Joint Functioning of Crane Rails and Industrial Buildings Crane Girders Under Local Crane Loads

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ScienceDirect Procedia Engineering 150 (2016) 1853 – 1860

International Conference on Industrial Engineering, ICIE 2016

Joint Functioning of Crane Rails and Industrial Buildings Crane Girders under Local Crane Loads V.F. Saburova, V.V. Ankudinova,* a

South Ural State University, 76, Lenin Avenue, Chelyabinsk, 454080, The Russian Federation

Abstract The paper describes the theory and calculation of a crane rail combined with the upper zone of a crane runway girder under a local torque. Rail is analyzed as a continuously long girder on a solid elastic base loaded with a local torque. The crane runway girder web is simulated as an elastic base. The expressions of the rail rotation angle and girder web as well as the value of full torque are obtained as a result of solving the differential equation of girder twisting on the elastic half-plane. It is shown that these parameters depend not only on the torsional stiffness of crane rail, but also on the upper zone of crane runway girder as well as the elastic characteristic of the lower layer. Obtained expressions can be used to analyze stress strain state of both crane rail and the upper zone of crane runway girder web. ©2016 2016The TheAuthors. Authors. Published by Elsevier © Published by Elsevier Ltd. Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016 Keywords: Crane rail; Crane runway girder; Local crane loads; Bending torsion; Compliance;

1. Introduction Crane rails serve two functions in the crane runways of industrial buildings. Firstly, rails are the runways of bridge cranes, and must possess the necessary strength and wear-resistance. Secondly, rails distribute considerable bridge crane wheel loads on the crane runway girder webs and should have necessary stiffness in bending and torsion. The realization of the second function requires the improvement of the structural form of domestic rails and related geometrical characteristics of efficiency, which in turn makes it necessary to evaluate their stress-strain state.

* Corresponding author. Tel.: +7-963-473-1748. E-mail address: [email protected]

1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016

doi:10.1016/j.proeng.2016.07.182

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The problem of the fatigue strength of the elements of welded crane girders arose in the 50s of XX century because of the mass transfer from riveted to weld connections. The crane rail is the only stress state regulator of the upper zone of the web in welded crane girders. The theoretical solution to the problem of distribution of crane wheel vertical pressure on the girder web is given in the monograph [1], where the rail is considered as an element that is freely supported by the girder flange. Further investigation showed that the load is transferred to the girder flange through microroughnesses, the so-called "contact patch" [2, 3]. The roughness problem is solved by the use of low-modulus pads under the lower flange of rail [4], but it does not solve the problem of the durability of crane runways as a system because the work of the crane rail in the new conditions of bearing has not been investigated. In the crane runways the crane rails take loads from the wheels of bridge cranes and are in a combined stress state, as indicated by different types of deterioration and wear [5]. The stress state of the rail is composed of the stresses of the overall bending and torsion, local underhead stresses caused by cross-sectional shape, and the contact stresses in the head. 2. Theoretical part The crane rail on distance between the joints can be represented as a continuously long girder on an elastic base which is loaded with local vertical and horizontal loads. The elastic base here is the crane girder web, which can be pre-written in the form of elastic half-plane [6]. Let us consider the crane rail as a continuously long girder on the elastic base, which is loaded with the torque Mt (Fig. 1) at the beginning of coordinates. It is conceived that the separation of rail lower flange from the girder flange does not occur at the torsion process due to the securing work of the rails. From this precondition it follows that the rail rotation angle matches with the rotation angle of girder flange. Elastic properties of the base characterize the compliance coefficient under the torsion kt.

Fig. 1. Calculation scheme of the crane rail for torsion.

