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Composites Science and Technology 32 (1988) 265-277

A Beam Theory for Thin-walled Composite Beams

L. C. Bank & P. J. Bednarczyk

Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

(Received 1 October 1987; accepted 1 February 1988)

A BSTRA CT

A beam theory & presented that &formulated in terms of the &-plane elastic. properties of the panels of the cross-section of a thin-walled composite beam. Shear deformation is accounted for by using a suitable form of the Timoshenko beam theory together with a modified form of the shear coefficient. The theory gives both the bending deflection and the shear deflection of a beam loaded by an applied transverse load. Numerical and graphical results obtained from a computer code show the effects of using different composite material systems and lay-ups in the panels of typical beams.

1 INTRODUCTION

Thin-walled fiber-reinforced composite material beams are being used in a variety of structural applications in the aerospace, mechanical and civil engineering disciplines.I-3 A significant benefit obtained from the use of composite materials in the panels or walls of the beam cross-section is the ability to tailor the mechanical properties of the beam to a specific application. In addition to the usual considerations of the beam cross- sectional shape, the loading conditions and the end conditions, the beam response will now also be affected by the differences in the mechanical properties of the panels of the beam. The total deflection of a beam, under applied static or dynamic transverse loads, will depend on the combination of composite material systems used and on their lay-ups in the panels of the beam.

265 Composites Science and Technology 0266-3538/88/$03-50 © 1988 Elsevier Applied Science Publishers Ltd, England. Printed in Great Britain

266 L. C. Bank, P. J. Bednarczyk

In general, the panels of the beam cross-section will have anisotropic mechanical properties. For most anisotropic composite material panels the ratio of the longitudinal modulus to the shear modulus (E/G) in the plane of the panel will be significantly higher than that for conventional isotropic metal panels. 4'5 Consequently, the deflection of a beam constructed of panels of these materials is more likely to be affected by the deformations due to shearing stresses in the beam cross-section. This shear deformation of the cross-section will cause the beam to deflect in the transverse plane which together with the bending deflection will give the total beam deflection.

The shear deformation beam theory first considered by Bresse 6 but commonly known as the Timoshenko beam theory 7 can be used to obtain both the bending and the shear deflection of the beam. In order to use the Timoshenko beam theory for the thin-walled composite beam a recently formulated modified shear coefficient 8 is obtained as a function of the beam cross-sectional geometry and the material properties of the panels. A brief account is given of the development of the modified shear coefficient and the form of the Timoshenko beam theory needed for the analysis of thin-walled composite beams. Results obtained using a dedicated computer code based on the theory are shown and discussed.

2 SHEAR DEFORMATION BEAM THEORY A N D SHEAR COEFFICIENT

The cross-section of the thin-walled composite beam considered lies in the xy plane, see Fig. 1, with the beam major axis parallel to the z coordinate. The beam deflects under the action of a transverse load in the xz plane which is a plane of symmetry for the loading and the cross-sectional geometry. The cross-section is constructed of mid-plane symmetric orthotropic panels that are placed in the cross-section with one of the panels' orthotropic axes parallel to the beam z-axis. This gives each panel one coordinate in the z-direction and the other in-plane coordinate in a generic direction's' which may be either 'x' for a vertical panel or 'y' for a horizontal panel. Each orthotropic panel is characterized by its in-plane longitudinal modulus, Ez, its in-plane shear modulus, Gsz, and its in-plane longitudinal Poisson's ratio, vsz. These properties are obtained for each orthotropic panel either by using unidirectional ply properties and classical lamination theory 9 or by performing tests on specimen coupons. The symmetry of the loading and of the cross-sectional construction, and the alignment of the panel and the beam axes are required to uncouple the transverse deflection response of the beam from the twisting response.

Beam theory for thin-walled composite beams 267

shear flow I r

Fig. 1.

r I shear flow

Y

laminated composite panel

Typical thin-walled beam cross-section.

