# Method to Determine the Allowable Deformation of Crane Runway Girders

### Technical Article

3 December 2019

001612

This article describes the different options to determine the allowable deformation of crane runway girders. Since multi-span beams and flexible lateral supports (sway bracing) are used in practice, this article will show how to select the correct method.

#### General

In addition to the ultimate limit state design, the serviceability limit state design is of particular importance for crane runway girders. Compliance with deformation limit values is not only important for the serviceability but also for the reduction of wear. Thus, large horizontal deformations can lead to an increased skewing of the crane and thus cause an increased wear of tracking means. Vertical deformations must also be avoided to the greatest possible extent in order to avoid excessive vibration of the crane during operation. Finally, it is also necessary to limit the inclination (slope) of the crane runway girder because otherwise the crane will not be able to move under full load.

#### Method 1: Deformation Related to Undeformed System

Method 1 can be used for single-span beams with fixed and rigid supports.

The following boundary conditions apply:
${\mathrm\delta}_{\mathrm y,\mathrm z}\;<\;\frac{\mathrm L}{\mathrm X}\;\mathrm{or}\;{\mathrm\delta}_{\mathrm y,\mathrm z}\;<\;25\;\mathrm{mm}$
The deformation is determined as follows:
${\mathrm\delta}_{\mathrm y,\mathrm z}\;=\;\left|\frac{{\mathrm U}_{\mathrm c}\;-\;{\mathrm U}_{\mathrm L}\;-\;({\mathrm U}_{\mathrm R}\;-\;{\mathrm U}_{\mathrm L})\;\cdot\;\mathrm x}{\mathrm L}\right|\;<\;\frac{\mathrm L}{\mathrm X}$

Uc ... Deformation of the cross-section
UL ... Deformation of the left support
UR ... Deformation of the right support
x ... Coordinate of the cross-section in the local axis system
L ... Distance of supports

The following applies:
${\mathrm U}_{\mathrm c}\;\neq\;0,\;{\mathrm U}_{\mathrm L}\;=\;0,\;{\mathrm U}_{\mathrm R}\;=\;0$

#### Method 2: Deformation Related to Deformed System

If you define spring constants for the supports to consider flexible supports, you can use Method 2 under Details. The example file 2, which you can download below this article, contains defined springs for the vertical supports. Figure 02 shows the difference between Method 1 and Method 2.

The following boundary conditions apply:
${\mathrm\delta}_{\mathrm y,\mathrm z}\;<\;\frac{\mathrm L}{\mathrm X}\;\mathrm{or}\;{\mathrm\delta}_{\mathrm y,\mathrm z}\;<\;25\;\mathrm{mm}$
The deformation is determined as follows:
${\mathrm\delta}_{\mathrm y,\mathrm z}\;=\;\left|\frac{{\mathrm U}_{\mathrm c}\;-\;{\mathrm U}_{\mathrm L}\;-\;({\mathrm U}_{\mathrm R}\;-\;{\mathrm U}_{\mathrm L})\;\cdot\;\mathrm x}{\mathrm L}\right|\;<\;\frac{\mathrm L}{\mathrm X}$
The following applies:
${\mathrm U}_{\mathrm c}\;\neq\;0,\;{\mathrm U}_{\mathrm L}\;\neq\;0,\;{\mathrm U}_{\mathrm R}\;\neq\;0$

The spring stiffnesses of the supports should have similarly large values when using this method.

#### Method 3: Deformation Related to the Inflection Points of the Deformed System

This method is used for continuous beams. Compared to a single-span beam, it does not make sense to use the distance of the supports to determine the allowable deformation for multi-span beams. This can lead to conservative, uneconomical results. In order to determine the governing length, the inflection points of the bending line are determined in Method 3.

The following condition is applied:
$\mathrm\omega''\;=\;-\;\frac{{\mathrm M}_{\mathrm y}}{\mathrm E\;\cdot\;{\mathrm I}_{\mathrm y}}$
At the inflection points:
$\mathrm\omega''\;=\;0$
The deformation is determined as follows:
${\mathrm\delta}_{\mathrm y,\mathrm z}\;=\;\left|\frac{{\mathrm U}_{\mathrm c}\;-\;{\mathrm U}_{\mathrm{Li}}\;-\;({\mathrm U}_{\mathrm{Ri}}\;-\;{\mathrm U}_{\mathrm{Li}})\;\cdot\;\mathrm x}{\mathrm L}\right|\;<\;\frac{\mathrm L}{\mathrm X}$

Uc ... Deformation of the cross-section
ULi ... Deformation of the left inflection point
URi ... Deformation of the right inflection point
x ... Coordinate of the cross-section in the local axis system
L ... Distance between left and right inflection points

Another advantage of this method is that the supports can also have different spring stiffnesses.

#### Crane Runway Girder with Cantilevers

For cantilevers, the bending line is similar to the half-inverted bending line of a single-span beam. Therefore, the following calculation is performed for Method 1:
${\mathrm\delta}_{\mathrm y,\mathrm z}\;=\;\left|{\mathrm U}_{\mathrm c}\right|\;<\;2\;\cdot\;\frac{\mathrm L}{\mathrm X}$
If Method 3 is activated, the limit deformation of a cantilever is checked by rotating the cantilever on the support about the local y-axis.

The limit condition is as follows:
${\mathrm\varphi}_{\mathrm y}\;<\;\frac1{200}$

Results cantilever of example file 4 with Method 1:
${\mathrm\delta}_{\mathrm z}\;=\;\left|7.618\;\mathrm{mm}\right|\;<\;2\;\cdot\;\frac{2.000}{600}\\{\mathrm\delta}_{\mathrm z}\;=\;\left|7.618\;\mathrm{mm}\right|\;<\;6.667\;\mathrm{not}\;\mathrm{met}\\\mathrm{In}\;\mathrm{other}\;\mathrm{words}:\\\frac{7.618}{2\;\cdot\;2.000}\;=\;525\;>\;600$

Results cantilever of example file 4 with Method 3:
${\mathrm\varphi}_{\mathrm y}\;=\;4.258\;\mathrm{mrad}\;=\;0.004258\;\mathrm{rad}\;<\;\frac1{200}\;\mathrm{rad}\\0.004258\;<\;0.005\;\mathrm{true}\\\frac{0.004258}{0.005}\;=\;0.851\;<\;1$

You can see that the allowable deformation according to Method 1 for the cantilever is not met. However, the German National Annex to EN 1993-6 specifies ${\mathrm\delta}_{\mathrm y,\mathrm z}\;<\;\frac{\mathrm L}{500}\;\mathrm{and}\;{\mathrm\delta}_{\mathrm y,\mathrm z}\;\leq\;25\;\mathrm{mm}$ for the allowable vertical deformation according to Table 7.2 row a).

#### Summary

To ensure correct functioning of a crane system, deformations and displacements must be limited. As a result, the wear is also limited. If the limit conditions of the serviceability limit state design are met, it is usually not necessary to carry out a separate vibration design for the crane runway girder.

#### Dipl.-Ing. (FH) René Flori

Head of Customer Support

Mr. Flori is the customer support team leader and also provides technical support for customers of Dlubal Software.

#### Reference

 [1] Seeßelberg, C.: Kranbahnen - Bemessung und konstruktive Gestaltung nach Eurocode, 4. Auflage. Berlin: Beuth, 2014

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