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Beam Theory
Composites Science and Technology 32 (1988) 265277
A Beam Theory for Thinwalled Composite BeamsL. C. Bank & P. J. BednarczykDepartment of Civil Engineering, Rensselaer Polytechnic Institute, Troy, NY 121803590, 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 crosssection of a thinwalled 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 layups in the panels of typical beams.
1 INTRODUCTION Thinwalled fiberreinforced composite material beams are being used in a variety of structural applications in the aerospace, mechanical and civil engineering disciplines.I3 A significant benefit obtained from the use of composite materials in the panels or walls of the beam crosssection is the ability to tailor the mechanical properties of the beam to a specific application. In addition to the usual considerations of the beam crosssectional 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 layups in the panels of the beam.265
Composites Science and Technology 02663538/88/$0350 1988 Elsevier Applied SciencePublishers Ltd, England. Printed in Great Britain
266
L. C. Bank, P. J. Bednarczyk
In general, the panels of the beam crosssection 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 crosssection. This shear deformation of the crosssection 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 thinwalled composite beam a recently formulated modified shear coefficient 8 is obtained as a function of the beam crosssectional 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 thinwalled composite beams. Results obtained using a dedicated computer code based on the theory are shown and discussed.
2 SHEAR D E F O R M A T I O N BEAM THEORY A N D SHEAR COEFFICIENT
The crosssection of the thinwalled composite beam considered lies in the x y 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 crosssectional geometry. The crosssection is constructed of midplane symmetric orthotropic panels that are placed in the crosssection with one of the panels' orthotropic axes parallel to the beam zaxis. This gives each panel one coordinate in the zdirection and the other inplane 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 inplane longitudinal modulus, Ez, its inplane shear modulus, Gsz, and its inplane 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 crosssectional 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 theoryfor thinwalled composite beams
267
r I shear flow I r
shear flowY
laminated composite panelFig. 1. Typical thinwalled beam crosssection.
Considering, initially, a thinwalled beam constructed of panels all having the same inplane properties the beam equations are written as,
Ej~z=MQ+
(1)
~3z
~b 
KAGs=
 
(2)
where E= is the longitudinal modulus of the panels,/is the second moment of the crosssection 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 crosssectional area and Gs, is the inplane 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,
w(z)= A(z)

