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Ultimate strength and twisting analyses analysis of cable ladders BVS-Niedax Norge AS

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TABLE OF CONTENTS

1 SUMMARY ............................................................................................................... 3

1.1 MAIN RESULTS AND CONCLUSIONS .................................................................................................... 3

2 INTRODUCTION ........................................................................................................ 6

2.1 OBJECTIVES AND SCOPE ............................................................................................................. 6

2.2 DESIGN BASIS ........................................................................................................................ 6

2.3 REFERENCES ......................................................................................................................... 6

2.3.1 Rule references .............................................................................................................. 6

2.3.2 Source data and project documents ...................................................................................... 6

3 ANALYSIS APPROACH ................................................................................................. 7

3.1 ANALYSIS PRINCIPLES ................................................................................................................ 7

3.2 LOADS ............................................................................................................................... 7

3.3 MATERIAL PROPERTIES ............................................................................................................... 8

3.4 ANALYSIS SOFTWARE ................................................................................................................ 8

4 ANALYSIS MODEL ...................................................................................................... 9

4.1 CO-ORDINATE SYSTEM ............................................................................................................... 9

4.2 DIMENSIONS ......................................................................................................................... 9

4.3 STRUCTURAL GEOMETRY ............................................................................................................. 9

4.4 FE MODEL .......................................................................................................................... 13

4.4.1 Ultimate strength model .................................................................................................. 13

4.4.2 Twisting analysis model ................................................................................................... 15

4.4.3 Definition of initial distortions for second stage twisting analysis model ....................................... 18

4.5 MODELLING OF THE LOADS.......................................................................................................... 18

4.5.1 Modelling of loads in ultimate strength model ....................................................................... 18

4.5.2 Modelling of loads for twisting analysis model ....................................................................... 19

4.6 MODELLING OF BOUNDARY CONDITIONS ............................................................................................. 20

4.6.1 Modelling of boundary conditions in ultimate strength model .................................................... 20

4.6.2 Modelling of boundary conditions for twisting analysis model .................................................... 21

5 RESULTS ................................................................................................................ 22

5.1 ULTIMATE STRENGTH ANALYSIS ..................................................................................................... 22

5.2 TWISTING ANALYSIS ................................................................................................................ 29



1 SUMMARY

In this document are presented the results of the ultimate strength analysis of four type cable ladders where

the width and plating thickness of the profiles has been varied. Considered ladders have the width of 500 and

600 mm and profile thickness of 1.5 mm and 1.25 mm. Applied pressure load on transverse profiles was

increased up to the collapse of the cable ladder.

Additionally, the cable ladder structures with various length where analysed under thermal loads that are

caused due to thermal expansion of material with temperature increase of 50 C0. The purpose was the

determination of maximum internal axial forces in continues ladder structures. For that the simplified linear

models were used. The ladder units where attached to each other by bolts and supported vertically with

specially designed hanger structures and rock bolts. In order to determine the ultimate limit condition 30 m

long detailed model was axially fixed at the ends and thermally loaded up to buckling state. This Structure

was initially distorted in order to see the influence of the misalignment due to installation. The main idea is

to determine the maximum length of the bolted ladders that will have no buckling or significant twisting in

case of temperature increase 50 C0.

1.1 Main results and conclusions

The analysis proves that the ultimate load carrying capacity of the ladders is not less than 6.15 kN by assuming

that the load is equally distributed on ladder transverse profiles and that the ladders are vertically supported

at the ends with the span of 3 m. The collapse of the ladder units starts from transverse profiles. Ultimate

strength values of different ladder units are presented in Table 1–1.

Table 1–1. Ultimate loads for ladder units.

Ladder

model

Ladder width

[mm]

wall

thickness

[mm]

Maximum

load

[kN]

Maximum

evenly

distributed

weight per

unit length

[kg/m]

Maximum

evenly

distributed

weight

[kg]

500A 500 1.5 8.09 275 825

500B 500 1.25 6.15 209 627

600A 600 1.5 8.01 272 816

600B 600 1.25 6.29 203 609



Twisting analysis proved that temperature increase of 50 C0 will not cause any remarkable out of plane

deformations in case of 30 m ladder structure 500A and 600A. In Figure 1–1 are presented elongation results

of various length cable ladders of type 500A that are deformed due to 50 C0 temperature increase. The ladder

length 30m, 99 m, 201 m and 300m gives corresponding elongation 23.9 mm, 75.0 mm, 131.2 mm and 162.4

mm.

