Bending of thin-walled beams of open section with influence of shear—Part II: Application

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

Contents lists available at ScienceDirect

Thin–Walled Structures journal homepage: www.elsevier.com/locate/tws

Full length article

Bending of thin-walled beams of open section with influence of shear—Part II: Application ⁎

Radoslav Pavazzaa, Ado Matokovićb, , Marko Vukasovića 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 FEM Comparisons Examples

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is valid for general open thin-walled cross-sections. The loads are reduced to the crosssection principal pole, i.e. the cross-section shear center. In part I, the TBTS is established, in this part II, the theory is used to analyze various beam cross-section shapes, loads and boundary conditions. Beams with the extreme low ratios of beam length to beam cross-section contour dimensions are analyzed. The stress predictions of the TBTS, stresses as functions of the longitudinal coordinate as well as the cross-section curvilinear coordinate, are compared to those obtained by finite elements computations (FEM) as well as to exact solutions of the theory of elasticity. The results of the TBTS are given in closed analytic forms, i.e. parametric forms, suitable for general studies of thin-walled beam behavior under transvers bending loads, as well in the early design stage of thin-walled structures.

1. Introduction The theory of bending of thin-walled beams with influence of shear (TBTS) established in part I [10] is based on the classical theories of thin-walled beams of open cross-sections [15] improved by the influence of shear, i.e. the cross-section warping due to shear [4,9,10,12,13,14,16]. The cross-section contour distortion is ignored [1,3,11], as well as Poisson's effect [2]. To illustrate the TBTS, this part II is devoted to analytical and numerical analyzes of simple supported and clamped beams, subjected to uniform transverse bending loads acting through the cross-section principal pole. The stresses and displacements are analyzed for nonsymmetric thin-walled cross-sections, mono-symmetrical and bi-symmetrical cross-sections. In the case of non-symmetric cross-sections, the beam will be subjected to bending with influence of shear and in addition to tension (compression) and torsion due to shear; in the case of mono-symmetric cross-sections with the transverse loads in the plane of symmetry, the beam will be subjected to bending with influence of shear in that plane and in addition to tension (compression); in the case of loads through the principal pole, orthogonal to the plane of symmetry, the beam will

⁎

be subjected to bending with influence of shear in that plane an in addition to torsion due to shear. In the case of bi-symmetric crosssection with load through the principal pole, the beam will be subjected to bending with influence of shear only. The obtained results will be compared to the results of the finite element calculations as well as to results of some simple examples of the theory of elasticity. Some results of from available literatures will be discussed also. The symbols used in this paper are those specified in the part I. 2. The theory 2.1. General case: bending with influence of shear with additional tension and torsion due to shear In the case of a non-symmetric open thin-walled cross-section, with the transverse loads reduced to the components qy and qz through the cross-section principal pole P, the normal stresses can be expressed as function of primary internal force components My = My (x ), Mz = Mz (x ) and secondary internal force components N y = N y (x ), N z = N z (x ), Myy , Myz (x ), Mzy (x ), Mzz (x ), By (x ) and Bz (x ):

Corresponding author at: Department of Professional Studies, University of Split, Kopilica 5, 21000 Split, Croatia. E-mail addresses: [email protected] (R. Pavazza), [email protected] (A. Matoković), [email protected] (M. Vukasović).

http://dx.doi.org/10.1016/j.tws.2016.08.026 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.2016.08.026

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

R. Pavazza et al.

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

⎛ ⎡ Ny Nz ⎤⎞ ⎜⎡ 0 ⎤ ⎢ A + A ⎥⎟ ⎡ 1 0 0 0 ⎤ ⎜ ⎢ Mz ⎥ ⎢ Mzy Mzz ⎥ ⎟ ⎢ 0 y 0 0 ⎥ ⎜ ⎢− Iz ⎥ ⎢− Iz − Iz ⎥ ⎟ ⎥⎜⎢ M ⎥ + ⎢ y =⎢ z ⎥⎟ ⎢ 0 0 z 0 ⎥ ⎜ ⎢ y ⎥ ⎢ My + My ⎥ ⎟ I I I y y ⎥⎟ ⎣0 0 0 ω⎦ ⎜ ⎢ ⎥ ⎢ y ⎜⎜ ⎣ 0 ⎦ ⎢ By Bz ⎥ ⎟ ⎟ + ⎢ ⎣ Iω ⎦⎠ Iω ⎥ ⎝ ⎡ ⎤ 0 ⎢ v ⎥ qy syj Sz* σ + I ∫ ds ⎥ t E ⎢⎢ xj z 0 ⎥ , σx = σxu + σxv + σxw + σxα. − G ⎢ σ w + qz ∫ szk Sy* ds ⎥ Iy 0 t ⎢ xk ⎥ ⎢⎣ ⎥⎦ 0

⎞ ⎟ ⎟ ⎥ ⎡ qy ⎤ ⎟ ⎥ ⎢⎣ qz ⎥⎦ ⎟ ⎟ ⎥ ⎟⎟ ⎥ ⎦ ⎠

⎛ ⎡ La κ ⎜⎡ 0 ⎤ ⎢ A xy ⎤ 0 0 0 ⎜ ⎢ Mz ⎥ ⎢ κyy − I ⎥ ⎥ ⎢ 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 ⎥ Iy 0 t ⎢ xk ⎥ ⎢⎣ ⎥⎦ 0

La κ ⎤ A xz ⎥ κyz ⎥

⎛ ⎡ La κ ⎜⎡ 0 ⎤ ⎢ A xy ⎢ 1 0 0 0 ⎤ ⎜ ⎢ Mz ⎥ − E ⎢ Ayy y 0 0 ⎥ ⎜ ⎢ Iz ⎥ ⎥⎜⎢ ⎥+ ⎢ 0 z 0 ⎥ ⎜ ⎢ My ⎥ G ⎢ 1 Azy Iy ⎦ ⎢ ⎥ ⎜ 0 0 ω ⎢ 1 ⎜⎣ 0 ⎦ ⎢ ⎣ WPy ⎝ ⎡ ⎤ 0 ⎢ v ⎥ qy syj Sz* σ + I ∫ ds ⎥ t E ⎢⎢ xj z 0 ⎥ − G ⎢ σ w + qz ∫ szk Sy* ds ⎥ Iy 0 t ⎢ xk ⎥ ⎢⎣ ⎥⎦ 0

⎞ La κ ⎤ ⎟ A xz ⎥ ⎟ 1 ⎥ Ayz ⎥ ⎡ qy ⎤ ⎟

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

A κzz A κωz WP

(1) or

where

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

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

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

(2)

where

κxy =

1 Iz La

∫A

A κyy = 2 Iz A κzz = 2 Iy =

WP Iy Iω

∫L

Ay* Sz* t2

dA ,

κxz =

1 Iy La

Az* Sy*

∫A

t2

⎛ Sz* ⎞2 A ⎜ ⎟ dA, κyz = κzy = A⎝ t ⎠ Iy Iz

∫

⎛ Sy* ⎞2 ⎜⎜ ⎟⎟ dA, A⎝ t ⎠ Sy* Sω* ds t

∫

κωy

W = P Iz Iω

∫L

∫L

dA , Sy* Sz* t

A* =

∫s*

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

Sz* Sω* ds, κωz t

(3)

∫s*

y dA* , Sy* =

∫s*

z dA* , Sω* =

z

σxjv = σxjv +1 (εxjv = εxv, j +1), σxjv + qy =

sy, j +1

∫0

szk

Sy*

E + qz GIy

∫0

t

∫s* ω dA* ,

E GIz

Sz* ds t

∫0

syj

Sz* ds t ;

w σxk

w w = σxw, k +1 (εxk = εxw, k +1), σxk

sy =−L y, j +1

= σxw, k +1 + qz

ds

syj = L yj

szk = L zk

E GIy

∫0

sz, k +1

Sy* t

ds

≤ szk ≤

Lzk+ ,

0 ≤ syj* ≤

L yj+,

*≤ 0 ≤ szk

Lzk+ ,

0 ≥ sy, j +1 ≥

−L y−, j +1,

0 ≥ sz, k +1 ≥

−Lz−, k +1,

0 ≥ sy*, j +1 ≥

−L y−, j +1,

0 ≥ sz*, k +1 ≥

−Lz−, k +1,

⎛ ⎡ La κ ⎡ ⎤ ⎢ A xy 0 0 0 ⎤⎜⎢ 0 ⎥ Qy ⎥⎜ ⎢ κyy Sz* 0 0 ⎥ ⎜ ⎢− Iz ⎥ E⎢ A ⎢ ⎥ + 0 Sy* 0 ⎥ ⎜ ⎢ Qz ⎥ G ⎢ κzy ⎥ ⎜ Iy ⎢ A ⎥ 0 0 Sω*⎥⎦ ⎜⎜ ⎢⎣ ⎢ κωy ⎦ 0 ⎣ WP ⎝ ⎤ ⎡ 0 ⎢ ⎞ ⎥ ⎛ ⎢ ∫ ⎜σ v + qy ∫ Sz* ds⎟ ds ⎥ xj * t I * s s z ⎢ yj ⎠ ⎥ ⎝ yj ∂ ⋅ ⎢ ⎥, ⎛ ∂x ⎢ Sy* ⎞ qz ⎥ v + d d σ s s ⎟ ⎜ ∫ ∫ Iy s* t ⎢ szk* ⎝ xk ⎠ ⎥ zk ⎥ ⎢ ⎦ ⎣ 0

