Higher order composite beam theory built on Saint-Venant’s solution. Part-II: Built-in effects...

Composite Structures 93 (2011) 567–581

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Composite Structures

journal homepage: www.elsevier .com/locate /compstruct

Higher order composite beam theory built on Saint-Venant’s solution. Part-II:Built-in effects influence on the behavior of end-loaded cantilever beams

Nejib Ghazouani, Rached El Fatmi ⇑Mécanique des structures, LGC, Ecole Nationale d’Ingénieurs de Tunis, B.P. 37, Campus universitaire, Le Belvédère, 1002 Tunis, Tunisia

a r t i c l e i n f o

Article history:Available online 25 August 2010

Keywords:Saint-Venant solutionBuilt-in effectsComposite sectionTorsionBendingTension

0263-8223/$ - see front matter � 2010 Elsevier Ltd. Adoi:10.1016/j.compstruct.2010.08.023

⇑ Corresponding author.E-mail addresses: [email protected] (N. Ghaz

rnu.tn (R. El Fatmi).

a b s t r a c t

The higher order composite beam theory (HOCBT), established in Part-I, is a refinement of the one-dimensional beam-like theory related to 3D Saint-Venant’s solution. HOCBT is based on a displacementmodel including in/out-of plane warpings and is devoted to symmetric and orthotropic composite beams.

In the present Part-II, HOCBT is applied to analyze the built-in effects influence on the structural behav-ior of end-loaded cantilever beams (torsion, tension and shear-bending). In the critical region, close to thebuilt-in section, the 3D (axial and shear) stresses calculated by the proposed theory are relevant and quitecomparable to those obtained by 3D-FEM computations. These results, obtained for a representative setof cross-sections, show that HOCBT is able to describe the built-in effects, and hence their influence onthe structural behavior of the beam.

As expected, moving from the built-in section, the results (displacements and stresses) tend towardsSaint-Venant’s solution in the interior part of the beam. It is shown that the extents of the built-in effectsare related to dimensionless constants that take into account the whole nature of the composite sectionand the loading case. Practically, regarding to Saint-Venant results, these constants allow to predict thebuilt-in effects expansion.

� 2010 Elsevier Ltd. All rights reserved.

1. Introduction

For a reference elastic beam problem, the exact 3D solution canbe viewed as the sum of Saint-Venant’s solution and an extremitysolution that contains the end-effects [4,6]. When the end-effectsare confined closely to the end section, SV’s solution (also calledthe central solution) is known to give a good description of the3D solution in a large interior part of the beam, and the correspond-ing one-dimensional beam-like theory (denoted SV-BT) provides agood description of the structural behavior of the beam. However,contrary to SV’s Principle, these end-effects are not always confinedclosely to the end-sections [9]. This depends on the cross-sectionnature and the boundary conditions. For instance, when stronglyanisotropic materials and/or thin walled open profiles are of con-cern, the end-effects can persist over distances comparable to thebeam length [5,11]. This leads to a global elastic behavior of thebeam notably different from that predicted by SV-BT.

In order to refine SV-BT, a higher order composite beam theory(HOCBT) has been proposed in Part-I of the present two-part-pa-per. HOCBT is built on a displacement model, that starts from thatof SV’s solution and allows to better satisfy the boundary condi-

ll rights reserved.

ouani), rached.elfatmi@enit.

tions. This theory is established for cross-section (CS) symmetricand made by orthotropic materials for which the principal materialcoordinates coincide with those of the beam (denoted hereafter byso-CS). HOCBT includes the out-of-plane warpings due to the shearforces and the torsional moment, and the in-plane warping (Pois-son’s effects) due to the axial force and the bending moments.

To illustrate the predictive capability of this beam theory, thepresent Part-II is devoted to the analytical and numerical analysesof cantilever beams made of different kinds of so-CS (solid-CS,walled open/closed-CS) and subjected to torsion, axial tension orshear-bending. The applications are limited1 to cantilever beamsbecause they are handled, in literature, as benchmark problems to as-sess such refined beam theories.

Built-in effects are intimately related to the restrained in/outwarping effects (RWE). Starting from the built-in section, it is ex-pected that RWE vanish and the results (displacements and stres-ses) tend toward the values predicted by SV’s solution. Theexpansion of the RWE in the interior part of the beam is analyzedfor each loading and section. Numerical results are given for the dis-tributions along the beam axis of the cross-sectional displacementsand for the three-dimensional (3D) stress distributions close to thebuilt-in section; these stresses are compared to those obtained bythree-dimensional finite element (3D-FEM) computations.

1 HOCBT equations established in Part-I consider distributed loadings.

568 N. Ghazouani, R. El Fatmi / Composite Structures 93 (2011) 567–581

The numerical applications of HOCBT need, for a given cross-section, to first compute all its SV-characteristics: the cross-sec-tional constants and the SV-warping functions. This is achievedusing the numerical method proposed by El Fatmi and Zenzri[2,3] for the computation of the 3D SV’s solution within the frame-work of the exact beam theory [7].

In the present paper, for convenience,2 the first section recallsthe main HOCBT equations needed for the developments presentedin this Part-II. It should also be noted that all the symbols used arethose introduced in Part-I.

2. HOCBT equations

This section recalls the main equations of the HOCBT needed forthe study of a cantilever composite beam subjected to a tip loading(Fig. 1, x is the beam axis and (y,z) are the sectional coordinates).

The displacement model, parametrized by (u = (ux,uy,uz),x = (xx,xy,xz) and a = [ax,ay,az], g = [gx,gy,gz]), is given by:

n ¼ uðxÞ þxðxÞ ^0yz

24 35þ axðxÞ0

Vxðy; zÞWxðy; zÞ

24 35þ ayðxÞ

0Vyðy; zÞWyðy; zÞ

24 35þ azðxÞ0

Vzðy; zÞWzðy; zÞ

24 35þ gxðxÞwxðy; zÞ

00

24 35þ gyðxÞ

wyðy; zÞ00

24 35þ gzðxÞwzðy; zÞ

00

24 35 ð1Þ

where (u,x) are the cross-sectional displacements, (ai,gi,i 2 {x,y,z})are the in and out warping parameters related to the in-plane andthe out-of-plane SV-warping functions ((Vi,Wi),wi), respectively.The generalized stresses associated with this displacement modelare:

R ¼ ðN; Ty; TzÞ Am ¼ Axm;A

ym;A

zm

� �Mw ¼ Mx

w; Tyw; T

zw

h iM ¼ ðMx;My;MzÞ Bs ¼ Ns;M

ys ;M

zs

� �Ms ¼ Mx

s ; Tys ; T

zs

� �9=; ð2Þ

where (R,M) are the classical cross sectional resultant and momentand (Am,Bs; Mw,Ms) are additional internal forces related to the

My

Mys

Aym

Mzw

Tz

Tzs

26666666664

37777777775¼

hz2K11i �ðhz2K11i � fEIyÞ 0 �az 0

�ðhz2K11i � fEIyÞ ðhz2K11i � fEIyÞ 0 az 00 0 gy 0 fz

�az az 0 gKIzw 0

0 0 fz 0 hGxzi0 0 ez 0 �ðhGxzi �gGA

2666666666664

Mz

Mzs

Azm

Myw

Ty

Tys

26666666664

37777777775¼

hy2K11i �ðhy2K11i � fEIzÞ 0 �ay 0

�ðhy2K11i � fEIzÞ ðhy2K11i � fEIzÞ 0 ay 00 0 gz 0 fy

�ay ay 0 gKIyw 0

0 0 fy 0 hGxy

0 0 ey 0 �ðhGxyi �gGA

2666666666664

2 To make the present Part-II self-sufficient.

in-plane and out-of-plane warpings. The latter are related to thestresses by:

