Bending of thin-walled beams of open section with influence of shear, part I: Theory

Thin--Walled Structures xx (xxxx) xxxx–xxxx

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Bending of thin-walled beams of open section with influence of shear, part I: Theory R. Pavazzaa, A. Matokovićb, a b

⁎

Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Ruđera Boskovića bb, 21000 Split, Croatia Department of Professional Studies, University of Split, Kopilica 5, 21000 Split, Croatia

A R T I C L E I N F O

A BS T RAC T

Keywords: Theory of thin-walled beams Bending Influence of shear Open sections Analytic Finite element method

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is based on the Vlasov’s general beam theory as well as on the Timoshenko’s beam bending theory. The theory is valid for general thin-walled open beam cross-sections. Part I is devoted to the theoretical developments and part II discusses analytical and obtained numerical results. The theory is based on a kinematics assuming that the cross-section maintains its shape and including three independent warping parameters due to shear. Poison’s effect is ignored, as well as warping constrains due to shear (as it known, those effect have small and for engineering praxis neglected influence on the stresses and displacements). Closed-form analytical results are obtained for three-dimensional expressions of the normal and shear stresses. Under general transverse loads, reduced to the cross-section principal pole, the beam will be subjected to bending with influence of shear and in addition to torsion due to shear with respect to the cross-section principal pole and to tension/compression due to shear, in the case of non-symmetrical cross-sections. The beam will be subjected to bending with influence of shear (i) in the plane of symmetry under the loads in that plane and in addition to tension/compression due to shear, (ii) in the plane through the principal pole orthogonal to the plane of symmetry under the loads in that plane and in addition to torsion with respect to the principal pole, in the case of the mono-symmetrical cross-sections. The beam will be subjected to bending with influence of shear in the principal planes, in the case of the bi-symmetrical cross-sections. The principal crosssection axes as well as the principal pole are defined by the classical Vlasov’s theory of thin-walled beams of open section. The analytical and numerical analyses presented in part II include comparisons with the classical beam theory, Euler-Bernoulli’ theory (EBBT) as well as comparisons with the finite element method (FEM).

1. Introduction The Euler-Bernoulli's beam theory as well as the Vlasov's thinwalled beam theory [24] do not take into account shear deformations due to shear forces. The shear effect, as well as Poisson's effect, can be included by methods of theory of elasticity [7], but in that case the problem is no longer one-dimensional. Thus, approximate methods to include the shear effect are developed; especially, in analyses of the beam displacements [19,26], by deriving adequate stiffness matrix [16,17]. For that purpose, the concept of shear factors, first introduced by Timoshenko [20,21], was used. The shear factor was defined by Timoshenko as the ratio of the maximum stress to the average shear stress over a cross section. Recent approaches to the problem are based on geometric assumptions [1,10,12–14,18,23,5,8] or shear energy relations [6,16,17]. Numerical examples comparing results obtained by different ap-

⁎

proaches can be fined in literatures [2,4], Comparisons to the results of the finite element analysis can be fined in the literature as well [12– 14,6]. In this paper, an approximate analytical solution for stresses along the beam cross-section contour as well as solutions for displacements along the beam length will be developed. Various types of transverse loading and boundary condition are assumed. The beams with general cross-section contour shapes and common symmetrical shapes, as special cases, will be investigated. The Poisson's effect is ignored. Its influence on the stresses as well as displacements in the case of common open thin-walled crosssections is small, even for extremely low ratios of beam length to cross section contour dimensions [9,17]. The shear warping effect, defined by “non-uniform warping bending theory” [27] will be ignored as well. It is shown that this effect remain very localised close to the clamped ends, where by this theory warping due to shear is restricted. The effect

Corresponding author. E-mail address: [email protected] (A. Matoković).

http://dx.doi.org/10.1016/j.tws.2016.08.027 Received 24 April 2016; Received in revised form 25 July 2016; Accepted 29 August 2016 Available online xxxx 0263-8231/ © 2016 Elsevier Ltd. All rights reserved.

Please cite this article as: Pavazza, R., Thin--Walled Structures (2016), http://dx.doi.org/10.1016/j.tws.2016.08.027

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

vS = v P − (z − az ) αP, wS = w P + ( y − a y ) αP,

of the cross-section distortion will be ignored as well [3,6,11,15,22,25].

(1)

. where vS = vS (x, s ) and wS = wS (x, s ) are the displacements of an arbitrary point S of the cross-section contour (the median line) in the y and z-directions, respectively, y = y (s ) and z = z (s ) are the rectangular coordinates, i.e. the rectangular axes in the cross-section planes, a y and az the coordinates of an arbitrary pole P, s is the curvilinear coordinate of the point S from the starting point M, v P = v P (x ) and w P = w P (x ) are the displacements of the pole P in the y and z-directions, respectively, i.e. the displacements of the cross-section contour as rigid contour, and αP = αP (x ) is the angular displacement of the contour as a rigid contour with respect to the pole P;

2. Stresses and strains 2.1. Strains and displacements Developed analytical model is based on the following assumptions: 1. The shape of the cross-section middle line is maintained during the beam deformations. 2. The normal stresses can be neglected, except the normal stresses in the longitudinal direction, which are uniformly distributed in the normal direction to the cross-section middle line. 3. The shear stresses can be neglected, except the shear stresses in the direction of the tangent on the cross-section middle line, which are uniformly distributed.

v∼S = vS cos φ + wS sin φ ,

(2)

where v∼S = v∼S (x, s ) is the displacement of the point S in the ξ -direction, Sx′ηξ is the rectangular coordinate system with respect to the point S, φ = φ (s ) is the angle between the tangent ξ and the axis y and Oxyz is the rectangular coordinate system with respect to the cross-section contour. Substitution of Eq. (1) into Eq. (2) gives

These assumptions are identical to the Vlasov's assumptions of the general thin-walled beam of open section theory, except the Vlasov's major assumption that shear deformation in the beam middle surface can be neglected (against the existing shear stresses in the direction of the tangent on the cross-section middle line). The following constraints are presumed:

⎡ vP ⎤ v∼S = [cos φ sin φ hP ] ⎢ w P ⎥ , ⎢⎣ αP ⎥⎦

1. The beam is a thin-walled structure with the ratio

(3)

where

t / d ≤ 1/10

hP = ( y − a y )sin φ − (z − az )cos φ ,

where t is the wall thickness and d is a part of the cross-section middle line (between junctions or between a junction and the free edge). Thus, the thin-walled beam can be considered as a thin-walled structure of a cylindrical or prismatic shell type, with an arbitrary cross-section consisting of the finite number of thin (flat or curved) plates. In Vlasov's beam theory such thin-walled structures are classified as long cylindrical shells, where

(4)

i.e.

⎡ dy v∼S = ⎢ ds ⎣

dz ds

⎡ vP ⎤ dω ⎤ ⎢ wP ⎥, ⎥ ds ⎦ ⎢ ⎣ αP ⎥⎦

(5)

where

h / l ≤ 1/10

dy dz dω , sin φ = , hP = , ds ds ds

cos φ =

where h is the height (or the width) or the beam cross-section. Here such a constraint is useless. The theory can be applied even for extremely short thin-walled structures, by assuming that the ratio t / d is small. 2. The cross-section must be stiffened against the cross-section distortion by intermediate diaphragms.

(6)

ω = ω (s ) is the sectorial coordinate with respect to the pole P and the point M: ω=

∫0

s

hP ds.

(7)

The relation between displacements and shear strains in the beam middle surface γxξ = γxξ (x, s ), can be expressed as

According to the assumption 1. the following expressions for the displacements can be written (Fig. 1):

γxξ =

∂uS ∂v∼ + S. ∂s ∂x

(8)

Substitution of Eq. (5) into Eq. (8) gives

∂uS ⎡ = ⎢− ⎣ ∂s

dy D ds

−

dz D ds

−

⎡ vP ⎤ dω ⎤ ⎢ D wP ⎥ ds ⎥ ⎦ ⎣⎢ αP ⎦⎥

+ γxξ,

(9)

d . dx

where D ≡ The solution of Eq. (9) can be expressed as follows

⎡ si γ u ds ⎤ ∫ ⎥ ⎡ uiu ⎤ ⎢ 0 xξ ⎡ uSu ⎤ ⎡ 1 ⎤ u 0 0 0 syj v ⎡ ⎤ ⎢ ⎥ M ⎢ v⎥ ⎢ v⎥ ⎢ ⎥ γ d s ∫ y⎥ 0 0 u ⎢ uS ⎥ = ⎢ 0 − y (sy )D ⎥ ⎢ v P ⎥ + ⎢ j ⎥ + ⎢ 0 xξ , ⎢ uSw ⎥ ⎢ 0 0 − z (sz )D 0 ⎥ ⎢ w P ⎥ ⎢ ukw ⎥ ⎢ ∫ szk γ w dsz ⎥ ⎢ ⎥ ⎢ ⎥ xξ ⎢ ⎥ ⎢ α⎥ ⎢ α ⎣ ⎦ 0 0 0 − ω (s )D ⎥⎦ P ⎥ ⎣ uS ⎦ ⎣ 0 ⎣ uiα ⎦ ⎢ si α ⎢⎣ ∫0 γxξ ds ⎥⎦ (10) where

Fig. 1. Displacements of an arbitrary point of the cross-section middle line.

2

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

⎡ εxiu ⎤ ⎤ ⎡ εxu ⎤ ⎡ D 0 0 0 ⎥ ⎡ uM ⎤ ⎢ v ⎥ ⎢ v ⎥ ⎢ 0 − y (s )D2 0 0 y ⎥ ⎢ v P ⎥ + ⎢ εxj ⎥ ⎢ εx ⎥ = ⎢ ⎥ ⎢ wP ⎥ ⎢ ε w ⎥ ⎢ εxw ⎥ ⎢ 0 0 − z (sz )D2 0 ⎥ ⎢⎣ αP ⎥⎦ ⎢ xkα ⎥ ⎢ α⎥ ⎢ 2 ε 0 0 − ω (s )D ⎦ ⎣ x ⎦ ⎣0 ⎣ εxi ⎦

uS = uSu + uSv + uSw + uSα , γxξ = γxξu + γxξv + γxξw + γxξα, y (syj = 0) = 0, z (szk = 0) = 0, ω (si = 0) = 0, i = 1, 2 … m, j = 1, 2 … n , k = 1, 2 … r . (11)

⎡ si γ u ds ⎤ ⎢ ∫0 xξ ⎥ ⎢ syj v ⎥ d γ s ∫ y ⎢ xξ ⎥ ∂ 0 + ⎢ szk w ⎥, ∂x ⎢ ∫ γ dsz ⎥ xξ 0 ⎢ si α ⎥ ⎢⎣ ∫0 γxξ ds ⎥⎦

Here uM = uM (x ) is the longitudinal displacements of the point M , where ω (s = 0) = 0 , i.e. the displacements of the cross-section contour in the longitudinal direction as a rigid contour, and γxξu = γxξu (x, s ), γxξv = γxξv (x, sy ), γxξw = γxξw (x, sz ) and γxξα = γxξα (x, s ) are the shear strain components in the middle surface with respect to the displacements uM , v P , w P and αP , respectively; uiu = uiu (x ), ujv = ujv (x ), ukw = ukw (x ) and uiα = uiα (x ) are the integration constants, i.e. the displacement components in the longitudinal direction of the starting points due to shear; m, n and r are the numbers of the starting points, with respect to the coordinate si , svj and swk , respectively. It is assumed displacements given by first term in Eq. (10) are displacements due to tension, bending and torsion with influence of shear, while the last terms represent the additional displacements due to the shear warping. Hence,

vMax (x ), vPb

where

εx =

∂uS ∂x

du M dx

∂uSv

+

Duiu

=

εxiu ,

Dujv

εxiu

εxiu (x ),

εxjv

∂x εxjv ,

=

+

∂uSw

+

∂x Dukw

εxjv (x ),

=

w εxk

∂uSα ∂x w εxk ,

= εxu + εxv + εxw + εxα , Duiα = εxiα ;

εxiα

w εxk (x )

(18)

εxiα (x )

= and are the strain com= = = ponents in the longitudinal direction of the starting points due to shear. Here D2 ≡

d2 dx 2

.

