The influence of elasticity of joints on the behaviour of thin-walled structures

Thin-Walled Structures 36 (2000) 67–88 www.elsevier.com/locate/tws

The influence of elasticity of joints on the behaviour of thin-walled structures Alexander Tesar *, Jana Kuglerova Institute of Construction and Architecture, Slovak Academy of Sciences, 842 20 Bratislava, Slovak Republic Received 26 February 1999; received in revised form 18 August 1999; accepted 18 August 1999

Abstract Application of modified technical torsion-flexural theory for analysis of slender thin-walled structures with elastic joints is treated in present paper. Theoretical approach and numerical assessments of the influence of elastic joints on the load-bearing capacity of thin-walled systems are presented. Theoretical approach under consideration of linear and non-linear influences possibly appearing is established. The numerical verifications of the problem specified are presented. Discussion and evaluation of the results and conclusions obtained are submitted.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Bridges; Elastic joints; FETM-method; Modified torsion-flexural theory; Shear components; Thin-walled structures; Torsion

1. Introduction For analysis and calculation of thin-walled structures it is assumed that the joints between the load bearing structural members are rigid. Such assumption suitably simulates the behaviour of thin-walled structure, with a lot of advantages dealing with analytical approaches and calculations performed as well as with the interactions of structural members adopted. However, in present design and structural engineering, thin-walled structures having elastic couplings in joints and connections adopted are often used. For example,

* Corresponding author. 0263-8231/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 9 ) 0 0 0 3 8 - 5

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the rigid cores of tall buildings with floor openings (see Fig. 1a) or composite steelconcrete cross-sections of thin-walled bridges (see Fig. 1b) have such properties and a more sophisticated analysis of their behaviour comes distinctly to the fore. The load-bearing capacity of thin-walled cross-section is reduced in such cases due to additional cross-sectional degrees of freedom when taking into account additional shear deformations possibly appearing in elastic joints. The fundamentals of the mechanics of the phenomenon described above are submitted below. The approach presented is based on the application of technical torsion-flexural theory of thin-walled structures adopting the refinements and modifications required in accordance with consideration of the elasticity of joints studied. Numerical approaches based on the adoption of parallel processing FETM-technique are presented. Theoretical backgrounds and numerical verifications of the approach developed are presented below.

2. Principal equations The resulting equilibrium state of thin-walled structures studied are analysed on the basis of the vector of deformations given by v⫽(vt,vp)⫽(vz,vx,vy,r,;v1,v2,%)T,

Fig. 1.

Examples of structures having elastic joints.

(1)

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

69

with deformations of the rigid cross-section vz,x,y coupled on the centre of torsion d in the direction of principal axes and with r as rotation around d, as well as with deformations ni of the cross-section due to corresponding distortional behaviour (see [1]) in accordance with the consideration of elastic couplings of structural components studied. The corresponding set of sectorial functions is given by w⫽(wt,wp)T⫽(z,x,y,w;w1,w2,%)T,

(2)

where z, x and y are principal cross-sectional co-ordinates coupled on the centroid and w is the sectorial characteristics of the cross-section in torsional behaviour. The functions wi are cross-sectional characteristics of distortional behaviour of the crosssection studied (see [2]). Such characteristics take into account the unit shear deformations occurring in the contact points of elastic couplings of thin-walled crosssection analysed. The diagrams of sectorial functions wp for cross-sectional degrees of freedom vp =1 are obtained over superposition of unit shear deformations dwi=1 in the points i of the elastic couplings of the cross-section studied. Corresponding warping relations wi describe the geometry of axial deformations of cross-sectional fibres due to shear displacements possibly appearing. When dealing with such functions it is advantageous to use the symmetry of the cross-section. Sectorial functions wi corresponding to unit shear deformations dwi =1 in the points of elastic couplings for some geometries of open and closed thinwalled cross-sections are submitted in Fig. 2. When dealing with the consequences of elastic deformations in the contact points

Fig. 2.

Cross-sectional functions wi for some geometries of thin-walled cross-sections.