Axm ¼ hsxyVx þ sxzWxi Mx

w ¼ hrxxwxi

Aym ¼ hsxyVy þ sxzWyi My

w ¼ hrxxwyi

Azm ¼ hsxyVz þ sxzWzi Mz

w ¼ hrxxwzi

Ns ¼ ryyVx;y þ rzzWx

;y þ syz Vx;z þWx

;y

� �D EMx

s ¼ sxywx;y þ sxzw

x;z

D EMy

s ¼ ryyVy;y þ rzzW

y;y þ syz Vy

;z þWy;y

� �D ETy

s ¼ sxywy;y þ sxzw

y;z

D EMz

s ¼ ryyVy;y þ rzzW

z;y þ syz Vy

;z þWz;y

� �D ETz

s ¼ sxywz;y þ sxzw

z;z

D E

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;ð3Þ

where h(�)i denotesR

Sð�ÞdS. An external surface force density H act-ing on an end section reduces to external (1D) generalized forces(F,C,Q,S) related to H by:

F ¼ hHi Q ¼ ½hHxwxi; hHxw

yi; hHxwzi�

C ¼ hX ^Hi S ¼ ½hHyVx þ HzWxi; hHyVy þ HzWyi; hHyVz þ HzWzi�

)ð4Þ

For a beam, in equilibrium only under tip forces, the 1D equilib-rium equations reduce to

R0 ¼ 0 M0w �Ms ¼ 0

M0 þ x ^ R ¼ 0 A0m � B ¼ 0

)ð5Þ

where (�)0 denotes the derivative with respect to x.1D behavior. Let K11,Gxy,Gxz,Gyz be the (local) mechanical con-

stant defined by Eq. (A.1) (see Appendix A), the (1D) structuralbehavior is given by the following uncoupled constitutiverelations:

N

Ns

Axm

264375 ¼ hK11i �ðhK11i � fEAÞ 0

�ðhK11i � fEAÞ hK11i � fEA 00 0 gx

26643775

cx

ax

a0x

264375

Mx

Mxs

Mxw

264375 ¼

gGIx �ðgGIx �fGJÞ 0

�ðgGIx �fGJÞ gGIx �fGJ 0

0 0 gKIxw

26643775

vx

gx

g0x

264375 ð6Þ

0

0ez

0

�ðhGxzi �gGAzÞ � eRzfz

zÞ � eRzfz hGxzi �gGAz þ eRzðfz � ezÞ

3777777777775

vy

ay

a0yg0zcz

gz

2666666664

3777777775ð7Þ

0

0ey

0

i �ðhGxyi �gGAyÞ þ eRyfy

yÞ þ eRyfy hGxyi �gGAy � eRyðfy � eyÞ

3777777777775

vz

az

a0zg0ycy

gy

2666666664

3777777775ð8Þ

Fig. 1. Cantilever composite beam, so-CS, and tip loadings.

3 This composite section is not a walled open section and if E2 = E1, it leads to arectangular homogeneous section. However, if E2� E1, the section behavior corre-sponds to that of a walled open section. HOCBT enables to take into account the wholenature of the section: shape-and-materials.

N. Ghazouani, R. El Fatmi / Composite Structures 93 (2011) 567–581 569

where (c = u0+ x ^x, v = x

0) define the classical cross-section

strains, fEA;gGAy ;gGAz ;fGJ; fEIy ; fEIz ; eRy ¼fGAyeEIz

; eRz ¼fGAzeEIy

� �the cross-sec-

tional constants that derive from SV-beam-theory (SV-BT),gKIxw ;gKIy

w ;gKIz

w

� �and (gx,gy,gz) the out-of-plane and the in-plane

warping rigidities, and (ay,ey, fy,az,ez, fz) the coupling constants be-tween in-plane and out-of-plane warpings in shear-bending behav-ior; this coupling will be denoted by I/O–W-coupling (like in/out-warping coupling).

The closed-form expressions of the 3D stresses that correspondto this beam theory are given in Appendix A by Eqs. (A.2), (A.3), andthe new cross-sectional constants gKIx

w ;gKIy

w ;gKIz

w ; gx; gy; gzay; ey; fy;�

az; ez; fzÞ are defined in Appendix B by Eq. (B.1).

3. Analytical analyzes of torsion, axial tension and shear-bending of a cantilever beam

Let us consider (Fig. 1) a cantilever beam, made of any so-CS,with a length to thickness ratio k = L/h. The section S0 at x = 0 isembedded and the section SL at x = L is loaded by a surface forcedensity H.

1D equations. The equilibrium equations are given by Eq. (5),the constitutive relations by Eqs. (6)–(8), and the boundary condi-tions are:

uð0Þ ¼ 0 RðLÞ ¼ ðhHxi; hHyiÞ; hHziÞÞxð0Þ ¼ 0 MðLÞ ¼ ðhzHy� yHzi; hzHxiÞ; h�yHxiÞÞgð0Þ ¼ 0 MwðLÞ ¼ ½hHxw

xi; hHxwyi; hHxw

zi�að0Þ ¼ 0 AmðLÞ ¼ ½hHyVxþHzW

xi; hHyVyþHzWyi; hHyVzþHzWzi�

9>>>=>>>;ð9Þ

Warpings are free on SL and restrained on S0. Thus, moving fromthe built-in section, it is expected to see the warpings increasingfrom zero to tend towards the values predicted by SV-theory whichis known to give a good description of the elastic behavior in theinterior part of the beam.

Three cases of loading will be considered: torsion, axial tensionand x-z-shear-bending of the beam. For each tip loading, the anal-ysis focuses on the RWE close to built-in section S0 and its expan-sion to the interior part of the beam by comparison with SV-results.

3.1. Torsion

Let us consider the tip loading H ¼ Cx

Ixð�zy þ yzÞ; ðIx ¼ hy2 þ z2iÞ.

In that case the 1D external tip loading reduces to C = Cxx whichcorresponds to a tip torque. It is clear, from the equations of thepresent theory, that only the torsional out-of-plane warping (wx)has to be considered for the case of torsion. The internal forces re-duce to Mx;Mx

s ;Mxw

� �and the corresponding strains are

vx ¼ x0x;gx;g0x�

. The equilibrium equations and the boundary con-ditions are:

ðMxÞ0 ¼ 0 xxð0Þ ¼ 0 MxðLÞ ¼ Cx

Mxw

� �0�Mx

s ¼ 0 gxð0Þ ¼ 0 MxwðLÞ ¼ 0

9=; ð10Þ

and the constitutive relations are given by Eq. (6) down. One canderive from these equations that the twisting rate vx ¼ x0x and

the torsional warping parameter obey the following similarequations:

�gKIxw

gGIxgGIx �fGJv00x|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

NUW�effect

þfGJvx|ffl{zffl}SV

¼ Cx

�gKIxw

gGIxgGIx �fGJg00x|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

NUW�effect

þfGJgx|ffl{zffl}SV

¼ Cx ð11Þ

where the contribution part of the non-uniform warping (NUW)effect is specified. The solutions are given by:

xx ¼CxfGJ

x|{z}SV

þ CxfGJ

Zxg

Kxg

sinh KxgðL� xÞ

� �cosh Kx

gL� � � tanh Kx

gL� �0@ 1A

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}NUW�effect

gx ¼CxfGJ|{z}SV

� CxfGJ

cosh KxgðL� xÞ

� �cosh Kx

gL� �

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}NUW�effect

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;ð12Þ

where the scalars Zxg and Kx

g are defined by:

Zxg ¼

gGIx �fGJgGIx

Kxg ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffifGJgKIxw

Zxg

vuut ð13Þ

Remark on Vlasov theory. The twisting rate is given by:

x0x ¼CxfGJ� Zx

gCxfGJ

cosh KxgðL� xÞ

� �cosh Kx

gL� � ð14Þ

Thus the classical assumption x0x � gx if justified if Zxg � 1. In that

case Eq. (11) (up) reduces to

�gKIxwx

000x þfGJx0x ¼ Cx ð15Þ

and may be seen as an extension to composite section (so-CS) of thefundamental equation of non-uniform torsion established forhomogeneous an isotropic thin-walled open profiles [10]. Indeed,for these profiles J� Ix(=Iy + Iz) and Zx

g � 1, but Eq. (15) is also validfor a composite section for which fGJ �gGIx , and this depends on thewhole nature of the composite section (shape and materials), andnot necessary3 an open walled one.

