring behavior as the glass fiber volume content Vf varies in the epoxy matrix. In the solutions investigated, Vf varied from 0.2 to 0-7 in angle-ply laminates having the fiber orientation ag = 0 °. Vf = 0.6 was selected as reference. Then, with Vf = 0.6, fiber orientation ag was varied from zero to _+30 °. Reference w a s O~g = 0 °. A n o t h e r class of solutions considered was combined glass- and c a r b o n - e p o x y layers. This investigation focused on a leaf spring having an inner plate at the m e a n plane of a constant-thickness c a r b o n - e p o x y layer (Fig. 8). A solution of this type can cause stiffening of the leaf spring in terms of transverse loads that affect vehicle behavior a r o u n d curves. In the reference solution, the presence of the c a r b o n - e p o x y layer was not considered. Different solutions were investigated on angle-ply glass- and c a r b o n - e p o x y laminates, both of which had Ctg ~---0 °, (Z c = 0 ° and Ve= 0"6. C a r b o n - e p o x y layer thickness was varied from 1 to 14 mm. Leaf spring structural behavior was also evaluated with varying angle a c from zero to _ 9 0 °, w i t h a g = 0 ° and carbon layer thickness tc = 5 ram. A final analysis

Composite beam design using FEM analysis

271

tc Fig. 8.

Leaf spring section with carbon-epoxy interposed.

2,0 i

1,5

g 1,0

"o era 8

e,.

0'5

ca

E o

0

0,0'

-0,51 0,2

0,3

0,4

O,S

0,6

0,7

Fiber Percentage

Fig. 9.

Compliance in the glass-epoxy leaf spring versus percentage in fiber volume Vf and maximum energy stored (t~, work; *, vertical; m, transverse).

was performed to evaluate the effects of varying the glass fiber angle ag, while maintaining all other parameters constant (ac = 0 ° and tc = 5 mm). The results show the effects of increasing the percentage of fiber volume in the composite on compliance and maximum storable energy in the 100% glass-fiber leaf spring. Logically, for both vertical and transverse loads, the maximum stiffness and the maximum storable energy are obtained at the highest values of Vf (Fig. 9). Note that, if the fiber lamination angle is varied instead, there are no marked variations in the two reference parameters for ag between 0 and _+5 ° (Fig. 10).

272

A. Corvi 1,2

1,0 ' ~7 e"

o

"~ 0.8" ca > 0

~

0,6

C II3

8C

m ~.

0,4

E o

o

0,2

0,0 '

0

Fig. I0.

5

10

15

alpha

20

25

30

35

Compliance in the glass-epoxy leaf spring versus the lamination angle and maximum energy stored (n, work; e, vertical; m, transverse).

0,00j = !

.

.

.

.

-0,05t co am e= >

8

e-

.=_

-O,lOt -o,15t -0,20]

E

o0

-0,25] -0,30]

Fig. 11.

thickness (ram) Compliance in the glass-carbon-epoxy leaf spring versus the carbon-epoxy layer thickness(D, vertical; e, transverse).

Composite beam design using FEM analysis

"~

273

°"°t ................

I

-o,oot

I

-0,05

=t -o,ot

/

1/

IL

lamination angle (D, vertical; ~, transverse). 3,0

....

2,5.

2,0. ~

1,5,

~

1,0,

~

0,5.

0,0

-0,5

Fig. 13.

I

I

!

!

15

30

45

60

alpha

'

~

I

75

90

Compliance in the glass-carbon-epoxy leaf spring versus the glass-epoxy lamination angle (t~, vertical; ~, transverse).

274

A. Corvi

The effects on compliance of varying the thickness of carbon fiber layers interposed in a carbon-epoxy laminate were also evaluated. They turned out to be negligible with respect to vertical loads and marked with respect to transverse loads (Fig. 11). The variation in the carbon lamination angle does not influence the behavior of the vertical loads, whereas stiffness effects are maintained for transverse loads only when a c does not exceed +60 ° (Fig. 12). The variation in the glass-layer lamination angle ag affects both types of behavior, albeit negligibly up to around + 10 ° (Fig. 13). While far from complete, an analysis of this type represents a first step in leaf spring design. It enables the designer to establish limits in design parameter variations such as lamination angles in glass and carbon and the effects of varying the thickness of the carbon-epoxy plate on several leaf spring behavior characteristics. However, it must be recalled that the analysis fails to investigate leaf spring torsional behavior or fatigue loads, two extremely important conditions which must necessarily be accounted for in the final design.