Considering, initially, a thin-walled beam constructed of panels all having the same in-plane properties the beam equations are written as,

E j ~ z = M (1)

Q - - + ~b - - - ( 2 ) ~3z KAGs=

where E= is the longitudinal modulus of the panels,/is the second moment of the cross-section about the y axis, ~b is the bending slope, M is the bending moment, w is the total deflection, Q is the shear force, K is the shear coefficient, A is the cross-sectional area and Gs, is the in-plane shear modulus of the panels. For specific loading and end conditions the above equations can be solved and the solution written in a general form as,

A(z) A(z) w(z) = - - ( 3 ) KAG~

wherefl(z ) andf2(z) depend on the loading and end conditions. For the case of the tip-loaded cantilever beam, the uniformly-loaded simply-supported beam, and the center-loaded simply-supported beam, these functions are easily calculated and are given in Table 1. The first term in eqn (3) is called the bending deflection, wb(z), and the second term in eqn (3) is called the shear deflection, Ws(Z ).

The shear coefficient, K, in eqns (2) and (3) depends on the cross-sectional


TABLE 1 Deflection Functions for Timoshenko Beams

Beam type f l(z) f2(z)

Tip-loaded cantilever P (N)

Uniformly-loaded simply-supported.q (N m - 1)

Center-loaded simply-suported P (N) (z < l/2)

P 3 2 ~( lz - z 3) Pz

~4(z'-2lz3+13z) 2 (lz-- Z2)

(312z -- 4z 3) ~ -

shape and on the material properties of the panels that form the cross- section. For cross-sections constructed of panels parallel and perpendicular to the plane of loading the shear coefficient, K, is given by Bank 8 following the method due to Cowper 1° who obtained the Timoshenko beam equations by integrating the equations of 3-dimensional elasticity. In the derivation due to Cowper, classical Saint Venant flexure theory, see Love, 11 is used to obtain the relationship between the shear stress distribution in the beam and the Saint Venant flexure function. As discussed by Cowper, exact solutions, exclusive o fead effects, can be found for the case of the tip loaded cantilever beam and for the case of uniformly loaded beam. For these two cases the distribution of transverse shear stress is identical. As further noted by Cowper, this distribution of shear stress also is used as an approximation of the exact shear stress distribution for general loadings. The method, therefore, involves approximations relating to the boundary conditions of the beam and to the nature of the loading considered, both of which are discussed in greater detail by Cowper. x° The method has been previously applied to materials with anisotropic elastic properties laA3 and has shown considerable success for use with thin-walled sections made of isotropic materials. ~,

The equation for Kgiven by Bank s allows one to find the shear coefficient by performing line integrals around the thin-walled cross-section, see Fig. 1, and is given as,

K = IE~/G~ ~f A f (4, + V~zs2tds +-~ xOtds

where t is the wall thickness and s is the arc length coordinate around the cross-section, which coincides with one of the in-plane coordinates of each panel, as previously described. The positive sign in the first term in the denominator is taken for integration in the y direction and the negative sign


for integration in the x direction. The function, ~O(s), in eqn (4), called the modified flexure function, is related to the axial deformation of the beam and is found from the following two equations,

Gsz O~k _ E j Ox Q cos~ %~ + (v~zx2) (5)

G~ O~k E j z~ +G~zvs~xy (6) ~-y =Q sin 0

Eqn (5) is used when integrating in the x direction along a 'vertical' panel parallel to the xz plane and eqn (6) is used when integrating in the y direction along a 'horizontal' panel parallel to the yz plane, z~z is the shear stress distribution around the cross-section which is assumed to be uniform across the wall thickness and is found by standard equilibrium methods. For the case of closed cell cross-sections having multiple cells the shear stress distribution cannot be found from equilibrium methods alone and compatibility conditions must be enforced to ensure the continuity of the warping displacements. ~5 As discussed by Jensen and Pedersen 14'16 the calculation of the shear coefficient for the closed cell cross-sections must use the correct shear stress distribution. When Poisson effects are included the correct form of the compatibility conditions 16'1v must be used to find the shear stress distribution. 0 is the inclination of a panel with respect to the x axis, see Fig. 1.

For the more general case in which the panels of the cross-section have different in-plane elastic properties a ' transformed section' is used and the beam equations are written as,

E'It ~ z = M (7)

Ow Q + - K * A E , (8)

and the solution as,

fl(z) f2(z) w(z) = E ~ t + - - (9) K * A E I

where Et is the representative longitudinal modulus of the transformed section, I, is the transformed second moment, and K* is a modified shear coefficient given as,

K * = I, 1 2 + f +_v,: tds f x Otds (10)


TABLE 2 Modified Shear Coefficient K* for Box, I- and T-Beams

Rectangular box beam

I-beam

T-beam

K* = 20(~ + 3rn)2/[~--~X(6OmZnZ+60~mn z)