The extent of the restrained warping effect (RWE). Let us introducethe dimensionless constant kx

g ¼ Kxgh; using the aspect ratio k = L/h,

the second equation of Eq. (12) may be recast:

Fig. 2. Variation from the built-in section of the warping gx (g in the figure) normalized by gSVx , for different (theoretical) values of kx

g (k in the figure).

0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

k

Fig. 3. The curve [d–k] solution of Eq. (17).

4 Sy – 0 because of the I/O–W-coupling.


gx ¼ gSVx 1�

cosh kxg k� x

h

� � �cosh kx

gk� �

0@ 1A ð16Þ

For k sufficiently large, Fig. 2 depicts the curves gxðxÞgSV

xfor different the-

oretical values of kxg. Starting from the built-in section, gx increases

from zero to reach, asymptotically, the value which corresponds tothe magnitude of the uniform torsional warping in SV-theory.

Let now dxg ¼ dx

g=h be the ratio that defines the distance dxg from

the built-in section to reach (for example 95% of) SV-result; dxg,

which may be seen as the ratio related to the extent (or the depth)of the RWE, is solution of the following non-linear equation:

coshðkðk� dÞÞ � 0:05 coshðkkÞ ¼ 0 ð17Þ

where, for convenience in the present paper, k and d have been usedinstead of kx

g and dxg, respectively. It is easy to numerically show that

the value of d, solution of Eq. (17), depends, in practice, only on thevalue of k. The curve [d-k] Fig. 3 has been obtained by solvingEq. (17) for different values of k.

Therefore, for a given composite section, regarding to SV-result,kx

g may be seen as the cross sectional constant that enables to know,a priori, if the torsional RWE will be localized or not, close to thebuilt-in section, by comparing dx

g with the length of the beam L.The inverse of kx

g may be seen as the decaying parameter of the tor-

sional RWE (in Chandra and Chopra [1], a similar parameter isintroduced and called the constrained warping parameter).

3.2. Axial tension

Let us consider the tip loading H ¼ Fx

A x, where A is the area ofthe section. In that case the 1D external tip loading reduces toF = Fxx which corresponds to a tip axial tension. For this axial ten-sion, only the in-plane warping (related to Poisson’s effect) has tobe considered. The internal forces reduce to N;Ns;A

xm

� and the cor-

responding strains are cx ¼ u0x;ax;a0x�

. The equilibrium equationsand the boundary conditions reduce to:

N0 ¼ 0 uxð0Þ ¼ 0 NðLÞ ¼ Fx

Axm

� 0 � Ns ¼ 0 axð0Þ ¼ 0 AxmðLÞ ¼ 0

)ð18Þ

and the constitutive relations are given by Eq. (6) up. One can notethat these equations are wholly similar to those of torsion andtherefore lead to similar solutions. In particular, the solution forthe in-plane warping parameter ax is:

ax ¼FxfEA|{z}SV

� FxfEA

cosh KxaðL� xÞ

� cosh Kx

aL� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

NUW�effect

ð19Þ

where the scalars Zxa and Kx

a are defined by:

Zxa ¼hK11i � fEAhK11i

Kxa ¼

ffiffiffiffiffiffiffiffiffiffiffiffifEAgx

Zxa

sð20Þ

As in torsion, starting from the built-in section, ax increases fromzero to reach, asymptotically, the value aSV

x ¼ FxeEAwhich corresponds

to the magnitude of the (uniform) in-plane warping in SV-theory.Similarly to the case of torsion, we define the dimensionless con-stant kx

a ¼ Kxah and the distance dx

a ¼ dxah from the built-in section

to reach (95% of) SV-result; dxa is solution of the same equation Eq.

(3) (where (k,d) represent now (kxa; d

xa)). Therefore, dx

a characterizeshow much the in-plane RWE extends to the interior part of the beam,and this is (in practice) related to the cross sectional constant kx

a.

3.3. Shear-bending

Let us consider the tip loading H ¼ Fz

A x, where A is the area of thesection. In that case the 1D external tip loading reduces to F = Fzzand4 S = [0,Sy,0], with Sy ¼ Fz hWyi

A . Only the in-plane warping (Vy,Wy)(related to My) and the out-of-plane warping wz (related to Tz) are

Fig. 4. The curves dy;0a � ky

a

� �and dy

a � kya

� �solution of Eqs. (17) and (28),

respectively.


considered. The internal forces reduce to My;Mys ;A

ym;M

zw;

�Tz; Tz

sÞ andthe corresponding strains are vy ¼ x0y;ay;a0y;g0z; cz ¼

�u0z þxy;gzÞ.

The equilibrium equations and the boundary conditions are:

ðTzÞ0 ¼ 0 uzð0Þ ¼ 0 TzðLÞ ¼ Fz

ðMyÞ0 � Tz ¼ 0 xyð0Þ ¼ 0 MyðLÞ ¼ 0

Mzw

� �0� Tz

s ¼ 0 ayð0Þ ¼ 0 AymðLÞ ¼ Sy

Aym

� 0 �Mys ¼ 0gzð0Þ ¼ 0 Mz

wðLÞ ¼ 0

9>>>>>=>>>>>;ð21Þ

and the behavior is given by Eq. (7). Eqs. (7)–(21) lead to coupled equa-tions between (vy,ay,cz,gz) for which there is no closed-form solutionto present. For the numerical applications, in the next section, theseequations will be solved by 1D finite element method. However, forthe moment, it is interesting to analyse the case for which theI/O–W-coupling effect is not significant (which will practically bethe case, see the numerical applications in the next section). In thatcase, Sy is eliminated from Eq. (21) and the constants (az,ez, fz) fromthe constitutive relations (Eq. (7)) which can be reduced and split in:

My

Mys

Aym

264375 ¼ hz2K11i �ðhz2K11i � fEIyÞ 0

�ðhz2K11i � fEIyÞ ðhz2K11i � fEIyÞ 00 0 gy

26643775

vy

ay

a0y

264375

Tz

Tzs

Mzw

264375 ¼ hGxzi �ðhGxzi �gGAzÞ 0

�ðhGxzi �gGAzÞ hGxzi �gGAz 0

0 0 gKIzw

26643775

g0zcz

gz

264375

9>>>>>>>>>>>=>>>>>>>>>>>;ð22Þ

It is worth noting that these constitutive relations have the sameform. Similarly to torsion and axial tension, one can derive fromEqs. (21) and (22) the following equations for the warping parameters:

�gyhz2K11i

hz2K11i � fEIy

a00y|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}NUW�effect

þ fEIyay|fflffl{zfflffl}SV

¼�FzðL� xÞ

�gKIzw

hGxzihGxzi �gGAz

g00z|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}NUW�effect

þgGAzgz|fflfflffl{zfflfflffl}SV

¼ Fz ð23Þ

which lead to the solutions:

ay ¼ �FzðL� xÞfEIy|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

SV

þ FzLfEIy

cosh KyaðL� xÞ

� cosh Ky

aL�

sinh Kyax

� Ky

aL cosh KyaL

� !|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

NUW�effect

ð24Þ

gz ¼FzgGAz|{z}SV

� FzgGAz

cosh KzgðL� xÞ

� �cosh Kz

gL

0@ 1A|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

NUW�effect

ð25Þ

where the cross-sectional constants Zya;K

ya

� and Zz

g;Kzg

� �are de-

fined by:

Zya ¼hz2K11i � fEIy

hz2K11iKy

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffifEIy

gyZy

a

sand

Zzg ¼hGxzi �gGAz

hGxziKz

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffigGAzgKIzw

Zzg

vuut ð26Þ

Remark. For a cantilever beam subjected to a pure bending C = Cyy,one can show that the in-plane warping parameter (denoted bya0

y for distinction) is given by:

a0y ¼

CyfEIy|{z}SV

� CyfEIy

cosh KyaðL� xÞ

� cosh Ky

aL

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

NUW�effect

ð27Þ

Similarly to torsion and axial tension, one can also define thedimensionless constants ky

a ¼ Kyah; kz

g ¼ Kzgh

� �, and (dy

a ¼ dyah; dz

g ¼dzgh and dy;0

a ¼ dy;0a h (for pure bending)) which represent the dis-

tances, from the built-in section, to reach (95% of) SV-results. dzg

and dy;0a are solution of the same Eq. (3); however, dy

a (for theshear-bending) is solution of the following non-linear equation:

k cosh kyaðk� dy

aÞ�

�sinh ky

adya

� ky

ak

!� 0:05 k� dy

a

� cosh ky

ak�

¼ 0

ð28Þ

Fig. 4 depicts the curves dy;0a � ky

a

� �(solution of Eq. (17)) and

dya � ky

a

� �(solution of Eq. (28)); these curves are very close which

shows that, in practice, dy;0a gives a good approximation of dy

a.

3.4. Conclusion on the extent of the restrained warpings effects (RWE)

The above analysis of the elastic solution given by the presentbeam theory for of a cantilever beam subjected to torsion, axialtension and bending, compared with SV-results, has clearly shownthe contribution part on the REW-effect related to the built-in sec-tion. For each loading, starting from the built-in section the warp-ing increases to reach asymptotically SV-result, on a distancedefined by a ratio d��

� (practically) related to a particular dimen-

sionless cross-sectional constant k��

that takes into account thewhole nature of the composite section (shape and materials).These constants k��

� are summarized in Table 1. Further, the vari-

ation of each ratio d��

with respect to its corresponding constantk��

is described by the same curve [d � k] given by Fig. 3. For in-stance, this curve indicates that the RWE is confined close to thebuilt-in section on a distance d < h for k > 3 (approximately). Final-ly, regarding to SV-results, the cross-sectional constants k��

� en-

able us to predict, for each kind of loading, the extent, from thebuilt-in section, of the RWE.

4. Numerical applications

For the application of the present theory, the first step is thecomputation, for any given so-CS, of all its constantsðfEA;gGAy ;gGAz ;fGJ; fEIy ; fEIzÞ and, in particular, its SV-(in-plane/out-of-plane)-warping functions ((Vx,Wx), (Vy,Wy),(Vz,Wz),wx,wy,wz).This is achieved by using the software tool designated by SECOPE(cross-SECtional OPErators) which is implemented in the finite

Table 1The sectional constants (k��) that are related to the extent of the in-plane or the out-ofplane RWE.

Axial tension Bending/y Bending/z

kxa

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieEAgx

hK11i� eEAhK11i

rkya

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifEIygy

hz2K11 i�fEIy

hz2K11i

rkza

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifEIzgz

hy2 K11i�fEIzhy2 K11i

rTorsion y-Shear-force z-Shear-force

kxg

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieGJfKIx

w

fGIx�eGJfGIx

skyg

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifGAyfKIy

w

hGxy i�fGAy

hGxyi

skzg

h ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifGAzfKIz

w

hGxzi�fGAzhGxzi

s

Table 3Properties of the composite (sandwich) sections with isotropic and orthotropicphases.

Es/Ec ms mc

c-iso-CS (01/0.3/0.3) 1 0.3 0.3c-iso-CS (20/0.3/0.3) 20 0.3 0.3c-iso-CS (40/0.3/0.3) 40 0.3 0.3c-iso-CS (01/0.2/0.4) 1 0.2 0.4

Skins Core

c-ort-CS (mat-06) mat-06 (0) mat-06 (90)c-ort-CS (mat-33) mat-33 (0) mat-33 (90)c-ort-CS (mat-66) mat-66 (0) mat-66 (90)


elements code CASTEM [8] (a quick description of SECOPE is pre-sented below). An upgrade of SECOPE has been realized in orderto compute all the supplementary constants introduced by thepresent HOCBT.

To illustrate the predictive capability of the present beam the-ory, this section is devoted to the numerical analyzes of cantileverbeams, with a length-to-thickness ratio L/h, made with differentkinds of so-CS (homogeneous or composite, solid-CS and walled-CS (open or closed)) and subjected to a unit torsion (Cx = 1), a unitaxial tension (Fx = 1) or shear-bending with a unit force (Fz = 1).

Numerical results are given for the distributions along the beamaxis of cross-sectional displacements (1D behavior); and, close tothe built-in section, the 3D stress predictions (using Eqs. (A.2)and (A.3)) of the HOCBT are compared to SV-stresses and to thoseobtained by 3D finite element (3D-FEM) computations. The com-parison will concern the main components of the stress tensor,for particular important points of the cross-section.

In the figures presented below, (�)3D, (�)1D and (�)SV upperscriptsstand for quantities obtained by the 3D-FEM computation, thepresent HOCBT and SV’s solution (computed with SECOPE),respectively.

Remarks on the finite element computations.

� The computation of the sectional characteristics uses the soft-ware SECOPE. This one has been developed conforming to thenumerical method proposed by El Fatmi and Zenzri [3] for thecomputation of the 3D SV’s solution for any composite section.This method consists in solving six elastic problems on a longi-tudinal slice of beam. The computation is immediate whenusing standard three-dimensional elasticity codes that affordthe quadratic 15-nodes triangular prism element or the 20-nodes rectangular prism element. The cross-section has to be

h

h/2 h/2 h/2

Fig. 5. Homogeneous and comp

Table 2Mechanical properties of the orthotropic materials.