By ignoring the normal stresses in the cross-section middle line direction (not neglected), according to the assumptions 2 and 3, Hooke's law for the state of plane stress can be written as

(12)

vPb (x )

wPb

σx = Eεx , τxξ = Gγxξ

wPb (x )

= = = where and are the displacements of the cross-sections as plane sections in the x, y and z-directions, respectively, αPt = αPt (x ) is the angular displacement of the crosssections as plane sections with respect to the pole P; uMa = uMa (x ), vPa = vPa (x ), wPa = wPa (x ) and αPa = αPa (x ) are the additional displacements due to shear, with respect to the displacements vMa , vPb , wPb and αPt , respectively. Eq. (10) can also be written as

(19)

where σx = σx (x, s ) is the normal stress in the longitudinal direction and τxξ = τxξ (x, s ) is the shear stress in the beam middle surface, E is the modulus of elasticity and G is the shear modulus. Thus, substitution of Eq. (17) into Eq. (19) gives

⎡ σxiu ⎤ ⎡D ⎤ ⎡ σxu ⎤ 0 0 0 ⎢ ⎥ ⎡ uM ⎤ ⎢ v ⎥ ⎢ v⎥ 2 0 0 ⎥ ⎢ v P ⎥ + ⎢ σxj ⎥ + E ⎢ σx ⎥ = E ⎢ 0 − y (sy )D ⎢0 ⎥ ⎢ wP ⎥ ⎢ σ w ⎥ G ⎢ σxw ⎥ 0 − z (sz )D2 0 ⎢ ⎥ ⎢⎣ αP ⎥⎦ ⎢ xkα ⎥ ⎢ α⎥ 2 0 0 − ω (s )D ⎦ ⎣ σx ⎦ ⎣0 ⎣ σxi ⎦

⎡ si γ u ds ⎤ u⎤ ⎢ ∫0 xξ ⎥ ⎡ ⎡ uSu ⎤ ⎡ 1 u ⎤ 0 0 0 ⎡ uM ⎤ ⎢ i ⎥ ⎢ syj v ⎥ ⎢ v⎥ ⎢ ⎥ v ⎢ γ ⎥ ⎢ uj ⎥ ⎢ ∫0 γxξ dsy ⎥ 0 0 ⎢ uS ⎥ = ⎢ 0 − y (sy ) ⎥⎢ + , ⎥+ ⎢ uSw ⎥ ⎢ 0 0 − z (sz ) 0 ⎥ ⎢− β ⎥ ⎢ ukw ⎥ ⎢ ∫ szk γ w ds ⎥ ⎢ α ⎥ ⎢ 0 xξ z ⎥ ⎢ α⎥ ⎢ ⎥ ⎣ ⎦ − ϑ 0 0 − ω (s ) ⎦ ⎥ ⎣ uS ⎦ ⎣ 0 ⎣ ui ⎦ ⎢ si α ⎢⎣ ∫0 γxξ ds ⎥⎦

⎡ si τ u ds ⎤ ⎢ ∫0 xξ ⎥ ⎢ syj v ⎥ d τ s ∫ y xξ ⎥ ∂ ⎢ ⋅ ⎢ 0szk ⎥, w ∂x ⎢ ∫ τxξ dsz ⎥ 0 ⎢ si α ⎥ ⎢⎣ ∫0 τxξ ds ⎥⎦

(13) where

(20)

where

⎡ γ ⎤ ⎡ vP ⎤ ⎢− β ⎥ = D ⎢ wP ⎥, ⎢ ⎥ ⎣⎢ αP ⎥⎦ ⎣− ϑ ⎦

u v w α σx = σxu + σxv + σxw + σxα, τxξ = τxξ + τxξ + τxξ + τxξ ;

(14)

w w Eεxiu = σxiu, Eεxjv = σxjv , Eεxk = σxk , Eεxiα = σxiα;

i.e.

⎡ γ ⎤ ⎡ γb ⎤ ⎡ γa ⎤ ⎢ β ⎥ = ⎢ β ⎥ + ⎢ β ⎥, ⎢ ⎥ ⎢ b⎥ ⎢ a ⎥ ⎣ ϑ ⎦ ⎣ ϑt ⎦ ⎣ ϑa ⎦

σxu

⎡ vPa ⎤ ⎡ γa ⎤ ⎢ a⎥ ⎢− β ⎥ = D ⎢ wP ⎥ . ⎢ a⎥ ⎢⎣ αPa ⎥⎦ ⎣− ϑa ⎦

σxu (x,

σxv

σxv (x,

σxw

(21)

σxw (x, sz )

σxα

σxα (x,

= sy ), = = s ) are the = s ), and normal stress components with respect to the displacements uM , v P , u u v v w w w P and αP , and τxξ = τxξ (x, s ), τxξ = τxξ (x, sy ), τxξ = τxξ (x, sz ) and α α τxξ = τxξ (x, s ) are the shear stress components with respect to the displacement components vP , wP and αP , respectively; σxiu = σxiu (x ), w w = σxw (x ) and σxiα = σxiα (x ) are the normal stress compoσxjv = σxjv (x ), σxw nent of the starting points due to shear. For the equilibrium of a portion of the wall in the longitudinal direction it can be written (Fig. 2)

(15)

where γb = γb (x ) and βb = βb (x ) are the angular displacements of the cross sections as plane sections with respect to the y and z-axes and ϑt = ϑt (x ) is the relative angular displacements with respect to the xaxis displacement, analogous to the classical theory of thin-walled beams of open cross-sections; γa = γa (x ), βa = βa (x ) and ϑa = ϑa (x ) are the additional angular displacements due to shear, with respect to the displacement γb , βb and ϑt , respectively, where

⎡vb⎤ ⎡ γb ⎤ ⎢ P⎥ ⎢− β ⎥ = D ⎢ wPb ⎥ , ⎢ b⎥ ⎢ b⎥ ⎣ − ϑt ⎦ ⎣ αP ⎦

=

2.2. Stresses in terms of displacements

⎡ u ax ⎤ ⎡ a ⎤ ⎡ uM ⎤ ⎢ M ⎥ ⎢ uM ⎥ ⎢ v P ⎥ ⎢ vPb ⎥ ⎢ vPa ⎥ ⎢ wP ⎥ = ⎢ b ⎥ + ⎢ w a ⎥, ⎢⎣ αP ⎥⎦ ⎢ wP ⎥ ⎢ Pa ⎥ ⎢⎣ αPt ⎥⎦ ⎣ αP ⎦

uMax

(17)

∂(τxξ t ) ∂(σx t ) + = 0, ∂x ∂s

(22)

where t = t (s ) is the wall thickness.. If u ∂τxξ

(16)

∂x εxiu σxiu

Thus, the strains in the longitudinal direction can now be expressed as 3

= const , = const , = const ,

v ∂τxξ

∂x εxjv = σxjv =

= const , const , const ,

w ∂τxξ

w εxk w σxk

∂x

= const ,

= const , = const ,

εxiα σxiα

α ∂τxξ

∂x

= const ,

= const , = const ,

(23)

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

Fig. 2. Equilibrium in the longitudinal direction of a portion of the beam wall.

the solution of equation Eq. (22), taking into account Eq. (20), (21) and (23), can be expressed as u⎤ ⎡ τxξ ⎡ A (s ) 0 0 0 ⎤ ⎡ u ⎤ ⎡ T0u ⎤ ⎢ v⎥ M ⎢ ⎥ ⎢ ⎥ 0 0 ⎥ 3 ⎢ v P ⎥ ⎢ T0v ⎥ ⎢ τxξ ⎥ E ⎢ 0 Sz (sy ) D ⎢ w ⎥ + ⎢ w ⎥, ⎢τ w ⎥ = ⎢ 0 P Sy (sz ) 0 0 ⎥ T0 t ⎢ xξ ⎥ ⎢ ⎥ ⎢⎣ αP ⎥⎦ ⎢ α ⎥ α⎥ S s 0 0 0 ( ) T0 ⎦ ⎣ ⎦ ⎢⎣ τxξ ⎣ ω ⎦

(24)

where s

sy

sz

s

Fig. 3. Equilibrium in the longitudinal direction of the cut-off portion of the beam wall.

A (s ) = ∫ dA Sz (sy ) = ∫ y dA, Sy (sz ) = ∫ z dA, Sω (s ) = ∫ ω dA, 0 0 0 0 dA = t ds; u v T0u = T0u (x ) = τxξ (x, s = 0) t (s = 0), T0v = T0v (x ) = τxξ (x, sy = 0) t (sy = 0), w α T0w = T0w (x ) = τxξ (x, sz = 0) t (sz = 0), T0α = T0α (x ) = τxξ (x, s = 0) t (s = 0);

(25)

D3 ≡

d3 dx 3

.

Equation Eq. (24) can also be written as u⎤ ⎡ τxξ ⎡ A* ⎢ v⎥ ⎢ τ ⎢ xξ ⎥ E ⎢ 0 = ⎢τ w ⎥ t ⎢0 ⎢ xξ ⎥ ⎢ α⎥ ⎢⎣ 0 ⎢⎣ τxξ ⎦

0 0 0⎤ ⎥ ⎡ uM ⎤ Sz* 0 0 ⎥ ⎢ v P ⎥ D3 , 0 Sy* 0 ⎥ ⎢ w P ⎥ ⎥ ⎢⎣ αP ⎥⎦ 0 0 Sω*⎥⎦

(26)

where

Sz* =

∫s* y dA* , Sy* = ∫s* z dA* , y

z

Sω* =

∫s* ω dA* ;

Fig. 4. Equilibrium of a portion of the beam wall.

(27)

∑ Fx = ∫ L

where Sy* = Sy* (sz ) = Sy* (sz*) and Sz* = Sz* (sy ) = Sz* (sy*) are moments of the cut-off portion of cross-section area with respect to the z and yaxes, Sω* = Sω* (s ) = Sω* (s*) is the sectorial moment of the cut-off portion of the cross-section area, where is assumed s* = 0 where τxξ = 0 , for the appropriate coordinates sy , sz and s, i.e. sy*, sz* and s* (Fig. 3). Referring to Eqs. (24) and (26)

∑ Fy = ∫ L ∑ Fz = ∫ L

∂x ∂(τxξ t ) ∂x

ds = 0,

cos φ dx ds + qy dx = 0,

sin φ dx ds + qz dx = 0, ∑ MP = ∫ L

∂(τxξ t ) ∂x

dx hP ds = 0. (30)

.

dA* (s*) = −dA (s ), dSz* (sy*) = −dSz (sy ), dSy* (sz*) = −dSy (sz ), dSω* (s*) = −dSω (s )

∂(σx t ) dx ∂x ∂(τxξ t )

Taking into account Eq. (6), it may be written (28)

∫L ∫L

2.3. Equilibrium equations

∂σx dAdx ∂x ∂(τxξ t )

= 0, ∫ L

∂(τxξ t ) ∂x

dy + qy = 0,

dz + qz = 0, ∫ L

∂x

∂(τxξ t ) ∂x

dω = 0.

(31)

By integrating by parts one has It is assumed the beam is loaded by the transverse forces per unit length qy = qy (x ) and qz = qz (x ) in the cross-section planes through the pole P

qy =

∫L py ds,

qz =

∫L pz ds,

∂(τxξ t ) ∂x

y

e2 e1

∂

− ∫ y ∂s [ L

∂(τxξ t ) ∂x

] ds + qy = 0,

∂(τxξ t ) e2 ∂x

z

e1

∂

− ∫ z ∂s [ L

∂(τxξ t ) ∂x

] ds + qz

= 0, ∂(τxξ t )

(29)

∂x

where py = py (x, s ) and pz = pz (x, s ) are the surface loads with respect to the y and z-axes, respectively and L is the cross-section middle line length. For a portion of the beam wall, the following equilibrium equations can be written (Fig. 4)

ω

e2 e1

∂ ∂(τxξ t ) ω ∂s [ ∂x ]ds L

− ∫

= 0. (32)

where e1 and e2 are the boundaries where τxξ = 0 , for the appropriate coordinates. Hence 4

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

∫L

∂σx ∂x

∂

dA = 0, ∫ y ∂s [ L

∫L z ∂∂s [

∂(τxξ t ) ∂x

∂(τxξ t ) ∂x

⎡ Iz D3 0 ⎡Qy ⎤ 0 ⎤ ⎡ vP ⎤ ⎥ ⎢ ⎢ ⎥ 3 E ⎢ 0 Iy D 0 ⎥ ⎢ w P ⎥ = −⎢Qz ⎥ , ⎢ ⎥ α ⎣ ⎦ ⎥ ⎢ P ⎢⎣ 0 ⎥⎦ 0 Iω D3⎦ ⎣ 0

] ds − qy = 0, ∂

] ds − qz = 0, ∫ ω ∂s [ L

∂(τxξ t ) ∂x

]ds = 0.

