Large torsion on elastic lamella grid structures

Proceedings of the IASS Annual Symposium 2019 – Structural Membranes 2019

Form and Force

7 – 10 October 2019, Barcelona, Spain

C. Lázaro, K.-U. Bletzinger, E. Oñate (eds.)

Copyright © 2019 by Jonas Schikore, Anna.M. Bauer

Published by the International Association for Shell and Spatial Structures (IASS) with permission.

Large torsion on elastic lamella grid structures

Jonas SCHIKORE*, Anna M. BAUERa, Rainer BARTHEL, Kai-Uwe BLETZINGERa

*Technische Universität München

Department of Architecture, Chair of Structural Design, Arcisstr. 21, 80333 München

[email protected] a Department of Civil-, Geo.- and Environmental Engineering, Chair of Structural Analysis

Abstract

Elastically curved grid structures pose great potential to simplify fabrication as the structure may

consists of straight elements [1]. However, for assembly, erection and load bearing the structures

restraint forces need to be considered. While bending stresses are well explored, nonlinear normal

stresses due to torsion (helix-strain) are often neglected. For large twists of lamella profiles in a practical

scale, the helix-strain effects need to be considered as they amount significantly to the lamellas torsional

stiffness and utilization. For lamella profiles, calculations according to G. Lumpe [2] can be simplified.

The “Inverse”-method allows simulations using the final shape as initial simulation geometry and

therefore reduces the complexity of static calculations. This paper presents a method to consider helix-

strain effects for the simulation of lamella grids using the IGA “Inverse”-method. This method has been

applied to simulate the behavior of a grid structure as central element of a canopy. A simplified

simulation offered valuable information about the flat state at assembly and necessary forces to erect

the grid structure.

Keywords: elastic gridshell, helix-torsion, Isogeometric Analysis, Formfinding, asymptotic networks, active

bending

1. Introduction

Grid structures are suitable for highly efficient lightweight constructions. Creating curved grid

geometries however, goes with constructional complex solutions. An elegant approach is to use elastic

deformation to create curvature [3]. The Multihalle (Mannheim 1972) is an impressive example for this

method. Frei Otto managed to create a doubly curved gridshell out of straight timber beams by bending

and torsion. Since then this method has fascinated architects and engineers and has been improved and

applied on later projects such as the Wealds downland gridshell or Saville Gardens.

The shape of elastically bent structures is subject to minimum strain energy laws and limited by material

and profile strain capacities. The local beam stiffness parameters directly specify these restrictions. A

prevalent approach is to keep the section stiffness for bending and torsion low to minimize these

restrictions.

New designs, such as the research pavilion “InsideOut” (2017), at the TUM (Figure 1) use an asymptotic

curve network, where lamella sections are deformation by bending on the weak axis and torsion [4]. A

first commercial application of a canopy for a business hotel is currently under construction. The

erection process benefits from these restrictions as the grid structure follows a predefined shape turning

from flat to spatial without the need for a form giving scaffolding.


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2

Figure 1: “Inside/Out” Pavilion, TUM [Foto by Felix Noe] and Hotel Canopy [Rendering by Denis Hitrec]

To control the elastic mechanisms, strain energy states and effects of large deformation need to be

carefully considered. Large torsional deformations of sections with large polar moment of inertia go

along with normal stresses according to the theory of helix-torsion [2]. Conventional FEM-Beam

elements do not consider those effects. For lamella profiles, the use of shell elements seems appropriate.

Figure 2: helix-strain due to torsion (red: compression, blue: tension)

Simulating the complete construction process of elastically bent grid structures, starting from straight

beam elements is very complex and time intensive. With non-regular grids, the beam elements need to

be stepwise connected during simulation. The “Inverse”-Method provides a faster solution.

In chapter 2.1, this paper evaluates the impact of helix-strain in a practical scale. Therefore, calculations

quantify resulting stresses, torsional moments and strain energy due to helix-strain of a lamella section.

Furthermore, in chapter 2.2, a new approach using IGA is presented to respect the effects of helix-strain

using the “Inverse”-Method for simulating elastically bent lamella grid structures. Finally, in chapter

2.3, this method was applied on real scale canopy to predict the structures behavior during erection. The

natural shape of the structure was evaluated.

2. Methods

2.1. Calculation of helix strain effects

To evaluate the impact of helix-strain in a practical scale, the torsional effect of a representative lamella

element has been investigated.