According to [7], the differential equation of girder twisting on the elastic base takes on the form:

EI I ˜4( X ) IV  GI d ˜4( X ) IV

m( X )

(1)

where EIij – rail stiffness under bending torsion; Ĭ(X) – rail rotation angle matching with the rotation angle of girder flange (Fig. 1); GId – rail stiffness under free torsion; m(X) – reactive moment of elastic base. Reactive moment m(X) can be represented as m( X )

 kt ˜ 4 ( X )

(2)

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Inserting (2) in (1) and dividing term by term into EIij, we will receive the differential equation of torsion of continuously long rail situated on the elastic base:

4( X ) IV  2 ˜ r 2 ˜4( X ) II  s 4 ˜4( X ) where r 2

GI d / 2 EI I ; s 4

0

(3)

kt / EI I

(4)

The solution of one differential equation (3) is sought in the form:

4( X )

eD X

(5)

where Į – unknown multiplier which needs to be determined. At the pre-assigned function of rotation angle (5) the differential equation (3) takes on the form:

D 4 ˜ eD X  2 ˜ r 2 ˜ D 2 ˜ eD X  s 4 ˜ eD X

(6)

0

Having reduced the equation (6) by the common multiple eD X , we obtain the characteristic equation for determination of the multiple Į

D 4  2 ˜ r 2 ˜D 2  s4

(7)

0

The roots of biquadratic equation (7) are calculated by the formulas (8) and depend on the correlation of parameters s and r:

D1...4

r r 2 r r 4  s4

(8)

Let us conduct the analysis of real correlations between s and r for welded crane runway girders and crane rails. From the expressions (4) it follows that the parameter value r is determined by the rail geometric characteristics at the torsion. It is taken into consideration that G E / 2(1  P ) , parameter r2 can be recorded as the following form: r2

I 1 ˜ d 4 1  P I I

(9)

where ȝ – the Poisson ratio of the girder material. Using the values Id and I from [8], and taking ȝ=0.3, we calculate the parameter r. The calculation results are presented in Table 1. Table 1. Characteristics of crane rails. Rail

CR50

CR60

CR70

CR80

CR100

CR120

CR140

Id, cm

78

137

253

387

765

1310

2130

Iij, cm6

1820

5920

14100

26400

76400

191000

384000

4

Id/Iij

0.043

0.023

0.018

0.015

0.010

0.007

0.006

r, cm-1

0.091

0.067

0.059

0.053

0.044

0.036

0.033

In order to find the parameter s it is necessary to know the compliance coefficient of elastic base kt. From the expression (2) it follows that the kt can be presented in the form:

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kt

(10)

 m( X ) / 4 ( X )

The rotation angle of girder flange Ĭ(X), according to [9], under the action of concentrated torque Ɇt, applied in the middle of the girder panel equals: ª J x  0,5 d  J x  0,5 d º Ɇt « e e  » & » J d 2p « e J 1  ¬« ¼»

4( X )





(11)

In the formula (11) the following notations are used: p

4 Dw ˜ hw  2 1  P ˜ D f b f  GI d 15

(12)

where Dw – web cylindrical stiffness; Df – flange cylindrical stiffness; hw – girder web height; tw – web thickness; bf – upper flange width; tf – upper flange depth; d – panel length; Ȗ = (p/q)0,5, where q

3 Dw ˜ hw3 D f ˜ b f   EI I 105 12

(13)

The calculations by formulas (12), (13) of typical crane runway girders showed that the distribution of stress in the web as the result of the action of local torque is implemented only by means of the torsion stiffness of the flange (6–15%) and crane rail. In accordance with the calculation scheme we believe that the reluctance torque m(X) is distributed along the length of the girder panel d in accordance with the law of changes in local bending stresses. To simplify the calculations we take a triangular diagram instead of a curved triangular diagram (Fig. 1). After such prerequisite, we can write 0,5·m(X)·d = Ɇt, whence m( X )

(14)

2˜ Mt / d

Inserting the values (11) and (14) in (10), we obtain the expression for the calculation of elastic compliance coefficient of the crane runway girder under the local torsion:

kt

ª º J 1  e J d 4˜ p « » d « x ˜ J 1  e J d  ªeJ  x  0,5 d  e J  x  0,5 d º » ¬ ¼ ¼» ¬«









(15)

From (15) it follows that x=0 and x=d kt ĺ’, and when x=0,5d (the middle of the panel) the compliance coefficient has a minimum value under the torsion.