Internal axial force in ladders 500A increase significantly when the length of the structure reaches over 99 m.

As can be seen in Figure 1–2 the increase of the structural length from 99 m to 201 m changes the internal

axil force from 2.6 kN to 8.8 kN. In Figure 1–3 is presented the situation where the temperature is increased

in 30 m long cable ladder structure so that the axial displacements at the ends of the structure are restrained.

In that case the maximum axial force in the ladder system will be around 9 kN after what the structure

buckles. This means that the safe length for installation is below 201 m. Otherwise, the internal axial forces

will become equal to buckling force and that is not recommended. Therefore, the recommended maximum

length for continues installation is 150 m. In that case the approximated out of plane displacement is 4 mm.

The same limit is applicable for type 600A ladder system as well.

Figure 1–1 Elongation of cable ladders of type 500A with various length.



Figure 1–2 Internal axial force various length cable ladders caused by temperature increase of 50 C0.

Figure 1–3. Axial force in cable ladder with the length of 30 m that is fixed at the ends and loaded thermally up to buckling.

Buckling of 30 m

long ladder

system 8.9 kN



2 INTRODUCTION

2.1 Objectives and scope

In the present report the ultimate strength of 3 m long cable ladders with the height of 75 mm is analysed in

case of bending. Two types of ladders with the width of 500 mm and 600 mm are considered. In addition to

the width the plate thickness of longitudinal and transverse profiles is varied as well where the plate thickness

of 1.5 mm and 1.25 mm are considered.

In addition to ultimate strength analysis the twisting analysis of ladder structures with various length (30 m,

99m, 201 m and 300 m) is carried out. It is assumed that the ladder structure is installed with “rock bolt“ to

tunnel ceiling so that the installation height of ladder structure is 1 m below the ceiling level. The analyses

are carried out in two stages. First the simplified linear models are used in order to determine the maximum

internal forces. In the second stage the initially distorted structure with the length of 30 m is analysed. There,

the ends of the structure are axially fixed and temperature is increased up to buckling of the structure. From

that analysis can be determined the maximum axial force that the structure can carry before buckling. Two

stages together give the understanding about the maximum safe continues installation of ladders that will not

buckle or deform significantly in case of temperature increase of 50 Co.

2.2 Design basis

The analysis follow the DNV Recommended Practice (DNV- RP-C208) Determination of Structural Capacity by

Non-linear FE analysis Methods, see reference [1]. Loads and loading cases are defined by client. The finite

element model is defined according the drawings and 3D models delivered by client, see references [2] and

[3] .

2.3 References

2.3.1 Rule references

[1] DNV-RP-C208 Determination of Structural Capacity by Non-linear FE analysis Methods. June 2013.

2.3.2 Source data and project documents

[2] Tunnel ladder 75 mm SS. BVS4053-4054.pdf

[3] Glide connector for 75 mm high ladder. BVS4062.pdf



3 ANALYSIS APPROACH

3.1 Analysis principles

The ultimate strength analysis are carried out with the non-linear finite element code. The single ladder unit

is supported by rigid supports and loading is applied to transverse elements. The analysis can consider the

effect of geometrical and material nonlinearities. The structure is loaded with in time increasing pressure

load applied to transverse elements. The load is increased up to collapse of the structure.

The twisting analysis are conducted in two stages where in the first stage the linear finite element code is

used. In the second stage the non-linear finite element code is used. There 10 ladder units are attached to

each other and fixed with help of rock bolts to the ceiling of the tunnel. Also the ends of the structure are

axial fixed. The loading is applied by the increase of temperature ΔT=50 C°. The thermal expansion for the

30 m ladder structure takes place and rock bolts together with end supports restrain the extension of ladder.