2.1.1 Calculation set up

The material parameters are set according to steel: E = 21000 kN/cm2 ; G = 8000 kN/cm2

A lamella profile with dimensions of t x h = 2,5 x 100 mm and 1000 mm length was twisted by 90° as

shown in Figure 3.


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Figure 3: lamella calculation model

2.1.2 Manual Calculation:

For a lamella profile, where h >> t, calculations regarding the helix torsion are simplified with the

following assumption:

Simplification of moments of inertia:

𝐼𝑟 = ∫ 𝑟4𝑑𝐴 ~ 𝐴

𝑡 ∫ 𝑧4𝑑𝑧 =𝑡ℎ5

80 ; 𝐼𝑝~𝐼𝑦

ℎ/2

−ℎ/2=

𝑡ℎ3

12 (1)

For a lamella profile with h >> t, the normal stresses due to helix-strain:

𝜎𝑥,𝐻𝑒𝑙𝑖𝑥−𝑠𝑡𝑟𝑎𝑖𝑛(𝑥, 𝑧) =1

2𝐸(𝑧2 − 𝑖𝑝

2)𝜗′2 (2)

The torsional moment of a lamella due to helix-strain:

𝑀𝑡,𝐻𝑒𝑙𝑖𝑥−𝑠𝑡𝑟𝑎𝑖𝑛 =1

2 𝐸(𝐼𝑟 − 𝑖𝑝

2𝐼𝑝)𝜗′3 (3)

The strain energy of a lamella due to helix-strain in normal direction:

𝐸𝑡,𝐻𝑒𝑙𝑖𝑥−𝑠𝑡𝑟𝑎𝑖𝑛𝐹 =

1

2 ∫

1

9𝐸𝐼𝑟𝜗′4𝐿

0 (4)

The strain energy of a lamella due to St.Venant torsional shear:

𝐸𝑡,𝑆𝑡.𝑉𝑒𝑛𝑎𝑛𝑡𝐹 =

1

2 ∫ 𝐺𝐼𝑡𝜗′2𝐿

0 (5)

2.1.3 FEM-Calculation

FEM-Calculation (R-FEM5) are performed in Th.III.O. using beam or shell components of element size

1cm. The edges of the lamella, i.e. endsections, are fixed to stay in plane, but overall elongation or

shortening is set free.

2.2. Implementation of Helix-strain to the “Inverse”-Method with IGA

The problem while designing structures with rather large torsion in the built state is that the different

elongations of the fibers over the cross section can only be considered if one models the full twisting in

a classical finite element approach. However, this is rather complicated for a structure consisting of

several lamellas. It is beneficial to draw the assembled structure and then compute backwards the

stresses and deformations. For the bending stresses this is simply possible by referencing the actual

curvature to an initial zero-curvature which results in respective bending moments [5]. This is valid for

100 mm


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initially flat stripes. Applying this method for the stresses related to the helix torsion is not directly

possible as the initial local length related to the finite element has to be known a priori. This is not

possible for an arbitrary discretization. In the case of initially straight lamellas with constant height, one

can assume that the initial local length is constant over the height. However, the fiber is getting longer

when measuring the fiber length outside of the centerline in the twisted configuration (see Figure 4). If

the computation starts from this geometry, the strain would be referred to a varying fiber length over the

cross section which is not in accordance with the assumed initial geometry of the lamella. By choosing

one local reference length over the cross section the stresses resulting from of helix torsion can be

represented. Note that this reference can vary over the length of the beam if the elements are not equal

sized as usually the case in complex models. Furthermore, the reference fiber is not necessarily restricted

to the center line.

For a twisted profile, where ends are not fixed in longitudinal direction, helix compression and tension

stresses will be in equilibrium. Therefore, the fiber of distance ip (radius or gyration) to the centerline

experiences no strain.

𝜎𝑥,𝐻𝑒𝑙𝑖𝑥−𝑠𝑡𝑟𝑎𝑖𝑛(𝑥, 𝑧) = 0 =1

2𝐸(𝑧2 − 𝑖𝑝

2)𝜗′2 → 𝑧 = 𝑖𝑝 = √1

12ℎ → 𝑣𝑝,𝑟𝑒𝑓 = 0,5 + √

1

12ℎ (6)

The reference fiber may be set to this position accordingly.

