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Eindhoven University of Technology

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Structural active facadea research into an adaptive facade that actively controls its deformations in response to itschanging loading environment over time by means of an active rotating connection

van Ruitenbeek, S.T.

Award date:2016

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Author S.T. (Sebastiaan) van Ruitenbeek Student number 0753735 Email [email protected] Master Architecture, Building and Planning Specialization Structural Design Course Graduation project Code 7PP37 Graduation Committee Chairman Prof. Dr. Ing. P.M. (Patrick) Teuffel Supervisor Ir. A.P.H.W. (Arjan) Habraken Supervisor Ir. H.M (Hans) Lamers Date 13 April 2016 University Technical University Eindhoven

STRUCTURAL ACTIVE FACADE A research into an adaptive facade that actively controls its deformations in

response to its changing loading environment over t ime by means of an active rotating connection.

PREFACE

Before you lies my final thesis of the specialization Structural Design of the master Architecture Building and Planning at the Technical University of Eindhoven. This research investigates active deformation control of the Structural Active Facade to minimize material use. I would like to thank my supervisors Patrick Teuffel, Arjan Habraken and Hans Lamers for their feedback and support in this project. Further on, I would like to thank Laurie van Krugten for reading and giving feedback on my writing and presentation. Finally, I would like to thank Eric Wijen and Toon van Alen of the laboratory for their help in constructing the experimental model. I hope you will enjoy reading this thesis. Sebastiaan 13 April 2016. Eindhoven

ABSTRACT

Nowadays, sustainability is a growing concern in the building industry. Hereby material use is the main contributor to climate impact of the building industry in the Netherlands in 2010 (Bijleveld, Bergsma, Krutwagen, & Afman, 2014). The design of building structures is based on the worst expected load scenario during a structure’s entire life time (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011b, p. 12; Senatore, Duffour, Hanna, Labbé, & Winslow, 2011). However, the real loads that a structure is subjected to are almost never at their maximum and are largely varying over time. This leads to large unused structural capacity during most of a structures lifetime, which is in sharp contrast to the goal of sustainability in material use. Adaptive structures are engineering structures that are able to alter their configuration, form or properties in response to their changing loading environment over time and are embed with integrated sensors, actuators and a control system (Korkmaz, 2011; Fisco & Adeli, 2011; Morales-Beltran & Teuffel, 2013; van Bommel, 2014; Soong & Spencer, 2002; University of Stuttgart, 2014). This research proposes a Structural Active Facade as adaptive structure, defined as follows: A Structural Active Facade is an adaptive facade that actively controls its deformations in response to

its changing loading environment over time by means of an active rotating connection. This could lead to a reduction in material use, since now only the strength requirements need to be taken into account. The deformation requirements are satisfied by means of actuation. The goal of this research is to find and test a method for deformation control of the Structural Active Facade. The main question thus sounds:

Main question: How can the deformations of a structural facade be actively controlled in response to its changing loading environment over time?

The method of this research consists of a database that connects the measured strains to the required rotations for the actuators. A variant study determines the best variant to implement the rotations in the facade. Then, it is verified whether the deformations are controlled numerically and experimentally by using this best variant, the database and a control algorithm. Also, the influence on response time and mass of the system is investigated. From the results it appeared that this method is not sufficient for deformation control of the Structural Active Facade. The main reason lies within the reconstruction of the deformations from the strains, which is too sensitive to imperfections and makes the system highly liable to errors. Determining the required rotations from the deformations works well via this method, however within the (linear) boundaries of this method. Rotating actuators can be used to implement the rotations and actively steer the deformations of the Structural Active Facade. Important is to realize a delay time of minimal 0.3 s. and to minimize the masses in the system. It is therefore recommended to build an identical nonlinear dynamical numerical model of the experimental test set up of the Structural Active Facade. This experimental test set up would contain stronger rotary actuators, hereby enabling a stiffer and more stable system, and faster rotary actuators, which in combination with a real time operating system would enable the control of dynamical deformations as well. By analysing the behaviour of both models, a relation can may be found between the rotations and deformations without the use of the database. This will make the translation from strains to deformations superfluous, which results in a less sensitive system. Subsequently, this relation can be transformed into a control algorithm that steers the deformations directly, whereby the deformations serve both as input- and control parameter. Then the system gets feedback on its control and can correct for small errors without error accumulation.

TABLE OF CONTENTS

1. Introduction ....................................................................................................................................... 9

1.1 Problem description ................................................................................................................. 9

1.2 Adaptive structures ................................................................................................................ 10

1.3 Structural Active Facade ........................................................................................................ 10

1.4 Research description ............................................................................................................. 12

2. Methodology ................................................................................................................................... 13

2.1 Construction of the database ................................................................................................. 14

2.1.1 Deformations by the environment ...................................................................................... 14

2.1.2 Deformation requirements ................................................................................................. 17

2.1.3 Active deformations ........................................................................................................... 17

2.1.4 Final deformations ............................................................................................................. 18

2.1.5 Solving the rotational angles ............................................................................................. 18

2.2 Construction and analysis of the numerical model ................................................................ 19

2.3 Construction and analysis of the experimental model ........................................................... 21

3. Results............................................................................................................................................ 23

3.1 Variant study .......................................................................................................................... 23

3.1.1 Variant one ......................................................................................................................... 23

3.1.2 Variant two ......................................................................................................................... 24

3.1.3 Variant three ...................................................................................................................... 27

3.1.4 Variant four ........................................................................................................................ 29

3.1.5 Variant five ......................................................................................................................... 30

3.2 Numerical test ........................................................................................................................ 32


3.2.2 Influence of response time ................................................................................................ 37

3.2.3 Influence of the mass......................................................................................................... 38

3.3 Experimental test ................................................................................................................... 41


3.3.2 Influence of time response ................................................................................................ 48

4. Conclusion and discussion ............................................................................................................. 49

4.1 Conclusion ............................................................................................................................. 49

4.2 Discussion ............................................................................................................................. 50

4.2.1 Interpretation results .......................................................................................................... 50

4.2.2 Limitations .......................................................................................................................... 50

4.2.3 Recommendations ............................................................................................................. 51