La κ ⎤ A xz ⎥ κyz ⎥

⎛ ⎡ La κ ⎤ ⎜⎡ ⎢ A xy 0 0 0 ⎤⎜⎢ 0 ⎥ ⎢ 1 Qy ⎥ Sz* 0 0 ⎥ ⎜ ⎢− Iz ⎥ E ⎢ Ayy ⎢ ⎥ + ⎜ ⎢ 0 Sy* 0 ⎥ ⎜ ⎢ Qz ⎥ G ⎢ 1 Azy ⎥ Iy ⎢ ⎥ ⎜ ⎢ 1 0 0 Sω*⎥⎦ ⎣ ⎜ 0 ⎦ ⎢ ⎣ WPy ⎝ ⎤ ⎡ 0 ⎢ ⎞ ⎥ ⎛ ⎢ ∫ ⎜σ v + qy ∫ Sz* ds⎟ ds ⎥ xj * Iz syj * t ⎠ ⎥ ∂ ⎢ syj ⎝ ⋅ ⎢ ⎥, ⎛ ∂x ⎢ Sy* ⎞ qz ⎥ v + d d σ s s ⎟ ⎜ ∫ ∫ Iy s* t ⎢ szk* ⎝ xk ⎠ ⎥ zk ⎥ ⎢ ⎦ ⎣ 0

La κ ⎤ A xz ⎥ 1 ⎥ Ayz ⎥

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

where

0 ≤ syj ≤

(7)

A κzz A κωz WP

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

or

, sz =−L z, k +1

(4)

L yj+,

⎤ ⎥ A ⎥ . κzz ⎥ ⎥ WP ⎥ κωz ⎦ A κyz

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

are the properties of the cut-off portions of cross-section area. w The stress components σxjv and σxk can be obtained from the compatibility conditions

E + qy GIz

(6)

The shear stresses can be expressed as

y

σxv, j +1

1 WPz

⎥⎢q ⎥⎟ ⎥⎣ z⎦⎟ ⎟ ⎥ ⎟ ⎥ ⎦ ⎠

where

ds,

are the shear factors;

dA* , Sz* =

1 Azz

0

(5)

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, both L 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 normal stresses can also be expressed as:

where 2

1 Azz 1 WPz

⎞ ⎟ ⎟ ⎡ qy ⎤ ⎟ E D ⎥ ⎢q ⎥⎟ + ⎥ ⎣ z⎦⎟ G ⎟ ⎥ ⎟ ⎥ ⎦ ⎠

(8)

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

R. Pavazza et al. u v w α τxξ = τxξ + τxξ + τxξ + τxξ .

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

(9)

The average shear stresses can be expressed (by assuming qy = const , qz = const ) as

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

1 ⎤ Ayz ⎥ ⎡Qy ⎤

⎢

⎥

1 ⎥ ⎣Q ⎦ z Azz ⎥ ⎦

⎡0⎤ ⎡D 0 ⎢ Mz ⎥ 0 0 ⎤ ⎡ uax ⎤ ⎢ ⎥ 2 1 ⎢ Iz ⎥ 0 0 ⎥ ⎢ vb ⎥ ⎢0 D ⎢ ⎥, = 0 ⎥ ⎢⎢ wb ⎥⎥ E ⎢ My ⎥ ⎢ 0 0 − D2 α ⎣ ⎦ ⎢⎣ 0 0 ⎢ Iy ⎥ 0 − D2 ⎥⎦ t ⎣0⎦

⎡ 1 ⎢ Ayy v ⎡ ⎤ b ⎡v⎤ 1⎢ 1 ⎢ ⎥ w w ⎢ ⎥ = b + ⎢ Azy ⎣ α ⎦ ⎢⎣ 0 ⎥⎦ G ⎢ ⎢ 1 ⎣ WPy

⎡ D2 0 0 0 ⎤ ⎡ uax ⎤ ⎢ ⎥⎢ v ⎥ 3 0 D 0 0 ⎢ ⎥⎢ b ⎥ 3 ⎢0 0 −D 0 ⎥ ⎢ wb ⎥ ⎣α ⎦ ⎢⎣ 0 0 0 − D3 ⎥⎦ t

⎡ 0 ⎤ ⎢ Qy ⎥ 1 ⎢− Iz ⎥ = ⎢ Q ⎥, E⎢ z ⎥ ⎢ Iy ⎥ ⎣ 0 ⎦

=−

b

b

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

(11)

v = v (x ) ,

w = w (x )

(17)

where “B” denote the beam right section. Clamped beam section:

⎡v⎤ ⎡v ⎤ ⎡ ⎤ ⎛ ⎡ v ⎤ ⎡ vb ⎤ ⎡ ⎤ ⎞ ⎡v ⎤ = ⎢ wb ⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ wA ⎥ = ⎢ Ab ⎥ = ⎢ 0 ⎥ ⎟⎟ , D ⎢ wb ⎥ ⎢w⎥ ⎣ ⎦x=x ⎣ b ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ A ⎦ ⎢⎣ wA ⎥⎦ ⎣ 0 ⎦ ⎠ ⎣ b ⎦x=x A A A ⎡ ⎤ ⎛⎡ γb ⎤ ⎡ ⎤⎞ ⎡ v ⎤ ⎡ v ⎤ = ⎢ 0 ⎥ ⎜⎜ ⎢ Ab ⎥ = ⎢ 0 ⎥ ⎟⎟ , ⎢ wB ⎥ = ⎢ wb ⎥ ⎣ 0 ⎦ ⎢⎣ β ⎥⎦ ⎣ 0 ⎦ ⎣ B⎦ ⎣ b ⎦ x = x B ⎝ A ⎠ ⎡ 1 1 ⎤ E ⎢ Ayy Ayz ⎥ 2 ⎡ (−vb x = x B + vb x = xA ) Iz ⎤ ⎥ + ⎢ ⎥D ⎢ GA ⎢ 1 1 ⎥ ⎣ (−wb x = x B + wb x = xA ) Iy ⎦ ⎣ Azy Azz ⎦

and

⎡ ⎤ = ⎢0⎥ ⎣0⎦

⎛ ⎡ 1 ⎜ ⎡ v B ⎤ ⎡ vb ⎤ 1 ⎢ Ayy = + ⎜ ⎢ w B⎥ ⎢ wb ⎥ ⎢ GA ⎢ 1 ⎜ ⎣ ⎦ ⎣ ⎦x=xB ⎣ Azy ⎝

1 ⎤ Ayz ⎥ ⎡−

(13)

⎢

1 ⎥⎣ Azz ⎥ ⎦

⎞ ⎡ ⎤⎟ ⎡ b ⎤ ⎡ ⎤ ⎛⎡ γb ⎤ ⎡ ⎤⎞ = ⎢ 0 ⎥ ⎜⎜ ⎢ Bb ⎥ = ⎢ 0 ⎥ ⎟⎟ ; = ⎢ 0 ⎥ ⎟, D ⎢ v b ⎥ ⎣ 0 ⎦ ⎟ ⎣ w ⎦x=x ⎣ 0 ⎦ ⎢⎣ β ⎥⎦ ⎣ 0 ⎦ B ⎝ B ⎠ ⎠

The additional displacement due to shear can be expressed as

1 ⎡ La ⎢ κxy G⎣A

⎡v⎤ ⎡v ⎤ = ⎢ wb ⎥ ⎢w⎥ ⎣ ⎦x=x ⎣ b ⎦x=x

A

u ⎡ u ⎤ ⎡ uax ⎤ ⎡⎢ a ⎤⎥ ⎢ v ⎥ = ⎢ vb ⎥ + ⎢ va ⎥ . ⎢ w ⎥ ⎢ wb ⎥ ⎢ wPa ⎥ ⎣ α ⎦ ⎢⎣ αt ⎥⎦ ⎣ α ⎦ a

ua = −

(16)

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

(12)

⎡ 1 ⎢ Ayy ⎢ 1 ⎤ ⎡Qy ⎤ ⎡− va ⎤ 1 La κ ⎢ ⎥ , ⎢ wa ⎥ = ⎢ Azy A xz ⎥ ⎦ ⎣Qz ⎦ ⎢⎣ αa ⎥⎦ G ⎢ ⎢ 1 ⎣ WPy

(uax = 0).