Ex (GPa) Ey (GPa) Ez (GPa) Gxy (GPa) Gxz

mat-06 53.78 17.93 17.93 8.96 8.96mat-33 148.00 8.37 8.37 4.40 4.40mat-66 206.80 5.17 5.17 3.10 3.10

discretized as necessary but only one element is required in thelongitudinal direction of the beam. Thus the cost is very eco-nomic. For the present work, the section being y-z-symmetric,only the quarter of the section is treated. The discretization doesnot exceed 50 elements and, to obtain all the characteristics of agiven section, the cpu-time is less than one minute on a current(mono-processor) personal computer.� For the 3D-FEM computations, different beam lengths has been

considered. This depends on the extent of the RWE. For eachcase, the length is chosen sufficiently large so that the RWE van-ish. The discretization uses the same meshing of the section and20 to 40 elements has been used in the longitudinal direction ofthe beam, refined close to the built-in section.� It should be noted that the numerical 3D-FEM results on the

stresses have to be considered (a priori) more qualitatively thanquantitatively when they concern the built-in section S0 or forsingular regions of a composite section; nevertheless, thesenumerical 3D-FEM results will be given and compared withthe HOCBT ones. Besides, the comparison will not concern thelocal region close to SL, but focuses on the region that start fromS0 right to the point where (as expected) 3D, 1D and SV resultsmeet.

4.1. Cross-sections and characteristics

4.1.1. Geometry and materialsTwo groups of section are considered (Fig. 5):

� The first group pertains to three homogeneous sections: bodyrectangular-CS (body-CS), walled closed-CS (clos-CS) andopen-CS (open-CS); the dimensions are (h h/2) and, for the

h/2 h/2

h/10

osite (sandwich) sections.

(GPa) Gyz (GPa) mxy mxz myz E/G

3.45 0.25 0.25 0.34 062.72 0.33 0.33 0.54 332.55 0.25 0.25 0.25 66

in-plane warpings out-of-plane warpings

N T yM y Mz T z Mx

SaintVenant

Fig. 6. In-plane and out-of-plane Saint-Venant warping functions.


walled section, the thickness is t=h/10. For these three sections,three orthotropic materials are considered: mat-06, mat-33 andmat-66 ; their mechanical properties are given in Table 2. Thesematerials are chosen because their ratio E/G (approximativelyequal to 6, 33 or 66) can significantly influences the amountof the extent of the RWE, and therefore the global behavior ofthe beam.� The second group pertains to composite (sandwich) section

with isotropic (c-iso-CS) or orthotropic (c-ort-CS) phases. Thedimensions of the sections are (h h/2) and the thickness ofthe outer layers is t=h/10. (Es,ms) and (Ec,mc) denoting Young’smodulus and Poisson’s ratio of the skins and the core, the mate-rials of the isotropic phases for the sections c-iso-CS are given inTable 3. For the sections c-ort-CS, the orthotropic materials arethose given above in Table 2, and the layers are h-oriented: 0 forthe skins and 90 for the core (Table 3).

These sections are investigated with the purpose to assess:

� for the homogeneous sections, the effects of the shape and theratio E/G of the orthotropic material;� and, for the composite sections, the contrast effect between dif-

ferently phased materials.

5 Similar results may be obtained for composite sections.6 The results for clos-CS are similar to those of body-CS.

4.2. Cross-sectional warpings and constants

For the numerical applications, the characteristic dimension ofthe sections is taken equal to unity (h=1). For each section defined

above the cross-sectional warpings and constants have been com-puted by the upgrade version of SECOPE. The in-plane and the out-of-plane SV-warping functions obtained for a representative set ofsection are depicted by Fig. 6, and the numerical values of thecross-sectional constants obtained for each cross-section are givenin Appendix B by Tables B.6 and B.7 ( in these tables, all dimensionsare given in the international unit system (SI); for instance, fEA isin[N], fEIy in [Nm2], etc).

Tables 4 and 5 give the constants k�� and d�� (where d�� are relatedto 95% of SV-result) obtained for the homogeneous sections and forthe composite ones, respectively (in these tables, the values ofd�� P 1 are in boldface character).

4.3. Torsion

The values of dxg in Tables 4 and 5 indicate that the torsional (out-

of-plane) RWE is not so localized close to the built-in section, but canextend to the free end of the beam if the length is not large enough.

1D Behavior. Using the analytical results obtained before (Eq.(12)) for the torsion, Fig. 7 depicts, from the built-in section, thevariations of the torsional warping parameter gx and the torsionalrotation xx for the homogeneous5 sections6 body-CS and open-CS,and for the three different orthotropic materials. For each section,xx and gx are normalized by xSV

x ðLÞ and gSVx , respectively. The results

Table 4Values of k�� d�� for the homogeneous sections.

body-CS(mat-06)

body-CS(mat-33)

body-CS(mat-66)

clos-CS(mat-06)

clos-CS(mat-33)

clos-CS(mat-66)

open-CS(mat-06)

open-CS(mat-33)

open-CS(mat-66)

kxa 7.63 8.91 6.54 6.49 7.57 5.55 6.39 7.46 5.47

dxa 0.393 0.336 0.458 0.462 0.396 0.539 0.469 0.402 0.548

Kyg 9.91 4.07 2.90 3.70 1.50 1.07 10.6 4.21 2.98

dyg 0.302 0.736 1.03 0.809 2.00 2.81 0.284 0.711 1.0

kzg 4.88 2.03 1.45 7.77 3.26 2.33 8.21 3.51 2.52

dzg 0.614 1.47 2.06 0.385 0.918 1.28 0.365 0.852 1.19

kxg 2.52 1.08 0.779 3.07 1.32 0.949 0.480 0.207 0.148

dxg 1.19 2.76 3.85 0.976 2.27 3.16 6.25 14.5 20.2

kya 12.0 14.1 10.3 10.2 11.9 8.71 13.8 16.1 11.8

dya 0.249 0.213 0.291 0.295 0.252 0.344 0.217 0.186 0.254

kza 6.19 7.29 5.35 1.51 2.10 1.34 3.93 4.63 3.40

dza 0.484 0.411 0.560 1.98 1.43 2.23 0.762 0.647 0.881

Table 5Values of k�� d�� for the composite sections.

c-iso-CS(01) (0.3/0.3)

c-iso-CS(20) (0.3/0.3)

c-iso-CS(40) (0.3/0.3)

c-iso-CS(01) (0.2/0.4)

body-CS (mat-06) (homog.)

c-ort-CS(mat-06)s

(0,90)s


c-ort-CS(mat-33)s

(0,90)s


c-ort-CS(mat-66)s

(0,90)s

kxa 8.44 6.11 5.94 9.01 7.63 8.94 8.91 6.23 6.54 5.03

dxa 0.355 0.491 0.504 0.332 0.393 0.335 0.336 0.481 0.458 0.596

kyg 15.7 12.5 12.6 14.1 9.91 13.2 4.07 15.9 2.90 20.4

dyg 0.191 0.239 0.239 0.212 0.302 0.227 0.736 0.188 1.03 0.147

kzg 7.13 7.73 5.99 6.61 4.88 5.11 2.03 6.81 1.45 9.17

dzg 0.420 0.388 0.500 0.453 0.614 0.586 1.47 0.440 2.06 0.327

kxg 3.41 1.23 0.937 2.82 2.52 2.59 1.08 3.01 0.779 4.31

dxg 0.879 2.43 3.20 1.06 1.19 1.16 2.76 0.995 3.85 0.696

kya 13.3 12.9 12.9 14.3 12.0 17.2 14.1 12.7 10.3 10.3

dya 0.225 0.232 0.232 0.210 0.249 0.174 0.213 0.236 0.291 0.292

kza 6.63 5.58 5.67 6.75 6.19 5.0 7.29 3.64 5.35 3.43

dza 0.452 0.536 0.528 0.444 0.484 0.599 0.411 0.822 0.560 0.874

Fig. 7. Variations, from the built-in section, of the torsional warping parameters gx and the torsional rotation xx for the homogeneous sections body-CS and open-CS, and forthe different orthotropic materials. gx and xx are normalized by gSV

x and xSVx ðLÞ, respectively.


show how the extent of the torsional RWE increases with respect to thevalues of the constant kx

g (indicated on each curve in a small box) whichtakes into account the whole nature of the section (shape7 and mate-

7 Note that, for a fixed shape, as expected, the extent of the RWE increases with theratio E/G.

rial). However, this effect appears more important for an open section.SV-theory gives a good approximation of xx for a body section but anon-uniform warping theory is needed for an open profile.