Therefore, the ladder structure may experience out of plane displacements as twisting of ladder profiles. The

simulation will indicate the size of out of plane displacements and possible buckling issues. In order to be

more conservative it is assumed that the ladder structure have initial distortions defined as antisymmetric 5%

of out of plane displacements (5% of descending degree) with respect to ladder width. So 500A ladder has

initial displacements of 25 mm and 600A ladder 30 mm.

3.2 Loads

The loads that are considered in ultimate strength analysis and in twisting analysis are presented in Table 3–

1.

Table 3–1 Loads used in analysis

Load name

Load components

Pressure Temperature load Acceleration

MPa ΔC° mm/s2

Pressure on transverse profiles in order to model

the weight of the cables Max 0.05

Gravity load applied to structural elements of

the ladder structure 9810

Temperature load 50



3.3 Material properties

Material properties of the stainless steel (316) used in analysis are as follows

Modulus of elasticity � � 210 ∙ 10��

Poisson ratio � � 0.3

Density � � 7800 �� ⁄

Linear thermal expansion coefficient � � 16.0 ∙ 10�� ∙ ��⁄

Yield stress �� 380�

Tensile strength �� 573�

Failure train !" � 0.48

The non-linear material model is presented in Figure 3–1.

Figure 3–1 Stress-strain curve of the non-linear material model.

3.4 Analysis software

Windows-native pre- and post- processor FEMAP v11.4.2 is used for FEA modelling and solver NX NASTRAN

10.1 for linear FE analysis. The non-linear analysis are carried out with LS.DYNA 3D R7.1.2.

Yield stress

380 MPa

Tensile strength

573 MPa



4 ANALYSIS MODEL

4.1 Co-ordinate system

• The FE model is based on a right handed Cartesian co-ordinate system. Co-ordinates are defined as

follows.

- X in, transverse direction

- Y in longitudinal direction

- Z in vertical direction,

4.2 Dimensions

The quality of the dimensions used in the modelling and in the analysis is presented in Table 4–1.

Table 4–1 Choice of model dimensions

Quantity Quality

Dimensions Mm

Forces N

Stresses N/mm2 = MPa

Masses t

Densities t/mm3=1012kg/m3

Accelerations mm/s2

4.3 Structural geometry

ultimate strength analysis

The ultimate strength analysis of the cable ladder is carried out for a single ladder unit by assuming that the

ladder is simply supported at the ends. For this analysis only single ladder structure including the longitudinal

profiles and transverse profiles is considered. For analysis are considered four type of ladders 500A, 500B,

600A and 600B. The first number describes the width of the ladder structure in mm and the letter A and B the

plate thicknesses of profiles which are correspondingly as 1.5 mm and 1.25 mm. So totally there are four

ultimate strength analysis carried out for ladders. The structure is presented in Figure 4–1.

twisting analysis

In the first stage analysis are used ladder structures of type 500A with the length of 30 m, 99m, 201 m and

300 m. Geometry consists rock bolts and ladders modelled with simplified geometry, see Figure 4–2.

For the second stage analyses the structure consists of 10 ladder modules of type 500A or 600A (plate thickness

of profiles 1.5 mm). The structure is fixed to ceiling with rock bolts with the length of 1 m. Rock bolts are



attached to specially design hangers consisting of two components hanger itself (made of 2 mm plate ) and

adapter (made of 3 mm plate ). Hanger is attached to ladder unit with four M10 bolts. Hanger and adapter

are attached to each other with two M10 bolts. Ladder units are attached also to each other. For that are

used 60x197x2 mm plates that are fixed to longitudinal profiles with four M10 bolts. For better understanding

see Figure 4–3.

Figure 4–1 Structural geometry of the cable ladder with the length of 3 m.

transverse profile

plate thickness

version

A - 1.5 mm

B - 1.25 mm

Rigid supports

Longitudinal profile

Plate thickness

version

A - 1.5 mm

B - 1.25 mm

Ladder width

version

500 - ladder width 500 mm

600 - ladder width 600 mm

Length of the ladder

module 3000 mm



Figure 4–2 Structural geometry of the cable ladder structure used for first stage analysis.