⎡ ⎤ ⎛ ⎡ MAz ⎤ ⎡ 0 ⎤ ⎞ = ⎢0⎥ ⎜ ⎢ ⎥ = ⎢ ⎥ ⎟; ⎣ 0 ⎦ ⎝ ⎣ MAy ⎦ ⎣ 0 ⎦ ⎠

La La ⎡ La κ ⎤ ⎡ La κ ⎤ ⎢ A xy A κxz ⎥ ⎢ A xy A κxz ⎥ ⎤ 0 ⎡ ua ⎤ ⎢− 1 − 1 ⎥ ⎢− 1 − 1 ⎥ ⎥ Ayz ⎥ ⎡ qy ⎤ Ayz ⎥ ⎢ Ayy 1 ⎢ Ayy 0 ⎥ ⎢ va ⎥ = , = κ ⎢ ⎥ ⎢ 1 ⎢ ⎥ 1 ⎥, 1 ⎣ qz ⎦ 0 ⎥ ⎢⎢ wa ⎥⎥ G ⎢ 1 ⎥ ⎢ Azy Azz Azy Azz ⎥ α ⎢ 1 ⎥ ⎢ 1 ⎥ − D2 ⎥⎦ ⎣ a ⎦ 1 1 ⎢ ⎥ ⎢ ⎥ WPz ⎦ WPz ⎦ ⎣ WPy ⎣ WPy

u = u (x ) ,

⎡ ⎤ κxz L a ⎤ Qy ⎥⎢ ⎥ A ⎦ ⎣Qz ⎦

⎡ vb ⎤ ⎡v⎤ ⎡v ⎤ ⎡ ⎤ ⎛ ⎡ v ⎤ ⎡ vb ⎤ ⎡ ⎤ ⎞ = ⎢ wb ⎥ = ⎢ 0 ⎥ ⎜⎜ ⎢ wA ⎥ = ⎢ Ab ⎥ = ⎢ 0 ⎥ ⎟⎟ , D2 ⎢ Ab ⎥ ⎢w⎥ ⎣ ⎦x=x ⎣ b ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ A ⎦ ⎣⎢ wA ⎥⎦ ⎣ 0 ⎦ ⎠ ⎣⎢ wA ⎥⎦ x = x A A A

Relations between the load components and the additional displacements due to shear ua = ua (x ), va = va (x ), wa = wa (x ) and αa = αa (x )are

displacements

1 ⎡ κxy La ⎢ G⎣ A

⎤ ⎥ 1 ⎥ ⎡− Mz + MAz ⎤ ⎥ , u = ua Azz ⎥ ⎢ M − M Ay ⎦ ⎥⎣ y 1 ⎥ WPz ⎦ 1 Ayz

For the hinged section it may be written

⎡ D3 0 0 0 ⎤ ⎡ uax ⎤ ⎢ ⎥⎢ v ⎥ 4 0 D 0 0 ⎢ ⎥⎢ b ⎥ 4 ⎢0 0 −D 0 ⎥ ⎢ wb ⎥ ⎣α ⎦ ⎢⎣ 0 0 0 − D 4 ⎥⎦ t

⎡ 0 ⎤ ⎢ qy ⎥ 1 ⎢ Iz ⎥ = ⎢ qz ⎥ . E ⎢− ⎥ I ⎢ y⎥ ⎣ 0 ⎦

The total α = α (x )are

(15)

The total displacements can then be expressed as (10)

Relations between the internal force components, the load components and the displacements of the plane sections uax = uax (x ), vb = vb (x ), wb = wb (x ) and αt = αt (x ) are

⎡D 0 0 ⎢ 2 0 ⎢0 D ⎢ 0 0 − D2 ⎢⎣ 0 0 0

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

MBz + MAz ⎤ ⎥ MBy − MAy ⎦

(18)

Free beam section:

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

⎡v ⎤ ⎡ ⎤ ⎛ ⎡ Mz B ⎤ ⎡ 0 ⎤ ⎞ 3 ⎡ vb ⎤ ⎡ ⎤ ⎛ ⎡Q By ⎤ ⎡ 0 ⎤ ⎞ D2 ⎢ wb ⎥ = ⎢0⎥ ⎜ ⎢ = ⎢ 0 ⎥ ⎜⎜ ⎢ ⎥ = ⎢ ⎥ ⎟, ⎥ = ⎢ ⎥ ⎟, D ⎢ w ⎥ ⎣ b ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ MyB⎦ ⎣ 0 ⎦ ⎠ ⎣ b ⎦x=x ⎣ 0 ⎦ ⎝ ⎣ Q Bz ⎦ ⎣ 0 ⎦ ⎟⎠ B

B

(19) (14)

where Cv , Cw and Cα are integration constants. Boundary conditions are: for the beam left end, section A:

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

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

2.2. Special case: bending with influence of shear with additional tension (compression) due to shear

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

In the case of cross-sections with one axis of symmetry, with the transverse loads reduced to the component qz in the beam plane of symmetry, the normal stresses, given by Eq. (1), can be expressed by the primary internal force component My = My (x ) and the secondary internal force components N z = N z (x ) and Myz (x ):

Hence 3

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

R. Pavazza et al.

σxw =

My Iy

Myz

z+

Iy

q E⎛ w ⎜⎜σxk + z G⎝ Iy

z−

∫0

⎞ Nz ds⎟⎟ , σxu = , t ⎠ A

w u τxξ = τxξ + τxξ .

Sy*

szk

(20)

The displacements, given by (16), become

or by the primary internal force component My = My (x )and the shear area Azz and the shear factor κxz :

q E E⎛ w z + qz z − ⎜⎜σxk + z σxw = Iy GAzz G⎝ Iy My

σxw

∫0

⎞ EL ds⎟⎟ , σxu = qz a κxz, t ⎠ GA

σx = σxw + σxu.

(21)

2.4. Special case: bending with influence of shear In the case of a cross-sections with two axis of symmetry, with the transverse loads qy and qz through the principal pole, the normal stresses, given by Eq. (1), can be expressed by the primary internal force components My = My (x ) and Mz = Mz (x ) and the secondary internal force components Myz (x )and Mzy (x ):

(22)

The shear stresses, given by (8), are

Iy t ds,

u τxξ

⎛

E Sy* E ∂ ⋅ ⋅ + ⋅ dx GAzz t G ∂x

dqz

+

Sy*

q

∫s* ⎜⎜⎝σxkv + Iyz ∫s* zk

t

zk

⎞ ds⎟⎟ ⎠

dq E A* La = z⋅ ⋅ κxz, dx G At

σxw =

(23)

w u τxξ = τxξ + τxξ .

, u = ua = −

Iy

q E⎛ w ⎜⎜σxk + z G⎝ Iy

z−

∫0

syj

∫0

szk

⎞ My M ds⎟⎟ , σxv = − z y + z y Iz Iz t ⎠

Sy*

Sz* ⎞ ds⎟ , t ⎠

(32)

szk S * ⎞ q E E⎛ w y + z ds⎟⎟ , σxv z − ⎜⎜σxk GAzz G⎝ Iy 0 t ⎠ syj S * ⎞ qy M E E⎛ z = − z y + qy y − ⎜σxjv + ds⎟ , Iz GAyy G⎝ Iz 0 t ⎠

σxw =

The displacements, given by Eq. (16), are

GAzz

Iy

Myz

z+

or by the primary internal force components My = My (x ) and Mz = Mz (x ) and the shear factors κzz and κyy :

(24)

My − MAy

My

qy E⎛ − ⎜σxjv + G⎝ Iz

w w = τxξ (x, sz )are the shear stresses due to bending with influence where τxξ u u = τxξ (x, s )are the shear of shear in the plane of symmetry and τxξ stresses due to tension due to shear:

w = wb +

(31)

Sy*

szk

σxw (x, sz )

Qz Sy*

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

v = vb +

are the normal stresses due to bending with where = influence of shear in the plane of symmetry and σxu = σxu (x ) are the normal stresses due to tension (combustion) due to shear;

w τxξ =

(30)

My Iy

∫

z + qz

∫

Qz La κxz, GA

(25)

(33)

where

σx = σxw + σxv. 2.3. Special case: bending with influence of shear with the additional torsion due to shear

The shear stresses are given by Eqs. (23) and (29): v =− τxξ

In the case of a cross-sections with one axis of symmetry, with the transverse loads reduced to the component qy orthogonal to the beam plane of symmetry, the normal stresses, given by Eq. (1), can be expressed by the primary internal force component Mz = Mz (x ) and the secondary internal force component and Mzy (x ) and the shear factor κωy :

σxv = −

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

∫0

syj

Sz* ⎞ α E ds⎟ , σx = qy κωy ω, t ⎠ GWP

∫0

syj

Qy Sz* Iz t

+

w ds, τxξ =

E Sz* E ∂ ⋅ + ⋅ dx GAyy t G ∂x

dqy

⋅

Qz Sy* Iy t

+

dqz

⎛

∫s * ⎜⎜σxjv + yj

⎝

Sy*

E E ∂ ⋅ + ⋅ dx GAzz t G ∂x ⋅

∫s*

zk

qy

∫ Iz s *

yj

Sz* ⎞ ds⎟⎟ t ⎠

⎛ q v ⎜⎜σxk + z Iy ⎝

∫s*

zk

⎞ ds⎟⎟ ds. t ⎠

Sy*

(35) where w v τxξ = τxξ + τxξ .

(26)

(36)

The displacements, given by Eqs. (25) and (31), become

or by the primary internal force component Mz = Mz (x ) and the shear area Ayy and the shear polar modulus WPy

qy M E E⎛ y − ⎜σxjv + σxv = − z y + qy Iz GAyy G⎝ Iz

(34)

w = wb +

Sz* ⎞ α E ds⎟ , σx = qy ω, t ⎠ GWPy

My − MAy GAzz

, v = vb +

−Mz + MAz , GAyy

(37)

(27) 3. The Finite Element Method (FEM)

where σxv = σxv (x, sy ) are the normal stresses due to bending with influence of shear in the principal plane orthogonal to the plane of symmetry and σxα = σxα (x, s ) are the normal stresses due to torsion due to shear;

σx = σxv + σxα.