~ ( 1 8 0 m 3 + 300am 2 + 144~Zrn + 24a 3) +

+ v l ( - 30mZn 2 - 50~mn 2) + vz(30m z + 6am - 4~z)j

b btl E2 where n = ~, m = h~2' a = E-~

K* = 20(~+ 3m)2/[~(6OmZn2+6Ootmn2)

+ ~--~12 (180m3 + 300~m 2 + 144c~2m + 24~ 3)

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

b 2bt t E 2 where n = ~, m = - - , ~ = - -

htw El

K* = 10~(~ + 4m)Z(~ + re)z~

~-(12~ 6 + 120~Srn + 480~4m z + 840a3rn 3

+ 660~2m 4 + 192am s)

~ (30mn2a(~ + m) 3) +

+ vx(- 2a 5 -- 5ct4m -- 150t3m / - 20ct2m 3 + 40ctm 4 + 48m 5)

+ v2(2Oot4mn 2 -- 15ct4m + 8 0 c t 3 m 2 n 2 _ 75c t3m 2

+ 120~tZm3n 2 -- 60~t2m 3 + 80~m4n 2 + 20mSn2)~ - I

b btr E2 w h e r e n = - m = - - ~ = - -

h' ht W' Ex

w h e r e At is the t r a n s f o r m e d a rea o f the c ross - sec t ion . N o t e t h a t in e q n (9) the ac tua l c ro s s - s ec t i ona l area , A, is used in the d e n o m i n a t o r o f the s e c o n d term. T o eva lua t e K * the expres s ions in eqns (5) a n d (6) a re a g a i n used t o f ind the m o d i f i e d f lexure f u n c t i o n b u t in this case the i n -p l ane shea r m o d u l u s , Gs=, a n d in -p l ane P o i s s o n ra t io , vs=, m u s t c o r r e s p o n d to the a p p r o p r i a t e pane l s u n d e r c o n s i d e r a t i o n , a n d the t r a n s f o r m e d s e c o n d m o m e n t I , m u s t be used.


t b t

r l i t2 Y

(a)

Fig. 2.

I

h tw -o

# t~

b I

E1 G1Vl ql-

=

Y

E2G2v i

(b)

b I

tf tw -~

E2G2v2

X

(c)

E1 G1 V1

Y

Cross-sections of thin-walled beams in Table 2(a) Rectangular box beam (b) I-beam (c) T-beam.

Formulas for K* for a rectangular box-beam and an I beam are given by Bank; a for a T-beam by Bednarczyk. Is They are repeated here for convenience in Table 2. The cross-sections are shown in Fig. 2. Note that in Table 2 the values of K* are given for sections transformed with respect to the longitudinal modulus of the horizontal panels, El.

3 RESULTS AND DISCUSSION

A computer code was developed 18 to analyze thin-walled fiber-reinforced composite beams using the shear deformation beam theory and the modified shear coefficient. The code allows the user to investigate the effects of varying the different parameters which affect the beam deflection. The input to the code consists of the beam shape and dimensions (box, I- or T-beam), the orthotropic lay-ups of the horizontal and vertical panels, and the end and loading conditions (cantilever or simply-supported with concentrated or uniform load). Multiple runs can be made for the same loading to investigate


TABLE 3 Lay-ups of Horizontal and Vertical Panels in the Example I-Beams

Case no. Horizontal panel Vertical panel

1 Graphite/epoxy Graphite/epoxy [75% (0), 25% (+45)] [75% (0), 25% (90)3

2 Glass/epoxy Graphite/epoxy [75% (0), 25% (+__45)] [75% (0), 25% (90)]

3 Graphite/epoxy Graphite/epoxy [50% (_ 15), 25% (0), 25% (90)3 [50% (_45), 25% (0), 25% (90)3

4 Graphite/epoxy Graphite/epoxy [50% (+ 15), 25% (0), 25% (90)] [33% (+45), 33% (+ 15), 17% (0),

17% (90)3

the effects of altering the lay-up of the panels. The output is in both graphical and numerical form. The graphical output consists of a set of plots which are generated for each case run, and a set of plots which show the comparison between the different cases run. The numerical output consists of the calculated values for the in-plane properties of the panels, the modified shear coefficient, the bending deflection, the shear deflection and the total deflection along the beam length for each case. To demonstrate the use of the beam theory the computer code is used to analyze two example cases.