(33)

Substitution of Eqs. (20) and (24), into Eq. (33), taking into account Eq. (23), gives

⎡ AD2 − Sz D3 − Sy D3 − Sω D3 ⎤ ⎢ ⎥ ⎡ uM ⎤ ⎡ 0 ⎤ Izy D 4 Izω D 4 ⎥ ⎢ v P ⎥ ⎢ qy ⎥ ⎢ − Sz D3 Iz D 4 E⎢ ⎥⋅⎢ ⎥ = ⎢ ⎥ , 3 Iyz D 4 Iy D 4 Iyω D 4 ⎥ ⎢ w P ⎥ ⎢ qz ⎥ ⎢ − Sy D ⎣ αP ⎦ ⎣ 0 ⎦ ⎢⎣− Sω D3 Iωz D 4 Iωy D 4 Iω D 4 ⎥⎦

where

cos φ dA = cos φ t ds = t dy, sin φ dA = sin φ t ds = t dz, hP dA = hP t ds = t dω;

∫L Sz* dy = Sz* y ee12 − ∫L y (−dSz ) = ∫A y 2dA = Iz,

(34)

∫L Sz* dz = Sz* z ee12 − ∫L z (−dSz ) = ∫A yz dA = Iyz = 0,

where

∫L Sz* dω = Sz* ω ee12 − ∫L ω (−dSz ) = ∫A yω dA = Izω = 0;

A = ∫ dA, Sz = ∫ y dA, Sy = ∫ z dA, Sω = ∫ ω dA, A A A A Iz = ∫ y 2 dA, Izy = ∫ zy dA, Izω = ∫ yω dA, A A A

∫L Sy* dy = Sy* y ee12 − ∫L y (−dSy ) = ∫A yz dA = Izy = 0,

Iy = ∫ z 2 dA, Iyω = ∫ zω dA, Iω = ∫ ω 2 dA, A A A

(35)

∫L Sy* dz = Sy* z ee12 − ∫L z (−dSy ) = ∫A z 2 dA = Iy, ∫L Sy* dω = Sy* ω ee12 − ∫L ω (−dSy ) = ∫A zω dA = Iyω = 0;

where

Izy = Izy, Iyω = Iωy, Izω = Iωz.

(36)

∫L Sω* dy = Sω* y ee12 − ∫L y (−dSω) = ∫A yω dA = Izω = 0,

If y, z and ω are the principal coordinates, i.e. the pole P is the principal pole, i.e. the shear centre, when

Sy = 0, Sz = 0, Izy = Izy = 0, Iyω = Iωy = 0, Izω = Iωz = 0,

∫L Sω* dz = Sω* z ee12 − ∫L z (−dSω) = ∫A zω dA = Iyω = 0, ∫L Sω* dω = Sω* ω ee12 − ∫L ω (−dSω) = ∫A ω2 dA = Iω.

(37)

Eq. (34) become

Referring to (38) and (40) gives

⎡ AD 2 0 0 0 ⎤ ⎡u ⎤ ⎡ 0 ⎤ ⎢ ⎥ M 4 0 0 ⎥ ⎢ v P ⎥ ⎢ qy ⎥ ⎢ 0 Iz D = ⎢ ⎥. E⎢ 0 0 Iy D 4 0 ⎥ ⎢ w P ⎥ ⎢ qz ⎥ ⎢ ⎥ ⎢⎣ αP ⎥⎦ ⎣ ⎦ 0 0 0 Iω D 4 ⎦ ⎣ 0

dQy dx

Qz =

∫A τxξ sin φ dA,

Mω =

dQz = −qz . dx

(41)

⎡ 0 ⎤ u⎤ ⎡ τxξ ⎢ v ⎥ ⎢ Qy Sz* ⎥ ⎢ τxξ ⎥ ⎢ Iz t ⎥ ⎢ τ w ⎥ = ⎢ Qz Sy* ⎥ . ⎢ xξ ⎥ ⎢ I t ⎥ ⎢ y ⎥ α⎥ ⎢⎣ τxξ ⎦ ⎢⎣ 0 ⎥⎦

Integration of the shear stresses over the cross-sections gives (Fig. 5)

∫A τxξ cos φ dA,

= −qy,

Thus, by substituting Eq. (40) into Eq. (26), the shear stresses can finally be expressed as

(38)

2.4. Shear stresses in terms of internal forces

Qy =

(40)

(42)

∫A τxξ hP dA = 0, (39) 2.5. Normal stresses in terms of internal forces

where Qy = Qy (x ) and Qz = Qz (x ) are the shear forces with respect to the y and z-axis, respectively and Mω = Mω (x ) is the sectorial moment of torsion with respect to the pole P, equal to zero by the loading conditions.. Substitution of Eq. (26), taking into account (30) and (31), into Eq. (39) gives

Integration of the normal stresses over the cross-sections gives (Fig. 5)

N=

∫A σx dA = 0, B=

Mz = −

∫A σx y dA,

My =

∫A σx z dA,

∫A σx ω dA = 0,

(43)

where My = My (x ) and Mz = Mz (x ) are the bending moments with respect to y and z-axes and B is the bimoment, which is equal to zero by the load conditions. Substitution of Eqs. (20) and (21) into Eq. (43), taking into account Eq. (42), gives

⎡ AD 0 0 0 ⎤ u ⎡ ⎤ ⎡ Ny ⎢ ⎥ ⎡ M⎤ ⎢ 0 ⎥ ⎢ y 2 I 0 D 0 0 z ⎥ ⎢ v P ⎥ = ⎢ Mz ⎥ + ⎢ Mz E ⎢⎢ 0 0 − Iy D2 0 ⎥ ⎢ w P ⎥ ⎢ My ⎥ ⎢ Myy ⎢ ⎥ ⎢⎣ αP ⎥⎦ ⎢⎣ ⎥⎦ ⎢ y ⎣ B 0 0 0 − Iω D2 ⎦ ⎣ 0 where

Fig. 5. Internal force components.

5

+ + + +

Nz ⎤ Mzz ⎥ ⎥, Myz ⎥ ⎥ Bz ⎦

(44)

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković v ⎡ ⎤ sy ∂τxξ ⎢ − ∫A dA ∫0 ∂x ds ⎥ ⎢ ⎥ ⎡ Ny⎤ ∂τ v ⎢ ∫ y dA ∫ sy xξ ds ⎥ ⎢ My⎥ E z x ∂ 0 ⎥, ⎢ ⎥= ⎢ L v sy ∂τxξ ⎥ ⎢ Myy ⎥ G ⎢ z A s − d d ∫ ∫ ⎢ ⎥ ⎢⎣ By ⎥⎦ ∂x A 0 v ⎢ ⎥ sy ∂τxξ ⎢⎣− ∫ ω dA ∫ ∂x ds ⎥⎦

A

⎡ AD 2 0 ⎡ Ny 0 0 ⎤ ⎡u ⎤ ⎡0⎤ ⎢ ⎥ M ⎢My 3 ⎢M ⎥ 0 0 ⎥ ⎢ vP ⎥ z z ⎢ 0 Iz D E⎢ = D⎢ ⎥ + D⎢ y ⎢ ⎥ 3 w ⎥ P M ⎢ My 0 0 − Iy D 0 y⎥ ⎢ ⎢ ⎥ ⎢⎣ αP ⎥⎦ ⎢⎣ y ⎢⎣ 0 ⎥⎦ B 0 0 − Iω D3 ⎦ ⎣ 0

w ⎡ ⎤ sz ∂τxξ ⎢ − ∫A dA ∫0 ∂x ds ⎥ ⎢ ⎥ ⎡ Nz ⎤ ∂τ w ⎢ ∫ y dA ∫ sz xξ ds ⎥ ⎢Mz⎥ E z x ∂ 0 ⎥; ⎢ z⎥ = ⎢ A w sz ∂τxξ ⎥ ⎢ My ⎥ G ⎢ z A s − d d ∫ ∫ ⎢ ⎥ ⎢⎣ Bz ⎥⎦ ∂x A 0 w ⎢ ⎥ sz ∂τxξ ⎢⎣− ∫ ω dA ∫ ∂x ds ⎥⎦

A

0

⎡ Ny + ⎡ 0 ⎤ ⎢My + ⎢− Q ⎥ y ⎥ + D ⎢ zy =⎢ ⎢ My + ⎢ Qz ⎥ ⎢⎣ y ⎢⎣ 0 ⎥⎦ B +

0

(45) where

+ + + +

Nz ⎤ Mzz ⎥ ⎥ Myz ⎥ ⎥ Bz ⎦

Nz ⎤ Mzz ⎥ ⎥, Myz ⎥ ⎥ Bz ⎦

(48)

and according to Eq. (38)

n

r

m

∑ j =1 N jv = 0, ∑k =1 Nkw = 0, ∑i =1 Niα = 0,

⎡ AD3 ⎢ ⎢0 E⎢ 0 ⎢ ⎣0

⎤ ⎡N y + Nz ⎤ ⎡0 ⎤ ⎥ ⎡ uM ⎤ ⎢My + Mz ⎥ ⎢ ⎥ M z⎥ ⎥ ⎢ vP ⎥ 2⎢ z⎥ 2⎢ z ⎥ ⎢ w P ⎥ = D ⎢ My ⎥ + D ⎢ Myy + Myz ⎥ = ⎥ ⎢⎣ αP ⎥⎦ ⎢⎣ y ⎥ ⎢⎣ 0 ⎥⎦ B + Bz ⎦ 0 0 −Iω D 4 ⎦ ⎡ N y + N z ⎤ ⎡0 ⎤ ⎡N y + Nz ⎤ ⎡0 ⎤ ⎢ M y + M z ⎥ ⎢q ⎥ ⎢My + Mz ⎥ ⎢− Q ⎥ y z⎥ y z⎥ 2⎢ z ⎥+D 2 ⎢ zy = D⎢ . z = ⎢ − q ⎥+ D + M M ⎢ ⎥ ⎢ Myy + Myz ⎥ Q z ⎢ z ⎥ y y ⎢ ⎥ ⎢⎣ y ⎥ ⎢ ⎥ ⎢⎣ 0 ⎥⎦ ⎣ B y + Bz ⎦ B + Bz ⎦ ⎣ 0 ⎦

w N jv = σxjv Ayj , Nkw = σxk Azk , Niα = σxiα Aωi

Ayj = A (sy ) sy = L + − A (sy ) sy =−L − , j = 1, 2, 3 … n, yj

yj

Azk = A (sz ) sz = L + − A (sz ) sz =−L − , k = 1, 2, 3 … r; zk

zk

N jv yj = 0 ( yj = 0), Nkw zk = 0 (zk = 0), Niα ωi = 0 (ωi = 0); Myy (N jv ) ≈ 0, By (N jv ) ≈ 0, Mzz (Nkw ) ≈ 0, Bz (Nkw ) ≈ 0, My (Niα ) ≈ 0, Mz (Niα ) ≈ 0.