A series of examples have been analysed by applying the FEM using Autodesk Algor Simulation Pro in order to compare the results with those obtained analytically with the presented theory (TBTS). In all examples shown below 4-nodded membrane elements with 2 DOF are used (Fig. 1a). The mid-surface geometry of 3D FEM model was meshed uniformly by 5 × 5 mm square elements (Fig. 1b). The beam’s uniformly distributed load per unit length is modelled by applying forces acting in nodes through beam length. The loads acting in directions of both y and z principal axes were analysed separately (Fig. 2a) and elastic constants used in all examples are E = 210 GPa and G = 80, 769 GPa . The loads are reduced to the cross-section principal pole, i.e. the cross-section shear centre. Due to symmetry, only one half of the beam is modelled. Fig. 2b shows the boundary conditions that are used: at the simply supported

(28)

The shear stresses, given by (8), become v =− τxξ

Qy Sz* Iz t

+

E Sz* E ∂ ⋅ + ⋅ dx GAyy t G ∂x

dqy

⋅

dq E S* α ds, τxξ = z⋅ ⋅ ω , dx G WPy v τxξ

⎛

∫s * ⎜⎜σxjv + yj

⎝

qy

∫ Iz s *

yj

Sz* ⎞ ds⎟⎟ t ⎠ (29)

v τxξ (x ,

= sy ) are the shear stresses due to bending with where influence of shear in the principal plane orthogonal to the plane of ω ω = τxξ (x, s ) are the shear stresses due to torsion due to symmetry and τxξ shear: 4

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

R. Pavazza et al.

Fig. 1. a) Membrane element with 2 DOF, b) mesh density.

end and at x = l /2 , at the clamped end and at x = l /2 . The check marks mean that certain displacements of the translation T, are constrained. Warping of the cross-section due to shear is not constrained at the clamped end of analytical model, while warping is constrained at the clamped end of numerical model. Consequently, certain discrepancies for normal stresses are obtained at the clamped end of the beam. These discrepancies are more pronounced for extremely short beams (l / h < 3). If the distributed load is reduced to act in y-direction through the principal pole of mono-symmetrical I-section (Example 1 – Appendix A) (Fig. 3b), the following condition must be satisfied

κyy =

=

6(1 + λ + ψ )(1 + λη 4 ) 6(1 − η2 )(1 + λη 4 ) , κωy = , κzz 2 2 5(1 + λη ) 5η2 (1 + λη2 )2 ⎡φ 2 2 φ 2 (1 + λ + ψ )3 ⎢ ψ (1 + λ2μ3) + 3 φ 2 (1 + λμ4 )+ 15 ψφ3 (1 + μ5) ⎣ +

1 2 ρ (1 12

+ λμ2 η2 )] 1

, κxz

2

1

[λ + 3 ψ (1 + λ + 4 ψ )] ⎡φ φ (1 + λ + ψ ) ⎢ ψ (1 − λ2μ2 ) + ⎣ +

FE⋅h2P = FB⋅h1P

=−

5 ψφ3 (1 24

1 2 ρ (1 12

5

− λμη2 ) + 6 φ 2 (1 − λμ3)

⎤ − μ4 ) ⎥ ⎦ 1

1

,

λ + 3 ψ (1 + λ + 4 ψ )

For the mono-symmetrical U-section; (Example 3 Appendix B) (Fig. 3d) the connection between forces acting in horizontal and vertical wall must be

(38) where

FD⋅hP = FB⋅b For the beam with non – symmetric U-section (Example 4) (Fig. 3e), to satisfy condition that resultant force acts through principal pole, connection between forces acting in both principal y and z axes must be

λ=

A2 h b A h b , μ = 2C , η = 2 , ψ = 0 , φ = 1C , ρ = 1 , La = h . A1 h1C b1 A1 h h

Normal stresses at the junctions of the web and the flanges, for qy = 0 , for the upper flange:

F1⋅zP′ = F2⋅yP′ σBx = σBwx + σBux = −Φ Bwu

My

h1C , Φ Bwu = 1 +

qz Iy

Iy My Azz Azz (3A1 + t0 h1C ) h1C κxz h ⎤ E⎡ ⎥; − ⋅ ⋅ ⎢1 − G⎣ 3Iy t0 κzz h1C ⎦

4. Examples Example 1. Mono-symmetrical I-sections (Appendix A). Shear factors, giving by Eq. (3), are:

for the lower flange:

Fig. 2. a) The uniformly distributed load in directions of principal axes, b) boundary conditions.

5

(39)

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

R. Pavazza et al.

Fig. 3. Bending loads that are reduced to the principal pole of the FEM models: a) to the plane of symmetry of the mono-symmetrical I-section; b) to the plane orthogonal to the plane of symmetry of the mono-symmetrical I-section; c) to the planes parallel to the plane of symmetry of the mono-symmetrical U-section; d) to the planes parallel and perpendicular to the plane of symmetry of the mono-symmetrical U-section; e) to the non-symmetrical U-section.

σEx = σEwx + σEux = Φ Ewu ⋅

My Iy

h2C, Φ Ewu = 1 +

A (3A2 + t0 h2C ) h2C E⎡ ⎢1 − zz G⎣ 3Iy t0

The displacements v of the flange b1, b1 > b 2 , for the beam midspan can be expressed as

qz Iy

My Azz κ h ⎤ ⎥. + xz ⋅ κzz h2C ⎦

⎛ Ayy h1P ηv − 1 ⎞ ⎟, ⋅ v1 = vb ηvα , ηvα = ηv ⎜1 + WPy ηv ⎠ ⎝

(40)

Normal stresses at the flange free edges, for b1 > b 2 , for qz = 0 :

σAx

qy Iz Mz b1 vα = + = ∓ ⋅ , ΦA = 1 − Iz 2 Mz Ayy Ayy (WPy b12 − 12Iz h1P ) ⎤ E⎡ ⎥, ⋅ ⎢1 − ⎥⎦ 12Iz WPy G ⎢⎣ σAv x

σAαx

The displacements u for the beam ends can be expressed as

ΦAvα

u (x = 0, l ) = ∓ wb ηwu , ηwu = (ηw − 1)

λ=

(42)

For simple supported and clamped ends, at the beam midspan, respectively, the bending moments are

5qy l 4

My =

384EIy

, vb =

qy l 4 384EIz

, ηw = 1 + 48k w, ηv = 1 + 48k v,

for the clamped beam, where

kw =

EIy GAzz l 2

, kv =

qz l 2 8

, My =

qz l 2 24

,

where l is the beam length. The membrane, 5 mm×5 mm, elements are used, where b1 = h = 400 mm, b 2 = 200 mm, l = 1200 mm/l = 2000 mm, t1 = t2 = t0= 10 mm. A half of beam length is modelled, where for x = l /2 : u = 0 (for web and flanges). For simply supported beams for x = 0 : v = 0 (for flanges), w = 0 (for web). For clamped beams for x = 0 : u = 0 (for web and flanges), v = 0 (for flanges), w = 0 (for web), The elastic constants are E = 210 GPa, G = 80.769 GPa. The correction factors Φwu , for the beam midspans, for the points B

for the simple supported beam;

qz l 4

3 1 2 1 , μ = , η = , ψ = 1, φ = , ρ = 1, 2 2 2 5

the shear factors are

48 48 , vb = ,η =1+ k w, ηv = 1 + k v, wb = 384EIy 384EIz w 5 5

wb =

(44)

κyy = 2.444, κzz = 2.757, κxz = −0.089, κωy = 2.933.

where for uniformly distributed loads

5qz l 4

4hκxz , lκzz

Comparison with the finite element method For

(41)

The factors Φwu and Φvα are correction factors with respect to the normal stresses according to the classical Euler-Bernoulli theory (EBBT). The total displacements w and v, according to Eq. (37), for the beam midspans, can be written as

w = wb ηw , v = vb ηv ,

(43)

EIz , GAyy l 2 6

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

R. Pavazza et al.

Table 1 Correction factors for normal stresses for mono-symmetrical I-sections, Φ Bwu / Φ Ewu (λ = 1/2 ,μ = 3/2 , η = 1/2 , ψ = 1, φ = 2/5, ρ = 1). Simple

1/3 1/5

Table 4 Correction factors for displacements α and u for mono-symmetrical I sections, ηvα and ηwu (λ = 1/2 , μ = 3/2 , η = 1/2 , ψ = 1, φ = 2/5, ρ = 1).