Considered length of the structure

30 m (10 ladder units)

99m (33 ladder units)



Rock bolts that fix the

ladder structure to the

ceiling of the tunnel

(length 1 m)

Ladder units

No initial distortions



Figure 4–3 Structural geometry of the cable ladder structure with the total length of 30 m (second stage analysis)

Length of the structure 30 m

(10 ladder units)

Initial antisymmetric out of plane displacements 5 % with respect to ladder width

25 mm for ladder 500A

30 mm for ladder 600A

Rock bolts that fix the

ladder structure to the

ceiling of the tunnel

(length 1 m)

Cable ladder type

500 A (width 500 mm thickness 1.5 mm)

600 A (width 600 mm thickness 1.5 mm)

Longitudinal profile of

ladder

Transverse profile of

ladder

Hanger structure

(plate thickness 2mm)

Hanger adapter for rock bolt


Plate connecting ladder units


Rock bolt



4.4 FE model

4.4.1 Ultimate strength model

The finite element model for the ultimate strength analysis is presented in Figure 4–4. The longitudinal and

transverse profiles are modelled with the shell elements with size of 5mm. Longitudinal on transverse profiles

are attached to each other using common nodes. Ladder structure is supported by two rigid supports that are

fixed in space and located at both ends. For supports the additional contact definition between the supports

and longitudinal profile of the ladder are defined. The mesh size of the model enables to described well all

opening and profile rounding’s that might have influence on global load carrying capacity. The number of

nodes and elements used in the ultimate strength model are presented in Table 4–2.

Table 4–2. Number of nodes and elements used in ultimate strength model.

Ladder

model

Number of

nodes

Number of

elements

500 A&B 69884 66613

600 A&B 73574 69500



Figure 4–4 FE-model of the cable ladder for ultimate strength analysis.

Rigid supports that

are fixed in space

Non-linear material model

Shell elements with the size of 5 mm

All structure is modelled with 4-node shell elements

Contact definition between the

supports and ladder structure

Load is applied to transverse

profilies as pressure prediscribed

in time



4.4.2 Twisting analysis model

In the first stage models the ladder units are modelled using two node beam elements with cross-section

properties equivalent to longitudinal and transverse profiles, see Figure 4–7.

c

Figure 4–5 Simplified FE-model for first stage analysis in order to determine maximum internal axial forces in

longitudinal elements.

Table 4–3. Number of nodes and elements used in twisting analysis model for first stage analysis.

Ladder

model 500A

length [m]

Number of

nodes

Number of

elements

30 m 1971 2118

99 m 6456 6948

201 m 13086 14088

300 m 19525 21022

The second stage twisting analysis model consists of 10 ladder units. As there the local failure of the plating

is not so important the more simplified approach in modelling was used. Rounded edges in profiles where

removed as well as smaller openings. General element size was taken as 8 mm. For twisting analysis two

models where developed for 500 mm ladder width with profile thickness of 1.5 mm (500A) and for 600 mm

ladder width with the profile of 1.5 mm (600A). In both cases profiles are modelled using 4-node shell

elements. Hanger structure together with hanger adapter and connecting plate are also modelled with 4-node

Length of the structure 30 m, 99 m, 201 m and 300 m.

All structural detailes modelled as 2 node beam

elements

Element size 50 mm

Element size 80 mm

Element size 100 mm



shell elements. Rock bolts are modelled with 2-node beam elements. For better understanding see Figure 4–

6. Total number of elements and nodes used in models are presented in Table 4–4.

Table 4–4. Number of nodes and elements used in twisting analysis model for second stage analysis.

Ladder

model

Number of

nodes

Number of

elements

500 A 254654 231011

600 A 270132 244018



Figure 4–6 FE-model of the cable ladder for twisting analysis.

Length of the structure 30 m (10 ladder units)

M10 bolts are modelled with 2–node beam elements.

Rock bolt that fix the ladder structure to the

ceiling of the tunnel are modelled with beam 2-

node beam elements.

Element size 8 mm

Longitudinal and transvers profiles are modelled with 4-node shell elements. (no small

opening and rounded edges)

Hanger plate, hanger adapter and

connection plate are modelled with 4-

node shell elements.



4.4.3 Definition of initial distortions for second stage twisting analysis model

For the definition of initial out of plane distortions the natural frequencies and normal modes of 30 m long

ladder structure are calculated. The mode number 17 that corresponds to the natural frequency of 12.9 Hz

for 500A ladder is presented in Figure 4–7. This natural mode is used for definition of initial deformations.