3.1 Example 1

In this example a 1000 mm long cantilever I-beam loaded by a concentrated force of 100N at its tip is considered. The dimensions of the beam, see Fig. 2(c), are, h = 1OO mm, b = 50 mm, tf --- t w = 2 mm. To demonstrate the effect of changing the lay-ups and composite material systems in the vertical (web) and the horizontal (flange) panels four different cases, detailed in Table 3, are considered. The lay-ups in the different panels that form the cross-section are given in terms of the percentage by thickness contribution of each ply type to allow for ease of interpretation since the in-plane properties depend only on

TABLE 4 Response of Tip-loaded Cantilever I-Beams; ! = 1000mm, P = 100N

Beam K* wJw s wb(mm ) ws(mm ) w(mm) description x 10- 3

Case 1 24.62 4-96 0.352 0.071 0-423 Case 2 38-54 4.22 0.849 0-201 1"050 Case 3 119.4 27.7 0-443 0"016 0"459 Case 4 94.01 19.9 0-418 0"021 0'439


E E

i

Z 0 I.- 0 W .J EL ILl 1:3

1 .2 -

1 .0 .

0.8

0.6

0 . 4

0.2

0.0 (b

(3-

1.2

CASE 1

- - - - - Bending deflection Total deflection

(a)

f

i g g (3 (3

LENGTH A L O N G BEAM-(mm)

E E lo - r -

z O 0.8

E £3 I.,U 0.6 _1 kl_ I..U t~l 0 4 - .,_1 ,< t- O 0.2- I'-

0.0

C A S E 1 (¢ )

. . . . . C A S E 2 • '

- - - - C A S E 3 r

. . . . . C A S E 4 /

J

e"

,,j I~°°S'° . • "

(3 (3 (3 ~ ~ (3 (3 . . . . . .(3~"

LENGTH A L O N G BEAM-(mm)

1.2

1.0

E" 0.8 E I

Z O ~_. 0.6

o i i i . J LL W 0.4, 0

0.2.

0.0 (3

(3.

30.0

~ " 25.0 < W 3- 03 20.0

~ " 15.0

Z

10.0 t.U m

~ " 5.0

0.0 (3 (3'

CASE 2 (b)

- - To ta lde f lec~ / - -Bend ing deflection /

'1 i ! (3 (3 (3 ~ (3

LENGTH ALONG B E A M - - ( m m )

C A S E 1 (d ) i

. . . . . C A S E 2 f /

- - - C A S E 3 /

. . . . . C A S E 4 /

/ t / / . /

/ / / /

/ / / / . /

/ . /

g g g ~ (3 . . . . (3 ~ •

LENGTH A L O N G BEAM-(mrn)

Fig. 3. Graphical output for the example I-beams; (a) total and bending deflection for case 1, (b) total and bending deflection for case 2, (c) total deflection for all four cases, (d) wb/w s for

all four cases.

the relative quantities of each ply type through the thickness of a balanced laminate. The particular lay-ups were chosen to demonstrate important aspects o f the theory.

The most significant fact that is seen from the numerical, see Table 4, and the graphical, see Fig. 3, results is that the shear deflection can have a large contribution to the total beam deflection. As the ratio o f the bending


deflection to shear deflection, Wb/W s, decreases the proportion of the deflection due to shear deformation of the beam cross-section increases. For a value of the ratio equal to 4, 20% of the beam deflection is due to shear. Also significant is the fact that both the bending deflection and the shear deflection are affected by the lay-ups in both the vertical and the horizontal panels. The contribution of the horizontal panels to the shear deflection is a result of the in-plane warping of these panels. The significance of this warping effect has been recognized, 19 but has not previously been incorporated into such a 'simple' beam theory.

As can be seen from a comparison between cases 1 and 2, the shear deflection in case 2 is greatly increased by changing the horizontal flange panels to glass/epoxy without changing the vertical web panel. This result shows how the proposed formulation for the modified shear coefficient accounts for the deformation of the whole cross-section due to shear and not only for the shear deformation of the vertical web of the section. In fact the Wb/W ~ ratio decreases slightly indicating a proportionally greater effect on the shear deflection than on the bending deflection. As is also expected for this case, the bending deflection is greatly increased due to the decrease in the flexural rigidity of the I-beam with the glass/epoxy flanges.