(46)

(49)

Substitution of Eq. (42) into Eq. (46) gives

The solution given by Eqs. (48) and (49) is approximate due to the assumption given by Eq. (23). The normal stresses given by Eqs. (20) and (21), referring to Eqs. (41), (42) and (44), can now be written as follows

⎡ ⎤ sz Sy* ⎢ ∫A dA ∫0 t ds ⎥ ⎢ ⎥ sz Sy* ⎢Mz⎥ Eqz ⎢− ∫A y dA ∫0 t ds ⎥ ⎢ zz ⎥ = ⎢ ⎥, sz Sy* ⎢ My ⎥ GIy ⎢ ⎥ z d A d s ∫ ∫ ⎢⎣ Bz ⎥⎦ ⎢ A ⎥ t 0 ⎢ ⎥ sz Sy* ⎢⎣ ∫A ω dA ∫0 t ds ⎥⎦

sy Sz* ⎡ ⎤ ⎢ ∫A dA ∫0 t ds ⎥ ⎢ sy Sz* ⎥ ⎢ My⎥ Eqy ⎢− ∫A y dA ∫0 t ds ⎥ ⎢ z⎥= ⎢ ⎥, ⎢ Myy ⎥ GIz ⎢ ∫ z dA ∫ sy Sz* ds ⎥ t ⎢⎣ By ⎥⎦ 0 ⎢ A ⎥ sy Sz* ⎢ ⎥ ω d A d s ∫0 t ⎥⎦ ⎢⎣ ∫A

⎡ Ny⎤

⎡ Nz ⎤

⎡ σxu ⎤ ⎢ v⎥ ⎢ σx ⎥ ⎢ σxw ⎥ ⎢ α⎥ ⎣ σx ⎦

i.e. integrating by parts

⎡ ⎤ Az* Sy* ⎢ ∫L t ds ⎥ ⎢ ⎥ S* S * ⎡ Nz ⎤ ⎢− ∫ z y ds ⎥ ⎢Mz⎥ L t Eqz ⎢ ⎥ ⎢ zz ⎥ = , 2 ⎢ My ⎥ GIy ⎢ ⎛⎜ Sy* ⎞⎟ dA ⎥ ⎢ ∫L t ⎥ ⎢⎣ Bz ⎥⎦ ⎠ ⎝ ⎢ ⎥ ⎢ ⎥ * Sy* Sω d s ∫ ⎢⎣ L t ⎥⎦

⎡ ⎤ Ay* Sz* ⎢ ∫L t ds ⎥ ⎡ Ny⎤ ⎢ ⎥ Sz* 2 ⎢ My⎥ Eqy ⎢− ∫L ( t ) dA ⎥ z ⎢ ⎥= ⎢ ⎥, Sy* Sz* ⎢ Myy ⎥ ⎥ GIz ⎢ d s ∫ ⎢⎣ By ⎥⎦ ⎢ L t ⎥ ⎢ ⎥ * Sz* Sω d s ∫ ⎢⎣ L t ⎥⎦

0 0 0 0 Iz D 4 0 0 −Iy D 4 0

⎡1 ⎢0 =⎢ ⎢0 ⎣0

0 y 0 0

⎛ ⎡ Ny Nz ⎤⎞ ⎜⎡ 0 ⎤ ⎢ A + A ⎥⎟ ⎡ ⎢ v Mzz ⎥ ⎟ 0 ⎤ ⎜ ⎢ Mz ⎥ ⎢ Mzy − − − ⎜ ⎟ ⎢ σxj + ⎢ ⎥ Iz Iz E 0 ⎥ ⎜ ⎢ Iz ⎥ ⎢ ⎥ ⎢ M ⎥+⎢ y z ⎥⎟ − M M y 0 ⎥⎜⎢ w ⎥ ⎢ y + y ⎥ ⎟ G ⎢ σxk + I Iy ⎥ ⎟ ⎢ ω ⎦ ⎜ ⎢ y ⎥ ⎢ Iy ⎢ ⎜⎜ ⎣ 0 ⎦ ⎢ By Bz ⎥ ⎟ ⎣ ⎢⎣ Iω + Iω ⎥⎦ ⎟⎠ ⎝

0 0 z 0

(47)

E

σxjv + qy GI ∫ 0

syj Sz* t

E

w σxk + qz GI ∫ 0

Ay* = ∫ * dA* , dA* = t dsy*; Az* = ∫ * dA* , dA* = t dsz*; s s

∫A dA ∫0

sy Sz* t

ds = −Ay* ∫

∫A dA ∫

∫A y dA ∫0 ∫A z dA ∫0

∫A z dA ∫0

sy Sz*

ds = −Sz* ∫

t

sy Sz*

t

ds = Sy* ∫

t

sz Sy* t

sz Sy* t

0

sy Sz*

ds = −Sω* ∫

0

e1 e2

ds e1

e1

ds e2

sy Sz* t

0

ds = Sω* ∫

ds

t

0

sz Sy*

e1 e2

sy Sz*

ds = −Sy* ∫

t

∫A ω dA ∫0

t

0

sz Sy* t

+ ∫ L

ds

t

sz Sy*

+ ∫ L

e2

sy Sz*

0

sz Sy*

∫A ω dA ∫0

= −Az* ∫

ds = −Sz* ∫

t

ds

t e1 e2 sz Sy* ds t 0 e1 0

sz Sy* ds t 0

∫A y dA ∫0

sy Sz*

ds e1 e2

ds e1

t

ds = ∫ L

t

S* 2

Sz* Sy*

+ ∫ L

t

syj = L yj εxw, k +1),

Sy* Sz*

+ ∫ L

t

Sy* Sy* t

+ ∫ L

+ ∫ L

* Sy* Sω t

ds = ∫ L

t

t

= σxw, k +1 + qz GI ∫ 0

ds

sz, k +1 Sy* t

y

; sy =−L y, j +1

ds

, sz =−L z, k +1

ds ,

0 ≤ syj ≤ L yj+, 0 ≤ syj* ≤ L yj+, 0 ≥ sy, j +1 ≥ −L y−, j +1, 0 ≥ sy*, j +1 ≥ −L y−, j +1

Sz* Sy* t Sy* Sz* t

(52)

L yj and L y, j +1 are the distances from the starting points, where z = 0 , to the free edges, where Sz* = 0 , i.e. to arbitrary points between starting points, and leL zk and L z, k +1 are the distances of the staring points, where y = 0 , to free edges, where Sy* = 0 . i.e. to arbitrary points between starting points. The components σxiu and σxiα according to (42) may be assumed. For instance, for r = 3:

ds , ds.

* Sz* Sω

ds = ∫ L

ds = ∫ L

t

* Sy* Sω t

(51)

+ + * ≤ L yk 0 ≤ szk ≤ L yk , 0 ≤ szk , 0 ≥ sz, k +1 ≥ −Lz−, k +1, 0 ≥ sz*, k +1 ≥ −Lz−, k +1.

⎛ Sy* ⎞2 ds = ∫ ⎜ t ⎟ dA , L⎝ ⎠

* Sz* Sω

szk = L zk

sy, j +1 Sz*

z

E

ds

t

= σxv, j +1 + qy GI ∫ 0

S* 2

ds = ∫ L

0

where

t Az* Sy*

szk

ds ,

+ ∫ ( tz ) dA = ∫ ( tz ) dA, L L

− ∫ L e2

t Az* Sy*

ds = ∫ L

E

ds

szk Sy*

y

z

Ay* Sz*

∫0

σxjv = σxjv +1 (εxjv = εxv, j +1),

w w σxk = σxw, k +1 (εxk =

Ay* Sz*

Iy

⎤ ⎥ s d ⎥ t ⎥. Sy* ⎥ s d t ⎥ ⎥⎦

syj Sz*

(50)

where

e2

Iz qz

∫0

w The stress components σxjv and σxk can be obtained from the compability conditions, with respect to the appropriate curvilinear coordinates:

z

y

0 qy

σxw1 = σxw2, E

σxw1 + qz GI ∫ 0

sz1 Sy*

y

ds ,

t

E

ds sz1= L z1

= σxw2 + qz GI ∫ 0

sz 2 Sy*

y

t

ds

, sz 2 =−L z 2

σxw2 = σxw3,

ds .

E

σxw2 + qz GI ∫ 0 y

sz 2 Sy* t

E

ds sz 2 = L z 2

= σxw3 + qz GI ∫ 0

σxw1 Az1 + σxw2 Az2 + σxw3 Az3 = 0,

From Eq. (44), according to Eq. (40), one has 6

y

sz3 Sy* t

ds

, sz3=−L z3

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

IP =

∫A hP2 dA

(55)

is the polar second moment of area with respect to the principal pole P;

WP =

IP h0

(56)

is the polar section modulus with respect to the principal pole P and h 0 is the distance of the tangent through the arbitrary principal starting point M 0 from the principal pole P; La is the arbitrary length of the cross-section middle line. The normal stresses given by Eq. (50), taking into account Eqs. (55) and (56) can also be written as

⎛ ⎡ La κ ⎜⎡ 0 ⎤ ⎢ A xy ⎤ ⎢ ⎥ 0 0 0 ⎜ M ⎢ κyy − Iz⎥ ⎥ ⎢ E y 0 0 ⎜ z ⎥ ⎜ ⎢ M ⎥ + ⎢ κAzy 0 z 0⎥ ⎢ y ⎥ G ⎢ ⎜ ⎢ A 0 0 ω ⎦ ⎜ ⎢ Iy ⎥ ⎢ κωy ⎜⎣ 0 ⎦ ⎣ WP ⎝ ⎡ ⎤ 0 ⎢ v ⎥ qy syj Sz* σ + I ∫ ds ⎥ t E ⎢⎢ xj z 0 ⎥. − G ⎢ σ w + qz ∫ szk Sy* ds ⎥ xk Iy 0 t ⎢ ⎥ ⎢⎣ ⎥⎦ 0

⎡ σxu ⎤ ⎡ ⎢ v ⎥ ⎢1 ⎢ σx ⎥ = ⎢ 0 ⎢ σxw ⎥ ⎢ 0 ⎢ α ⎥ ⎣0 ⎣ σx ⎦

⎞ ⎟ ⎟ q ⎥⎡ y⎤⎟ ⎢ ⎥ ⎥ ⎣ qz ⎦ ⎟ ⎟ ⎥ ⎟⎟ ⎥ ⎦ ⎠

La κ ⎤ A xz ⎥ κyz ⎥ A κzz A κωz WP

(57)

The shear stresses given by Eq. (42) can now be rewritten, taking into account compability. Condition given by Eq. (22), Eqs. (24), (25), (48) and (55): ⎛ ⎡0 ⎤ ⎤⎜⎢ Q ⎥ 0 ⎥ ⎜ ⎢− y ⎥ 0 ⎥ ⎜ ⎢ Iz ⎥ + ⎥ ⎜ ⎢ Qz ⎥ 0 ⎥⎜ I Sω (s ) ⎦ ⎜ ⎢ y ⎥ ⎜ ⎢⎣ 0 ⎥⎦ ⎝ ⎤ ⎡0 ⎢ ⎛ ⎞ ⎥ ⎡ u⎤ ⎢ ∫ syj ⎜σ v + qy ∫ syj Sz* ds ⎟ ds ⎥ ⎢T0 ⎥ Iz 0 t 0 ⎝ xj ⎢ ⎠ ⎥ ⎢T0v ⎥ E ∂ − G ⋅ ∂x ⎢ ⎥ + ⎢ w⎥ ⎛ * S s s q ⎢ zk ⎜σ v + z zk y ds ⎞⎟ ds ⎥ ⎢T0 ⎥ ∫ α ⎢ ∫0 ⎝ xk Iy 0 t ⎠ ⎥ ⎣T0 ⎦ ⎥ ⎢ ⎦ ⎣0

Fig. 6. Example of three starting points with appropriate coordinates.

u⎤ ⎡ τxξ ⎡ A (s ) ⎢ v⎥ ⎢ ⎢ τxξ ⎥ 1 ⎢0 ⎢τ w ⎥ = − t ⎢0 ⎢ xξ ⎥ ⎢ α ⎣0 ⎣⎢ τxξ ⎦⎥

where

Az1 = A (sz1) sz1= L + − A (sz1) sz1=−L − , z1

z1

Az2 = A (sz2 ) sz2 = L + − A (sz2 ) sz2 =−L − , z2

z2

Az3 = A (sz3) sz3= L + − A (sz3) sz3=−L − . z3

z3

Thus, one has five unknowns, and five equations. The unknowns are σxw1, σxw2, σxw3, Lz−,1, Lz+3 (Fig. 6). The internal forces given by Eq. (47) can also be written as