5. References ..................................................................................................................................... 53

6. Appendices ..................................................................................................................................... 57

Appendix A: Indication material savings ............................................................................................ 57

Appendix B: Derivation ∆l variants ..................................................................................................... 57

Variant 1 ......................................................................................................................................... 57

Variant 2 ......................................................................................................................................... 58

Appendix C: Maximum wind pressure Eurocode ............................................................................... 58

Appendix D: Design check strength ................................................................................................... 59

Check bending ............................................................................................................................... 59

Check shear ................................................................................................................................... 60

Check normal force ........................................................................................................................ 60

Appendix E: Formula’s deformations by the environment ................................................................. 61

Numerical test ................................................................................................................................ 61

Experimental test ............................................................................................................................ 62

Appendix F: Numerical test ................................................................................................................ 65

Appendix G: Secondary tests ............................................................................................................ 72

Maximum torque actuator............................................................................................................... 72

Bending stiffness plates ................................................................................................................. 73

Relation moment and strains ......................................................................................................... 74

Appendix H: Experimental test........................................................................................................... 75

Test 1 .............................................................................................................................................. 76

Test 2 .............................................................................................................................................. 80

Test 3 .............................................................................................................................................. 84

Test 4 .............................................................................................................................................. 89

Test 5 .............................................................................................................................................. 92

Test 6 .............................................................................................................................................. 95

LIST OF FIGURES AND TABLES

Figure 1: (a) Bruto end energy use in the Netherlands in 2012, categorized per sector (Energieonderzoek Centrum Nederland, 2014); (b) Climate impact of the building industry in the Netherlands in 2010, per kton CO2-equivalents (Bijleveld et al., 2014) .................................................. 9 Figure 2: Wind velocity measurements over time in Eindhoven from 1971-2015 (KNMI, 2013a) .......... 9 Figure 3: Histogram of specific wind velocities measured in Eindhoven from 1971-2015 (KNMI, 2013b) ............................................................................................................................................................... 10 Figure 4: Structural Active Facade (a) Schematic representation; (b) Possible practical implementation ................................................................................................................................................................ 11 Figure 5: Structural Active Facade covering a monumental building (Habraken, 2014) ........................ 11 Figure 6: Method of deformation control of the Structural Active Facade ............................................. 13 Figure 7: (a) Deformations by the environment; (b) Deformation requirements; (c) Active deformations; (d) Final deformations ............................................................................................................................ 14 Figure 8: (a) uniformly distributed load q (a=1, b=H); (b) non-uniformly distributed load q (a=10, b=H/10); (c) random wind pressure distribution (a=1000, b=H/1000); (d) local wind pressure and suction (a=3, b=H/50); ........................................................................................................................... 16 Figure 9: Numerical model of the Structural Active Facade .................................................................. 20 Figure 10: (a) Linear actuator (electric) (Kuroda Jena Tec, 2012); (b) Rotary actuator (pneumatic) (Parker Pneumatic, 2015, p. A49) ......................................................................................................... 23 Figure 11: Variant one (a) Conceptual design; (b) Geometric representation ....................................... 23 Figure 12: Variant two (a) Conceptual design; (b) Geometric representation ....................................... 25 Figure 13: Moment distribution in the connection for variant two .......................................................... 25 Figure 14: The influence of the design parameter β on the force F for l=0.3, l=0.4 and l=0.5 m. (dark to light) ....................................................................................................................................................... 26 Figure 15: The influence of the design parameter β on the required change in length ∆l for l=0.3, l=0.4

and l=0.5 m. (dark to light) ..................................................................................................................... 26 Figure 16: Conceptual design of variant three ...................................................................................... 28 Figure 17: Force distribution in node C of variant three for (a) β = 45° and l = 0.400 m; (b) an adjusted length of the attachment ........................................................................................................................ 28 Figure 18: Conceptual design of variant four ........................................................................................ 29 Figure 19: Variant four (a) with actuator; (b) without linear actuator; (c) forces to make an angle of 1° 29 Figure 20: Conceptual design of variant five ......................................................................................... 30 Figure 21 Parker Digiplan (a): ML2340A servo motor; (b): BLHX30 servo regulator ............................ 31 Figure 22: Wind spectrum of 120 seconds with spectra for q1-q5 ......................................................... 33 Figure 23: Static linear analysis with maximum wind load (a) Deformations by wind; (b) Active deformations; (c) Final deformations ..................................................................................................... 34 Figure 24: Static linear deformations in the middle of the facade under wind load spectrum ............... 34 Figure 25: Static nonlinear analysis with maximum wind load (a) Deformations by wind; (b) Active deformations; (c) Final deformations ..................................................................................................... 35 Figure 26: Static nonlinear deformations in the middle of the facade under wind load spectrum ......... 36 Figure 27: Dynamic linear deformations in the middle of the facade under wind load spectrum .......... 36 Figure 28: Dynamic linear final deformations in the middle of the facade under wind load spectrum .. 37 Figure 29: Static linear final deformations in the middle of the facade under wind load spectrum for delay times............................................................................................................................................. 37 Figure 30: Dynamic linear final deformations in the middle of the facade under wind load spectrum for delay times............................................................................................................................................. 38 Figure 31: Static nonlinear deformations by wind in the middle of the facade under wind load spectrum for masses ............................................................................................................................................. 39 Figure 32: Dynamic linear deformations by wind in the midst of the facade under wind load spectrum for different masses ............................................................................................................................... 39 Figure 33: (a) maximum-torque test of the actuator ML2340A; (b) three-point-bending test of the plate ............................................................................................................................................................... 41 Figure 34: Experimental model of the Structural Active Facade ........................................................... 42 Figure 35: Final deformations in the middle of the facade under load spectrum for test one to three .. 43 Figure 36: Strains in the three plates for test one ................................................................................. 44 Figure 37: Time-specific (t) and constant (c) rotations for actuators B and C for test one .................... 44 Figure 38: Strains in the three plates for test one, including the expected values based on the real measured deformations ......................................................................................................................... 45 Figure 39: Produced moment per actuator for test one ........................................................................ 46