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A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

of thin-walled box cross-sections (see Fig. 2b), the shape of sectorial function wp is influenced by the shear flow given by qp = qp t = G f vp and is corresponding to the unit shear deformation dwp =1. For a single box cross-section as shown in Fig. 3, the deformation condition given by

冕 s

fp 1/t ds⫽dwp⫽1,

(3)

0

is established. When taking into account Bernoulli’s assumption of plane displacements of thinwalled members of the cross-section studied, the total warping function of crosssectional fibres is given by u⫽wTv⬘⫽wTt v⬘t⫹wTp v⬘p,

(4)

with corresponding strains e⫽u⬘⫽wv⬘.

(5)

The deformation of elastic joints in the cross-section studied is then given by du⫽dwv⬘⫽dw1v⬘1⫹dw2v⬘2⫹% .

(6)

Normal stress s⫽Ee

(7)

and elastic resistance D⫽Cdu,

(8)

Fig. 3.

Shear functions for single box cross-section.

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

71

correspond then to the deformation du in the joint studied. The parameter C determines the elasticity of joint assuming the shear flow causing the deformation du =1. For structure in equilibrium, the total balance of virtual works for arbitrary virtual deformation is given by

冕冋冕 L

dp⫽

0

A

0

冘 n

sdedA⫹

0

冕 A

册冕 L

Dd(du)dz⫹ tpdgpdA ⫺ prdvdz⫽0. 0

(9)

0

After some treatment of Eq. (5) and (8) the following relation is obtained:

冕 L

dp⫽ [dvTEJv⬘⫹dvTCdKv⬘⫹dv⬘GKov⬘⫹drgvr⬘⫺dvp]dz⫽0,

(10)

0

where E J=E兰 A0w wT dA is the sectorial rigidity of thin-walled cross-section, C dK=C⌺A0 dw dw is the elastic rigidity of joints studied, G Ko=G兰 A0f/t fT/t dA is the Bredt’s torsional rigidity of the box members of thin-walled cross-section analysed,

Fig. 4.

Deformations of thin-walled open cross-section.

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GKv=1/3 G⌺n0 b t3 is the Saint-Venant’s rigidity of open members of thin-walled cross-section studied and p = pe r is the load parameter assumed, with external load pe defined and corresponding direction vector r = (0, sin a, cos a, rd, 0, 0,...)T specified in accordance with Fig. 4, which describes the location of the load pe submitted in relation to the principal axes x, y and z as well as to the centre of torsion d. Double integration of Eq. (10) gives

冕冋 冋

冉

L

dv EJvIV⫺G

0

冊册 册

C dK⫹Ko⫹Kv vII⫺p dz⫽0. G

(11)

Because Eq. (11) has zero value in the equilibrium state for each virtual deformation dvi, the corresponding system of principal simultaneous differential equations is given by II E JvIV i ⫺G(dK⫹Ko⫹Kv)vi ⫽pi,

(12)

Present principal equations are further treated below.

3. Norming of principal equations

Differential Eq. (12) represents the simultaneous system of differential equations for determination of the components of deformations vt for rigid cross-section and deformations vp dealing with distortional behaviour of the cross-section when considering the elastic couplings possibly appearing in the joints of the structure studied. For open cross-section Eq. (12) is given by Az 0

0

0

Jz1 Jz2 . . .  vz 

0

Jxx 0

0

Jx1 Jx2 . . . vx

0

0

Jyy 0

Jy1 Jy2 . . . vy

 

   

0 0 0 Jw Jw1 Jw2 . . . r E   J1z J1x J1y J1w J11 J12 . . . v1 J2z J2x J2y J2w J21 J22 . . .

v2

.

.

.

.

.

.

. . . .

.

.

.

.

.

.

. . .  . 

IV

(13)

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

0 0 0 0 0

0

. . .  vz 

pz 

0 0 0 0 0

0

. . . vx

px

0 0 0 0 0

0

. . . vy

py

 

73

       

0 0 0 K 0 0 . . . r pw ⫺G   ⫽  . 0 0 0 0 dK11 dK12 . . . v1 0 0 0 0 0 dK21 dK22 . . . v2

0

. . . . .

.

. . . .

.

. . . .

.

. . .  . 

.

.



When using the transformation matrix AI of the first stage of norming

AI⫽

冤

冥

1

0

0

0

0 0 . . .

0

1

0

0

0 0 . . .