Clamped

Analytic

Analytic

FEM

Analytic

FEM

1.1463/1.0741 1.0527/1.0267

1.1453/1.0711 1.0525/1.0265

1.4389/1.2222 1.1580/1.0800

1. 4264/1.2010 1.1571/1.0786

Table 2 Correction factors for normal stresses for mono-symmetrical I sections: ΦAvα /Φ Dvα (λ = 1/2 , μ = 3/2 , η = 1/2 , ψ = 1, φ = 2/5, ρ = 1).

h /l

1/3 1/5

Simple

Simple

h /l

1/3 1/5

1.2773/ − 1.0998/ −

1/3 1/5

0.029 0.006

Analytic

FEM

Analytic

FEM

1.0385/1.0096 1.0139/1.0035

1.0386/1.0085 1.0139/1.0023

1.1156/1.0289 1.0416/1.0104

1.1155/1.0277 1.0415/1.0092

Iy

vb, max =

5qy

384EIz

ηvα 1.2813 2.3867/– 1.1004 1.4992/– ηwu ; x/l=0.25 0.022 0.143 0.006 0.031

FEM

2.3782 1.4950 0.066 0.026

q Iy My h ⋅ = −σEx , Φ Bw = 1 + z Iy 2 My Azz M A (6A1 + A0 ) h ⎤ E⎡ ⎥ , σAx = σAv x = −ΦAv z ⋅ ⎢1 − zz 12Iy t0 Iz G⎣ ⎦ qy Iz E ⎛ Ayy b 2 ⎞ b ⎟. = σAx , ΦAv = 1 − ⋅ ⎜1 − 2 12Iz ⎠ Mz Ayy G ⎝

(46)

Comparison with the finite element method The correction factors Φ , in comparison with the finite element method calculation, for the beam midspans, for uniformly distributed loads, are given in Tables 5 and 6; E / G = 2.6 . The correction factors ηw , ηv are given by Eq. (42): Table 7.

h1C .

Normalised displacements v B / vb, max along the beam length of the upper flange for the simply supported and clamped beam (qz = 0 ) and for the ratio l / h = 3 is presented in Fig. 5, where

l4

Analytic

σBx = σBwx = −Φ Bw

and E, are given in Table 1 and Φvα , for the points A and D in Table 2. The correction factors ηw , ηv and ηvα , ηwu , in comparison with the finite element method, are given in Tables 3 and 4. Normalised normal stresses σxwu / σxb, max along the upper flange, web and lower flange contour for clamped beam (qy = 0 ) and for the ratio l / h = 3 are presented in Fig. 4, where

My, max

FEM

Clamped

⋅

σxb, max =

Clamped

Example 3. Mono-symmetrical U-sections (Appendix B) The shear factors given by Eq. (3) are:

κyy =

.

= Example 2. Bi-symmetrical I sections (Appendix A Fig. A1). For I section with two axis of symmetry:

λ = 1, μ = 1, η = 1, ψ =

4[1 +

ψ 3

1 ψ

1 + ρ2 (1 + 2ψ ) , κωy 4(1 + 2ψ )(2 + ψ )

[18ψ + ρ2 (1 + 6ψ )2]⋅[10(5 + 6ψ ) − 2ρ2 ] , 20ρ2 (2 + 3ψ )(1 + 6ψ )2

(47)

where

⎞ 1 1 + 30 ψ + 6 ρ2 ⎟ ⎠

(2 +

, κzz

3[2(8 + 55ψ + 140ψ 2 + 160ψ 3 + 80ψ 4 + 16ψ 5) + 5ψρ2 (1 + 2ψ )3] 20ψ (1 + 2ψ )2 (2 + ψ )2

=−

Thus, the shear factors given by Eqs. (96) are

6⎛ 1 ⎞ κyy = ⎜1 + ψ ⎟ , κzz = 5⎝ 2 ⎠

5(1 + 6ψ )2

κxz =

A0 1 b , φ = , ρ = 1. A1 2 h

⎛1 (2 + ψ )3 ⎜ 3 + ⎝

⎞ ⎛ ψ 6(1 + 2ψ ) ⎜1 + 10ψ + 20 2 + 30ψ 2⎟ ρ ⎠ ⎝

ψ 2 )] 4

ψ=

, κxz = 0,

A0 b , ρ = , Ls = h . A1 h

(45) 1

For ψ = 2 , ρ = 2 :

For ψ = 1, ρ = 1:

κyy = 2.400, κzz = 3.264, κxz = 0.450, κωy = −1.173.

κyy = 1.80, κzz = 3.380. The normal stresses, given by Eqs. (39) and (41), are

The normal stresses at the free edges and at the junctions of the web and the flange can be expressed as

Table 3 Correction factors for displacements v and w for mono-symmetrical I sections, ηw /ηv (λ = 1/2 , μ = 3/2 , η = 1/2 , ψ = 1, φ = 2/5, ρ = 1). Simple

h /l

1/3 1/5

σAx = σAwx + σxu = −ΦAwu ⋅

Clamped

Analytic

FEM

Analytic

FEM

2.3253/1.2542 1.4771/1.0915

2.3535/1.2577 1.4806/1.0919

7.6267/ 2.2711 3.3856/1.4576

7.6392/2.2631 3.3709/1.4536

My Iy

h 0C, ΦAwu = 1 +

qz Iy My Azz

My A h2 κ h ⎤ E⎡ ⎢1 − zz 0C − xz ⎥ , σBx = σBwx + σxu = Φ Bwu h¸1C, Φ Bwu 3Iy Iy G⎣ κzz h 0C ⎦

=1+

qz Iy My Azz

⋅

2 2 A (3h 0C − h1C ) κ h ⎤ E⎡ ⎢1 − zz + xz ⎥ , G⎣ κzz h1C ⎦ 6Iy

(48)

The normal stresses at the left web, sy = b /2 , can be expressed as 7

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

R. Pavazza et al.

Fig. 4. Normal stresses σxwu /σxb, max at the beam midspan, for the clamped beam: a) along the upper flange, b) along the web, c) along the lower flange.

σAx = σAv x + σAαx = − ΦAvα ⋅

qy Iz Mz b1 vα ⋅ , ΦA = 1 − Iz 2 Mz Ayy

Table 5 Correction factors for normal stresses for bi-symmetrical I-sections, Φ Bw = Φ Ew (ψ = 1, ρ = 1).

Ayy [WPy (6A0 b + A1 b + 6t1 h2 ) + 12t1 (h − hP ) Iz ] ⎫ E⎧ ⎨1 − ⎬; σBx 12t1 Iz WPy G⎩ ⎭

= σBvx + σBαx = − Φ Bvα

Mz b1 ⋅ , Φ Bvα = 1 − Iz 2 Mz Ayy

Ayy [bWPy (6A0 + A1) − 12t1 hP Iz ] ⎫ E⎧ ⎬. ⋅ ⎨1 − 12t1 Iz WPy G⎩ ⎭

Simple

h /l

qy Iz

1/3 1/5

Clamped

Analytic

FEM

Analytic

FEM

1.1706 1.0614

1.1702 1.0613

1.5117 1.1842

1. 4982 1.1838

(49)

Comparison with the finite element method The correction factors Φwu and Φvα , in comparison with the finite element method, for the beam midspans, are given in Tables 8 and 9, respectively (E / G = 2.6 ) The correction factors ηw , ηv , ηvα and ηwu , in comparison with the finite element method, for the beam midspans, are given in Tables 10 and 11, respectively (E / G = 2.6 ). Normalised normal stresses σxwu / σxb, max of the horizontal wall (at point B) along the beam length and along the horizontal wall contour at x / l = 0.5 for simply supported beam (qy = 0 ) is presented in Fig. 6.

Table 6 Correction factors for normal stresses for bi-symmetrical I-sections,ΦAv = Φ Dv (ψ = 1, ρ = 1). Simple

h /l

1/3 1/5

Clamped

Analytic

FEM

Analytic

FEM

1.0385 1.0139

1.0384 1.0138

1.1156 1.0416

1.1154 1.0494

Fig. 5. Displacements v B /vb, max along the beam length, for of the upper flange for the: a) for the simply supported beam, b) for the clamped beam.

8

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

R. Pavazza et al.

longitudinal edges are presented by Fourier series by cosine mode shapes for simply supported beams, where at x = ± l /2 : σx = 0 , σy = 0 , v = 0 , τxy ≠ 0 , u ≠ 0 , and by sine mode shapes for clamped beams, where at x = ± l /2 : τxy = 0 , u = 0 , σy ≠ 0 , v ≠ 0 . For simply supported and clamped beams, respectively, one has ([6,7]):

Table 7 Correction factors for displacements v and w for bi-symmetrical I sections, ηw /ηv (ψ = 1, ρ = 1). Simple

h /l

Clamped

Analytic

1/3 1/5

FEM

2.8225/1.2773 1.6561/1.0998

2.8576/1.2812 1.6605/1.1002

Analytic

FEM

10.1124/2.3867 4.2805/1.4992

1,3... 1

10.0511/2.3779 4.2472/1.5029

π

2,4... 1 nπ πb ∑n n2 ϑn sin n 2 πb ∑n n ϑn cos 2 ⋅ 1,3... 1 , Φ1w = ⋅ 2,4... 1 , π nπ l ∑ l ∑ 3 sin n 2 cos

Φ1w =

n

n

2

n

(50)

2

n

where Table 8 Correction factors for normal stresses for mono-symmetrical U sections, ΦAwu /Φ Bwu (ψ = 1/2 , ρ = 2 ). Simple

b /l

1/3 1/5

ϑn =

The results of comparison for E = 210 GPa , G = 80.77GPa are presented in Table 12.