⎡ La κxy ⎤ ⎢ I ⎥ ⎡ Ny⎤ ⎢− Az κyy ⎥ ⎢ My⎥ E ⎥, ⎢ z ⎥ = q ⎢ Iy ⎢ Myy ⎥ G y ⎢ A κzy ⎥ ⎢ ⎥ ⎢⎣ y ⎥⎦ B ⎢ Iω κωy ⎥ ⎣ WP ⎦

⎡ La κxz ⎤ ⎢ Iz ⎥ ⎡ Nz ⎤ ⎢− A κyz ⎥ ⎢Mz⎥ E ⎥, ⎢ zz ⎥ = q ⎢ Iy z ⎢ My ⎥ G ⎢ A κzz ⎥ ⎢ I ⎥ ⎢⎣ Bz ⎥⎦ ⎢ ω κωz ⎥ ⎣ WP ⎦

κyy = κωy =

1 Iz Ls A Iz2

∫A

Ay* Sz* t2

dA, κxz =

1 Iy Ls

* 2

∫A

S ∫A ( tz ) dA, κyz = κzy =

WP Iz Iω

∫L

* Sz* Sω t

ds, κωz =

WP Iy Iω

Az* Sy*

A Iy Iz

∫L

t2

* Sy* Sω t

Sy* Sz* t

ds;

ds, κzz =

⎡ La κ ⎢ A xy ⎢ κyy E⎢ A G ⎢ κzy ⎢A ⎢ κωy ⎢⎣ WP

⎞ La κ ⎤ ⎟ A xz ⎥ κyz ⎥ ⎟ ⎥ ⎡ qy ⎤ ⎟ A D ⎢ ⎥ κzz ⎥ ⎣ qz ⎦ ⎟ ⎟ ⎥ A ⎟ κωz ⎥ ⎟ ⎥ WP ⎦ ⎠

−

or taking into account Eqs. (26) and (29)

⎛ ⎡ ⎤ 0 ⎤ ⎜ ⎢0 ⎥ ⎥ ⎜ − Qy 0 ⎥ ⎜ ⎢ Iz ⎥ ⎢ ⎥+ 0 Sy* 0 ⎥ ⎜ ⎢ Qz ⎥ ⎥ ⎜ Iy ⎥ 0 0 Sω*⎥⎦ ⎜⎜ ⎢⎣ ⎦ ⎝ 0 ⎤ ⎡0 ⎥ ⎢ ⎞ ⎛ v qy * ⎢ ∫ ⎜σ + ∫ Sz ds ⎟ ds ⎥ xj * Iz syj * t s ⎠ ⎥ E ∂ ⎢ yj ⎝ + G ⋅ ∂x ⎢ ⎥. ⎞ ⎥ ⎛ * Sy qz ⎢ v + d d σ s s ⎟ ⎜ ∫ ∫ xk Iy s* t ⎢ szk* ⎝ ⎠ ⎥ zk ⎥ ⎢ ⎦ ⎣0 u⎤ ⎡ τxξ ⎡ A* ⎢ v⎥ ⎢ τ ⎢ xξ ⎥ 1 ⎢ 0 = ⎢τ w ⎥ t ⎢0 ⎢ xξ ⎥ ⎢ α⎥ ⎢⎣ 0 ⎢⎣ τxξ ⎦

(53)

dA,

∫L

0 0 S y (s ) 0

(58)

where

κxy =

0 Sz (s ) 0 0

A Iy2

⎛ S * ⎞2

∫A ⎜ ty ⎟ dA, ⎝ ⎠

(54)

κxy is the shear factor with respect to the uM displacements during the v P displacements, κxz — during the w P displacements;. κzz is the shear factor with respect to the w P displacements; κzy — during the v P displacements;. κyy is the shear factor with respect to the v P displacements; κyz — during the w P displacements;. κωy is the shear factor with respect to the αP displacements during the v P displacements, κωz — during the w P displacements;

0 0 Sz* 0

⎡ La κ ⎢ A xy ⎢ κyy E⎢ A G ⎢ κzy ⎢A ⎢ κωy ⎣ WP

⎞ ⎟ ⎟ q ⎡ ⎤ ⎥D y ⎟ + ⎢ ⎥ ⎥ ⎣ qz ⎦ ⎟ ⎟ ⎥ ⎟⎟ ⎥ ⎦ ⎠

La κ ⎤ A xz ⎥ κyz ⎥ A κzz A κωz WP

(59)

3. Differential equations with separated displacements Eq. (44), taking into account Eqs. (53) and (54), can be expressed as follows 7

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

⎡ La κ La κ ⎤ ⎡0⎤ ⎢ A xy A xz ⎥ ⎡D 0 ⎢ Mz ⎥ 0 0 ⎤ ⎡ uM ⎤ κyz ⎥ ⎢ κyy ⎢ ⎥⎢ v ⎥ ⎢ ⎥ 2 1 Iz 1 ⎢− A − A ⎥ ⎡ qy ⎤ 0 0 ⎥ P ⎢0 D ⎢ ⎥ = + ⎥. κzz ⎥ ⎢ ⎣ qz ⎦ 0 ⎥ ⎢⎢ w P ⎥⎥ E ⎢ My ⎥ G ⎢ κzy ⎢ 0 0 − D2 A A ⎥ ⎢ Iy α ⎣ ⎦ P 2 ⎢⎣ 0 0 ⎥ ⎢ ⎥ 0 −D ⎦ κωz ⎥ ⎢ κωy ⎣0⎦ ⎣ WP WP ⎦

⎡ vb ⎢ ⎢ Dvb 2 Ay = ⎢ D vb ⎢ 3 ⎢ D vb ⎢ D 4v b ⎣ ⎡1 ⎢1 ⎢1 ⎢ EI Cy = ⎢ z ⎢− 1 ⎢ EIz ⎢1 ⎣ EIz

(60)

Approximately, for qy ≠ const , qz ≠ const , taking into account Eq. (48),

⎡ La κ La κ ⎤ ⎡ 0 ⎤ ⎢ A xy A xz ⎥ ⎡ D2 0 ⎢ Qy ⎥ 0 0 ⎤ ⎡ uM ⎤ κyz ⎥ ⎢ κyy ⎢ ⎥ − 3 0 0 ⎥ ⎢ v P ⎥ = 1 ⎢⎢ Iz ⎥⎥ + 1 ⎢− A − A ⎥ D ⎡ qy ⎤ , ⎢0 D ⎥ ⎢ ⎥ κ κzz ⎥ ⎢ ⎣ qz ⎦ ⎢ 0 0 − D3 0 ⎥ ⎢ w P ⎥ E ⎢ Qz ⎥ G ⎢ zy A ⎥ ⎢ A Iy ⎣ αP ⎦ ⎢⎣ 0 0 ⎥ 3 ⎢ ⎥ ⎦ 0 −D κωz ⎥ ⎢ κωy ⎣ 0 ⎦ ⎣ WP WP ⎦

0 γb

0 0

Dγb

Mz

0 0 0

D2γb DMz

Qy

D3γb D2Mz DQy

0⎤ ⎡ vb ⎥ ⎢ 0⎥ ⎢0 ⎥ 0 , By = ⎢ 0 ⎥ ⎢0 0⎥ ⎢ ⎥ ⎢⎣ 0 qy ⎦

0 0 Mz 0 0

0 0 0 Qy 0

0⎤ ⎥ 0⎥ 0 ⎥, 0 ⎥⎥ qy ⎥⎦

0⎤ 0⎥ ⎥ 0⎥ ⎥; −1 1 0 ⎥ ⎥ 1 −1 1 ⎥ ⎦

0 1

0 0 1

1 EIz 1

− EI

z

1

− EI

z

0 0 0

(68)

Az = Bz Cz ,

(61)

where

i.e.

⎡ D3 0 0 ⎢ 4 0 ⎢0 D ⎢ 0 0 − D4 ⎢⎣ 0 0 0

⎡ La κ La κ ⎤ ⎡ 0 ⎤ ⎢ A xy A xz ⎥ ⎢ qy ⎥ 0 ⎤ ⎡ uM ⎤ κyz ⎥ ⎢ κyy ⎥⎢ v ⎥ 0 ⎥ P = 1 ⎢ Iz ⎥ + 1 ⎢− A − A ⎥ D2 ⎡ qy ⎤ . ⎢ q⎥ ⎢q ⎥ ⎢ ⎥ κ κzz ⎥ ⎣ z⎦ 0 ⎥ ⎢ w P ⎥ E ⎢− z ⎥ G ⎢ zy A ⎥ Iy ⎢ A ⎣ αP ⎦ ⎥ 4 ⎦ ⎢ ⎥ −D κωy κωz ⎥ ⎢ ⎣ 0 ⎦ ⎣ WP WP ⎦

⎡ wb 0 0 ⎢ D w β 0 b b ⎢ 2 Az = ⎢ D wb Dβb My ⎢ 3 ⎢ D wb D2βb DMy ⎢ 4 3 2 ⎣ D wb D βb D My ⎡1 0 0 0 ⎢−1 1 0 0 ⎢ 1 1 1 0 ⎢− EI EI y y Cz = ⎢ 1 1 ⎢− 1 1 ⎢ EIy EIy ⎢ 1 1 ⎢⎣ EIy − EIy −1 −1

(62)

Referring to Eq. (12), equations Eq. (60) can be separated as follows

⎡0⎤ ⎡D 0 ⎢ Mz ⎥ 0 0 ⎤ ⎡ ua ⎤ ⎢ ⎥⎢ v ⎥ ⎢I ⎥ 2 1 0 D 0 0 b ⎢ ⎥ ⎢ ⎥ = ⎢ Mz ⎥ 2 w b 0 ⎥⎢ ⎥ E ⎢ y ⎥ ⎢0 0 − D α ⎢⎣ 0 0 ⎢ Iy ⎥ 0 − D2 ⎥⎦ ⎣ t ⎦ ⎣0⎦

0 0 0 Qz DQz

0⎤ ⎡ wb ⎥ ⎢ 0⎥ ⎢0 0 ⎥ , Bz = ⎢ 0 ⎥ ⎢ 0⎥ ⎢0 ⎥ ⎢⎣ 0 qz ⎦

0 βb 0 0 0

0 0 My 0 0

0 0 0 Qz 0

0⎤ ⎥ 0⎥ 0 ⎥, ⎥ 0⎥ qz ⎥⎦

0⎤ 0⎥ ⎥ 0⎥ ⎥. 0⎥ ⎥ ⎥ 1⎥ ⎦

In Eq. (64) (63)

duax = 0, uax = const . dx

and

⎡ La κ La κ ⎤ ⎢ A xy A xz ⎥ ⎡D 0 0 0 ⎤ ⎡ ua ⎤ κyz ⎥ ⎢ κyy ⎢ ⎥ 2 1 ⎢− A − A ⎥ ⎡ qy ⎤ 0 0 ⎥ ⎢ va ⎥ ⎢0 D = ⎥, κzz ⎥ ⎢ ⎣ qz ⎦ 0 ⎥ ⎢⎢ wa ⎥⎥ G ⎢ κzy ⎢ 0 0 − D2 A A ⎢ ⎥ α ⎣ ⎦ ⎢⎣ 0 0 0 − D2 ⎥⎦ a κωz ⎥ ⎢ κωy ⎣ WP WP ⎦

⎡− κyy − κyz ⎤ ⎡ Cv ⎤ A ⎥ ⎢ A ⎡− va ⎤ κ 1 ⎡L κzz ⎥ ⎡− Mz ⎤ ⎢ ⎥ ⎢ wa ⎥ = 1 ⎢ zy + C , u = − ⎢ Aa κxy ⎢ ⎥ A ⎥ My ⎦ ⎢ w ⎥ a ⎣ ⎣ ⎢⎣ αa ⎥⎦ G ⎢ κA G κωz ⎥ ⎣ Cα ⎦ ⎢ ωy ⎣ WP WP ⎦

(64)

where Cv , Cw and Cα are the integration constants with respect to displacements va , wa and αa , respectively. This can also be written as (65)