Figure 40: Final deformations in the middle of the facade under load spectrum for test four to six ..... 47 Figure 41: Principle of deformation control for test four just before actuation at t=130 s. (a) top view; (b) front view .......................................................................................................................................... 47 Figure 42: Principle of deformation control for test four just after actuation at t=146 s. (a) top view; (b) front view ............................................................................................................................................... 48

Table 1: Input parameters for the calculation of material savings ......................................................... 12 Table 2: Results of material savings by using actuation (Staalprijzen.nl bv, 2015) ............................... 12 Table 3: The three structural deformation states and the deformation requirements ............................ 18 Table 4: Set up of the database for controlling the deformations .......................................................... 19 Table 5: Analysis of the numerical model concerning deformation control ........................................... 20 Table 6: Parameter values for the database and the numerical model ................................................. 32 Table 7: Analysis of the numerical model concerning deformation control ........................................... 33 Table 8: Parameter values for the database and the experimental model ............................................ 41

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1. INTRODUCTION

1.1 Problem description Nowadays, sustainability is a growing concern in the building industry. This sector is the largest single contributor to greenhouse gas emissions and accounts for 40% of total energy use around the world (Confederation of International Contractors’ Associations, United Nations Environment Programme, & Division of Technology, Industry and Economics, 2002). In the EU, 33% of the total annual waste is generated by construction and demolition (Herczeg et al., 2014). In the Netherlands, at least one third of the total bruto end energy use comes from the building industry, as visualized in Figure 1a (Energieonderzoek Centrum Nederland, 2014). Figure 1b shows that material use is the main contributor to climate impact of the building industry in the Netherlands in 2010 (Bijleveld et al., 2014).

Figure 1: (a) Bruto end energy use in the Netherlands in 2012, categorized per sector (Energieonderzoek

Centrum Nederland, 2014); (b) Climate impact of the building industry in the Netherlands in 2010, per kton CO2-equivalents (Bijleveld et al., 2014)

Conventionally, the design of building structures is based on the worst expected load scenario that can take place during a structure’s entire life time (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011b, p. 12; Senatore et al., 2011). Structural dimensions and properties are designed according this load scenario, meaning that the structure is able to withstand the maximum design load. However, the real loads that a structure is subjected to are almost never at their maximum and are largely varying over time. When considering wind loads, their magnitude can be calculated from the wind velocity, which is plotted against time from 1971-2015 per hour for Eindhoven in Figure 2 (KNMI, 2013a). The wind velocities in this figure are the extreme values that are measured in that specific hour. Figure 2 shows that the maximum wind velocity is only reached at sporadic moments.

Figure 2: Wind velocity measurements over time in Eindhoven from 1971-2015 (KNMI, 2013a)

When the frequency of these wind velocities are plotted per velocity interval, a histogram (Figure 3)

0

5

10

15

20

25

30

35

1-1-1971 1-1-1976 1-1-1981 1-1-1986 1-1-1991 1-1-1996 1-1-2001 1-1-2006 1-1-2011 1-1-2016

v [m

/s]

T [h]

Structural Active Facade

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is obtained. As can be seen from the graph, a wind velocity of 3 m/s occurs more often than one of 10 m/s. The average wind velocity normally used in design is 26,1 m/s1, to make sure that all possible wind velocities are captured within a certain probability. However, these high wind velocities almost never occur, resulting in large unused structural capacity during most of a structures lifetime.

Figure 3: Histogram of specific wind velocities measured in Eindhoven from 1971-2015 (KNMI, 2013b)

So there is a discrepancy between the building structure, which is static, and its real working environmental loads over time, which are dynamic. This leads to structural dimensions and properties being largely over dimensioned during most of a structures lifetime. As stated above, material use in the building industry is one of the key contributors to CO2-emissions and has a large influence on climate impact. Unused structural material is thus in sharp contrast to the sustainable goal of reducing CO2-emissions and energy consumption.

1.2 Adaptive structures Since the environment over time cannot be made static, an opportunity lies within structures that are able to dynamically respond to their surrounding environment over time. Adaptive structures are engineering structures that are able to alter their configuration, form or properties in response to their changing loading environment over time (Korkmaz, 2011; Fisco & Adeli, 2011; Morales-Beltran & Teuffel, 2013; van Bommel, 2014). These structures are embed with three integrated main components (Fisco & Adeli, 2011; Soong & Spencer, 2002; Korkmaz, 2011; Morales-Beltran & Teuffel, 2013; University of Stuttgart, 2014):

Sensors to measure the external environmental parameters and/or monitor the structural response variables

Actuators to produce the required forces or to change the geometry or main characteristics of the elements or system to achieve the desired performance

Control system to process the measured incoming information from the sensors and to compute the necessary control actions of the actuators, through a given control algorithm

1.3 Structural Active Facade This research proposes a Structural Active Facade as adaptive structure, which is defined as follows: A Structural Active Facade is an adaptive facade that actively controls its deformations in response to

its changing loading environment over time by means of an active rotating connection.

1 This value is calculated for wind area III, a reference height of 10 m. and a roughness factor of 1.065 (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011c)

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

f [%

]

v [m/s]

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Figure 4a visualizes a schematic representation of the Structural Active Facade, with in this case four integrated actuators. Figure 4b shows a possible practical implementation of the Structural Active Facade, whereby the millions of Figure 4b can be replaced by the ones of the Structural Active Facade (Figure 4a). This could lead to a decrease in material use and an increase in transparency.

Figure 4: Structural Active Facade (a) Schematic representation; (b) Possible practical implementation

Another example of an application of the Structural Active Facade is the covering of a monumental building, which is visualized in Figure 5. The monumental building is now able to sustain the maximum wind load that occurs.

Figure 5: Structural Active Facade covering a monumental building (Habraken, 2014)

Active deformation control enables the structure to satisfy the structural requirements at times when that is necessary without having large unused structural capacity. Namely, actuation that can be turned on or off, while the energy of extra material is always present in the structure. Moreover, with active deformation control the structure only needs to be designed for strength instead of stiffness and strength, since the deformations are now controlled by means of actuators. To get an indication of these material savings, the required profile for both a conventional and active beam is calculated,


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hereby using the input parameters of Table 1. Table 2 summarizes the results and shows that 64% of the used material can be saved by using actuation. A detailed calculation can be found in Appendix A.