The results:, also show the importance of obtaining the optimum combination of bending and shear deflection of the beam in order to obtain the least total deflection. It is not necessarily the beam with the least shear deflection that will have the least total deflection. In the cases considered, case 1 has the least total deflection while case 3 has the least shear deflection. The influence of placing angle plies, especially at +_ 45 °, in the web is clearly an important, but not an exclusive, factor in reducing the shear deflection of the beam. Cases 1, 3 and 4 all have similar total deflections but have very different Wb/W s ratios. Neglecting the shear deflection may be acceptable in case 3 but will not be acceptable in case 1.

In the above four cases the different lay-ups were chosen to demonstrate certain features of the beam behavior. The optimal lay-up for this beam is not necessarily case 1 but may well be found with a different lay-up. The computer code allows the user to perform these changes to the beam in a systematic manner and to compare the graphical output to get an immediate 'feel' for the changes made.

3.2 Example 2

A 1000 mm long cantilever box beam is loaded at its tip by a concentrated force of 100N. The thin-walled composite beam, with wall thickness t -- 1 mm, has a rectangular cross-section. Two different aspect ratios (b/h) are considered; b/h = 4-0 and b/h = 0.25. The panels of the cross-section are

Beam theory for thin-walled composite beams

T A B L E 5 Response o f Tip-Loaded Cantilever Box Beams; l = 1000 ram, P = 100N

275

Beam description P K* x 10-3 wJw s wb(mm ) ws(mm ) w(mm)

Type 1 (b/h = 4) b = 100 m m h = 25 m m 0 3"719 9"16 5.440 0.594 6"034

Horizontal panels 25 5'205 13"0 5-529 0"425 5"954

All 0 ° plies 50 5-872 15"0 5"624 0-376 6"000

Vertical panels 75 6'286 16'3 5.723 0-352 6'075

% _+45 ° = P %0 ° = 100 - P 100 6"598 17-4 5'826 0"335 6-161

Type 2 (b /h=4) b = 1 0 0 m m h = 2 5 m m 0 3.719 9-16 5.440 0"594 6"034

Horizontal panels 25 6-594 15"9 6.743 0'424 7-167

% _ 45 ° = P %0 : = 100 - P 50 9"637 22.5 8-959 0"399 9'358

Vertical panels 75 14"39 31" ! 13"39 0"430 13"82

All 0 ° plies 100 24"77 41-2 26-58 0.645 27'23

Type 3 (b/h = 0"25) b - 2 5 m m h = 1 0 0 m m 0 29.33 8.41 0.631 0"075 0-706

Horizontal panels 25 82.19 26"6 0.717 0"027 0.744

All 0 ° plies 50 157.7 59.6 0-834 0"014 0"848

Vertical panels 75 271" 1 124'8 0"998 0"008 1"006

% +__45 ° = P % 0 = 1 0 0 - P 100 456.7 244.8 1.224 0.005 1-229

Type 4 ( b/h = 0.25) b = 25 m m h = 100 m m 0 29'33 8"41 0-631 0-075 0"706

Horizontal panels 25 30"54 7.54 0.694 0-092 0'786

% + 4 5 ° = P %0 ° = 100 - P 50 31.52 6"33 0.772 0"122 0-894

Verticalpanels 75 32'36 4"57 0'872 0'191 1"063

All 0 ° plies 100 32"94 2-06 1-001 0"485 1"486

constructed of 0 ° plies and _+ 45 ° plies of graphite/epoxy T300/5208. The percentage, P, of the _+45 ° plies is varied in either the vertical or the horizontal panels of the cross-section and is given in Table 5 together with the numerical results. Graphical results are not shown for example 2.

In Table 5 the effect of altering lay-up configurations and aspect ratios on a rectangular box beam is shown. It is again seen that the shear deflection in these thin-walled composite beams is significant and must be accounted for. By altering the lay-up of the panels of the beam both the magnitude of the shear deflection and of the bending deflection is affected. In order to obtain the least total deflection for the beam a lay-up must be found which 'optimizes' the two deflection components. In beam Type 1, for example, the least total deflection is obtained when 25% of the plies in the vertical panels are at -t-45 °. In beam Types 1 and 3 it is seen that the shear deflection decreases as the percentage of _+45 ° plies increases in the vertical panels.