⎡ 1 ⎢ Ayy ⎡ va ⎤ ⎢ ⎢ wa ⎥ = 1 ⎢ 1 ⎢⎣ αa ⎥⎦ G ⎢ Azy ⎢ 1 ⎣ WPy

Integration of equations Eq. (64), taking into account Eqs. (16) and (41), gives

⎡ La κ La κ ⎤ ⎢ A xy A xz ⎥ 0 0 0 ⎤ ⎡ ua ⎤ ⎡ ua ⎤ κyz ⎥ ⎢ κyy ⎢ ⎥ γ D 0 0 ⎥ ⎢ va ⎥ = ⎢ a ⎥ = − 1 ⎢− A − A ⎥ ⎡Qy ⎤ . ⎥ ⎥ κ κzz ⎥ ⎢ 0 − D 0 ⎥ ⎢ wa ⎥ ⎢ βa ⎥ G ⎢ zy ⎣Qz ⎦ ⎢ ⎥ A ⎥ ⎢ A 0 0 − D ⎦ ⎣ αa ⎦ ⎢⎣ ϑa ⎥⎦ κωz ⎥ ⎢ κωy ⎣ WP WP ⎦

⎤ ⎥ ⎡ Cv ⎤ 1 ⎥ ⎡− Mz ⎤ ⎢ ⎥ + Cw , ⎢ ⎥ ⎥ Azz ⎣ My ⎦ ⎢ ⎥ ⎥ ⎣ Cα ⎦ 1 ⎥ WPz ⎦ 1 Ayz

(71)

where

⎡ A ⎡ Ayy Ayz ⎤ ⎢ κyy ⎢ ⎥ ⎢ A ⎢ Azy Azz ⎥ = ⎢ κzy ⎢⎣WPy WPz ⎥⎦ ⎢ WP ⎢κ ⎣ ωy

(66)

The integration constants are ignored. It assumed the angular displacements γa , βa and ϑa as well as the displacements ua do not depend on the boundary conditions. Eq. (64) represent the well known equations of the classical theory of thin-walled beams. The second and third equations represent the displacements due to bending in two beam principal planes:

Ay = By C y ,

⎤ ⎡Qy ⎤ La κ ⎢ ⎥, A xz ⎥ ⎦ ⎣Qz ⎦ (70)

uM ≡ u , v P ≡ v , w P ≡ w , αP ≡ α , uMa ≡ ua , vPb ≡ vb, wPb ≡ wb, αPt ≡ αt , uMa ≡ ua , vPa ≡ va, wPa ≡ wa, αPa ≡ αa.

(69)

Integration of the second, third and fourth Eq. (66) gives

where due to simplicity

⎡1 ⎢0 ⎢ ⎢0 ⎣0

0 γb 0 0 0

⎤ ⎥ A ⎥ , κzz ⎥ ⎥ WP ⎥ κωz ⎦ A κyz

(72)

where Ayy is the shear area with respect to the v displacements, Ayz — during the w displacements;. Azy is the shear area with respect to the w displacements during the v displacements, Azz — with respect to the w displacements;. WPy is the shear polar modulus of area during the v displacements; WPz — during w displacements. The stresses, given by Eqs. (57) and (59), may now be written as

(67)

where 8

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

⎛ ⎞ ⎡ La κ La κ ⎤ ⎜ ⎡0 ⎤ ⎟ ⎢ A xy A xz ⎥ ⎤ ⎢ ⎥ ⎜ ⎟ 1 ⎢ 1 ⎥ 0 0 0 M − Iz⎥ ⎥ ⎢ ⎜ A A q ⎢ ⎥ ⎡ y⎤⎟ yy yz y 0 0 z E ⎥⎜⎢ ⎥+ ⎢ ⎥ ⎢q ⎥ ⎟ 1 ⎣ z ⎦⎟ 0 z 0 ⎥ ⎜ ⎢ My ⎥ G ⎢ 1 Azy Azz ⎥ Iy ⎥ ⎟ 0 0 ω⎦ ⎜ ⎢ ⎢ 1 ⎥ 1 ⎜ ⎣0 ⎦ ⎟ ⎢ ⎥ W W ⎣ ⎦ P y P z ⎝ ⎠ ⎡0 ⎤ ⎢ v ⎥ qy syj Sz* ⎢ σxj + Iz ∫0 t ds ⎥ E ⎥, − G⎢ ⎢ σ w + qz ∫ szk Sy* ds ⎥ xk Iy 0 t ⎢ ⎥ ⎢⎣ 0 ⎥⎦ ⎛ ⎞ ⎡ La κ La κ ⎤ ⎟ u⎤ ⎢ A xy A xz ⎥ ⎡ τxξ ⎡ A* 0 0 0 ⎤ ⎜ ⎡ 0 ⎤ ⎟ 1 ⎢ 1 ⎥ ⎢ v⎥ ⎢ ⎥ ⎜ ⎢ Qy ⎥ Ayz ⎥ ⎡ qy ⎤ ⎟ ⎢ τxξ ⎥ 1 ⎢ 0 Sz* 0 0 ⎥ ⎜ ⎢− Iz ⎥ E ⎢ Ayy ⎥ D ⎢q ⎥ ⎟ + ⎢ τ w ⎥ = t ⎢ 0 0 S * 0 ⎥ ⎜ ⎢ Qz ⎥ + G ⎢ 1 1 ⎣ z ⎦⎟ y ⎢ Azy ⎢ xξ ⎥ Azz ⎥ ⎢ ⎥ ⎜ ⎢ Iy ⎥ α ⎢ ⎥ ⎟ ⎢ 1 ⎥ ⎢⎣ 0 0 0 Sω*⎥⎦ ⎜ ⎢⎣ τxξ ⎥⎦ 1 ⎟ ⎜ ⎣0 ⎦ ⎢ ⎥ ⎣ WPy WPz ⎦ ⎠ ⎝ ⎡0 ⎤ ⎢ ⎛ ⎞ ⎥ ⎢ ∫ ⎜σ v + qy ∫ Sz* ds ⎟ ds ⎥ xj * I t * s s z ⎠ ⎥ yj E ∂ ⎢ yj ⎝ + G ⋅ ∂x ⎢ ⎥. ⎞ ⎥ ⎛ v Sy* qz ⎢ + d d σ s s ⎟ ⎜ ∫ ∫ ⎢ szk* ⎝ xk Iy s* t ⎠ ⎥ zk ⎢ ⎥ ⎣0 ⎦

Hence, taking into account Eq. (75),

⎡ σxu ⎤ ⎡ ⎢ v ⎥ ⎢1 ⎢ σx ⎥ = ⎢ 0 ⎢ σxw ⎥ ⎢ 0 ⎢ α ⎥ ⎣0 ⎣ σx ⎦

dU =

⎞ ⎤ ⎛ κzy κyz ⎞ κ dx ⎡ ⎛ κyy ⎢ ⎜ Qy + Qz⎟ Qy + ⎜ Qy + zz Qz⎟ Qz ⎥ , ⎝ A 2G ⎣ ⎝ A A ⎠ ⎦ A ⎠

i.e.

dU =

⎞ κzy κyz κ dx ⎛ κyy Qy Qz + zz Qz Qz⎟ , Qz Qy + ⎜ Qy Qy + ⎠ 2G ⎝ A A A A

i.e., according to the Maxwell's reciprocity theorem,

dU =

κyz κzz 2⎞ dx ⎛ κyy 2 Qz ⎟ , ⎜ Qy + 2 Qy Qz + 2G ⎝ A A ⎠ A

(80)

where

κyz = κzy, κzω = κωz, κωy = κyω. By equating Eqs. (78) and (80), the shear factors can be obtained, as are already obtained and given by Eq. (54).

5. Boundary conditions (73) It is assumed for the beam left section, A:

⎡ va ⎤ ⎡ 0 ⎤ ⎢ wa ⎥ = ⎢ 0 ⎥ . ⎢⎣ αa ⎥⎦ ⎢⎣ 0 ⎥⎦

4. Shear strain energy According to Hooke's low, referring to Eq. (19), the average shear stresses with respect to the displacements va and wa may be expressed as

⎡ κyy v ⎡γ v ⎤ ⎡ τxξ ⎤ ⎡ v ⎤ ⎡ γa ⎤ , av xξ, av ⎢A ⎢ w ⎥ = G ⎢ w ⎥ = GD⎢ a ⎥ = ⎢ G = ⎥ ⎢ κzy ⎣ wa ⎦ ⎣− βa ⎦ ⎢⎣ τxξ, av ⎥⎦ ⎢⎣ γxξ, av ⎥⎦ ⎣A

Hence, referring to Eq. (71)

⎡ 1 ⎢ Ayy ⎡ Cv ⎤ 1⎢ 1 ⎢ ⎥ ⎢Cw ⎥ = − G ⎢ Azy ⎢ 1 ⎣ Cα ⎦ ⎢ ⎣ WPy

κyz ⎤

⎡ ⎤ A ⎥ Qy , κzz ⎥ ⎢Q ⎥ ⎣ z⎦ A ⎦

(74)

where γxξv , av and γxξw, av are the average shear strains with respect to the displacements va and wa , respectively. u The average shear stress with respect to the displacements ua , τxξ , av is assumed equal to zero. v w The average shear stresses τxξ , av and τxξ, av may also be expressed as

⎡ 1 v ⎡ τxξ ⎤ ⎢ Ayy , av ⎢ w ⎥=⎢ ⎢⎣ τxξ, av ⎦⎥ ⎢ 1 ⎣ Azy

⎢

⎡ v ⎤ ⎡ vb ⎤ ⎢ w ⎥ = ⎢ wb ⎥ + ⎣ α ⎦ ⎢⎣ 0 ⎥⎦

⎥, (75)

∫A τxξ2 dA,

⎡ 1 ⎢ Ayy 1⎢ 1 ⎢ G Azy ⎢ 1 ⎢ ⎣ WPy

1 ⎡κ L u = ua = − G ⎢ xy a ⎣ A

The shear energy of the beam element can be expressed as

dx dU = 2G

⎤ ⎥ 1 ⎥ ⎡− MAz ⎤ ⎥. Azz ⎥ ⎢ M ⎥ ⎣ Ay ⎦ 1 ⎥ WPz ⎦ 1 Ayz

(82)

The total displacements can then be expressed as

1 ⎤ Ayz ⎥ ⎡Qy ⎤ 1 ⎥ ⎣Q ⎦ z Azz ⎥ ⎦

(81)

⎤ ⎥ 1 ⎥ ⎡− Mz + MAz ⎤ ⎥, Azz ⎥ ⎢ M − M Ay ⎦ ⎥⎣ y 1 ⎥ WPz ⎦ ⎡ ⎤ κxz L a ⎤ Qy ⎥ ⎢ ⎥ (uax = 0). A ⎦ ⎣Qz ⎦ 1 Ayz

(83)

(76) For the beam hinged sections it may be written

i.e.

dU =

dx 2G

∫A [(τxξv )2 + 2τxξv τxξw + (τxξw )2]dA.