Clamped

Analytic

FEM

Analytic

FEM

1.0520/1.1483 1.0187/1.0534

1.0518/1.1479 1.0187/1.0533

1.1560/1.4449 1.0562/1.1602

1.1520/1.4335 1.0561/1.1601

Example 4. Non-symmetric U-sections (Appendix C) For the cross-section scantlings (Fig. C1a) b1 = 200 mm, b 2 = 100 mm, h = 250 mm, t1 = t2 = t0 = 5 mm. The cross section properties are y1 = 45.46 mm, A = 2750mm2 , z1 = 102.27 mm, φ0 = 17.11∘ , Iy = 30539339mm4 , Iz = 7306305mm4 , The shear factors, given by Eq. (3) are κxz = −0.0928, κxy = −0.1634 , κyy = 2.6719 , κyz = κzy = −0.2162 , κzz = 2.7512 , κωy = 0.3894 , κωz = −1.4581. Normalised normal stresses σx / σxb, max for the point A along the beam length for clamped beam are presented in Fig. 10. Normalised displacements wb / wb, max and vb / vb, max for the point B along the beam length for clamped beam are presented in Fig. 11.

Table 9 Correction factors for normal stresses for mono-symmetrical U sections, ΦAvα /Φ Bvα (ψ = 1/2 , ρ = 2 ). Simple

b /l

1/3 1/5

Clamped

Analytic

FEM

Analytic

FEM

1.0509/1.0426 1.0183/1.0154

1.0509/1.0425 1.0183/1.0153

1.1527/1.1279 1.0550/1.0461

1.1527/1.1277 1.0549/1.0460

Example 5. I-section and concentrated loads For a beam clamped at one end and subjected to bending by a couple at the other simply supported end (Fig. 12) according to method of initial parameters [8,9]

Table 10 Correction factors for displacements v and w for mono-symmetrical U-sections, ηw /ηv (ψ = 1/2 , ρ = 2 ). Simple

b /l

1/3 1/5

Clamped

Qz = Q (0), My = My (0) + Qz (0) x, βb = βb (0) +

Analytic

FEM

Analytic

FEM

1.2357/ 2.1093 1.0849/ 1.3994

1.2327/ 1.0844/

2.1787/ 6.5467 1.4243/ 2.9968

2.0458/ 1.3945/

Simple Analytic

1/3 1/5

1.9756 / − 1.3512 / −

1/3 1/5

− 0.0108 − 0.0023

Analytic

ηvα 1.9976 5.8781 / − 1.3541 2.7561 ηwu ; x/l=0.25 − 0.0106 − 0.0542 − 0.0023 − 0.0117

EIy

x+

Qz (0) EIy

⎞ My (0) x 2 Q (0) ⎛ x 2 x2 − k w l 2x ⎟ , , w = w0 − βb (0) x + ⋅ + z ⎜ ⎠ 2 EIy 2 EIy ⎝ 6 EIy , kw = GAzz l 2 where

βb (0) = 0, wb (0) = 0, wu (0) = 0, My (l ) = M . Thus, the following equations may be written

Clamped FEM

My (0)

⋅

Table 11 Correction factors for displacements α and u for mono-symmetrical U-sections, ηvα and ηwu (ψ = 1/2 , ρ = 2 ).

b /l

1 + chα n nπb , αn = , α n + shα n l

FEM

My (l ) = My (0) + Qz (0) l = M , wt (l ) = −

5.9668 2.7648

⎞ My (0) l 2 Qz (0) ⎛ l 3 = 0. ⎜ − ks l 3⎟ − ⎠ 2EIy EIy ⎝ 6

Hence

Qz (0) = −

− 0.0459

3M 1 M 1 − 6k w ⋅ , My (0) = − ⋅ . 2l 1 + 3k w 2 1 + 3k w

0.0115

Thus, Normalised displacements w / wb, max and ua / wb, max along the beam length for simply supported beam (qy = 0 ) are presented in Fig. 7. Normalised normal stresses σxvα / σxb, max along the left vertical wall contour for clamped beam (qz = 0 ) are presented in Fig. 8. Normalised displacements v D / vb, max of the horizontal wall (at point D) along the beam length for simply supported beam and clamped beam (qz = 0 ) are presented in Fig. 9. Comparison with the exact solution of the theory of elasticity The solution of the plane theory of elasticity for the bottom wall of the mono-symmetrical U-section is compared to the analytical solution and to the solution of the finite element analysis is presented in Table 12. The linearly distributed shear forces along of the bottom wall

Qz =

⎛ 3M 1 1 M x⎞ Ml ⋅ , My = − ⋅ ⎜1 − 6k w − 3 ⎟ , β = − 2l 1 + 3k w 2 1 + 3k w ⎝ 4EIy l⎠ ⎛ 1 x⎞ x Ml 2 1 ⋅ ⎜2 − 12k w − 3 ⎟ , w = 1 + 3k w ⎝ l⎠l 4EIy 1 + 3k w ⎤ x ⎡⎛ x⎞ x ⋅ ⎢ ⎜1 − 6k w − ⎟ + 6k w ⎥ . l ⎣⎝ l⎠l ⎦ ⋅

(51)

Comparisons of the normalised normal stresses according to (21), for qz = 0 , and (51), σxw / M , the finite element analysis and the classical theory of bending σxFEM / M and classical theory of bending σxb / M , for the junction of the web and the lower flange, along the beam length, are presented in Fig. 13, where 9

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

R. Pavazza et al.

Fig. 6. Normal stresses σxwu /σxb, max for simply supported beam: a) of the horizontal wall (at point B) along the beam length, b) along the horizontal wall at x /l = 0.5.

Fig. 7. Normalised displacements along the beam length for simply supported beam: a) w /wb, max , b) ua /wb, max .

Table 12 Comparison with the finite element method, Φ1w (ψ = 1, ρ = 1). Simple

b /l Analytic

FEM

TE (n)

Analytic

FEM

TE (n)

1/3

1.148

1.148

1.148 (35)

1.445

1.434

1/5

1.053/ 1,0267

1.053

1.053 (33)

1.160

1.160

1.432/1.444 (448/ 450) 1.147/1.154 (448/ 450)

σxb = Fig. 8. Normal stresses σxvα /σxb, max along the left vertical wall for clamped beam.

Clamped

My ⎛ h ⎞ M⎛ x⎞ ⎜ − ⎟ , My = − ⎜1 − 3 ⎟ Iy ⎝ 2 ⎠ 2 ⎝ l⎠

Ratios of the displacements according to (51) and the finite element method, for l / h = 5, is presented in Table 13, where is

Fig. 9. Displacements v D /vb, max of the horizontal wall (at point D) along the beam length for: a) simply supported beam, b) clamped beam.

10

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

R. Pavazza et al.

h = 50 mm, t = 6 mm), with the other available factors in literatures [5] are given in Table 15. Comparisons of the shear factors for non-symmetric U section (for b1 = 40 mm, b 2 = 20h = 100 mm, t = 5 mm), with the other available factors in literatures [5] are given in Table 16.

7. Conclusion The theory of bending with influence of shear of thin-walled beams of open sections 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. The theory is based on the Vlasov's thin-walled beam theory of open sections as well as on Timoshenko's bending beam theory. The transverse loads are reduced to the shear centre/bending/torsion centre defined by Vlasov's theory as pure geometrical property. By proposed engineering approach more general loads conditions as well cross-section shapes can be considered. Neglecting the Poisson's effect as well as the “shear warping effect” (“non-uniform warping bending theory”), various engineering problems of thin-walled structures can be considered. The results can be obtained in analytic forms. The shear factors are defined with respect to the transverse displacements of the cross-sections as plane sections (factors κyy and 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 bisymmetrical cross-section the beam will be subjected to bending with influence of shear only (factors κωy ,κωz , κxy , κxz vanish). The comparison to the results of the finite element method by several typical examples has shown a high agreement of the obtained results.

Fig. 10. Normal stresses σx /σxb, max for the point A along the beam length for the clamped beam.

wb =

2 Ml 2 ⎛ x ⎞ ⎛ x⎞ ⎜ ⎟ ⎜1 − ⎟ 4EIy ⎝ l ⎠ ⎝ l⎠

The couple is modeled by concetrated forces acting in mash nodes:

Fi =

My (l ) Iy

zi tΔh =

MtΔh zi Iy

where Δh is the ditance between nodes, zi the coordinate of the i-th node. 5. Influence of Poisson's factor For bi-symmetrical I-sections Cowper's shear factor read [2]

1 10(1 + ν )⋅(1 + 3m )2 = 2 κzz (12 + 72m + 150m + 90m3) + ν (11 + 66m + 135m2 + 90m3)

κ=

+ 30n2 (m + m2 ) + 5νn2 (8m + 9m2 ) (52) where

m=

2bt1 , t0

n=

b . h

Comparison with analytical solution given by Eq. (45), for t1 = t0 , b = 2h , is presented in Table 14. 6. Comparison of shear factors Comparisons of the shear factors for U section (for b = 100 mm,

Fig. 11. Normalised displacements for the point B along the beam length for clamped beam: a) wb /wb, max , b) vb /vb, max .

11

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

R. Pavazza et al.

Table 13 Ratios of displacements w /wb along the beam length. x/l

Analytic

FEM

1/5 2/5 1/2 3/5 4/5

0.37 2.21 1.91 1.71 1.45

0.35 2.18 1.90 1.72 1.48

Fig. 12. A beam clamped at one end and subjected to bending by a couple at the other Table 14 Comparison of the analytical solution given by Eq. (45) and the Couper's solution.

b /h

Eq. (45)

Cowper Eq. (52) ν=0

Cowper Eq. (52) ν = 0, 3

1/2 1 2

2.1188 3.3796 6.4260

2.1188 3.3796 6.4260

2.0960 3.3829 6.5428

Table 15 Comparison of the shears factors for U section. TBTS

Ayy = Azz =

Fig. 13. The normalised normal stresses σxw /M , the finite element analysis and the

WPy =

classical theory of bending σxFEM /M and the classical theory of bending σxb /M , for the junction of the web and the lower flange, along the beam length.