The obtained displacement field is scaled so that the maximum out of plane displacement of the ladder is

equal to 5% form the ladder width of 500 mm which is 25 mm. Thereafter, the initial mesh was updated

according to the scaled displacement field. The same approach was used for 600A. There obtained

displacement field is scaled so that the maximum out of plane displacement of the ladder is equal to 30 mm.

Figure 4–7 Normal mode that is used for definition of out of plane initial misalignment.

4.5 Modelling of the loads

4.5.1 Modelling of loads in ultimate strength model

In ultimate strength model the loading is applied on transverse profiles of a ladder unit, see Figure 4–9.

Pressure value is increased in time using prescribed loading function that is presented in Figure 4–8.



Figure 4–8 Loading function for in time prescribed pressure value.

Figure 4–9. Pressure load applied on transverse profiles having the maximum value of 7.70 kN at time 1.0 sec.

4.5.2 Modelling of loads for twisting analysis model

In twisting analysis, the model is loaded with gravity load applied on ladder structure and temperature load

that causes thermal expansion of the ladder structure. The gravity load 9810 mm/sec2 is applied in 0.2

seconds. The temperature load is applied on all nodes in a model within 1 second. The increase of the

temperature during this time is 50 Co. The load proportionality factors for the gravity and temperature load

is presented in Figure 4–10.

Pressure is applied on transverse

profiles.



Figure 4–10 Loading functions for in time prescribed temperature and gravity.

4.6 Modelling of boundary conditions

4.6.1 Modelling of boundary conditions in ultimate strength model

During the loading the ladder unit is supported by two rigid bodies that are located near the ends of the

ladder. Therefore, the contact definition between the rigid supports and longitudinal profiles of the ladder

unit are defined including the friction of coefficient of 0.3. For better understanding see Figure 4–11.

Figure 4–11 Boundary conditions for the ladder unit.

Rigid supports fixed in space will

support the ladder in case of loading. For

this the contact definition of longitudinal

profiles and rigid supports is defined.



4.6.2 Modelling of boundary conditions for twisting analysis model

For the first and second stage twisting analysis the boundary conditions are applied at the upper ends of the

rock bolts. At these nodes the displacements and rotations of are fixed, see Figure 4–12 and Figure 4–13. In

addition to that in the second stage analysis the ends of the ladder unit are fixed axially.

Figure 4–12 Boundary conditions for the ladder structures used in first stage analysis.

Figure 4–13 Boundary conditions for the 30 m ladder structure used in second stage analysis.

Upper ends of the rock bolts are fixed.

Upper ends of the rock bolts are fixed.

Ends of the ladder structure are axially

fixed.



5 RESULTS

5.1 Ultimate strength analysis

Four ladder units were analysed. Ladder 500A (ladder width 500 mm and plate thickness of profiles 1.5 mm) has the highest ultimate strength 8.09 kN. Ladder unit 600A (ladder width 600 mm and plate thickness of profiles 1.5 mm) has the ultimate strength equal to 8.01 kN. Ladder units 500B (ladder width 500 mm and plate thickness of profiles 1.25 mm) and 600B (ladder width 600 mm and plate thickness of profiles 1.25 mm) have the ultimate strength equal to 6.15 kN and 6.29 kN, see Figure 5–1 and



Table 5–1.

As a conclusion the ladders with the width of 500 mm have the same ultimate strength compared to 600 mm

ladders. The explanation is the buckling of the longitudinal profile that is the same for both ladder types. In

all cases the ladder fails only due to collapse of the longitudinal profiles. For better understanding see Figure

5–2 to Figure 5–5.

Figure 5–1 Total force versus displacement curves of ladder units in case of pressure load.



Table 5–1. Ultimate loads for ladder units.

Ladder

model

Maximum

load

[kN]

Maximum

evenly

distributed

weight

[kg]

500 A 8.09 825

500 B 6.15 627

600 A 8.01 816

600 B 6.29 609



Figure 5–2. Von-mises equivalent stress in ladder unit 500A loaded with pressure.