0

0

1

0

0 0 . . .

0

0

0

1

0 0 . . .

kz1 kx1 ky1 kw1 1 0 . . . kz2 kx2 ky2 kw2 0 1 . . .

,

(14)

with corresponding parameters ki given by Jz1 Jz2 Jz1 Jx2 kz1⫽⫺ ; kz2⫽⫺ ; kx1⫽⫺ ; kx2⫽⫺ Az Az Jxx Jxx Jy1 Jy2 Jw1 Jw2 ky1⫽⫺ ; ky2⫽⫺ ; kw1⫽⫺ ; kw2⫽⫺ Jyy Jyy Jww Jww Eq. (13) is modified by Az 0

0

0

0

0

. . .  vz 

0 Jxx 0

0

0

0

. . . vx

Jyy 0

0

0

. . . vy

0

. . . r   . . . v1

 

0 0

   

0 0 E 0 0

0

Jww 0

0

0

J11 J12

0 0

0

0

J21 J12 . . . v2

.

.

.

.

.

.

. . . .

.

.

.

.

.

.

. . .  . 

IV

(15)

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A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

0 0 0 0 0

0

. . .  vz 

0 0 0 0 0

0

. . . vx

px

0 0 0 0 0

0

. . . vy

py

 

II

pz 

       

0 0 0 K 0 0 . . . r pw ⫺G    ⫽  , 0 0 0 0 dK11 dK12 . . . v1 p1 0 0 0 0 dK21 dK22 . . . v2

p2

. . . . .

.

. . . .

.

. . . .

.

. . .  . 

.

.



On the basis of analysis of the roots of corresponding eigenvalue equation of pth degree det[Jp⫺lidKp]⫽0,

(16)

the eigenvalues li of the problem studied are stated. The eigenvalues are used in the matrix equation [Jp⫺lidKp] [Ip]⫽0,

(17)

obtaining hereby the vector lp of the transformation matrix AII for the second stage of norming 1 0 0 0 0

0

. . .

0 1 0 0 0

0

. . .

0 0 1 0 0

0

. . .

 

 

0 0 0 1 0 0 . . . AII⫽   0 0 0 0 l11 l12 . . . 0 0 0 0 l21 l22 . . . . . . . .

.

. . .

. . . . .

.

. . .

(18)

corresponding with eigenfunctions of the characteristic system of the homogeneous differential equations of the problem studied (see [2]). Using the matrix multiplication’s A⫽AIIAI; w¯⫽Aw; v¯⫽ATv; J¯⫽AJAT; dK¯⫽AdKAT; r˜⫽Ar,

(19)

resulting diagonally orthogonalised, simultaneous Eq. (13) are obtained, given by ¯ ¯ IIi⫽p¯i, EJ¯iiv¯IV i ⫺GdKiiv

(20)

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75

for p-th components of additional cross-sectional deformations vi of elastically coupled members of thin-walled open cross-section studied. Eq. (20) is formally equivalent with the equation of laterally and axially loaded beam on elastic supports and the analogies with the problems of thin-walled structures can be adopted (see [2]). When analysing, for example, the stress state of thin-walled bridges with closed cross-sections having elastic contact points in the connections between the loadbearing members and roadway slabs, the solution approach is principally the same as by the above analysis of thin-walled structures with an open cross-section. However, due to additional shear rigidity parameters in the box members of the crosssections studied, in accordance with interaction of shear components appearing by pure torsion, by torsion-bending and by warping of elastic joints of the structure, the corresponding differential equations of the problem studied are given by Az 0

0

0

Jz1 Jz2 . .  vz 

0

Jxx 0

0

Jx1 Jx2 . . vx

0

0

Jyy 0

Jy1 Jy2 . . vy

 

IV

   

0 0 0 Jww Jw1 Jw2 . . r E   J1z J1x J1y J1w J11 J12 . . v1 J2z J2x J2y J2w J21 J22 . .

(21)

v2

.

.

.

.

.

.

. . .

.

.

.

.

.

.

. .  . 

0 0 0 0

0

0

. .  vz 

vz 

0 0 0 0

0

0

. . vx

vx

0 0 0 0

0

0

. . vy

vy

 

       

0 0 0 K0+Kv K01 K02 . . r vw ⫺G   ⫽  , 0 0 0 K10 dK11+K11 dK12+K12 . . v1 v1 0 0 0 K20

dK20+K21 dK22+K22 . . v2

v2

. . . .