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kt

ª J 1  e J d 4˜ p « d « 0,5d ˜ J 1  e J d  1  e J d ¬«









º » | 8˜ p » d2 ¼»

(16)

In the Fig. 2 (a) the dependence of the coefficient kt on the parameter p is presented, in the Fig. 2 (b) – on the panel length d. From Fig. 2 (b) it follows that with increasing length of the web panel d the web compliance coefficient for torsion decreases. Therefore, it is rational to take into account the web stiffness characteristics in the calculation only at the distance between the strong backs of not more than 2 meters.

Fig. 2. (a) dependence of the coefficient kt on the parameter p; (b) dependence of the coefficient kt on the length of the panel d.

Taking into account the expressions (16) the parameter s4 takes on the from: s4

8˜ p d ˜ EI I

(17)

2

The calculations by the formula (17) showed that for weld crane runway girders the value s fluctuates over the range 0.041–0.047. Let us calculate the roots of the biquadratic equation (7): ­ °D1 ® °¯D 3

r 2  r 4  s 4 ;D 2

 r 2  r 4  s4

r 2  r 4  s 4 ;D 4

 r 2  r 4  s4

(18)

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Knowing the roots of characteristic equations, we can record the solution of homogeneous differential equation (5) in the form:

4( X )

C1 ˜ eD1 X  C2 ˜ eD 2 X  C3 ˜ eD3 X  C4 ˜ eD4 X

or

C1 ˜ eD1 X  C2 ˜ eD1 X  C3 ˜ eD3 X  C4 ˜ eD3 X

4( X )

(19)

where ɋ1, ɋ2, ɋ3, ɋ4 – constants of integration determined from boundary conditions. Having the expression for the common integral of the rotation angle Ĭ(X) (19), we can obtain common integrals for the deplanation ĬI and inner force factors of cross section of the rail – bimoment ȼ and full torque ɇ, using the following correlations: ­4 I w4 / wx; °° II ® B  EI I ˜4 ( X ) ; ° III I °¯ H  EI I ˜4 ( X )  GI d ˜4 ( X )

(20)

Now we will write down the expressions (20) using the expression (19) and form the boundary conditions:

4I

C1D1eD1 X  C2D1eD1 X  C3D 3 eD3 X  C4D 3 eD3 X

(21)

B

 EI I ª¬C1D12 eD1 X  C2D12 eD1 X  C3D 32 eD3 X  C4D 32 eD3 X º¼

(22)

H

ªC1D13eD1 X  C2D13 eD1 X  C3D 33 eD3 X  º  EII « »  GI d 3 D 3 X »¼ ¬« C4D 3 e

(23)

ªC1D1eD 1x  C2D1eD 1x  C3D 3eD 3 x  º « » D 3 x »¼ ¬« C4D 3 e

x when x = 0 ĬI(X) = 0, H = – 0,5Ɇt (a); x when x ĺ ’ Ĭ(X) = 0, B=0, H=0 (b). From the condition (b) it follows that the constants of integration ɋ2 and ɋ4 in the expressions (19), (21)–(23) turn out into null due to the multiple eD X , and two summands turn out into null only when ɋ3=0. The constants of integration ɋ2 and ɋ4 will be found from the boundary (a).