4 CONCLUSION A program for the structural analysis of composite beams has been described. The program has demonstrated its usefulness in carrying out preliminary design, by providing rapid analysis of the effects of varying design parameters on component performance. It is based on composite mechanics equations by which the properties of layers and laminates may be computed and on an FEM analysis of beam behavior. The element used in the program was developed according to Timoshenko beam theory with reference to the transformed section in the case of thin-walled beams. The program, applied to the preliminary design of an automotive leaf spring, was successful in evaluating the effects of varying design parameters on leaf spring behavior.

REFERENCES 1. Rouse, N. E., Design optimization goes commercial. Machine Design, 23 (1896) 84-7. 2. Tsai, S. W. &Hann, H. T., Introduction to Composite Materials. Technomic, Pennsylvania, 1980. 3. Timoshenko, S. & Goodier, J., Theory of Elasticity. McGraw-Hill, New York, 1934.

Composite beam design usingFEM analysis

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4. Bauchau, O. A., A beam theory for anisotropic materials. Journal of Applied Mechanics, 52 (1985) 416-22. 5. Toll, G., Saint-Venant bending of an orthotropic beam. Composite Structures, 4 (1985) 1-14. 6. Hu, M. Z., Kolsky, H. & Pipkin, A. C., Bending theory for fiber-reinforced beams. Journal of Composite Materials, 19 (1985) 235-49. 7. Bank, L. C., Shear coefficient for thin-walled composite beams. Composite Structures, 8 ( 1987) 47-61. 8. Bank, L. C. & Bednarczyk, E J., A beam theory for thin-walled composite beams. Composites Science and Technology, 32 (1988) 265-77. 9. Bank, L. C. & Melehan, T. E, Shear coefficient for multicelled thin-walled composite beams. Composite Structures, 11 (1989) 259-76. 10. Skudra, A. M. & Bulavs, E Y., Composite beam structures. In Handbook of Composites, Vol. 2, Structures and Design, ed. C. T. Herakovich & Y. M. Tarnopol'skii. Elsevier Science Publishers, Amsterdam, 1989, pp. 395-462. 11. Stephen, N. G., Timoshenko's shear coefficient from a beam subjected to gravity loading. Journal of Applied Mechanics, 47 (1980) 121-7. 12. Jensen, J. J., On the shear coefficient in Timoshenko's beam theory. Journal of Sound and Vibration, 87 ( 1983) 621- 35. 13. Carmignani, C. & Conti, E, MACOMP: un programma agli elementi finiti per ranalisi in campo elestico di laminati in materiale composito, Atti I Convegno Nazionale di Meccanica Computazionale, Milano, Italy, 1986. 14. Kassimali, A., Craddock, J. N. & Matinrad, M., Bending and transverse shear stresses in fiber-composite beams by the transformed-section method. Composite Structures, 5 (1986) 33-49. 15. Beardmore, P., Composite structures for automobiles. Composite Structures, 5 (1986) 163-76. 16. Beardmore, E & Johnson, C. F., The potential for composites in structural automotive applications. Composites Science and Technology, 26 (1986) 251-81. 17. Morris, C. J., Composite integrated rear suspension. Composite Structures, 5 (1986) 233-42. 18. Watson, P. & Pullen, J., Composite springs for automotive suspensions. Proc. 2nd Seminar on Innovation in Materials and Application in the Transportation Industry, Vol. 1. ATA, Turin, Italy, 1989, pp. 471-9. 19. Faroni, E. & Quaranta, S., Composite leaf springs for light commercial vehicles. Proc. 2nd Seminar on Innovation in Materials and Application in the Transportation Industry, Vol. 2. ATA, Turin, Italy, 1989, pp. 75-83. 20. Yu, W. J. & Kim, H. C., Double tapered FRP beam for automotive suspension leaf spring. Composite Structures, 9 ( 1988) 279-300.