.

.

. . .

.

. . . .

.

.

. .  . 

. 

where members dK = Σ dw dw are multiplied by C/G. Using the transformation matrix A⫽AIIAI,

(22)

with matrices AI and AII as mentioned earlier and applying the matrix multiplication’s (19), the resulting differential equations are then given by

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A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

Az 0

0

0

0

0

. .  vz 

0 Jxx 0

0

0

0

. . vx

Jyy 0

0

0

. . vy

0

. . r   . . v1

 

0 0

IV

   

0 0 E 0 0

0

Jww 0

0

0

J11 0

0 0

0

0

0

J22 . . v2

.

.

.

.

.

.

. . .

.

.

.

.

.

.

. .  . 

(23)

0 0 0 0

0

0

. .  vz  pz 

0 0 0 0

0

0

. . vx

px

0 0 0 0

0

0

. . vy

py

 

0 0 0 K0+Kv 0 0 ⫺G  0 0 0 0 dK11+K11 0

II

       

. . r pw    , . . v1 p1

0 0 0 0

0

dK22+K22 . . v2

p2

. . . .

.

.

. . .

.

. . . .

.

.

. .  .  . 

for cross-section studied and additional shear rotations appearing. Similar situations pay there for the warping deformations too.

4. Solution of normed differential equations

For the solution of the present equations, the analogy with the solution of the beam on elastic foundation subjected to the action of lateral and axial forces can be applied (see [2]). Utilisation of the present analogy allows the approximate analysis of spatial behaviour of thin-walled bridge structures subjected to various load influences when taking into account the elasticity parameters of contact points between load-bearing members and slabs of composite bridges. For precise analysis of the above problem the numerical approaches on the basis of finite element, finite strips, boundary element or transfer matrix methods are needed. Up-to-date parallel processing approach using the FETM-method is applicable for sophisticated analysis of advanced problems of linear and non-linear mechanics of thin-walled structures with elastic joints (see [3]). The problems of ultimate static and dynamic behaviour of multibody simulations of slender thin-walled structures there can be efficiently treated in such a way (see, for example, [3,4,8]).

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

77

5. Numerical studies Studied is the influence of elastic joints between steel load-bearing members and concrete slabs of two thin-walled bridge structures with open and closed cross-sections as shown in Fig. 5. The span of the bridge studied is 70 m and the areas of the open and closed crosssections adopted are equivalent. Assumed value of the elastic couplings in the joints studied is C=831.5 MPa. The ratio C/G assumed is 831.5/81000=0.010265. The radius of curvature for both cases is 500 m. 5.1. Behaviour of the open cross-section as shown in Fig. 5a The values of sectorial parameters for the rigid cross-section, as shown in Fig. 6, are: Az⫽0.7806 m2; Jxx⫽8.562983; yo⫽0.9271 m; Jyy⫽1.465688 m4; Jxw ⫽9.753702 m5; yd⫽⫺1.139054 m; Jww⫽15.748542 m6. Sectorial parameters of the cross-section when considering the unit shear deformations dw=1 in elastic joints, as shown in Figs. 7 and 8, are: J11⫽2(3.3 0.014 1.5 12⫹0.004 1.5 12⫹0.05 12)⫽0.3006 m4 J22⫽2(3.3 0.014 12⫹0.004⫹12⫹0.0512)⫽0.2004 m4,

Fig. 5.

Cross-sections of thin-walled bridges studied.

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A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

Fig. 6.

Cross-sectional parameters of studied bridge with open cross-section.

J1z⫽j11⫽0.3006 m3, J1y⫽0.445004 m4, J2x⫽⫺0.9018 m4, J2w⫽⫺1.143872 m5. dK11⫽3 12⫽3.0 m2, dK11⫽0.030795 m2, dK22⫽2 12⫽2.0 m2, dK22 ⫽0.020530 m2. (Despite that concrete slab is elastically coupled in three points, due to the symmetry of the cross-section there may be assumed only two deformation components in flexure and torsion). When assuming the midspan load P=1 kN acting on the first of main girders and when considering the torsional rigidity of thick-walled slab given by G Kv⫽

冘

bt3⫽G/5.25 (9.00.283⫹3.00.123)⫽0.012873

where nG is the load coefficient for shear influences assumed by the value 5.25. (the same coefficient for normal influences is assumed by the value 6.0) then the set of differential Eq. (13) in actual shape is given by

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

Fig. 7.