Figure 4: Twisted lamella with different fibers over the height and evaluation point

𝒙(𝑢𝑝, 𝑣𝑝) with related base vector point at the mid 𝒙(𝑢𝑝, 𝑣𝑚).

Isogeometric Analysis applies Non-Uniform Rational B-Splines (NURBS) for the representation of

finite elements within the structural analysis. This implies that a lamella can be represented by a surface

𝑺(𝑢, 𝑣) with one parameter space described by u and v.

𝑺(𝑢, 𝑣) = ∑ ∑ 𝑅𝑖𝑗,𝑝𝑞(𝑢, 𝑣)𝑚𝑗 ∙𝑛

𝑖 𝑷𝑖𝑗 with 𝑢 ∈ [𝑢0, 𝑢1] 𝑎𝑛𝑑 𝑣 ∈ [𝑣0, 𝑣1] (7)

The computation of the reference length for a certain reference fiber can easily be derived from the

parameter space as only one coordinate has to be varied. It is independent of the element discretization

and storing information about neighbor elements and/or reference elements as needed for classical FEM

can be avoided.

However, the following assumptions have to be made. Every lamella is supposed to be straight in the

unstressed configuration. The cross section has to be constant and the discretization in cross section

direction has to be perpendicular to the center line in order to consider helix-torsional effects.

𝒙(𝑢𝑝, 𝑣𝑝)

𝒙(𝑢𝑝, 𝑣𝑚)

L=1000.9 mm

L=1.0 m

L=1000mm


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5

Furthermore, the center line is restricted to zero curvature around the strong axis of the lamella as it is

for asymptotic lines.

2.3. Simulation and erection of a canopy structure

The canopy structure consists of a lamella grid with a stiff steel frame. The grid is designed by finding

an asymptotic curve network on an anticlastic design surface. The top part is covered with an arch

supported membrane. The lamella grid is shaped by elastic deformation of initially straight lamella

strips. Therefore, the grid is assembled flat and pulled into its spatial position. In flat state, lamellas are

bent around their weak axis, in spatial state, lamellas are additionally bent and twisted. The lamella

section parameters are chosen under the specification to deform mainly elastically. Stainless steel of

with Rp,0.2 = 300 N/mm2 was used for the lamella profiles.

To control the erection process, it is essential to determine the shape, which the structure is likely to

adopt. In this context an analysis of minimum strain energy was executed using the IGA “Inverse”-

Method. The design grid was supported statically determined to avoid any restraints. The lamella strips

are connected only in translation at the intersecting lines thus a scissor hinge is modeled. In a first step

the self-weight was turned off. In a second simulation self-weight was respected and the design grid was

supported according to the erection set up. This simulation gave information about the necessary forces

to induce the design shape. On site, the necessary force to erect the structure was measured to verify

calculations.

Figure 5: a) construction b) static system of “pulled up” state simulation

The simulation of the erected position may be seen as rough prediction of necessary pull up forces. The

nodes of the real structure have slots of 50 mm within the lamellas to create centric nodes. These slots

are weakening of the profiles capacities and therefore affect the grids behavior. The simulation model

does not include those slots. The lamella thickness, however, has been chosen with the specification,

that the slotted sections are capable to perform elastic when applying the necessary curvature to the

lamellas. This consideration has been necessary to avoid kinks due to plastification at the nodes.

3. Results

3.1. Evaluation of helix-strain impact

The displayed graphs (Figure 6) show torsional moments and strain energy due to St. Venant shear

stresses (bottom portion, grey) and normal helix shear stresses (top portion,blue).

a) b)

Frame

Lamella

Grid


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Figure 6: torsional moment and strain energy due to lamella torsion up to 90°

The effects due to helix-strain are nonlinear (Figure 6). A 90° lamella twist results in a torsional moment

of 12,25 kNcm with a ratio of 54% due to St. Venant and 46% due to helix-strain. The ratio of induced

strain energy due to helix-strain and St. Venant shear is 70% to 30% respectively. Besides the amount

of torsion, the lamellas height affects the helix-strain significantly (see eq. 1)

FEM-Calculations verify these effects and show that simplifications lead to negligible differences.

3.2. Evaluation of the “Inverse” Method in IGA

The “Inverse”-method of considering helix torsion results in stresses and Moments (Figure 7).

Figure 7: normal stresses of 90° twisted lamella using IGA “Inverse”-Method

The distribution of membrane stresses along the section hight matches the expected quality. The lamella

does not significantly deform as expected for a constantly twisted lamella. A comparison with the

calculation methods used in chapter 2.1 verifies the “Inverse”-Method as deviations to the other methods

results are small.