KIM

TBTS − KIM ⋅100% KIM

A κyy

= 0.76229 m2

A3 ≡ Ayy = 0.76229 m2

0%

A

= 0.97471 m2

A2 ≡ Azz = 0.97471 m2

0%

κzz WP κωy

= −5.77375 m3

A3r ≡ −WPy = 5.77375

m3

Appendix A. Mono-symmetric I-section (z-axis) (Fig. A1a) See Fig. A1–A3 Cut-off portions of area and second moments of cut-off portions of area for the web (Fig. A2a):

t0 2 t (h1C − sz2 ), 0 ≥ sz ≥ −h1C ; Az* = b 2 t2 + (h2C − sz ) t0, Sy* = h2C A2 + 0 (h 22C − sz2 ), 0 ≤ sz ≤ h2C . 2 2

Az* = −b1 t1 + (−h1C − sz ) t0, Sy* = h1C A1 +

Cut-off portions of area and moments of cut-off portions of area for the flanges — odd coordinates (Fig. A2a; b):

⎞ ⎛b ⎞ ⎛b ⎞ ⎛b ⎞ ⎛b b b Az* = ⎜ 1 − sz⎟ t1, Sy* = −h1C t1 ⎜ 1 − sz⎟ , 0 ≤ sz ≤ 1 ; Az* = ⎜ 2 − sz⎟ t2, Sy* = h2C t2 ⎜ 2 − sz⎟ , 0 ≤ sz ≤ 2 ; ⎠ ⎝2 ⎠ ⎝2 ⎠ ⎝2 ⎠ ⎝2 2 2 Moments of cut-off portion of area for the flanges — even coordinates (Fig. A2c):

⎞ ⎞ t1 ⎛ b12 b t ⎛b2 b − sy2⎟ , 0 ≤ sy ≤ 1 , Sz* = − 2 ⎜ 2 − sy2⎟ , 0 ≤ sy ≤ 2 . ⎜ 2⎝ 4 2 2⎝ 4 2 ⎠ ⎠

Sz* =

a)∫

0

sz

(Sy*/ t ) ds ;

sy

b)∫ (Sz*/ t ) ds . 0 Integrals of the second moment of cut-off portion of area:

∫0

sz Sy* ds t

∫0 ∫0

sz

t

sy

h12C (3A1 + t 0 h1C ) 3t 0

, 0 ≤ sz ≤

b1 , 2

even coordinate;

Sy* s2 ⎞ ⎤ s ⎡ t ⎛ ds = z ⎢h2C A2 + 0 ⎜⎜h 22C − z ⎟⎟ ⎥ , −h1C ≤ sz ≤ h2C ; t0 ⎢⎣ 2⎝ 3 ⎠ ⎥⎦ t

sz Sy*

∫0

1

= − 2 h1C sz (b1 − sz ) −

1

ds = 2 h2C sz (b 2 − sz ) +

h22C (3A2 + t 0 h2C ) 3t 0

, 0 ≤ sz ≤

sy2 ⎞ Sz* 1 ⎛b2 b b ds = sy ⎜⎜ 1 − ⎟⎟ , − 1 ≤ sy ≤ 1 , 2 ⎝ 4 3⎠ 2 2 t

∫0

sy

b2 2

even coordinate;

sy2 ⎞ Sz* 1 ⎛b2 b b ds = − sy ⎜⎜ 2 − ⎟⎟ , − 2 ≤ sy ≤ 2 . 2 ⎝ 4 3⎠ 2 2 t

12

0%

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

R. Pavazza et al.

Table 16 Comparison of the shears factors for non-symmetric U section. TBTS

Ayy =

A κyy

Azz =

A κzz

Ayz =

A κyz

WPy =

WP κωy

WPz =

WP κωz

KIM

TBTS − KIM ⋅100% KIM

= 151.452 mm2

A2 ≡ Ayy = 151.452 mm2

= 445.879 mm2

mm2

A3 ≡ Azz = 446.252

0% −0,08%

A23 ≡ −Ayz = 1847.808 mm2

0,22%

= 113178.97 mm3

A2r ≡ −WPy = −113178.97 mm3

0%

= −144462.91 mm3

A3r ≡ WPz = −145931.35 mm3

−1,01%

= −1851.900

mm2

Fig. A1. Mono-symmetric I-section: a) dimensions; b) middle line; c) y-coordinate; d) z-coordinate; e) ω -coordinate.

Here

A = A0 + A1 + A2 , A0 = ht0, A1 = b1 t1, A2 = b 2 t2. 1

Iy = A1 h2

1

λ + 3 ψ (1 + λ + 4 ψ ) 1+λ+ψ

, Iz = A1 h2

ρ2 (1 + λη2 ) λη2ρ2 λ (1 + λη 4 ) , Iω = A1 h4 , IP = A1 h2 , 2 12 12(1 + λη ) (1 + λη2 )2 1

WP =

λ + 2ψ 1 + λη 4 λη2 1 IP = A1 h 2 , h1P = h , h2P = h , A = A1 (1 + λ + ψ ), h1C = h , h2C 2 2 2 1 + λη 1 + λη 1+λ+ψ h1P η (1 + λη ) 1

=h

1 + 2ψ 1+λ+ψ

−

∫A

Az* Sy* t2

;

⎛ Sy* ⎞2 ⎡φ ⎤ 2 2 1 2 ⎟⎟ dA = A1 h 4φ 2 ⎢ (1 + λ2μ3) + φ 2 (1 + λμ4 ) + ψφ3 (1 + μ5) + ρ (1 + λμ2 η2 ) ⎥ , ⎦ ⎣ψ 3 15 12 t ⎠

∫A ⎜⎜⎝

⎛ Sz* ⎞2 ρ4 (1 + λη 4 ) , ⎟ dA = A1 h4 120 t ⎠

∫A ⎜⎝

⎤ ⎡φ 1 2 5 5 dA = A1 h3φ ⎢ (1 − λ2μ2 ) + ρ (1 − λμη2 ) + φ 2 (1 − λμ3) + ψφ3 (1 − μ4 ) ⎥ . ⎦ ⎣ψ 12 6 24

Bi-symmetric I-section Properties for I-section with two axis of symmetry can be obtained from the I-section with one axis of symmetry, for b1 = b 2 = b , t1 = t2 = t . 13

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

R. Pavazza et al.

Appendix B. Mono-symmetric U-section (Fig. B1) See Figs. B1–B3 Cut-off portions of area and moments of cut-off portion of area — odd coordinates (Fig. B2a; Fig. B2b):

⎛b ⎞ ⎛b ⎞ 1 b A 1 Az* = (h 0C − sz ) t0, Sy* = − (h 02C − sz2 ) t0, 0 ≤ sz ≤ h 0C Az* = ⎜ − sz⎟ t1, Sy* = h1C t1 ⎜ − sz⎟ , 0 ≥ sz ≥ Az* = 1 + (h1C − sz ) t0, Sy* = (h 02C − sz2 ) t0, 0 ⎝2 ⎠ ⎝2 ⎠ 2 2 2 2 ≤ sz ≤ h1C ; Moments of cut-off portions of area — even coordinate (Fig. B2c):

Sz* =

⎞ bt0 bt A b 1 ⎛ b2 b b (h − sy ), 0 ≤ sy ≤ h , Sz* = 0 (h + sy ), 0 ≥ sy ≥ h ; Sz* = 0 + ⎜ − sy2⎟ t1, − ≤ sy ≤ ; ⎠ 2 2 2 2⎝ 4 2 2

a)∫

0

sz

(Sy*/ t ) ds ;

sy

b)∫ (Sz*/ t ) ds . 0 Integrals of moments of cut-off portions of area - even coordinate:

∫0

sz

Sy* t

ds = −

s2 ⎞ sz ⎛ 2 ⎜⎜h 0C − z ⎟⎟ , −h1C ≤ sz ≤ h 0C ; 2⎝ 3⎠

∫0

sz

Sy* t

ds =

h2 ⎞ h1C sz h ⎛ b (b + sz ) + 1C ⎜h 02C − 1C ⎟ , 0 ≤ sz ≤ ; 2 2 ⎝ 3 ⎠ 2

∫0

sz

Sy* t

ds =

s2 ⎞ sz ⎛ 2 ⎜⎜h 0C − z ⎟⎟ , −h 0C ≤ sz ≤ h1C . 2⎝ 3⎠

Fig. A2. Cut-off portions of area and second moments of cut-off portions of area of the mono-symmetric I-section: a) Az*; b) Sy*; c) Sz*.

Fig. A3. Integrals of moments of cut-off portions area of the mono-symmetric I-section:.

14

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

R. Pavazza et al.

Fig. B1. Mono-symmetric U-section: a) dimensions; b) middle line; c) y-coordinate; d) z-coordinate; e) ω -coordinate.

Fig. B2. Cut-off portions of area and second moments of cut-of portions of area of the mono-symmetric U-section: a) Az*; b) Sy*; c) Sz*.