79

Unit axial deformations assumed.

Fig. 8. Normed sectorial functions due to shear deformations assumed.

冤

E

0.7805 0

0

0

0.3006

0

0

8.562983 0

0

0

−0.9018

0

0

1.465688 0

0

0

0

0.3006 0 0

−0.9018

15.748542 0

0.445004 0 0

0.445004 0 0.3006

−1.143872 0

冥冤 冥 vz

vx vy

−1.143872 r 0 0.200400

v1 v2

IV

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A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

冤

⫺G

0 0 0 0

0

0

0 0 0 0

0

0

0 0 0 0

0

0

0 0 0 0.012873 0

0

0 0 0 0

0.030795 0

0 0 0 0

0

冥冤 冥 冤 冥 vz

II

0

0

vx vy r

⫽1

1

4.5

v1

0

0.2053 v2

0

.

Adopting the matrix

冤 冥 1 0 0 0 0 0

0 1 0 0 0 0

AI⫽

0 0 1 0 0 0 0 0 0 1 0 0 k1 0 k1 0 1 0

0 k2 0 k2 0 1

and using the multiplications J⫽AI J ATI, K⫽AI K ATI, the resulting normed matrix J in diagonal shape is obtained, given by

冤

J⫽

0.780600 0

0

0

0

0

0

0.85629830 0

0

0

0

0

0

1.465688 0

0

0

0

0

0

15.748642 0

0

0

0

0

0

0.049733 0

0

0

0

0

0

0.378455

冥

.

The normed sectorial functions of shear deformations in elastic joints are given by w⫽AI w, e.g., wi⫽w1⫹k11⫹k2y⫽w1⫺0.3006/0.78061⫺0.445004/1.465688 y, w2⫽w2⫹k2x⫹k2w⫽w2⫹0.9018/8.562983 x⫹1.143872/15.748542 w The diagrams of the above functions are shown in Fig. 8. Adopting the load vector p = AI p = P (0 0 1 1 k1y 4.5 k2w)T the following equations pay there for assessment of the influence of elastic joints in the example studied

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

81

II E 0.049733 vIV 1 ⫺G 0.030795 v ⫽⫺0.445004/1.465688 P, II E 0.378554 vIV 2 ⫺g 0.020530 v ⫽1.143872/15.748542 P 4.5.

The first equation approaches the influence of elastic couplings when dealing with vertical flexure whereas the second equation approaches the same problem when treating the torsional behaviour of the bridge structure studied. For assumed span L =70 m and for symmetric midspan load P the torque acting is given by

冉冊 冉 冊

L 70 ⫽P ⫽P 17.5 kNm 2 4

My and for

b1⫽√(GdK11/(EJ11))⫽√(81 0.030795)/(210 0.049733)⫽0.488710, [1/m] the resulting flexural moment is given by

冉 冊

M1(L/2)⫽P(⫺0.303624/(2b1))th b1

L ⫽P(⫺0.310618). [kNm] 2

The resulting stress state for vertical flexure is then given by s⫽sy⫹s1⫽My/Jyyy⫹M1/J11w1⫽P(17.5/1.46588 y⫺0.310618/0.049766 w1) Similarly, for the eccentric midspan load acting on the first main girder there may be written b1⫽√(G Kv/(E Jw))⫽√81 0.012873/(210 15.748542)⫽0.017756 1/m and the corresponding moment is given by Mw

冉冊

L ⫽P 4.5/(2 0.017756)0.552144⫽P 69.966434. [kNm2] 2

For b2⫽√G dK22/(E J22)⫽√81 0.02053/(210 0.378455)⫽0.144651 [1/m] there pays M2

冉冊

L ⫽P 4.5 0.072633/(2 0.144651)0.999920⫽P 1.129700 kNm 2

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A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