3D stresses. Let us recall first that in SV-torsion, rSVxx ¼ 0 and sSV

is x-constant, for any so-CS. Figs. 8 and 9 depict, from the built-insection, the variations of the axial stress rxx and the shear s, for

Fig. 8. Variations, from the built-in section of the axial stress rxx and the shear s, for the points A and B of the section. Comparison between 3D-FEM (3D), present HOCBT (1D)and Saint-Venant (SV) results. For each section, the stresses are normalized by sSV(B).


particular points A and B of the following sections: body-CS, open-CS, c-iso-CS (40/03/03) and (1/02/04)), and c-ort-CS (mat-33/(0;90)s). For the composite section, A and B are in the skin or inthe core. For each section, all the stresses are normalized by themagnitude of sSV(B).

For the different sections, 3D and 1D results for rxx and s arequite comparable; and, as expected, they tend toward SV-resultsin the interior part of the beam. Besides, moving from the interiorpart of the beam towards the built-in section, it is worth notingthat the shear strongly decreases (or vanishes) in favour of the ax-ial stress induced by the restrained warping. Moreover, this axialstress can reach several times the magnitude of the shear obtainedin the interior part of the beam.

4.4. Axial tension

Tables 4 and 5 indicate that dxa is smaller than h for all the

sections; therefore the in-plane RWE related to axial tensionseem to be always confined close to the built-in section. Onlytwo sections will be analysed for the axial tension: the homo-geneous section body-CS (mat-66) and the composite sectionc-ort-CS mat-66-(0,90) for which the values of dx

a=h are closerto 1.

1D-behavior. Using the analytical results obtained before (Eq.(19)) for the axial tension, Fig. 10 depicts, from the built-in section,the variations of the in-plane warping parameter ax and the axialdisplacement ux for the body and the composite sections. For eachsection, ux and ax are normalized by uSV

x ðLÞ and aSVx , respectively.

The results show that, SV-theory is sufficient to predict the axialdisplacement.

3D-stresses. For the SV-axial-tension, ~r1 ¼ ðrxx;ryy;rzz; syzÞ is x-constant and the shear ~r2 ¼ 0 (Eqs. (A.2) and (A.3)). Because of therestrained Poisson’s effect, it is expected that the transverse stres-ses (ryy,rzz,syz), negligible in the interior part of the beam, becomeimportant close to the built-in section. The numerical results indi-cate that only syz remains negligible and (rzz � ryy) can notably in-crease; so that we define the transverse stress rt = (ryy + rzz)/2 tobe compared with the axial stress rxx. Fig. 11 depicts, from thebuilt-in section, the variations of the transverse stress rt, for thepoint A of the section (for the composite section, A is in the skinor in the core). For each section, the stresses are normalized bythe value of rSV

xx ðAÞ.

For both sections, 3D and 1D results are comparable and tendtoward SV-results in the interior part of the beam. The transversestress rt induced by the restrained warping is quite negligible forthe homogeneous body-CS but not for the composite one for whichrt can reach 25% of rxx.

4.5. Shear-bending

Tables 4 and 5 indicate that the extent dya of the in-plane RWE

(related to the bending moment) is always smaller than h/2, andthat of out-of-plane RWE (related to the shear force) dz

g may ex-ceed 2h, (which corresponds to the case of the homogeneoussection body-CS (mat-66)). Two sections will be analyzed:body-CS (mat-66) and c-ort-CS (mat-66) (0-90)s. Besides, to as-sess the importance of the I/O–W-coupling in the shear-bendingbehavior, the numerical simulations will be done with and with-out coupling (az = fz = ez = 0). A relatively short beam is consid-ered, L = 6h.

1D Behavior. The solution of the 1D problem (Eqs. (21) and (22))is computed by 1D finite element method, using cubic (Hermite-type) interpolation (shape) functions for each of (uz,xy,ay,gz).Fig. 12 depicts, from the built-in section, the variation of thetransverse displacement uz for the sections body-CS-(mat-66)and c-ort-CS-(mat-66)-(0,90)s. For each section, uz is normalizedby uSV

z ðLÞ; and, for comparison, the curve that corresponds toNavier Bernoulli (NB) theory has been also given. The results showthat SV-theory is sufficient to predict the transverse displacement.

For both sections, Fig. 13 depicts, from the built-in section, thevariations of the out-of-plane warping parameter gz and the in-plane warping parameter ay, taking into account the I/O–W-cou-pling or not. For each section, ay and gz are normalized by aSV

y ð0Þand gSV

z ðLÞ, respectively. The results show that, the I/O–W-couplinghas an effect on ay for the homogeneous section and on gz for thecomposite section, but this does not affect significantly the extentsof the RWE.

3D-stresses. For SV-theory, rSVxx (due to the bending moment) is

x-linear and sSV is x-constant (Eqs. (A.2) and (A.3)). Fig. 14 depicts,from the built-in section, the variations of the axial stress rxx andthe shear s, for a particular point A of the section (for the compos-ite section, As is in the skin and Ac in the core). For both sections,3D and 1D results are comparable and tend toward SV-results inthe interior part of the beam. Moving from the interior part of

Fig. 9. Variations, from the built-in section of the axial stress rxx and the shear s, for the points A and B for the composite sections (A and B are in the skin or in the core).Comparison between 3D-FEM (3D), present HOCBT (1D) and Saint-Venant (SV) results. For each section, the stresses are normalized by sSV(B).


the beam towards the built-in section, the shear decreases infavour of an increase of the magnitude of the axial stress (initiallydue to the bending moment). This increase of rxx, which is inducedby the out-of-plane RWE is important for the body-CS and for the

composite section (in the skins), and can reach about 30%-40% (thisamount can be more important for a shorter beam). Besides, theinfluence of the I/O–W-coupling is nil for the body-CS, and is notso important for the composite section.

Fig. 10. Variations, from the built-in section, of the in-plane warping parameters ax and the axial displacement ux, for the homogeneous section body and a composite section.

Fig. 11. Variation, from the built-in section of the transverse stress rt = (ryy + rzz)/2, for the point A of the section (for the composite section, A is in the skin or in the core). Foreach section, the rt is normalized by rSV

xx ðAÞ. Comparison between 3D-FEM (3D), present HOCBT (1D) and Saint-Venant (SV) results.


Fig. 12. Variation, from the built-in section of the transversal displacement uz for the sections body-CS (mat-66) and c-ort-CS (mat-66) (0,90)s. Comparison between Navier–Bernoulli (NB), present HOCBT (1D) and Saint-Venant (SV) results.

Fig. 13. Variation, from the built-in section of the in-plane warping parameter ay, for the homogeneous section body-CS-(mat-66) and the composite section c-ort-(mat-66)-(0,90)s. Comparison between Navier-Bernoulli (NB), present HOCBT (1D) and Saint-Venant (SV) results.


Fig. 14. Variations, from the built-in section of the axial stress rxx and the shear s for the points A and B of the homogeneous section body-CS-(mat-66) and the compositesection c-ort-mat-66-(0,90)s. Comparison between 3D-FEM (3D), Saint-Venant (SV) and the present HOCBT (1D) results (‘1D’ (with) and ‘1D-0’ without the I/O–W-coupling).