Table 1: Input parameters for the calculation of material savings

Parameter Unit Value

σy;steel [kN/m2] 3.55E+05

Esteel [kN/m2] 2.10E+08

q [kN/m] 1.5

H [m] 15

δallowed [m] 0.060

Table 2: Results of material savings by using actuation (Staalprijzen.nl bv, 2015)

Results Unit Conventionally Actively

Istiffness [m4] 7.85E-05 Wstrength [m3]

1.19E-04

Profile [-] HEA260 HEA140

A [m2] 8.68E-03 3.14E-03

Material savings [%] -63.9

Some materials have a high strength and other favourable properties (low self-weight), but cannot or hardly be used in constructional engineering due to their unfavourable stiffness. Especially these materials have large potential to be used in the Structural Active Facade. Namely, the deformation requirements are now taken care of by the actuators, so that their unfavourable stiffness is not that important anymore. Examples of such materials are aluminium or composites.

1.4 Research description This research aims at finding and testing a method for deformation control of the Structural Active Facade, which will be explained in detail in the next Chapter. Deformation control could lead to a reduction in material use in constructional engineering, but that will not be the main focus of this research. This graduation project is of social value because this research gives insight into a real application of adaptive structures, which can lead to material savings in the building industry. Since material use is one of the main contributors to climate impact, this could make the environment more sustainable. Furthermore, people could be hesitant when it comes down to structures that have the ability to move. The real experimental application could prove to people that adaptive structures not only have potential for material savings, but also guarantee safety. Finally, adaptive structures could inspire architects and engineers in their designs, which could lead to alternative designs of more challenging and lightweight structures. The scientific value of this graduation project lies within a new way of actively controlling structural deformations. This will be an active connection, which is not yet investigated throughout literature and thus will be an addition to scientific research. Although there is much theoretical research done into adaptive structures, real experiments are less often explored. Moreover, real applications that have direct relevance with the building industry are scarce. Therefore the experimental scale model of the structural active facade will be of scientific value. Further on, the preceding research into adaptive structures at the Technical University Eindhoven exists of trussed structures, arch structures and a real application into a trussed steel building, which were all concerned with controlling the load path in the structure. This graduation project will extend this research field towards the control of deformations in a structure that cannot adopt another load path. Moreover, this graduation project will create a base for future research into adaptive structures on static shape control at the chair of Innovative Structural Design.

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2. METHODOLOGY

This chapter describes the main steps to fulfil the aims and objectives of this research. The main question exists of five sub-questions, whereby sub-questions one and two are used to support sub-questions four and five theoretically. Sub-question three is a study to find the best variant to implement in the model of sub-questions four and five. Main question:

How can the deformations of a structural facade be actively controlled in response to its changing loading environment over time?

Sub-questions:

1. What are the deformations by the changing loading environment over time? 2. What are the deformation requirements for a structural facade? 3. How can the deformations of a structural facade be actively altered? 4. How can the deformations of a structural facade be actively controlled numerically? 5. How can the deformations of a structural facade be actively controlled experimentally?

Figure 6 presents a method that answered sub-questions four and five and thus the main question. There were five main phases that were distinguished for both the numerical as the experimental test, namely the initiation, the input for both the database and the model, the model itself and finally the results. The initiation phase described the design of the facade, as explained in Chapter 1.3, the load measurements and an assumed variation of the load over height. The design of the facade and the load measurements determined the profile dimensions, which served together with the other parameters of the initiation phase as input for the database. After set up, the database formed the input for the model, together with the model parameters and the applied load. Finally, an analysis of the model was used to verify whether the final deformations were controlled and to understand the influence of time response of the actuators. For the numerical model the influence of mass of the actuators was analysed as well.

Figure 6: Method of deformation control of the Structural Active Facade


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There were three processes in the methodology that need to be highlighted, namely: 1. Construction of the database 2. Construction and analysis of the numerical model 3. Construction and analysis of the experimental model

2.1 Construction of the database The database played a key role in controlling the deformations by coupling strains to rotational angles. Hereby, the strains were used to recognize the deformations of the structure, which is in analogy with preceding research (Neuhäuser et al., 2013; van Bommel, 2016). The rotational angles were required to steer the deformations with the actuators. To set up the database, three structural states were distinguished, namely the deformations caused by the changing load environment, the active deformations steered by the actuators and the final deformation after actuation (Figure 7a, c and d respectively). The required rotational angles were found by solving for the deformation requirements (Figure 7b). Hereafter, the three states and the requirements are explained individually.

Figure 7: (a) Deformations by the environment; (b) Deformation requirements; (c) Active deformations; (d) Final

deformations

2.1.1 Deformations by the environment The main load type for this research that caused the structural facade to deform was wind. To determine its deformation shape and magnitude, a method called ‘Direct Integration’ was used. This structural analysis method measured the internal shear, moment, rotation and deformation of a beam by integrating for each measure, starting from the applied load (Fenner & Reddy, 2012). However, the applied load needed to be a continuous function of the beam length.

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In reality, the facade wassubjected to a random wind load, which fluctuated over height. To approximate this behaviour, the load on the facade was divided into a number of uniform loads a (Figure 8), with length b. By means of suitable step functions these loads were made continuous over the facade height H. A Heaviside step function was of the continuous form: 𝑓(𝑥) = ⟨𝑥 − 𝑎⟩ (1) The angle brackets made that the function has the following properties: ⟨𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒⟩ → 𝑓(𝑥) = 0 ⟨𝑧𝑒𝑟𝑜 𝑜𝑟 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒⟩ → 𝑓(𝑥) = 1 Thus the term between the triangular brackets defined the properties of the Heaviside step function: 𝑓(𝑥) = {

0 𝑥 < 𝑎1, 𝑥 ≥ 𝑎

(2)

The Heaviside step function was used to make the function of a partial load qn, ranging from 0 to for

instance 𝐻

2 ,continuous over the facade height:

𝑞n(𝑧) = q (⟨𝑧⟩ − ⟨𝑧 −𝐻

2⟩) (3)