KAG~
A(z) 
(3)
wherefl(z ) andf2(z) depend on the loading and end conditions. For the case of the tiploaded cantilever beam, the uniformlyloaded simplysupported beam, and the centerloaded simplysupported 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 crosssectional
268
L. C. Bank, P. J. BednarczykTABLE 1 Deflection Functions for Timoshenko Beams
Beam typeTiploaded cantilever P (N) Uniformlyloaded simplysupported.q (N m  1) Centerloaded simplysuported P (N) (z < l/2)
f l(z)P( 3 lz 2  z 3) ~
f2(z)
Pz2 (lz Z2) ~
~4(z'2lz3+13z)(312z  4z 3)
shape and on the material properties of the panels that form the crosssection. For crosssections 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 3dimensional 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 f e a d 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 thinwalled 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 thinwalled crosssection, see Fig. 1, and is given as,
K=
IE~/G~ ~f + V~zs2tds+~ A f xOtds
(4,
where t is the wall thickness and s is the arc length coordinate around the crosssection, which coincides with one of the inplane 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
Beam theoryfor thinwalled composite beams
269
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 _Ox
Q cos~ %~ +
Ej
(v~zx2)
(5) (6)
G~ O~k
~y = Q sin 0
Ej
z~ +G~zvs~xy
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 crosssection which is assumed to be uniform across the wall thickness and is found by standard equilibrium methods. For the case of closed cell crosssections 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 crosssections 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 crosssection have different inplane elastic properties a 'transformed section' is used and the beam equations are written as,
E'It ~ z = MOw+
(7)
Q K*AE, (8)
and the solution as,
fl(z) f2(z) w(z) = E ~ t + K *A E I
(9)
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*=1 2
I,
f +_v,:tds + f xOtds
(10)
270
L. C. Bank, P. J. BednarczykTABLE 2 Modified Shear Coefficient K* for Box, I and TBeams
Rectangular box beam
K* = 20(~ + 3rn)2/[~~X(6OmZnZ+60~mn z) + ~ ( 1 8 0 m 3 + 300am 2 + 144~Zrn + 24a 3) + v l (  30mZn2  50~mn 2) + vz(30m z + 6am  4~z)j
where n = ~, m = h~2' a = E~ Ibeam K* = 20(~+ 3m)2/[~(6OmZn2+6Ootmn2) + ~~12 (180m3 + 300~m 2 + 144c~2m+ 24~ 3) + vl (60m2n z + 40~mn 2) + v2(30m2 + 6~m  4~2)] where n = ~, m =   , Tbeam
b
btl
E2
b
2bt t E2 ~ = htw El
K* = 10~(~ + 4m)Z(~ + re)z~ ~(12~ 6 + 120~Srn + 480~4m z + 840a3rn 3 + 660~2m4 + 192am s)
~
(30mn2a(~ + m) 3) +
+ vx( 2a 5  5ct4m  150t3m/  20ct2m3 + 40ctm4 + 48m 5)
+ v2(2Oot4mn2  15ct4m +
80ct3m2n 2 _ 75ct3m 2
+ 120~tZm3n2  60~t2m3 + 80~m4n 2 + 20mSn2)~I
b btr E2 wheren= m=~=h' ht W' Ex
w h e r e At is t h e t r a n s f o r m e d a r e a o f t h e c r o s s  s e c t i o n . N o t e t h a t in e q n (9) t h e a c t u a l c r o s s  s e c t i o n a l area, A, is u s e d in t h e d e n o m i n a t o r o f t h e s e c o n d term. T o e v a l u a t e K * t h e e x p r e s s i o n s in e q n s (5) a n d (6) a r e a g a i n u s e d t o find t h e m o d i f i e d flexure f u n c t i o n b u t in this c a s e t h e i n  p l a n e s h e a r m o d u l u s , Gs=, a n d i n  p l a n e P o i s s o n ratio, vs=, m u s t c o r r e s p o n d t o t h e a p p r o p r i a t e p a n e l s u n d e r c o n s i d e r a t i o n , a n d t h e 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.
Beam theoryfor thinwalled composite beams
271
t
b
t
r
lit2
Y
(a)
I
b
IE1G1Vl
I
b
tftw ~
E1 G1 V1
h
tw o
ql=
Y E2G2v2
Y# t~
E2G2vi
X (b)
(c)
Fig. 2. Crosssectionsof thinwalledbeams in Table 2(a) Rectangular box beam (b) Ibeam
(c) Tbeam. Formulas for K* for a rectangular boxbeam and an I beam are given by Bank; a for a Tbeam by Bednarczyk. Is They are repeated here for convenience in Table 2. The crosssections 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 thinwalled fiberreinforced 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 Tbeam), the orthotropic layups of the horizontal and vertical panels, and the end and loading conditions (cantilever or simplysupported with concentrated or uniform load). Multiple runs can be made for the same loading to investigate
272
L. C. Bank, P. J. BednarczykTABLE 3
Layups of Horizontal and Vertical Panels in the Example IBeamsCase no. Horizontal panel Vertical panel
1 2 3 4
Graphite/epoxy [75% (0), 25% (+45)] Glass/epoxy [75% (0), 25% (+__45)] Graphite/epoxy [50% (_ 15), 25% (0), 25% (90)3 Graphite/epoxy [50% (+ 15), 25% (0), 25% (90)]
Graphite/epoxy [75% (0), 25% (90)3 Graphite/epoxy [75% (0), 25% (90)] Graphite/epoxy [50% (_45), 25% (0), 25% (90)3 Graphite/epoxy [33% (+45), 33% (+ 15), 17% (0), 17% (90)3
the effects of altering the layup 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 inplane 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 1In this example a 1000 m m long cantilever Ibeam loaded by a concentrated force of 100N at its tip is considered. The dimensions of the beam, see Fig. 2(c), are, h = 1OOmm, b = 50 mm, tf  t w = 2 mm. To demonstrate the effect of changing the layups and composite material systems in the vertical (web) and the horizontal (flange) panels four different cases, detailed in Table 3, are considered. The layups in the different panels that form the crosssection are given in terms of the percentage by thickness contribution of each ply type to allow for ease of interpretation since the inplane properties depend only onTABLE 4
Response of Tiploaded Cantilever IBeams; ! = 1000mm, P = 100NBeam description K* x 10 3 wJw s wb(mm) ws(mm) w(mm)
Case Case Case Case
1 2 3 4
24.62 3854 119.4 94.01
496 4.22 27.7 19.9
0.352 0.849 0443 0418
0.071 0201 0"016 0"021
0423 1"050 0"459 0'439
Beam theory for thinwalled composite beams1.2
273
CASE 11.0.
(a)
1.2
CASE 21.0 
(b)/

 
E Ei
Bending deflection Total deflection
 Totaldeflec~/Bending deflection
Z 0 I.0 W .J EL ILl 1:3
0.8
E" 0.8 E IZ O ~_. 0.6
0.6
o0.4
LL W 0.4,0
iii .J
0.2 f
0.2. i0.0
0.0 (b (3
g
g
(3
(3
(3 (3.
'1 (3
(3
i
(3
!
~
(3
LENGTH ALONG BEAM(mm)1.2 30.0 CASE 1 () ' r / J 0.6 ~" < W 303 25.0
LENGTH ALONG BEAM(mm)
CASE 1 ..... 20.0 ..... CASE 2 CASE 3 CASE 4 / / / f
rz O 3 I.,U _1 kl_ I..U t~l .,_1 ,