Total load 4.04 kN

50% from ultimate

Total load 6.07 kN

75% from ultimate

Ultimate load 8.09 kN

Failure



Figure 5–3. Von-mises equivalent stress in ladder unit 500B loaded with pressure.

Total load 3.07 kN

50% from ultimate

Total load 4.61 kN

75% from ultimate


Failure



Figure 5–4. Von-mises equivalent stress in ladder unit 600A loaded with pressure.

Total load 4.00 kN

50% from ultimate

Total load 6.00 kN

75% from ultimate


Failure



Figure 5–5. Von-mises equivalent stress in ladder unit 600B loaded with pressure.

Total load 3.14 kN

Total load 4.72 kN

75% from ultimate


Failure



5.2 Twisting analysis

First stage analysis

The first stage analysis show that the internal axial force increases significantly when the length of the ladder

structure increases. In Figure 5–6 and Figure 5–7 are presented the internal axial for and axial displacement

as function of x-co-ordinate for the structures 30 m, 99m, 201 m and 300 m. Summarised results are presented

in Figure 5–8 and Figure 5–9. So the maximum internal force for various length 30m, 99m, 201m and 300m are

correspondingly 0.3 kN, 2.6 kN, 8.8 kN, 15.2 kN. Corresponding elongations are 23.9 mm, 75.0 mm,131.2 mm,

162.4 mm.

Figure 5–6 Internal axial force for various length ladders of type 500A.



Figure 5–7. Axial displacement for various length ladders of type 500A.

Figure 5–8. Internal axial force for various length ladders of type 500A.



Figure 5–9. Axial displacement for various length ladders of type 500A.

Second stage analysis detailed model with the length of 30 m

Twisting analysis show that the thermal expansion which corresponds to 50 Co is not causing any twisting or

other type problematic deformations. The elongation of 500A and 600A ladders is almost 20 mm in longitudinal

direction (see Figure 5–10) and stresses stay in general below 50 MPa, see Figure 5–11 and Figure 5–12. Out of

plane displacements are smaller than 2 mm. So as conclusion no twisting occurs in case of thermal expansion.

The thermal expansion of 30 m cable ladder made of steel is corresponding to analytical prediction and also

to simplified model prediction, see Figure 5–9 (length of the ladder structure 30 m). According to analytical

formulation the thermal expansion of the ladder structure can be estimated as

∆%& � � ∙ ∆' ∙ ( � 16 ∙ 10�� ∙ 50 ∙ 30000 � 24��



Figure 5–10. Total longitudinal displacements of end points of ladders 500A and 600A.



Figure 5–11. Von-mises equivalent stress in 30 m ladder structure made of 500A ladder units.

end B

end A



Figure 5–12. Von-mises equivalent stress in 30 m ladder structure made of 600A ladder units.

end B

end A



Second stage analysis to determine the compression strength of the ladder structure.

The elongation analysis for 500A type ladder structure with the length of 30 m is done for axially supported

ends. This situation presents the behaviour of the ladder structure in the middle of longer continues ladder

system. The analysis results are presented in Figure 5–13, Figure 5–14 and Figure 5–15. The analysis prove that

the structure cannot hold more internal axial force that 9 kN. In that case the maximum out of plane

displacements is around 38 mm. After this stage the ladder structure buckles and distortions will be well seen.

So as in first stage analysis the axial force for 99 m long ladder structure in case of temperature increase is

2.6 kN and for 201 m long structure 8.8 kN. Therefore, it can be assumed that structure with the maximum

continues length of 150 m will not have significant out of plane displacements and therefore also significant

distortions in case of temperature increase 50 Co. Using linear interpolation the 150 m long structure will have

the internal force equal to 5.67 kN which gives the out of plane displacement as 5.6 mm

Figure 5–13. Internal axial force in thermal loading when ends are axially supported.

Internal force that

corresponds to ladder length

201 m

Internal force that


150 m

Internal force that


99 m



Figure 5–14. Out of plane displacement of the ladder structure.

Maximum out of plane

displacement



Figure 5–15. Deformations of ladder structure in various loading stages.

Internal axial force 2.6

kN

Internal axial force 8.8

kN

After buckling

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