For a position z over the facade height H the value of qn was:

𝑞n(𝑧) = {𝑞, 𝑧 ≤

𝐻

2

0, 𝑧 ≥𝐻

2

𝑓𝑜𝑟 0 ≤ 𝑧 ≤ 𝐻 (4)

With this tool a sum function for the total load qtotal over the facade height H was established:

𝑞𝑡𝑜𝑡𝑎𝑙(𝑧) = ∑ 𝑞𝑘+1

𝑎−1

𝑘=0

(⟨𝑧 −1

2∙

𝐻 ∙ (1 + 2 ∙ 𝑘)

𝑎−

1

2∙ 𝑏⟩ − ⟨𝑧 −

1

2∙

𝐻 ∙ (1 + 2 ∙ 𝑘)

𝑎+

1

2∙ 𝑏⟩) (5)

With: a = the number (consecutive integers) of partial uniform loads q b = the height per partial uniform load q qk+1 = the load at a particular place in the facade z = position over the facade height H H = facade height An increase in value for the parameter a resulted in more non-uniformly distributed over the facade height. For a=10 and b=H/10, the load became as in Figure 8b. When a became 1000 and b=H/1000, the load on the structural facade approached a random wind pressure distribution over the facade height (Figure 8c). Moreover, local wind pressure or suction was also approximated with this function, as shown in Figure 8d for a=3 and b=H/50. The specific values for qn resulted from the load measurements from the initiation phase, as explained in the methodology of Figure 6.


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Figure 8: (a) uniformly distributed load q (a=1, b=H); (b) non-uniformly distributed load q (a=10, b=H/10); (c) random wind pressure distribution (a=1000, b=H/1000); (d) local wind pressure and suction (a=3, b=H/50);

With a continuous function of the load over the height of the facade, the load q was integrated to obtain the shear forces over the facade height H:

𝑉(𝑧) = ∫ − 𝑞𝑡𝑜𝑡𝑎𝑙(𝑧)𝑑𝑧 + 𝐶𝑠ℎ𝑒𝑎𝑟 (6)

A second integration resulted in a function for the moment distribution:

𝑀(𝑧) = ∫ 𝑉 𝑑𝑧 + 𝐶𝑚𝑜𝑚𝑒𝑛𝑡 (7)

Both constants were determined by filling in the boundary conditions: 𝑀(0) = 0 → 𝐶𝑚𝑜𝑚𝑒𝑛𝑡 𝑀(𝐻) = 0 → 𝐶𝑠ℎ𝑒𝑎𝑟 A third and fourth integration provided for the rotation and deformation at any position in the facade respectively, whereby the section modulus I resulted from the profile dimensions (Figure 6):

𝜑𝑤𝑖𝑛𝑑 =1

𝐸𝐼∫ 𝑀 𝑑𝑧 + 𝐶𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 (8)

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𝛿𝑤𝑖𝑛𝑑 = ∫ 𝜑𝑤𝑖𝑛𝑑 𝑑𝑧 + 𝐶𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛 (9)

The constants was solved by applying the following boundary conditions: 𝛿𝑤𝑖𝑛𝑑(0) = 0 → 𝐶𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛

𝛿𝑤𝑖𝑛𝑑(𝐻) = 0 → 𝐶𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 With these equations the shear, moment, rotation and deformation became functions of the facade height H, so that their precise values were known at any position in the facade. The method of direct integration can also be used when the wind is another function over the facade height H, as long as it is continuous. The wind load can cause a deformation shape as visualized in Figure 7a. An advantage was that the beam deformation can always be constructed as long as the load is a continuous function over the facade height H. This made this method easy to use, even for complex patterns of wind loads, and it gave a fast understanding of the deformation shape of the beam. Therefore this method was used for this research.

2.1.2 Deformation requirements The deformation requirements were directly related to the appearance of the structure. Since there were no secondary constructional elements connected to the millions of the facade, its functioning is of less importance. Article A1.4.3 remark 4 considers the appearance of a structure, which resulted in the following requirement for the deformation of structural elements (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011a):

𝛿𝑎𝑙𝑙𝑜𝑤𝑒𝑑 = −𝐻

250 (10)

However, equation (10) did not set the deformation requirements for each position z over the facade height H. Therefore, the equation for the deformation of a simply supported beam subjected to a uniformly distributed load was used, which can be found in literature (Fenner & Reddy, 2012, p. 321):

δ (𝑧) = −𝑞

𝐸𝐼(

𝑧 ∙ (𝑧3 − 2 ∙ 𝐻 ∙ 𝑧2 + 𝐻3)

24) (11)

In this equation 𝑞

𝐸𝐼 was seen as some constant to fit for the maximum deformation of equation (10). By

doing so, the following relation was established:

δ (𝐻

2) = −

5 ∙ 𝑞 ∙ 𝐻4

384 ∙ 𝐸𝐼= −

𝐻

250 (12)

Which resulted in the following constant: 𝑞

𝐸𝐼=

192

625 ∙ 𝐻3 (13)

By implementing the value for 𝑞

𝐸𝐼 in equation (11) the formula for the allowed deformations became:

δ𝑎𝑙𝑙𝑜𝑤𝑒𝑑 (𝑧) = ±8

625

𝑧 ∙ (𝑧3 − 2 ∙ 𝐻 ∙ 𝑧2 + 𝐻3)

𝐻3 (14)

Depending on the side of the facade, δmax had a positive or a negative value. The function of equation (14) set the boundaries for the deformations, as visualized in Figure 7b, and was used to solve for the required angles of the actuators.