This physically intuitive result serves to demonstrate that the proposed formulation for K* and the form of the beam theory used, do in fact provide reasonable predictions for the beam response. In beam Types 3 and 4 the total deflection predicted for the two beams is quite similar. However, the magnitude of the components of this total deflection in these two cases is very different. Neglecting the shear deflection in beam Type 4 would give a total deflection which would be up to 30% less than the actual deflection.

The above examples demonstrate the use of the shear deformation beam theory for thin-walled composite material beams. The lay-up optimization shown in the examples considers only the beam deflection as the optimization criterion. In actual design of a thin-walled composite beam many other factors, such as strength and weight, may also be optimized. In addition, the overall design of the beam will have to consider the fabrication technologies available for producing the desired lay-ups./When designing for strength the magnitudes of the axial and shear stresses will be required. The design of thin-walled composite beams based on these other considerations are separate topics 2'3 and are not addressed in this paper.

4 CONCLUSION

The effect of shear deformation on the total deflection of a thin-walled fiber- reinforced composite beam can be found by using a suitable shear deformation beam theory together with a modified shear coefficient. Formulas for the evaluation of the modified shear coefficient for different cross-sections make the theory especially easy to apply to the design of composite beams. Due to the anisotropic nature of the composite material panels in the cross-sections of a thin-walled composite beam it is recommended that this beam theory be used in initial investigations instead of Bernoulli-Euler beam theory.

REFERENCES

1. Green, A. & Bisarnsin, T., Building construction materials. In Advanced Thermoset Composites, ed. J. M. Margolis. Van Nostrand Reinhold Company, New York, 1986, pp. 249-71.

2. Bicos, A. S. & Springer, G. S., Design of a composite boxbeam. J. Comp. Mat. 20 (1986) 86-109.

3. Evans, P. R. & Slepetz, J. M., Graphite composite box beams for U.S. Army mobile bridging. In Proceedings of the Fourth Conference on Fibrous Composites in Structural Design, San Diego, CA, Nov. 14--17, 1978, pp. 687-700.

4. Dudek, T. J., Young's and shear moduli of unidirectional composites by a resonant beam method. J. Comp. Mat. 4 (1970) 232-41.


5. Zweben, C., Smith, W. S. & Wardle, M. W., Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates, composite materials: testing and design (Fifth Conference). A S T M STP 674, ed. S. W. Tsai, 1979, pp. 228-62.

6. Bresse, J. A. C., Courses de Mechanique, 2nd Edn, Part 1, Gauthier-Villars, Paris, 1866.

7. Timoshenko, S. P., On the correction for shear of the differential equation for transverse vibration of prismatic bars. Phil. Mag., 41 (1921) 744-6.

8. Bank, L. C., Shear coefficients for thin-walled composite beams. Composite Structures, 8 (1987) 47-61.

9. Tsai, S. W. & Hahn, H. T., Introduction to Composite Materials. Technomic Inc., Pennsylvania, 1980.

10. Cowper, G. R., The shear coefficient in Timoshenko's beam theory. J. Appl. Mech., 33 (1966) 335-40.

1 I. Love, A. E. H., A Treatise on the Mathematical Theory of Elasticity, Fourth Edition. Dover Publications, New York, 1944.

12. Dharmarajan, S. & McCutchen Jr, H., Shear coefficients for orthotropic beams. J. Comp. Mat., 7 (1973) 530-5.

13. Teh, K. K. & Huang, C. C., Shear deformation coefficient for generally orthotropic beam. Fib. Sci. and Tech., 12 (1979) 73-80.

14. Jensen, J. J., On the shear coefficient in Timoshenko's beam theory. J. Sound Vib., 87 (1983) 621-35.

15. Oden, J. T. & Ripperger, E. A., Mechanics of Elastic Structures, Second Edition. Hemisphere Publishing Corp., New York, 1981.

16. Jensen, J. J. & Pedersen, P. T., Recent advances in beam models used for ship hull vibration analysis. Paper presented at the International Symposium on Ship Vibrations, Genova, 22-24 May, 1984.

17. Bhat, S. U. & de Oliveira, J. G., A formulation for the shear coefficient of thin- walled prismatic beams. J. Ship Research, 29 (1985) 51-8.

18. Bednarczyk, P. J., Shear deflection in thin-walled fiber-reinforced composite beams. MS thesis, Department of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY, May 1987.

19. Bauchau, O. A., A beam theory for anisotropic materials. J. Appl. Mech., 52 (1985) 416-22.