Table 1 Resulting stress components for rigid cross-section and for the cross-section having elastic joints in the contact points between thin-walled members and roadway slabs for the first case studied (thin-walled bridge with open cross-section) MPa

a

bd

ce

b⬘ d⬘

a⬘

1

sy

⫺1.8449

s1

0.1078

3

sy+s1

⫺1.7371

4

sw

⫺5.0605

5

s2

0.0674

6

sw+s2

⫺4.9931

7

sy+sw

⫺6.9054

8

sy+sw+s1+s2 ⫺6.7302

⫺1.8449 ⫺11.0694 28.3319 0.1078 ⫺5.5986 0.6591 ⫺1.7371 ⫺16.6680 28.9910 0 0 0 0 0 0 0 0 0 ⫺1.8449 ⫺11.0694 28.3319 ⫺1.8449 ⫺11.0694 38.3319

⫺1.8449 ⫺11.0694 28.3319 0.1078 ⫺5.5986 0.6591 ⫺1.7371 ⫺16.6680 28.9910 3 .7954 22.7721 ⫺43.2020 ⫺0.0506 2.6817 ⫺0.5379 3.7448 25.4538 ⫺43.7399 1.9505 11.7027 ⫺14.8701 2.0077 8.7858 ⫺14.7489

⫺1.8449

2

⫺1.8449 ⫺11.0694 28.3319 0.1078 ⫺5.5986 0.6591 ⫺1.7371 ⫺16.6680 28.9910 ⫺3.7954 ⫺22.7721 43.2020 0.0506 ⫺2.6817 0.5379 ⫺3.7448 ⫺25.4538 43.7399 ⫺5.6403 ⫺33.8415 71.5339 ⫺5.4819 ⫺42.1218 72.7309

0.1078 ⫺1.7371

5.0605 ⫺0.0674

4.9931

3.2156

3.256

Resulting stress state for torsional behaviour is then given by s⫽sw⫹s2⫽Mw/Jww⫹M2/J22w2⫽P(69.966434 w/15.748542 ⫹1.129700 w2/0.378542). In accordance with above relations in Table 1 are compared the stresses in the nodes of the cross-section studied as shown in Fig. 9.

Fig. 9.

Nodal points where the stress components have been calculated.

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

83

Fig. 10. Cross-sectional parameters of the box cross-section studied.

5.2. Behaviour of the closed cross-section as shown in Fig. 5b The values of sectorial parameters of the rigid cross-section in accordance with the set of input data specified in Fig. 10, are given by Az⫽0.780680 m2, Jxx⫽7.633492 m4, yo⫽0.9273 m, Jyy⫽1.461390 m4, fT ⫽0.049389 m2, KT⫽2.933757 m4, Jwx⫽⫺5.387285 m5, yd⫽0.705743 m, Jww ⫽0.543518 m6. Sectorial parameters due to unit shear deformations dw =1 assumed in the joints correspond with warping functions as shown in Fig. 11.

Fig. 11.

Warping functions due to unit shear displacements in joints assumed.

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A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

For dw1 =1 assumed there pay J11⫽0.300680 m4, J1z⫽0.30068 m4, J1y⫽0.445133 m4. In the analogy with aforementioned considerations is there derived the matrix equation of the problem studied given by

冤

0.76080

0

0

0

0.300680

0

7.633493

0

0

0

0

0

1.466139

0

0.445133

0

0

0

0.543552

0

0.300680

0

0.445133

0

0.300680

0

0

−0.288.140

0

−0.100510

0

0.063565

E

冤

⫺G

0 0 0

0

0

0

0 0 0

0

0

0

0 0 0

0

0

0

0

0.09878

0.030795

0

0 0 0 2.946630 0 0 0

0

0 0 0 0.098780

0

0

冥冤 冥 vz

I

−0.288140 vx 0

vy

−0.100510 r

v1 v2

冥冤 冥 冤 冥 vz

II

0

0

vx vy

1

⫽

r

4.5

v1

0

0.023856 v2

0

as well as the transformation matrix for the first stage of norming

AI⫽

冤

冥

1

0

0

0 0 0

0

1

0

0 0 0

0

0

1

0 0 0

0

0

0

1 0 0

−0.385151

0

0

0.037747

−0.309609 0 1 0 0

0 0 1

With partially normed matrix J = A J is there obtained the frequency equation given by det