ɋ2 C4

Mt 2 EI I ˜ D1 ª¬D 32  D12 º¼ 

Mt 2 EI I ˜ D 3 ª¬D 32  D12 º¼

Inserting (24) in (19) and (20), we obtain the expressions for the calculations of parameters in question. Rotation angle

(24)

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§ eD1 X eD3 X ·  ¨ ¸. D3 ¹ 2 EII ¬ªD 32  D12 ¼º © D1 Mt

4( X )

(25)

Deplanation

4 I(X )

Mt 2

ªeD3 X  eD1 X º . ¬ ¼

2 1

2 EII ¬ªD 3  D ¼º

(26)

Bitorque

B( X )

Mt 2

2 ¬ªD 3  D12 ¼º

ªD 3 ˜ e D3 X  D1 ˜ eD1 X º . ¬ ¼

(27)

Full torque

H( X )

Mt 2

2 ¬ªD 3  D12 ¼º

 ª¬D

2 1



˜ eD1 X  D 32 ˜ eD3 X º¼  2 ˜ r 2 ª¬eD3 X  eD1 X º¼ .

(28)

Using the expressions (25) and (26), the values of rotation angle and bimoment have been calculated in the rail CR80, set on the girders by the bay of 6 and 12 m under the action of concentrated moment Ɇt=10 kN·m. From diagrams presented in the Fig. 3 it follows that the rotation angle is distributed to more distance than the panel plate, and bimoment load has a pronounced local nature.

Fig. 3. The distribution of bimoment B over the length of crane rail CR80 under the action Mt=10 kN·m: 1. Rail on the girder l=6 m (kt=9,7×10-2 kN/rad); 2. Rail on the girder l=12 m (kt=21,2×10-2 kN/rad).

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From the Fig. 3 it is clear that the bimoment load has a local nature and is damped already at the distance of 10– 20 cm from the application place of the local moment. The calculation value of the local torque can be calculated with the accountancy of recommendations [10]. 3. Conclusion

The accountancy of the work of rail and flange of the crane runway girder allows approximating the calculation model to the real work of the rail in crane runways of industrial buildings. The given theory of torsion of crane rail can be used in the analysis of its stress-strain state and for the stress-stain state of fuel pins of the web upper zone of crane girders. References [1] B.M. Broude, The Distribution of Localized Pressure in Metallic Girders, Stroiizdat, Ɇoscow, 1950. [2] Iu.I. Kudishin, The Distribution of Localized Pressure in the Crane Runway Girder Web Under the Irregularities on the Rail and Top Flange Upper Contact Surfaces, Materials on Metallic Constructions. 12 (1950) 123–129. [3] G. Maas, New Research of the Fatigue Strength of Crane Runways, Ferrous Metals. 19 (1971). [4] O. Steinhardt, U. Schulz, A local web stress centric loaded crane runway girder using elastically bedded crane rails with, The Civil. 44 (1969). [5] V.F. Saburov, Laws of Wearing Process and Peculiarities of Stress Behavior of Crane Rails of Crane Runways of Industrial Buildings, in: Proceeding of Metallic Constructions: Look in the Future and the Past, Materials of the VIII Ukraine Scientific and Technological Conference, Kiev. (2005) 624–630. [6] V.F. Saburov, The Use of Models of Elastic Basis for the Analysis of the Local Pressures Distribution in the Web of Hybrid Beams, YuUrGU Vestnik, Series Construction and Architecture. 4 (2014) 15–20. [7] V.Z. Vlasov, N.N. Leontev, Girders, Plates and Building Envelope on the Elastic Basis, Fizmatgih, Ɇoscow, 1960. [8] E.A. Mitiugov, To the Determination of Rotational Inertia of Crane Rails, Construction Mechanics and Structure Calculation. 5 (1968) 46–47. [9] E.A. Mitiugov, The Twisting of Top Flange of Crane Runway Girders, Metallic Constructions, in: Proceeding of Collected Works of MCEI named after V.V. Kuibyshev, Moscow. 85 (1970) 60–67. [10] Design and Construction Specifications 16.13330.2011, Steel Constructions, Revised Edition of SNiP II-23-81*, TsPP, Moscow, 2011.