5. Conclusion

The beam theory established in Part-I (HOCBT) has been appliedto analyze the influence of built-in effects on the 3D/1D behavior ofelastic cantilever beams, for a representative set of sections andloadings.

In the critical region that starts from the built-in section, the 3Dstresses (axial and shear stresses) predictions of the present HOCBTare relevant and quite comparable (qualitatively and quantita-tively) to those obtained by 3D-FEM computations. Moreover, asexpected, moving from the built-in section, these results tend(asymptotically) towards SV’s solution in the interior part of thebeam. Thus, we can draw from these results that this theory is ableto capture a significative part of the built-effects (which are re-strained warping effects RWE); therefore, one can trust the corre-sponding beam theory to analyze the 1D structural behavior ofthe beam.

Furthermore, regarding to Saint-Venant results, the 1D analyti-cal analysis of the extents of the built-in effects shows that theseextents are related to particular constants that take into accountthe whole nature of the composite section (shape and materials).

Even if the aim of this theory was not to characterize the decaylengths of the extremity solution (built-in effects), HOCBT providesconstants that allow to predict the expansion of these built-ineffects.

The present version of HOCBT is devoted, as a first step, tocomposite beam with y–z-symmetric section and made of x–y–z-orthotropic materials. However, it is worth noting that thereis no theoretical or numerical difficulties to extend this theoryto an arbitrary composite section. This will be done in the nearestfuture.

Built-in section is a severe boundary condition for a beam the-ory (especially for SV-BT that assumes the end-sections free towarp) and the 3D results given by the proposed theory (HOCBT)look really promising, but other boundary conditions must be ana-lyzed to assess its real relevance.

Appendix A. 3D-stresses

Let Hooke’s law split into r1 = K1 e1 and r2 = s = K2 e2 with(K1,K2,r1,r2,e1,e2) defined by


K1 ¼

K11 K12 K13 0

K12 K22 K23 0

K13 K23 K33 0

0 0 0 Gyz

266664377775; K2 ¼

Gxy 0

0 Gxz

" #

r1 ¼

rxx

ryy

rzz

syz

266664377775; r2 ¼

sxy

sxz

" #; e1 ¼

exx

eyy

ezz

2eyz

266664377775; e2 ¼

2exy

2exz

" #ðA:1Þ

The 3D stresses corresponding to this beam theory are given by thefollowing closed-form expressions:

r1 ¼ K1D1eK1

NMy

Mz

24 35|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

Saint�Venant

þK1D1eK1

Ns

Mys

Mzs

24 35þ K1

wx

000

26643775Mx

wgKIxw

þ K1

0Vx;y

Wx;z

Vx;z þWx

;y

� �266664

377775 Ns

hK11i � fEA

þ K1

wy 00 Vz

;y

0 Wz;z

0 Vz;z þWz

;y

� �266664

377775gKIy

w ay

ay hy2K11i � fEIz

" #�1My

w

Mzs

�

þ K1

wz 00 Vy

;y

0 Wy;z

0 Vy;z þWy

;y

� �266664

377775gKIz

w az

az hz2K11i � fEIy

" #�1Mz

w

Mys

�

ðA:2Þ

Table B.6The cross-sectional constants of the homogeneous sections (G1).

body-CS(mat-06)

body-CS(mat-33)

body-CS(mat-66)

clos-CS (mat-06)

clos33)

fEA10�9 26.890 74.0 103.40 13.983 38.4gGAy 10�9 3.7033 1.8325 1.2916 0.51933 0.26gGAz 10�9 3.7331 1.8333 1.2917 1.5643 0.76fGJ10�6 256.13 125.78 88.615 195.58 96.0fEIy 10�9 2.2408 6.1667 8.6167 1.5524 4.27fEIz 10�9 0.56021 1.5417 2.1542 0.46340 1.27

hK11i10�9 28.702 76.036 103.83 14.925 39.5hGxyi10�9 4.4800 2.2000 1.5500 2.3296 1.14hGxzi10�9 4.4800 2.2000 1.5500 2.3296 1.14gGIx 10�6 466.67 229.17 161.46 335.85 164

hz2K11i10�9 2.3919 6.3363 8.6527 1.6571 4.38hy2K11i10�9 0.59797 1.5841 2.1632 0.49464 1.31gKIx

w 10�6 18.229 48.290 65.944 8.6746 22.9

gKIyw 10�6 6.5330 18.497 25.572 29.420 91.1

gKIzw 10�6 26.178 73.966 102.24 8.5086 23.7

gx 10�6 29.167 24.956 10.091 20.991 17.9gy 10�6 0.97846 0.83631 0.33810 0.94817 0.81gz 10�6 0.92219 0.77625 0.31320 12.818 7.72fy 10�6 �18.831 �12.125 �6.4634 �21.506 �13ey 10�6 17.201 11.876 6.4127 20.551 13.7ay �0.0965 �2.1605 �2.4333 �0.0968 0.05fz 10�6 10.126 3.8584 1.7844 167.67 90.0ez 10�6 �4.0299 �2.9356 �1.5966 �129.78 �76az �0.0013 0.0015 �0.0007 0.0218 0.02

r2 ¼ K2D2eK2

Mx

Ty

Tz

24 35|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}

Saint�Venant

þK2D2eK2

Mxs

Tys � eRyAz

m

Tzs þ eRzAy

m

264375

þ K2wx;y

wx;z

� Mx

sgGIx �fGJþ K2

Vx

Wx

� Ax

mgx

þ K2wy;y Vz

wy;z Wz

� hGxyi �gGAy � eRyðfy � eyÞ ey

ey gz

� �1Ty

sAz

m

�

þ K2wz;y Vy

wz;z Wy

� hGxzi �gGAz þ eRzðfz � ezÞ ez

ez gy

" #�1Tz

sAy

m

� ðA:3Þ

with

D1 ¼

1 z �yVx;y Vy

;y Vz;y

Wx;z Wy

;z Wz;z

Vx;zþWx

;y

� �Vy;zþWy

;y

� �Vz;zþWz

;y

� �26664

37775; eK1 ¼

1eEA0 0

0 1eEIy

0

0 0 1eEIz

2666437775

D2 ¼�zþwx

;y

� �1þwy

;y� eRyVz� �

wz;yþ eRzVy

� �yþwx

;z

� �wy;z� eRyWz

� �1þwz

;zþ eRzWy� �24 35; eK2 ¼

1eGJ0 0

0 1fGAy

0

0 0 1fGAz

2666437775

9>>>>>>>>>>>>>=>>>>>>>>>>>>>;ðA:4Þ

and where (�),y and (�),z denote the partial derivative with respectto y and z, respectively. Eqs. (A.2) and (A.3) clearly show thecontribution part of SV-stresses and the contribution of eachadditional internal forces induced by the non-uniformity of thewarpings.

Appendix B. The cross-sectional constants

Tables B.6 and B.7.