2.1.3 Active deformations Four actuators were placed in-between the five elements to steer the deformations. A variant study pointed out which type of active connection was best to alter the structural deformations. The actuators influenced the structural deformation shape by rotating the active connections in the structure, which was described by the following functions over the facade height H:


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𝑓𝑜𝑟 0 ≤ 𝑧 ≤𝐻

5 𝛿𝑎𝑐𝑡;𝐴−𝐵 = 𝑧 ∙ tan(𝜑𝐴) (15)

𝑓𝑜𝑟 𝐻

5≤ 𝑧 ≤

2𝐻

5 𝛿𝑎𝑐𝑡;𝐵−𝐶 =

𝐻

5∙ tan(𝜑𝐴) + (𝑧 −

𝐻

5) ∙ tan(𝜑𝐴 + 𝜑𝐵) (16)

𝑓𝑜𝑟 2𝐻

5≤ 𝑧 ≤

3𝐻

5 𝛿𝑎𝑐𝑡;𝐶−𝐷 =

𝐻

5∙ (tan(𝜑𝐴) + tan(𝜑𝐴 + 𝜑𝐵)) + (𝑧 −

2𝐻

5) ∙ tan(𝜑𝐴 + 𝜑𝐵 + 𝜑𝐶) (17)

𝑓𝑜𝑟 3𝐻

5≤ 𝑧 ≤

4𝐻

5

𝛿𝑎𝑐𝑡;𝐷−𝐸 =𝐻

5∙ (tan(𝜑𝐴) + tan(𝜑𝐴 + 𝜑𝐵) + tan(𝜑𝐴 + 𝜑𝐵 + 𝜑𝐶)) + (𝑧 −

3𝐻

5) ∙

tan(𝜑𝐴 + 𝜑𝐵 + 𝜑𝐶 + 𝜑𝐷) (18)

𝑓𝑜𝑟 4𝐻

5≤ 𝑧 ≤ 𝐻

𝛿𝑎𝑐𝑡;𝐸−𝐹 =𝐻

5∙ (tan(𝜑𝐴) + tan(𝜑𝐴 + 𝜑𝐵) + tan(𝜑𝐴 + 𝜑𝐵 + 𝜑𝐶) +

tan(𝜑𝐴 + 𝜑𝐵 + 𝜑𝐶 + 𝜑𝐷)) + (𝑧 −4𝐻

5) ∙ tan(𝜑𝐴 + 𝜑𝐵 + 𝜑𝐶 + 𝜑𝐷 + 𝜑𝐸)

(19)

From equations (15) till (19) it was recognized that the horizontal deformations are the tangent of every rotational angle plus the horizontal displacements of the preceding rotations. The deformations by actuation will take on the shape as visualized in Figure 7c.

2.1.4 Final deformations The final deformations consisted of the summation of the deformations by the environment and the active deformations: 𝛿𝑤𝑖𝑛𝑑 + 𝛿𝑎𝑐𝑡 = 𝛿𝑓𝑖𝑛𝑎𝑙 (20)

Table 3 summarizes Chapters 2.1.1 till 2.1.4 for variable z from 0 to H, with two extra columns at the end containing the difference between the final and allowed deformations on both sides of the facade.

Table 3: The three structural deformation states and the deformation requirements

Height Wind Actuation Final Requirements

z qwind Vwind Mwind φwind δwind δactuators δfinal δ-alowed δ+alowed δfinal-

δ-alowed δ+alowed-

δfinal

[m] [kN/m] [kN] [kNm] [m/m] [m] [m] [m] [m] [m] [m] [m]

0

….…

H

2.1.5 Solving the rotational angles The last step is to solve the required angles that were captured within the formulas for the active deformations. The rotational angles of connection A, B, E and F were solved by means of a Solver function in Excel 2013, which were then used to solve the rotational angles of the last two connections C and D. The Solver function contained two targets:

1. −𝛿𝑎𝑙𝑙𝑜𝑤𝑒𝑑 ≤ 𝛿𝑓𝑖𝑛𝑎𝑙 ≤ 𝛿𝑎𝑙𝑙𝑜𝑤𝑒𝑑

2. min(𝜑𝑚𝑎𝑥;1,2)

The first target was achieved by allowing only positive values in the last two columns of Table 3 and assured that the final deformations satisfy the deformation requirements. The second one was reached by minimizing the sum of the two largest values of the rotational angles of the active connections B,C,D and E. This assured that not only the total sum of the angles B,C,D and E were minimized, but also that the largest angle on itself was minimized. Namely, sometimes it may be better to enlarge some smaller angles, so that the largest angle can be further minimized. Then it requires less time for actuation, since the actuators can rotate simultaneously. Having solved the rotational angles of connection A, B, E and F, the angle of rotation in connection C followed from the fact that all angles together should form 0° again:

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𝜑𝐶 = 𝜑𝐹 − (𝜑𝐴 + 𝜑𝐵 + 𝜑𝐷 + 𝜑𝐸) (21) Secondly, the displacement at the last connection F was 0 m in the x-direction due to the support, which made that: 𝛿𝑓𝑖𝑛𝑎𝑙;𝐹 = 𝛿𝑤𝑖𝑛𝑑;𝐹 + 𝛿𝑎𝑐𝑡;𝐹 = 0 (22)

And using this relation together with equation (21), the angle of rotation in connection D became: 𝜑𝐷 = 𝜑𝐹 − 𝜑𝐸 + tan−1(tan(𝜑𝐴) + tan(𝜑𝐴 + 𝜑𝐵) − tan(−𝜑𝐹 + 𝜑𝐸) + tan(𝜑𝐹)) (23) So for every possible load q, the rotational angles of the active connections were solved, which steered the deformations to satisfy the deformation requirements. By repeating this process over and over again, a database was constructed as visualized in Table 4. This database contained five strains in the midst of every element and four rotational angles under which the actuators could rotate the structure in the active connections. To find these strains, the measured moments in the midst of every element were multiplied by a constant, which is dependent on the properties of the material and profile. For a linear elastic material and a symmetric profile the stress at the top (or bottom) was:

𝜎𝑡𝑜𝑝 =𝑀𝑦 ∙ ℎ𝑝𝑟𝑜𝑓𝑖𝑙𝑒

2 ∙ 𝐼𝑦

(24)

The same stress made that the material deforms, hereby following Hook’s law: 𝜎𝑡𝑜𝑝 = 𝐸 ∙ 휀 (25)

Combining equations (24) and (25) resulted in the constant to convert the measured moments into the strains of the database:

휀 = 𝐶 ∙ 𝑀𝑦 𝑤𝑖𝑡ℎ 𝐶 =ℎ𝑝𝑟𝑜𝑓𝑖𝑙𝑒

2 ∙ 𝐸 ∙ 𝐼𝑦

(26)

For nonlinear elastic materials or stresses above the yield stress, Hook’s law could no longer be applied. Then the stress strain curve of that particular material needed to be used in order to convert the moments (stresses in the outer fibres) into strains.