⎡v ⎤ ⎡v ⎤ ⎡ ⎤ ⎛ ⎡v ⎤ ⎡vb ⎤ ⎡ ⎤ ⎞ = ⎢ wb ⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ wA ⎥ = ⎢ Ab ⎥ = ⎢ 0 ⎥ ⎟⎟ , ⎢w⎥ ⎣ ⎦x=x ⎣ b ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ A ⎦ ⎢⎣ wA ⎥⎦ ⎣ 0 ⎦ ⎠ A A

(77)

Taking into account Eq. (42), it may further be written

⎡ 2 dx ⎢ Qy dU = 2G ⎢⎣ Iz2

⎛ Sz* ⎞2 Qy Qz ⎜ ⎟ dA + 2 Iz Iy A⎝ t ⎠

∫

∫A

Sz* Sy* t2

dA +

Qz2 Iy2

⎡vb ⎤ ⎡ ⎤ ⎛ ⎡ MAz ⎤ ⎡ 0 ⎤ ⎞ D2 ⎢ Ab ⎥ = ⎢0⎥ ⎜ ⎢ ⎥ = ⎢ ⎥ ⎟; ⎣ 0 ⎦ ⎝ ⎣ MAy ⎦ ⎣ 0 ⎦ ⎠ ⎢⎣ wA ⎥⎦ x=x

⎛ Sy* ⎞2 ⎤ ⎜⎜ ⎟⎟ dA⎥ . ⎥⎦ A⎝ t ⎠

∫

A

⎡v ⎤ ⎡v ⎤ ⎡ ⎤ ⎛ ⎡v ⎤ ⎡v b ⎤ ⎡ ⎤ ⎞ = ⎢ wb ⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ wB ⎥ = ⎢ Bb ⎥ = ⎢ 0 ⎥ ⎟⎟ , ⎢w⎥ b ⎣ ⎦x=x ⎣ ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ B⎦ ⎢⎣ wB ⎥⎦ ⎣ 0 ⎦ ⎠ b b

(78) The shear energy can also be written by the average shear deformations, as follows

dU =

⎡v b ⎤ ⎡ ⎤ ⎛ ⎡ MBz ⎤ ⎡ 0 ⎤ ⎞ D2 ⎢ Bb ⎥ = ⎢0⎥ ⎜ ⎢ ⎥ = ⎢ ⎥ ⎟, ⎣ 0 ⎦ ⎝ ⎣ MBy ⎦ ⎣ 0 ⎦ ⎠ ⎢⎣ wB ⎥⎦ x=x

⎡Qy ⎤ ⎡Qy ⎤ ⎡Qy ⎤ dx v dx dx v w w [ γa − βa ] ⎢ ⎥ = [ γxξ, av γxξ, av ] ⎢ ⎥ = [ τxξ, av τxξ ⎥, , av ] ⎢ 2 2 ⎣Qz ⎦ ⎣Qz ⎦ 2G ⎣Qz ⎦

A

where “B” is the beam right section. Clamped beam sections:

(79) 9

(84)

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

⎡v ⎤ ⎡v ⎤ ⎡ ⎤ ⎛ ⎡v ⎤ ⎡vb ⎤ ⎡ ⎤ ⎞ = ⎢ wb ⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ wA ⎥ = ⎢ Ab ⎥ = ⎢ 0 ⎥ ⎟⎟ , ⎢w⎥ ⎣ ⎦x=x ⎣ b ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ A ⎦ ⎢⎣ wA ⎥⎦ ⎣ 0 ⎦ ⎠ A A

with respect to the z-axis (axis of symmetry), Sz* = Sz* (sy )Sω* = Sω* (s ) are the even coordinate. The total displacements given by Eq. (83) read

⎡v ⎤ ⎡ ⎤ ⎛ ⎡γ b ⎤ ⎡ ⎤ ⎞ D ⎢ wb ⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ Ab ⎥ = ⎢ 0 ⎥ ⎟⎟ ; ⎣ b ⎦x=x ⎣ 0 ⎦ ⎢⎣ β ⎥⎦ ⎣ 0 ⎦ A ⎝ A ⎠ ⎡ 1 1 ⎤ ⎡ (−vb x = x B + vb x = xA ) Iz ⎤ ⎡ 0 ⎤ ⎡ vB ⎤ ⎡ vb ⎤ E ⎢ Ayy Ayz ⎥ ⎥=⎢ ⎥ + G ⎢ 1 1 ⎥ D2 ⎢ ⎢ wB ⎥ = ⎢ wb ⎥ (−wb x = x B + wb x = xA ) Iy ⎦ ⎣ 0 ⎦ ⎣ ⎦ ⎣ ⎦x=x B ⎢⎣ Azy Azz ⎥⎦ ⎣ ⎛ ⎞ ⎡ 1 1 ⎤ ⎡ ⎤ ⎡0⎤ ⎟ ⎜ ⎡ vB ⎤ ⎡ vb ⎤ 1 ⎢ Ayy Ayz ⎥ − MBz + MAz + ⎢ ⎥ = ⎢ ⎥ ⎟, ⎜ ⎢ wB ⎥ = ⎢ wb ⎥ ⎥⎢ ⎜ ⎣ ⎦ ⎣ ⎦ x = x B G ⎢ A1 A1 ⎥ ⎣ MBy − MAy ⎦ ⎣ 0 ⎦ ⎟ ⎣ zy zz ⎦ ⎝ ⎠ ⎡ b⎤ ⎡ ⎤ ⎛ ⎡γ b ⎤ ⎡ ⎤ ⎞ D⎢v b⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ Bb ⎥ = ⎢ 0 ⎥ ⎟⎟ ; ⎣ w ⎦x=x ⎣ 0 ⎦ ⎢⎣ β ⎥⎦ ⎣ 0 ⎦ B ⎝ B ⎠

⎡ 1 ⎢ Ayy ⎡ v ⎤ ⎡ vb ⎤ ⎢ 1 ⎢ w ⎥ = ⎢ wb ⎥ + ⎢ 0 ⎣ α ⎦ ⎢⎣ 0 ⎥⎦ G ⎢ 1 ⎢W ⎣ Py

(91)

(85)

σxw =

My Iy

B

w τxξ =

Qz Sy* Iy t

(86)

B

+

E Sy* E ∂ ⋅ ⋅ + ⋅ dx GAzz t G ∂x

dqz

⎛ ⎡ 0 ⎜⎡ 0 ⎤ ⎢ ⎡ σxu ⎤ ⎡ ⎤ ⎜ ⎢ Mz ⎥ 1 0 0 0 ⎢ 1 ⎢ v⎥ ⎢ ⎥ ⎢− ⎥ ⎢ σx ⎥ = ⎢ 0 y 0 0 ⎥ ⎜⎜ ⎢ Iz ⎥ + E ⎢ Ayy ⎢ σxw ⎥ ⎢ 0 0 z 0 ⎥ ⎜ ⎢ My ⎥ G ⎢ 0 ⎢ ⎢ α ⎥ ⎣ 0 0 0 ω ⎦ ⎜ ⎢ Iy ⎥ ⎢ 1 ⎣ σx ⎦ ⎣ ⎦ ⎜ 0 ⎢W ⎣ Py ⎝ ⎡ ⎤ 0 ⎢ v ⎥ qy syj Sz* σ + I ∫ ds ⎥ t E ⎢⎢ xj z 0 ⎥; − G ⎢ σ w + qz ∫ szk Sy* ds ⎥ xk I t 0 y ⎢ ⎥ ⎢⎣ ⎥⎦ 0

⎛

Sy*

q

∫s* ⎜⎜⎝σxkv + Iyz ∫s* zk

t

zk

⎞ ds ⎟⎟ ds, ⎠ (93)

⎛ ⎤ ⎜⎡ 0 ⎤ ⎜ ⎢0 ⎥ Qy ⎥ 0 ⎥ ⎜ ⎢− Iz ⎥ ⎥+ ⎜⎢ 0 Sy* 0 ⎥ ⎜ ⎢ Qz ⎥ ⎥ Iy ⎥ 0 0 Sω*⎥⎦ ⎜ ⎢⎣ ⎜ 0 ⎦ ⎝ ⎤ ⎡0 ⎥ ⎢ ⎞ ⎛ v qy * ⎢ ∫ ⎜σ + ∫ Sz ds ⎟ ds ⎥ xj * Iz syj * t s ⎠ ⎥ E ∂ ⎢ yj ⎝ + G ⋅ ∂x ⎢ ⎥, ⎞ ⎥ ⎛ * Sy qz ⎢ v + d d σ s s ⎟ ⎜ ∫ ∫ xk * Iy s* t ⎢ szk ⎝ ⎠ ⎥ zk ⎥ ⎢ ⎦ ⎣0

σxv = −

v =− τxξ

(87)

(94)

qy Mz E E⎛ y + qy y − ⎜σxjv + Iz GAyy G⎝ Iz

∫0

syj

Sz* ⎞ α E ds⎟ , σx = qy ω; t ⎠ GWPy

Qy Sz* Iz t

+

E Sz* E ∂ ⋅ + ⋅ dx GAyy t G ∂x

dqy

⋅

⎛

∫s * ⎜⎜σxjv + yj

⎝

dqy E Sω* α = ⋅ ⋅ . τxξ dx G WPy

qy

∫ Iz s *

yj

Sz* ⎞ ds ⎟⎟ ds, t ⎠ (96)

The displacements (91) are

v = vb +

−Mz + MAz −Mz + MAz , α = αa = . GWPy GAyy

(97)

The beam is subjected to bending with influence of shear in the principal plane orthogonal to the plane of symmetry and in addition to torsion due to shear. The cross-section with two axis of symmetry. In the case of the beam with the cross-section with two axis of symmetry, the normal stresses given by Eq. (87) become

⎡ σxu ⎤ ⎡ ⎢ v ⎥ ⎢1 ⎢ σx ⎥ = ⎢ 0 ⎢ σxw ⎥ ⎢ 0 ⎢ α ⎥ ⎣0 ⎣ σx ⎦

(88)

(89)

0 y 0 0

0 0 z 0

⎛⎡ 0 ⎤ ⎡0 ⎢ 1 0 ⎤ ⎜ ⎢ Mz ⎥ − E ⎢ Ayy 0 ⎥ ⎜ ⎢ Iz ⎥ ⎥⎜⎢ M ⎥ + ⎢ 0 ⎥⎜⎢ y ⎥ G ⎢ 0 ω ⎦ ⎜⎜ ⎢ Iy ⎥ ⎢ ⎣0 ⎝⎣ 0 ⎦

⎡ ⎢ v σ + E ⎢⎢ xj − G ⎢σw + ⎢ xk ⎢⎣

where

∫A

κxz Qz La ⋅ . G A

(95)

where due to symmetry

Sy*⋅Sω* = 0.

, u = ua = −

the shear stresses given by (88) become

⎞ La ⎡0 κ ⎤ ⎟ A xz ⎥ ⎢ ⎟ ⎢ 1 0 ⎥ ⎥ ⎡ qy ⎤ ⎟ E ⎢ Ayy D ⎥ ⎢q ⎥ ⎟ + 1 G ⎢0 ⎣ z ⎦⎟ ⎢ Azz ⎥ ⎟ ⎢ 1 ⎥ ⎟ ⎢W 0 ⎥ ⎣ Py ⎦ ⎠

κxy = 0; κyz = κzy = 0: Ayz = Azy = ∞; κωz = 0: WPz = ∞;

GAzz

The beam is subjected to bending with influence of shear in the plane of symmetry and in addition to tension/compression due to shear. The case qz = 0 . The normal stresses given by Eq. (87) become

⎞ ⎟ ⎟ 0 ⎥ ⎥ ⎡ qy ⎤ ⎟ ⎥⎟ 1 ⎥⎢ ⎣ qz ⎦ ⎟ Azz ⎥ ⎟ ⎥ 0 ⎥ ⎟ ⎦ ⎠

La κ ⎤ A xz ⎥

the shear stresses become

0 0 Sz* 0

My − MAy

w = wb +

1) The cross-sections with one axis of symmetry. For the z-axis of symmetry the normal stresses given by Eq. (73) become

∫A

⎞ E L ds⎟⎟ , σxu = qz ⋅ a κxz; t ⎠ G A

Sy*

The total displacements given by Eq. (91) become

6. Special cases

∫A

szk

dq E A* La = z⋅ ⋅ κxz. dx G At

u τxξ

Sy* Sz* = 0,

∫0

the shear stresses given by Eq. (88) become

⎡v ⎤ ⎡ ⎤ ⎛ ⎡Q By ⎤ ⎡ 0 ⎤ ⎞ D3 ⎢ wb ⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ ⎥ = ⎢ ⎥ ⎟. ⎣ b ⎦x=x ⎣ 0 ⎦ ⎝ ⎣Q Bz ⎦ ⎣ 0 ⎦ ⎟⎠

Ay* Sz* = 0,

q E E⎛ w z − ⎜⎜σxk + z GAzz G⎝ Iy

z + qz

(92)

⎡ vb ⎤ ⎡ ⎤ ⎛ ⎡ MBz ⎤ ⎡ 0 ⎤ ⎞ = ⎢0⎥ ⎜ ⎢ ⎥ = ⎢ ⎥ ⎟, ⎢ wb ⎥ ⎣ ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ MBy ⎦ ⎣ 0 ⎦ ⎠

u⎤ ⎡ τxξ ⎡ A* ⎢ v⎥ ⎢ ⎢ τxξ ⎥ 1 ⎢ 0 = ⎢τ w ⎥ t ⎢0 ⎢ xξ ⎥ ⎢ α ⎢⎣ 0 ⎣⎢ τxξ ⎦⎥

⎡ ⎤ κxz L a ⎤ Qy ⎢ ⎥. A ⎥ ⎦ ⎣Qz ⎦

The beam is subjected to bending with influence of shear in two planes of symmetry and, in addition to, torsion and tension/compression due to shear. The case qy = 0 . The normal stresses given by Eq. (87) become