-CS (mat- clos-CS (mat-66)

open-CS(mat-06)

open-CS(mat-33)

open-CS(mat-66)

80 53.768 9.6804 26.640 37.224

626 0.18846 0.84089 0.41410 0.29179

820 0.54123 0.80378 0.39471 0.27809

46 67.668 5.5753 2.7378 1.9289

23 5.9696 1.3230 3.6408 5.0873

53 1.7819 0.11563 0.31820 0.44462

39 53.993 10.333 27.373 37.38040 0.80600 1.6128 0.79200 0.5580040 0.80600 1.6128 0.79200 0.55800.93 116.20 239.68 117.70 82.925

98 5.9946 1.4122 3.7410 5.108604 1.7894 0.12342 0.32696 0.4464880 31.381 23.669 62.702 85.624

17 127.24 3.6116 11.125 15.642

20 32.704 5.9825 16.029 21.939

60 7.2624 14.980 12.818 5.1828100 0.32791 0.43997 0.37643 0.1522165 4.1169 0.47194 0.39758 0.16030.929 �7.4333 �9.4857 �6.1472 �3.280983 7.4036 9.2184 6.1064 3.2725382 0.4032 �0.1258 1.4719 �3.812030 55.760 11.897 7.3850 3.9085.808 �47.294 �8.4650 �6.8676 �3.803332 0.0019 0.0001 �0.0043 0.0050

Table B.7Cross-sectional constants for the composite sections.

Constants c-iso-CS(01) (0.3/0.3)

c-iso-CS(20) (0.3/0.3)

c-iso-CS(40) (0.3/0.3)

c-iso-CS(01) (0.2/0.4)

body-CS(mat-06)(homog.)

c-ort-CS(mat-06)s

(0,90)s


c-ort-CS(mat-33)s

(0,90)s


c-ort-CS(mat-66)s

(0,90)sfEA10�9 50.000 12.000 11.000 50.095 26.890 8.9650 74.0 4.1850 103.40 2.5850gGAy 10�9 15.086 3.7125 3.4332 13.994 3.7033 3.4620 1.8325 1.4177 1.2916 1.1790gGAz 10�9 16.018 0.9646 0.4865 14.894 3.7331 1.4375 1.8333 1.1323 1.2917 1.0625fGJ10�6 1099.4 89.652 51.288 1051.3 256.13 115.65 125.78 85.209 88.615 75.943fEIy 10�9 4.1667 2.1400 2.0867 4.1847 2.2408 0.7471 6.1667 0.3488 8.6167 0.2154fEIz 10�9 1.0417 0.2500 0.2292 1.0427 0.5602 0.1868 1.5417 0.0872 2.1542 0.0539

hK11i 10�9 67.308 16.154 14.808 96.825 28.702 10.595 76.036 6.0328 103.83 2.7645hGxyi 10�9 19.231 4.6154 4.2308 18.452 4.4800 4.4800 2.2000 2.2000 1.5500 1.5500hGxzi 10�9 19.231 4.6154 4.2308 18.452 4.4800 1.7250 2.2000 1.3588 1.5500 1.2750gGIx 10�6 2003.2 919.23 890.70 1993.6 466.67 409.27 229.17 211.64 161.46 155.73

hz2K11i10�9

5.6090 2.8808 2.8090 6.8307 2.3919 0.8829 6.3363 0.50274 8.6527 0.2304

hy2K11i10�9

1.4022 0.3365 0.3085 2.0172 0.5980 0.2207 1.5841 0.1257 2.1632 0.0576

gKIxw 10�6 42.747 53.363 55.054 62.578 18.229 12.409 48.290 5.6140 65.944 2.0983

gKIyw 10�6 13.150 4.6270 4.1037 17.193 6.5330 4.5338 18.497 1.9879 25.572 0.6795

gKIzw 10�6 52.712 12.773 11.987 65.784 26.178 9.1822 73.966 4.0706 102.24 2.1080

gx 10�6 180.29 82.731 80.164 297.57 29.167 17.266 24.956 33.034 10.091 6.6419gy 10�6 6.0635 3.2911 3.2065 7.9761 0.97846 0.3875 0.8363 0.6617 0.3381 0.1329gz 10�6 6.0994 2.0612 1.8311 11.055 0.9222 1.1498 0.7763 2.0115 0.3132 0.2978fy 10�6 �98.504 �77.144 �77.331 �129.12 �18.831 �9.7836 �12.125 �12.229 �6.4634 �5.3126ey 10�6 75.194 75.661 76.583 102.23 17.201 9.0380 11.876 10.081 6.4127 4.6570ay 103 0.0001 �0.0063 �0.0167 �13785 �0.0001 � 0.0000 �0.0022 0.0020 �0.0024 �0.0002fy 10�6 101.95 16.576 12.697 158.28 10.126 19.124 3.8584 30.849 1.7844 6.2154ey 10�6 �13.617 14.033 14.736 �10.763 �4.0299 2.1897 �2.9356 1.8585 �1.5966 0.30482ay 103 0.0000 0.0000 �0.0000 318.47 �0.0000 0.0000 0.0000 0.0000 �0.0000 �0.0000fz 10�6 �98.504 �77.144 �77.331 �129.12 �18.831 �9.7836 �12.125 �12.229 �6.4634 �5.3126ez 10�6 75.194 75.661 76.583 100.73 17.201 9.0380 11.876 10.081 6.4127 4.6570az 103 0.0001 �0.0063 �0.0167 �173.81 �0.0001 �0.0000 �0.0022 0.0020 �0.0024 �0.0002


gKIxw ¼ hK11ðwxÞ2i gx ¼ hGxyðVxÞ2 þ GxzðWxÞ2igKIyw ¼ hK11ðwyÞ2i gy ¼ hGxyðVyÞ2 þ GxzðWyÞ2igKIzw ¼ hK11ðwzÞ2i gz ¼ hGxyðVzÞ2 þ GxzðWzÞ2i

ay ¼ hyK11wyi az ¼ h�zK11w

ziey ¼ hGxyw

y;yVz þ Gxzw

y;zWzi ez ¼ hGxyw

z;yVy þ Gxzw

z;zW

yify ¼ hGxyVzi fz ¼ hGxzWyi

9>>>>>>>>>>>=>>>>>>>>>>>;ðB:1Þ

References

[1] Chandra R, Chopra I. Experimental and theoritical analysis of composite i-beams with-elastic couplings. AIAA J 1991;29(12):2197–206.

[2] El Fatmi R, Zenzri H. On the structural behavior and the Saint-Venant solutionin the exact beam theory. Application to laminated composite beams. ComputStruct 2002;80:1441–56.

[3] El Fatmi R, Zenzri H. A numerical method for the exact elastic beam theory.Applications to homogeneous and composite beams. Int J Solids Struct2004;41:2521–37.

[4] Giavotto V, Borri M, Mantegazza P, Ghiringhelli G, Carmaschi V, Maffioli GC, et al.Anisotropic beam theory and applications. Comput Struct 1983;16:403–13.

[5] Horgan CO, Simmonds JG. Saint Venant end effects in composite structures.Compos Eng 1994;4(3):279–86.

[6] Ladevèze P, Sanchez P, Simmonds JG. On application of the exact theory ofelastic beams. In: Durban D, Givoli D, Simmonds J, editors. Advances in themechanics of plates and shells. Kluwer Academic Publishers; 2001.

[7] Ladevèze P, Simmonds J. New concepts for linear beam theory with arbitrarygeometry and loading. Eur J Mech 1998;17(3):377–402.

[8] Le Fichoux E. Présentation et utilisation de CASTEM. Rapport, DMT/SEMT/LAMS/RT/98-014-A. France: CEA; 1998.

[9] Toupin RA. Saint-Venant’s principle. Arch Ration Mech Anal 1965;18:83–96.[10] Vlasov VZ. Thin walled elastic beams. 2nd ed. English translation published for

US Science Foundation by Israel Program for Scientific Tranlations, Jerusalem;1961.

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Higher order composite beam theory built on Saint-Venant’s solution. Part-II: Built-in effects...

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