0

2

4

6

8

10

12

14

0 10 20 30 40 50 60 70 80 90

[kN

cm]

Twist [°]

Torsional Moment

Helix-strain

St. Venant

0

1

2

3

4

5

6

7

8

0 10 20 30 40 50 60 70 80 90

[kN

cm]

Twist [°]

Strain Energy due to Torsion

Membrane stresses in

u-direction, σmiddle Mt,tot

σedge


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An overwie of the results according to their calculation method is shown in Table 1:

Table 1: Result overview of 90° single lamella twist

manual

calculation FEM beam FEM shell IGA (Inverse)

Mt,St.V. [kNcm] 6,61 6,44 not measured not measured

Mt,Helix [kNcm] 5,64 - not measured not measured

Mt,total [kNcm] 12,25 6,44 12,92 12,132

σedge [kN/cm2] 43,2 0 44,9 42,2

σmiddle [kN/cm2] -21,6 0 -22,8 -21,6

3.3. Evaluation of the canopy simulation and erection

The grid structure found its energy minimum in the flat state. Besides torsion, also bending decreased

from erected to flat state. The flat grid layout was used to assemble the grid on the floor. Figure 8 a) and

b) shows the flat layout in simulation and assembly.

a) c)

b) d)

Figure 8: a) Simulation of grid structure relaxation, b) Photo: flat lamella grid, c) Pull up simulation, d) Photo:

Pull up of structure


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The second simulation (Figure 8c) showed, that for a chosen pull-up setting a force of ~1,2 kN is

necessary to erect the lamella grid. This result matches about the measured erection force of ~1,3 kN

(Figure 8d). The calculation shows, that no fixation in downward direction on the floor is needed to

keep the grid partly on the floor as self-weight was sufficient enough.

4. Conclusion

The resistance of a lamella profile to torsion due to helix-strain effects rises to a significant portion for

large twists. For conventional torsional deformation, those effects may be small. For large deformations

as for elastically bent grid structures the effects may be unneglectable. The impact can be quantified on

a reference element or even for a whole design grid by measuring the geometric values of curvature

along the profiles. Calculation methods according to G. Lumpe and V. Gensichen [1] may be simplified

for lamella sections with minor deviation.

Conventional beam elements do not consider helix-strain effects and are therefore questionable if used

for the simulation of elastically twisted profiles, or formfinding of elastic grids. The “Inverse”-method

has several advantages in the design process as the static simulations can start from the assembled state.

The described method allows to consider helix-strain effects for the “Inverse”-method with minor

deviation to conventional simulation strategies.

The simulation of a grid structures state of minimum strain energy is useful to characterize the elastic

structures behavior during assembly and erection. While for some cases, the flat state is the “natural”

state of the grid (as for the canopy), for other grid geometries the structure may tend to adopt to spatial

shape.

5. Acknowledgements

We thank Dr. Eike Schling for the architectural design of the canopy structure and collaboration. We

also thank Thomas Brandl and his experienced staff for technical assistance and assembly of the canopy.

References

[1] E. Schling, “Repetitive Structures, Design and construction of curved support structures”, Munich,

Germany, 2018

[2] G. Lumpe, V. Gensichen, “Evaluierung der linearen und nichtlinearen Stabstatik in Theorie und

Software” Bauingenieur Praxis, Berlin, Aug. 2014.

[3] J. Lienhard, “Bending-Active Structures. Form-finding strategies using elastic deformation in

static and kinematic systems and the structureal potentials theriein”, Stuttgart, Germany, 2014

[4] E. Schling, M. Kilian, H. Wang, J. Schikore, H. Pottmann, “Design and Construction of Curved

Support Structures with Repetitive Parameters”, Advances in Architectural Geometry 2018,

Gothenburg, Sweden, September, 2018

[5] A. M. Bauer, R. Wüchner and K.-U. Bletzinger, `` Isogeometric Analysis in the Design Process of

Lightweight Structures,'' in Form and Force: Proceedings of the IASS Annual Symposium 2019 -

Structural Membranes 2019, Barcelona, Spain, October 7-10, 2019 C. Lázaro, K.-U. Bletzinger,

and E. Onate Eds., 2019.

Documents

Large torsion on elastic lamella grid structures