Beam free sections:

D2

0⎤ ⎥ 1⎡ 1 ⎥ ⎡− Mz + MAz ⎤ ⎥ , u = ua = − ⎢ 0 Azz ⎥ ⎢ M − M A y y ⎦ G⎣ ⎥⎣ 0⎥ ⎦

while

(90)

According to (27), Sy* = Sy* (sz ) and Ay* (sz ) are the odd coordinates, 10

0 qy Iz qz Iy

∫0 ∫0 0

⎤ ⎥ ds ⎥ t ⎥, Sy* ⎥ ds t ⎥ ⎥⎦

⎞ 0⎤ ⎟ ⎥ 0 ⎥ ⎡q ⎤ ⎟ y ⎥ q ⎥⎟ 1 ⎢ ⎥⎣ z⎦⎟ Azz ⎟⎟ ⎥ 0⎦ ⎠

syj Sz* szk

(98)

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

the shear stresses become u⎤ ⎡ τxξ ⎡ A* ⎢ v⎥ ⎢ ⎢ τxξ ⎥ 1 ⎢ 0 = ⎢τ w ⎥ t ⎢0 ⎢ xξ ⎥ ⎢ α ⎢⎣ 0 ⎣⎢ τxξ ⎦⎥

⎛⎡ ⎤ 0 ⎤ ⎜ ⎢0 ⎥ ⎥ ⎜ − Qy 0 ⎥ ⎢ Iz ⎥ ⎜⎢ ⎥+ Sy* 0 ⎥ ⎜ ⎢ Qz ⎥ ⎥ ⎜ Iy ⎥ 0 Sω*⎦⎥ ⎜ ⎢⎣ ⎝ 0 ⎦

0 0 Sz* 0 0 0

κzz ) and by secondary shear factors: (i) with respect to the transverse displacements (factors κyz = κzy ) and (ii) angular displacements of the cross-section as plane sections in their planes (factors κωy and κωz ) and (iii) longitudinal displacements of the cross-sections as plane sections (factors κxy and κxz ). Thus, in a general case of non-symmetric cross-sections, with transverse loads reduced to the shear centre, the beam will be subjected to bending with influence of shear and in addition to torsion and tension/compression due to shear. In the case of mono-symmetrical cross-sections with transverse loads in the plane of symmetry, the beam will be subjected to bending with influence of shear and in addition to tension/compression due to shear (factors κωy , κωz vanish); in the case of loads through the shear centre, orthogonal to the plane of symmetry, the beam will be subjected to bending with influence of shear and to torsion due to shear (factors κxy , κxz vanish). Finally, in the case of bi-symmetrical cross-section, the beam will be subjected to bending with influence of shear only (factors κωy ,κωz , κxy , κxz vanish). To illustrate this theory, a number of example are studied in Part II (Application), compared to the finite element method, as well as to solutions available in literatures. The novelty of the presented theories is in an analytic approach to the bending of thin-walled beams on the basis of classical theories of plane sections, by including shear deformations of the beam middle surface. The intention is to derive the analytic expressions in the closed form, suitable for engineering practice, especially in the early design stage of thin-walled structures when many parameters need to be optimized.

⎞ ⎡0 0 ⎤ ⎟ ⎢ 1 ⎥ 0 ⎢ ⎥ ⎡ qy ⎤ ⎟ E Ayy D ⎢ ⎥ ⎢q ⎥ ⎟ + G 1 ⎢0 A ⎥ ⎣ z ⎦ ⎟ zz ⎟⎟ ⎢ ⎥ ⎣0 0 ⎦ ⎠

⎤ ⎡0 ⎥ ⎢ ⎞ ⎛ v qy * S ⎢ ∫ ⎜ σ + ∫ z ds ⎟ ds ⎥ xj * I t * s s z ⎠ ⎥ yj E ∂ ⎢ yj ⎝ + G ⋅ ∂x ⎢ ⎥, ⎞ ⎥ ⎛ * S q y ⎢ z v + d d σ s s ⎟ ⎥ ∫ Iy s* t ⎢ ∫szk* ⎜⎝ xk ⎠ zk ⎥ ⎢ ⎦ ⎣0

(99)

where additionally due to the y-axis of symmetry

κxz = 0; κωy = 0: WPy = ∞;

(100)

where

∫A Az* Sz* = 0, ∫A Sz*⋅Sω* = 0.

(101)

Thus M

E

σxv = − I z y + qy GA y − z

σxw =

My Iy

yy

E

z + qz GA z − zz

E⎛ v ⎜σ G ⎝ xj

E⎛ w ⎜σ G ⎝ xk

+

+

qz Iy

qy Iz

∫0

∫0

⎞ ds ⎟ , ⎠ Sy* ⎞ ds⎟ , t ⎠

syj Sz*

szk

t

(102)

and

References

⎞ ⎛ qy S* E E ∂ v τxξ = − I t + dx ⋅ GA ⋅ t + G ⋅ ∂x ∫ * ⎜σxjv + I ∫ tz ds ⎟ ds, * syj ⎝ z syj z yy ⎠ ⎞ ⎛ * * * Q S S S d q q z y E E ∂ y y w v τxξ = I t + dxz ⋅ GA ⋅ t + G ⋅ ∂x ∫ * ⎜σxk + I z ∫ t ds ⎟ ds. * szk ⎝ y zz y szk ⎠ Qy Sz*

dqy

Sz*

[1] U. Bhat, J.G. de Oliveira, A formulation for the shear coefficient of thin-walled prismatic beams, J. Ship Res. 29 (1985) 51–58. [2] E. Carrrera, G. Giunta, M. Petrolo, Beam Structures, Classical and Advanced Theories, John Wiley & Sons, Ltd, United Kingdom, 2011. [3] S. Chen, B. Diao, Q. Guo, S. Cheng, Y. Ye, Experiments and calculation of U- shaped thin-walled RC members under pure torsion, Eng. Struct. 106 (2016) 1–14. [4] F.A. Charney, H. Iyer, P.W. Spears, Computation of major axis shear deformations in wide flange steel girder and columns, J. Constr. Steel Res. 61 (2005) 1525–1558. [5] G.R. Cowper, The shear coefficient in Timoshenko's beam theory, J. Appl. Mech. 33 (1966) 335–340. [6] L.A. Duan, J. Zhao, S. Liu, B-splines based nonlinear GBT formulation for elastoplastic analysis of prismatic thin-walled members, Eng. Struct. 110 (01) (2016) 325–346. [7] P. Ladeveze, J. Simmonds, New concepts for linear beam theory with arbitrary geometry and loading, Eur. J. Mech. 17 (1998) 377–402. [8] W.E. Mason, L.R. Herrmann, Elastic shear analysis of general prismatic beams, J. Eng. Mech. 94 (1968) 965–983. [9] R. Pavazza, An approximate solution for thin rectangular orthotropic/isotropic strips under tension loads, Int. J. Solids Struct. 37 (2000) 4353–4375. [10] R. Pavazza, On the load distribution of thin-walled beams subjected to bending with respect to the cross-section distortion, Int. J. Mech. Sci. 44 (2002) 423–442. [11] R. Pavazza, B. Blagojević, On the cross-section distortion of thin-walled beams with multi-cell cross-sections subjected to bending, Int. J. Solids Struct. 42 (2005) 901–925. [12] R. Pavazza, S. Jović, A comparison of the analytical methods and the finite method in the analysis of short thin-walled beams subjected to bending by couples, Trans. FAMENA 30 (2006) 21–30. [13] R. Pavazza, S. Jović, A comparison of approximate analytical methods and the finite element method in the analysis of short clamped thin-walled beams subjected to bending by uniform loads, Trans. FAMENA 31 (2007) 37–54. [14] R. Pavazza, A. Matoković, B. Plazibat, Bending of thin-walled beams of symmetrical open cross-sections with influence of shear, Trans. FAMENA 37 (2013) 17–30. [15] R. Pavazza, B. Plazibat, Distortion of thin-walled beams of open section assembled of three plates, Eng. Struct. 57 (2013) 189–198. [16] W.D. Pilkey, Analysis and Design of Elastic Beams, Computational Methods, John Wiley & Sons, New York, 2002. [17] U. Schramm, L. Kitis, W. Kang, W.D. Pilkey, On the shear deformation coefficient in beam theory, Finite Elem. Anal. Des. 16 (1994) 141–162. [18] I. Senjanović, Y. Fan, The bending and shear coefficient of thin-walled girders, Thin Walled Struct. 10 (1990) 31–57. [19] N.G. Stephen, Timoshenko's shear coefficient from a beam subjected to gravity load, J. Appl. Mech. 47 (1980) 121–127. [20] S.P. Timoshenko, On the correction for shear of the differential equation for transverse vibrations of prismatic bars, Philos. Mag. 41 (1921) 744–746. [21] S.P. Timoshenko, G.H. MacCullough, Element and Strength of Materials, van Nostrand, New York, 1949.

(103)

The displacements given by Eq. (83) become

⎡ 1 ⎢ Ayy ⎡ v ⎤ ⎡ vb ⎤ 1 ⎢ w ⎥ = ⎢ wb ⎥ + ⎢ 0 ⎣ α ⎦ ⎢⎣ 0 ⎥⎦ G ⎢ ⎢⎣ 0

0⎤ ⎥⎡ − Mz + MAz ⎤ 1 ⎥⎢ ⎥, Azz ⎥ ⎣ My − MAy ⎦ ⎥⎦ 0

(104)

My − MAy Mz − MAz , w = wb + . GAzz GAyy

(105)

i.e.

v = vb −

The beam is subjected to bending in two plane of symmetry with respect to the cross-section centroid. 7. Conclusion An approximate novel theory of bending of thin-walled beams of open sections with influence of shear (TBTS) is derived in this paper. The accuracy of introduced assumptions is discussed by comparisons to the results of the theory of elasticity and to the finite element method (in Part II). The theory is based on the Vlasov's thin-walled beam theory of open sections as well as on the Timoshenko's bending beam theory. The transverse loads are reduced to the shear centre/bending/ torsion centre, defined by the Vlasov's theory, as pure geometrical property. By proposed, engineering approach more general load conditions as well cross-section shapes can be considered. Neglecting the Poisson's effect as well as the “shear warping effect” at the boundary conditions, various engineering problems of thin-walled structures can be considered. The results can be obtained in closed analytic forms. The shear factors are defined with respect to the transverse displacements of the cross-sections as plane sections (factors κyy and 11

Thin--Walled Structures xx (xxxx) xxxx–xxxx

R. Pavazza, A. Matoković

Translation, Israel, 1961. [25] L. Zhang, Z. Zhu, G. Shen, G. Cao, A finite element for spatial static analyses of curved thin-walled rectangular beams considering eight cross-sectional deformation modes, Arab. J. Sci. Eng. 40 (2) (2015) 3731–3743. [26] C.M. Wang, J.N. Reddy, K.H. Lee, Shear Deformable Beams and Plates, Elsevier, New York, 2000. [27] R. El Fatmi, Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory, Int. J. Solids Struct. 44 (2007) 5912–5929.

[22] R.F. Vieira, F.B. Virtuoso, E.B.R. Pereira, A higher order model for thin-walled structures with deformable cross-sections, Int. J. Solids Struct. 51 (2014) 575–598. [23] F. Vlak, R. Pavazza, M. Vukasović. An approximate analytic solution for the stresses and displacements of thin-walled orthotropic beams subjected to bending, in: Proceedings of the 16th European Conference on Composite Materials ECCM16Conference Proceedings-Seville, Spain: University of Seville, Spain, Paris, Federico (red.). Seville, 2014. [24] V.Z. Vlasov, Thin-Walled Elastic Beams, 2nd ed., Program for Scientific

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