冋

0.543552 −l2.946630 −0.100509 −l0.098780

−0.100509 −l0.098780 0.052689 −l0.023856

册

⫽0,

with roots calculated and given by l1=3.004916 and l2=0.101903. The matrix multiplication

冋

册冋

0.543552 −l12.946630 −0.100509 −l10.098788

册

1 k21

−0.100509 −l10.098780 0.052689 −l10.023856 k21 1

⫽0,

A. Tesar, J. Kuglerova / Thin-Walled Structures 36 (2000) 67–88

85

there states the transformation matrix for second stage of norming given by AII⫽

冋

册冋

1 k21

k21 1

⫽

册

1

20.916437

−0.454515

1

.

The resulting transformation matrix A = AI AII is then given by

A⫽

冤

冥

1

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0

0

0

0

−0.789533

0

1

0 20.916437

−0.385151

0

−0.303609

0

1

0

0

0.037747

0

0.454515 0

1

which gives the principal equations of the problem studied by using the matrix multiplication’s J = A J AT and K = A K AT in their normed configuration as follows

冤

E

0.780680

0

0

0

0

0

0

7.633492

0

0

0

0

0

0

1.466139

0

0

0

0

0

0

27.799238

0

0

0

0

0

0

0.049726

0

0

0

0

0

0

冤

⫺G

0 0 0

0

0

0

0 0 0

0

0

0

0 0 0

0

0

0

0

0 0

0 0 0 7.733041 0 0 0

0

0.030795

0 0 0

0

0

冥冤 冥 vz

IV

vx vy r

v1

0.073612 v2

冥冤 冥 冤 冥 vz

pz

vx

px

vy r

⫽

py

pw

v1

p1

0.722376 v2

p2

.

The co-ordinates of transformed sectorial functions are given by multiplication w = A w (see Fig. 12). The right sides p of normed set of independent differential equations of the problem studied are specified by use of transformed direction vector r = (0 0 1 b 0 0)T which corresponds to the action of concentrated vertical load P midspan of the left main girder of studied bridge structure, e.g.,

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Fig. 12.

Calculated torsional and distortional characteristics of thin-walled cross-section studied.

冤 冥 0 0

r⫽

1 b

, with p⫽Pr.

−0.303609 1 0.454515 b

Using above expressions, the mechanics of thin-walled cross-section with elastic joints studied is given by E 27.799238 rIV⫺G 7.733041 rII⫽P 4.5, ⫽P (⫺0.303609),

II E 0.049726 vIV 1 ⫺G 0.030795 v1

II E 0.073620 vIV 2 ⫺G 0.722376 v2 ⫽P 2.045317.

First and third equations specify the influence of elastic couplings by torsional behaviour and the second equation specifies the same influence for vertical flexure of the structure studied. When dealing with rigid cross-section the flexural moment My(1/2) = P 17.5 kNm is needed. With the parameter b⫽√G 2.946630/(E 0.543552)⫽1.446023, the value of bimoment is there given by Mw(L/2) P 4.5/(2 1.446023)⫽P 1.555986 kNm2

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87

The resulting stress state of thin-walled box cross-section provided with elastic joints for vertical flexure with the parameters b1⫽√G 0.0303795/(E 0.049726)⫽0.488744 and M1(L/2)⫽P(⫺0.303609)/(2 0.488744)v1⫽P(⫺0.310601) is then given by s⫽sy⫹s1⫽My/Jyyy⫹Mi/J11w1⫽P(17.5/1.466139 y⫺0.303609/0.049726 w1) ⫽P(11.9361 y⫺6.1056) The stress state due to torsional behaviour of thin-walled structure studied and provided with elastic joints is then given by s⫽sw⫹s2⫽Mw/Jww⫹M2/J22w2⫽P(6.868971/27.799286w ⫹0.525642/0.073620w2)⫽P(0.2470w⫹7) Using above relations there were calculated and compared the stress states of thinwalled box structure in the nodal points studied as shown in Fig. 9. The comparisons and resulting stress states are summed up in Table 2.