Table 4: Set up of the database for controlling the deformations

ε1 ε2 ε3 ε4 ε5 φB φC φD φE

[μm/m] [μm/m] [μm/m] [μm/m] [μm/m] [rad] [rad] [rad] [rad]

… … … … … … … … …

2.2 Construction and analysis of the numerical model The numerical model is one of the two models that tested whether the deformations were controlled, for which the program Oasys GSA 8.7 was used. Due to the large amount of lists and repetition, Excel 2013 was used as well to prepare the input, which then could be copied directly into GSA. The structure consisted of eleven nodes, representing both the supports, the four actuators and the five strain sensors (Figure 9). The millions were modelled as beam elements and the actuators as mass elements. The density of the material was somewhat enlarged to account for the distributed weight of the attached facade panels. The same height, centre to centre distance of the millions and profile dimensions were used as for the database. The wind loads were modelled statically as beam loads and dynamically as load curves. The gravitational loading had a value of -1 in the z-direction.


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Figure 9: Numerical model of the Structural Active Facade

For every case the analysis was run twice. The first run was used to obtain the moments in the midst of every element, which were then converted into strains via the constant as explained in Chapter 2.1.5. Using Matlab 2015, the best match was found between the five measured strains and the first five columns of the database. The founded index then gave the required rotational angles in the last four columns, which were implemented in the program again. Consequently, the second run resulted in the final deformations. The types of analyses, wind loads and the variance in response time and mass of the actuators for deformation control are summarized by Table 5.

Table 5: Analysis of the numerical model concerning deformation control

Deformations

Analysis type Wind load Response time Mass

SL qmax 0.0 s mall

SNL qspectrum 0.3 s mall-act

DL 0.5 s m-

1.0 s

The numerical model was analysed static linearly, static nonlinearly and dynamic linearly, whereby the structural facade was subjected to a maximum wind load and a wind spectrum of one minute. These wind loads were based on wind measurements in the past by the KNMI Hydra Project and at the Auditorium of the Technical University of Eindhoven. The influence on the final deformations of the response time of the actuators was considered for delay times of 0.0, 0.3, 0.5 and 1.0 seconds. The influence on the final deformations of the mass of the actuators was considered for three variations of the mass, namely all the mass, all the mass without that of the actuators and no mass at all.

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2.3 Construction and analysis of the experimental model The second way of testing deformation control was an experimental model, set up in the Pieter van Musschenbroek laboratory of the Technical University of Eindhoven. Since the main goal was to test whether the final deformations were controlled, the deformations should have been clearly visible. Also, the response time of the actuators was determined to see their influences on deformation control. The height of the experimental model was 2.5 m, making that every element is 0.5 m in height. The design of the connection with the type of actuator followed from the variant study. The programming was done in Labview and the strain sensors were placed in the midst of every element. A cable was attached around the same position as the sensors and kept on tension by a weight in combination with a pulley. By varying the weights, the wind load were varied for every element. For every weight combination, the deformation by wind, the active deformations and the final deformations were measured in the midst of the facade. The response of the structure was analysed, including the response time of the actuators, and it was verified whether the final deformations were controlled.


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3. RESULTS

The results will be presented according the methodology, hereby starting with the variant study. The variant study is followed by the results of the numerical and finally the experimental test.

3.1 Variant study Structural deformations can be altered by rotating specific points in the structure with for example actuators. Mainly used actuators are of the electric, hydraulic or pneumatic type. The advantage of electric devices over hydraulic or pneumatic ones is that they require the least amount of equipment. Hydraulic and pneumatic actuators on the other hand can deliver larger forces and moments (van Bommel, 2014). There are two types of actuators, namely linear and rotary actuators (Indramat Products, 2015), which are visualized in Figure 10a and b respectively. Linear actuators deliver an axial force in combination with a linear motion, which is achieved by a piston that is driven out of a long sleeve bearing by a motor. The circular motion of the motor is hereby conversed to linear motion, for example through a ball screw (Sclater & Chironis, 2007, p. 25). Rotating actuators on the other hand deliver a moment in combination with rotary motion. Frequently, a gear box is fixed to the rotating axis to increase the moment. In this chapter, different variants are presented to actively alter the structural deformations. Hereby, one actuator will be used per connection and the aim is to keep one side of the million straight for (glass) panel attachment. Each variant will be compared with the best variant so far, so that in the end the best variant remains, hereby answering sub-question three.

Figure 10: (a) Linear actuator (electric) (Kuroda Jena Tec, 2012); (b) Rotary actuator (pneumatic) (Parker

Pneumatic, 2015, p. A49)

3.1.1 Variant one The first variant exists of a hinge in combination with a linear actuator that is connected to two profiles, as visualized in Figure 11a. The actuator transfers the bending moment in the connection, which makes that the structure can be considered continuous. Moreover, the connection also has a rotational possibility, since the linear actuator can rotate both sides around the hinge by varying its length.

Figure 11: Variant one (a) Conceptual design; (b) Geometric representation


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To gain more specific insight in its behaviour, this variant is reduced to its geometric representation (Figure 11b), expressed by means of the following design parameters:

- Initial bar length l of the actuator - Design angle β, situated opposite to l

By expressing the forces in and the required change in length of the actuator by means of these design parameters, their influences can be investigated per rotational angle. Starting with the first one, the force in the actuator multiplied with the arm (a) must be equal to the moment of the connection: 𝑀𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑜𝑛 = 𝐹𝑎𝑐𝑡𝑢𝑎𝑡𝑜𝑟 ∙ 𝑎 (27) Subsequently, the magnitude of the moment at the connection can be derived via integration as explained in Chapter 2.1.1. Since this connection can be seen as an infinitely small part of a continuous beam, it is infinitely stiff and thus the rotation infinitely small. Therefore, the moment equilibrium can be considered for the undeformed state, just like moment equilibrium in a normal beam. The arm a can be expressed in the design parameters β and l of the connection according to:

tan(1

2𝛽) =

12

𝑙

𝑎

(28)