Fig. B3. Integrals of moments of the cut-off portions area of the mono-symmetric U-section:.

15

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

R. Pavazza et al.

odd coordinates:

∫0

sy

Sz* A b2 b3 bh b ds = 0 + + sy − sy2, 0 ≤ sy ≤ h ; t 4t1 24 2 4

∫0

sy

Sz* A b2 b3 bh b ds = − 0 − + sy + sy2, 0 ≥ sy ≥ −h . t 4t1 24 2 4

Here

A = 2A0 + A1 , A0 = ht0, A1 = bt1. ⎛ 1+ψ ψ 3ψ 2+ψ 1 + 6ψ 18ψ + ρ2 (1 + 6ψ )2 1⎞ A = A 0 ⎜ 2 + ⎟ , h 0C = h , h1C = h , h 0 = hP = h , Iy = A0 h2 , Iz = A0 h2ρ2 , IP = A0 h2 , WP 1 + 2ψ 1 + 2ψ 1 + 6ψ 3(1 + 2ψ ) 12ψ 2(1 + 6ψ )2 ψ⎠ ⎝ = A0 h

18ψ + ρ2 (1 + 6ψ )2 , h 0 = hP , 6ψ (1 + 6ψ )

⎛ Sz* ⎞2 ⎞ A h 4ρ 4 ⎛ ψ ⎟ dA = 0 ⎜1 + 10ψ + 20 2 + 30ψ 2⎟ , 120ψ ⎝ t ⎠ ρ ⎠

∫A ⎜⎝

⎛ Sy* ⎞2 A0 h 4 ⎟⎟ dA = [2(8 + 55ψ + 140ψ 2 + 160ψ 3 + 80ψ 4 + 16ψ 5) + 5ψρ2 (1 + 2ψ )3], t ⎠ 60(1 + 2ψ )5

∫A ⎜⎜⎝

∫A

Az* Sy* t2

dA = A0 h3

1 + ρ2 (1 + 2ψ ) , 12(1 + 2ψ )2

Appendix C. Non-symmetric U-section (Fig. C1) See Figs. C1–C3 Cut-off portions of area

⎞ ⎛ z ⎞ ⎛ z zB zC Az* = (−b1 − sz ) t1, 0 ≥ sz ≥ −b1; Az* = −b1 t1 + ⎜ B − sz⎟ t0, 0 ≥ sz ≥ ; Az* = b 2 t2 + ⎜ C − sz⎟ t0, 0 ≤ sz ≤ ; Az* = (b 2 − sz ) t2, 0 cos φ cos φ cos φ cos φ0 ⎠ ⎝ ⎠ ⎝ 0 0 0 ≤ sz ≤ b 2 .

Fig. C1. Non-symmetric U-section: a) dimensions; b) middle line; c) y-coordinate; d) z-coordinate; e) ω -coordinate.

16

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

R. Pavazza et al.

Fig. C2. Cut-off portions of area and second moments of cut-off portions of area of the non-symmetric U-section: a) Az*; b) Sy*; c) Ay*; d) Sz*; e) Sω*.

Fig. C3. Integrals of the moments of the cut-off portions of area of the non-symmetric U-section: a) ∫

sz

0

17

(Sy*/t ) ds ; b) ∫

sy

0

(Sz*/t ) ds .

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

R. Pavazza et al.

⎞ ⎛ ⎛ ⎞ ⎛ yC ⎞ y y y ⎟ t0 + ⎜ − B − sy⎟ t1, 0 ≤ sy Ay* = ⎜ − A − sy⎟ t1, 0 ≥ sy ≥ − A ; Ay* = ⎜h − m − cos φ0 sin φ0 ⎠ ⎠ ⎝ cos φ0 ⎝ ⎠ ⎝ cos φ0 ⎞ ⎛ y ⎞ ⎛ ⎞ ⎛ yC yC y y yB ⎟ t0 + ⎜ B − sy⎟ t1, 0 ≥ sy ≥ ≤ − B ; Ay* = ⎜h − m − ; Ay* = b 2 t2 + ⎜ C − sy⎟ t0, 0 ≤ sy ≤ ; Ay* = (b 2 − sy ) t2, 0 ≤ sy ≤ b 2. cos φ0 sin φ0 ⎠ sin φ0 sin φ0 ⎠ ⎝ sin φ0 ⎠ ⎝ sin φ0 ⎝ Moments of cut-off portions of area:

⎞ ⎛ z2 ⎞ ⎛ 1 1 1 zB t1 sin φ0 (b12 − sz2 ), 0 ≥ sz ≥ −b1; Sy* = b 2 t2 ⎜zD + b 2 sin φ0⎟ + t0 cos φ0 ⎜ 2C − sz2⎟ , ≤ sz ⎠ ⎝ 2 2 2 ⎠ cos φ0 ⎝ cos φ0 zC 1 ≤ ; Sy* = (b 2 − sz ) zD t2 + t2 sin φ0 (b22 − sz2 ), 0 ≤ sz ≤ b 2. cos φ0 2

Sy* = −(b1 + sz ) zB t1 +

⎞ ⎛ y2 ⎛ y2 ⎞ ⎞ ⎛ yB y y 1 1 1 ≤ sy Sz* = − t1 cos φ0 ⎜⎜ 2A − sy2⎟⎟ , − A ≤ sy ≤ − B ; Sz* = b 2 t2 ⎜ yC + b 2 cos φ0⎟ + t0 sin φ0 ⎜⎜ 2C − sy2⎟⎟ , ⎠ ⎝ 2 cos φ0 2 2 ⎝ cos φ0 ⎠ cos φ0 ⎠ sin φ0 ⎝ sin φ0 yC 1 ≤ ; Sz* = (b 2 − sy ) yC t2 + t2 cos φ0 (b22 − sy2 ), 0 ≤ sy ≤ b2. sin φ0 2 ⎞ ⎛1 ⎡ h ⎤ 1 Sω* = −t1 ⎢ (b1 h1 − h 01 h2 ) sα + 1 sα2⎥ , 0 ≥ sα ≥ −b1; Sω* = b1 t1 ⎜ h1 b1 − h 01 h2⎟ − t0 h2 h 01 (h 01 + sα ) + t0 h2 (h 01 + sα )2 , −h 01 ≤ sα ⎣ ⎠ ⎝2 2 ⎦ 2 ⎞ ⎛1 1 1 ≤ h 02 ; Sω* = b1 t1 ⎜ h1 b1 − h 01 h2⎟ − ht0 h2 + ht0 h2 h + t2 h2 h 02 sα − t2 (h − h1) sα2, 0 ≤ sα ≤ b 2 . ⎠ ⎝2 2 2

References [10] [1] 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. [2] G.R. Cowper, The shear coefficient in Timoshenko's beam theory, J. Appl. Mech. 33 (1966) 335–340. [3] 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. [4] R. El Fatmi, No-uniform warping inclouding the effects of torsion and shear forces. Part II: Analytical and numerical applications, Int. J. Solids Struct 44 (2007) 5930–5952. [5] N. Kim, M. Kim, Exact dynamic/static stiffness matrices of non-symetric thinwalled beams considering coupled shear deformation effects, Thin-Walled Struct. 43 (2005) 701–734. [6] A.A. Kurdyumov, A.Z. Lokashin, P.A. Josifov, V.V. Kozlyakov, Stroitel’naya mexanika korablya i teoriya uprugosti 2, Sudostrojenie, Leningrad ,1963. [7] R. Pavazza, An approximate solution for thin rectangular orthotropic/isotropic strips under tension by line loads, Int. J. Solids Struct. 37 (2000) 4353–4375. [8] 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. [9] R. Pavazza, S. Jović, A comparison of approximate analytical methods and the finite

[11] [12] [13] [14]

[15] [16]

18

element method in the analysis of short clamped thin-walled beams subjected to bending by uniform loads, Trans. FAMENA 31 (2007) 37–54. R. Pavazza, A. Matoković, B. Plazibat, Bending of thin-walled beams of symmetrical open cross-sections with influence of shear, Transaction of FAMENA 37 (2013) 17–30. R. Pavazza, B. Plazibat, Distortion of thin-walled beams of open section assembled of three plates, Eng. Struct. 57 (2013) 189–198. W.D. Pilkey, Analysis and Design of Elastic Beams, Computational Methods, John Wiley & Sons, New York, 2002. S.P. Timoshenko, G.H. MacCullough, Element and Strength of Materials, van Nostrand, New York, 1949. F. Vlak, R. Pavazza, M. Vukasović. An approximate analytic solution for stresses and displacements of thin-walled beams subjected to bending, in: Proceedings of the 16th European Conference on Composite Materials ECCM16-Conference Proceedings-Seville, Spain: University of Seville, Spain, 2014./ Paris, F. (r.), Seville: University of Seville, 2014, pp. 1–8. V.Z. Vlasov, Thin-Walled Elastic Beams, Oldbourne Press, London, 1961. M. Vukasović, R. Pavazza. An approximate solution for stresses and displacements of thin- walled composite beams with mono-symmetric cross-sections subjected to bending, in: Proceedings of the 20th International Conference on Composite Materials, (ICCM20 Programme and book of abstracts/Thomsen, T. (r.), Copenhagen: MCI Copenhagen, 2015, pp. 52-52.