6. Conclusions The problem studied comes to the fore when dealing with thin-walled long-span composite structures used nowadays. The stress states calculated illustrate the influence of elastic joints by open and closed thin-walled cross-sections studied. Present theoretical approaches are based on the adoption of technical torsional and flexural theory of thin-walled structures with deformable and non-deformable cross-sections. Further development of the present approach into the branch of non-linear static and dynamic problems taking into account elasticity of joints of thin-walled structures is significant in accordance with developments of new slender types of thinwalled structures made of combinations and interactions of different or new materials adopted, such as steel, light-weight concrete, laminated wood, glass, aluminium, glass-fibre, plastics, kevlar, etc. Ultimate behaviour of such structures appears today as challenge for structural engineering of future. Present concept allows fundamental theoretical approaches and hand-made calculations, verifications and assessments of the problem studied. Further developments and applications of present fundamental approach for numerical analysis of advanced problems of ultimate static and dynamic behaviour of thin-walled structures with elastic joints, new materials and with structural inhomogeneities were submitted, for example, in references [1–8].

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Table 2 Resulting stresses for cross-section with rigid and elastic joints between thin-walled members and roadway slabs (thin-walled bridge with box cross-section) MPa

a

bd

ce

b⬘ d⬘

a⬘

1

s1

⫺1.8447

s1

0.1079

3

sy+s1

4

sw

0.0321

5

sw

⫺0.3132

6

s2

0.4729

7

sw+s2

0.1597

8

sy+sw

⫺1.7982

9

sy+s1+sw+s2 ⫺1.5771

⫺1.8447 ⫺11.0684 28.3208 0.1079 ⫺5.5990 0.6591 ⫺1.7368 ⫺16.6674 28.9799 0 0 0 0 0 0 0 0 0 0 0 0 ⫺1.8447 ⫺10.0684 28.3208 ⫺1.7368 ⫺16.6674 28.9799

⫺1.8447 ⫺11.0684 28.3208 0.1079 ⫺5.5990 0.6591 ⫺1.7368 ⫺16.6674 28.9799 0.4730 2.8381 ⫺6.3456 0.3080 ⫺3.3202 ⫺2.0871 0.1671 8.1436 ⫺5.0676 0.4751 4.8234 ⫺7.1547 ⫺1.3573 ⫺8.1440 22.0604 ⫺1.2617 ⫺11.8440 21.8252

⫺1.8447

2

⫺1.8447 ⫺11.0684 28.3208 0.1079 ⫺5.5990 0.6591 ⫺1.7368 ⫺16.6674 28.9799 ⫺0.4730 ⫺2.8381 6.3456 ⫺0.3080 3.3202 2.0871 ⫺0.1671 ⫺8.1436 5.0676 ⫺0.4751 ⫺4.8234 7.1547 ⫺2.3033 ⫺13.8202 34.7516 ⫺2.2119 ⫺21.4911 36.1346

⫺1.7368

0.1079 ⫺1.7368 ⫺0.0321

0.3132 ⫺0.4729 ⫺0.1597 ⫺1.8624 ⫺3.5992

References [1] Tesar A. Influence of diaphragms on distortional behaviour of thin-walled structures. Computers and Structures 1998;4:1234–40. [2] Tesar A. Advanced analysis of thin-walled structures in bridge and structural engineering. Lectures for PhD-seminar, Espoo-Oulu, Helsinki University of Technology, Laboratory of Bridge Engineering, June 1992. [3] Tesar A. Advanced analysis of thin-walled structures in bridge and structural engineering. Lectures for PhD-seminar, Espoo, Helsinki University of Technology, Laboratory of Bridge Engineering, December 1998. [4] Tesar A, Fillo L. Transfer matrix method. Dordrecht/Boston/London: Kluwer Academic Publishers, 1988. [5] Tesar A. Tuned vibration control of long span bridges. Building Research Journal 1998;1:12–26. [6] Tesar A. Shear lag in the behaviour of thin-walled box bridges. Computers and Structures 1996;59(4):764–72. [7] Paavola H, Loikkanen P, Jutila A. Sillanrakennustekniikan perusteet. Espoo: Otapaino, 1984. [8] Tesar A. Parallel processing approach for analysis of corrosion damages. Building Research Journal 1997;1:24–42.