Combining equations (27) and (28) results in the force of the actuator for variant one:

𝐹𝑎𝑐𝑡;𝑣1 =2 ∙ 𝑀𝑐𝑜𝑛 ∙ tan(

12

𝛽)

l

(29)

The required change in length ∆l in order to rotate the structure can also be expressed in the design

parameters by means of the geometric representation. In contrast to the situation of moment equilibrium, the rotation now has a specific value, which makes that the structure needs to be considered in the deformed state. The final formula for the required change in length ∆l is expressed in

the following equation, of which the derivation can be found in Appendix B:

∆𝑙𝑎𝑐𝑡;𝑣1 = l (1 − cos (𝜑

2) +

sin (𝜑2

)

tan (𝛽2

)) (30)

From equation (30) it can be stated that an increase in:

- the initial bar length l results in a larger required change in length ∆l per rotational angle φ

- the design angle β results in a smaller required change in length ∆l per rotational angle φ

An advantage of this variant are its symmetric deformations as a result of wind, causing a clear flow of forces through the structure.

3.1.2 Variant two Just like variant one, variant two also makes use of a linear actuator, as can be seen in Figure 12a. However, only now the lower element has an attachment, whereby the upper element is used as a push-off by the actuator.

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Figure 12: Variant two (a) Conceptual design; (b) Geometric representation

This structure leads to an advantageous moment distribution in the upper element, since the internal bending moment is reduced by the force in the actuator. The moment distribution as linearly analysed with GSA 8.7 is presented in Figure 13, which indeed shows this reduction in bending moment. Due to the force in the actuator, the shear forces in the upper part increase as well. However, it is expected that the influence of these forces on the structural dimensions are less significant than that of the bending moment.

Figure 13: Moment distribution in the connection for variant two

Using the geometric representation (Figure 12b), the force of the actuator for variant two becomes:

𝐹𝑎𝑐𝑡;𝑣2 =M𝑐𝑜𝑛 ∙ tan(𝛽)

l (31)

The required change in length ∆l per rotational angle is defined by the following formula:


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∆𝑙𝑎𝑐𝑡;𝑣2 = 𝑙 ∙ (1 −

√1 − 2 cos(β − 𝜑) ∙ cos(𝛽) + cos2(𝛽)

sin(𝛽)) (32)

From equation (32) it can be stated that an increase in::

- the initial bar length l results in a larger required change in length ∆l per rotational angle φ

The design angle β is captured in multiply equations of sine and cosine, which makes it hard to predict

its influence. Anyway, a turning point for the required change in length ∆l when β = φ is expected,

since then the equation cos(β- φ) will become positive. The influence of the design parameters β and l on the force F in the actuator is expressed graphically in Figure 14, hereby using equations (29) and (31). Equations (30) and (32) were used to form Figure

15, which visualizes the influence of the design parameters on the required change in length ∆l.

Variant one and two are described for three values of the initial bar length l. The boundaries for the design angle β are dependent on the design, making that variant one varies from 0 to 180° and for variant two from 0 to 90°. The moment in the connection is set at a value of 40.176 kNm, which follows from the distributed moment calculated with the information of Appendix C. The rotational angle is assumed at a value of 15°, which is based on the maximum expected active deformations.

Figure 14: The influence of the design parameter β on the force F for l=0.3, l=0.4 and l=0.5 m. (dark to light)

Figure 15: The influence of the design parameter β on the required change in length ∆l for l=0.3, l=0.4 and l=0.5

m. (dark to light)

0

500

1000

1500

2000

2500

0 30 60 90 120 150 180

F [k

N]

β [°]

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180∆l [

m]

β [°]

Variant 2 Variant 1

Variant 1

Variant 2

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From Figure 15 it can be stated that in general, the larger the design angle β, the higher the force F in

and the lower the required change in length ∆l of the actuator. Furthermore, by increasing the initial

bar length l, the force F becomes lower and the required change in length ∆l higher. This is in

agreement with the geometric representation, since a larger design angle β and a shorter initial bar length both imply a shorter arm a and thus a larger force F to transfer the moment. A shorter arm a also results in a smaller change in length per rotating angle, which can be seen from the geometric representation as well. These statements are also in agreement with equations (29) to (32). Considering variant one and two, it can be seen from Figure 14 that their asymptotes are situated around 90° and 180° respectively. For these angles the arm is zero, which makes the force infinitely large. The rotating angle can be recognized from the summit in the graph of Figure 15, located precisely on the dotted line at β = 15°. So comparing variant one and two, it can be stated that variant one has a smaller force in the actuator, whereas variant two needs a smaller change in length to rotate the structure. The force in the actuator is directly correlated with its speed. When the force is smaller, the speed increases, which implies a smaller response time for the structure. However, a smaller force requires a larger change of length, which on its turn increases the response time of the structure again. So it depends on the precise specifications for force and speed of the linear actuator which variant suits best. For known design parameters, these graphs can be used to directly check the required force and length change, making the selection procedure for the right actuator easier. And when the design has to be made for a chosen actuator, these graphs can be used as well, but then the other way around. The length b of the geometric representation is an indication of the amount of material needed for the active connection. To compare variant one and two, a design angle β and initial bar length l is chosen whereby both variants have about the same value. By choosing β = 45° and l = 0.4 m, which is in accordance with the length of an actuator strong enough for these forces, b becomes:

𝑏𝑣1 =𝑙

2 ∙ sin(𝛽2

)=

0.4

2 ∙ sin(452

)= 0.523𝑚 (33)

𝑏𝑣2 =𝑙

sin(𝛽)=

0.4

sin(45)= 0.566𝑚 (34)

Since variant one needs two bars of length b for the connection, this variant is less efficient regarding material use in the connection. Moreover variant two has a safe mechanism in case wind suction is present and the actuators cannot hold the structure. The extended pin with length b then provides a lock mechanism that prevents the structure from collapsing. For these reasons and together with the more advantageous moment distribution in the connection as explained earlier, variant two is preferred over variant one.

3.1.3 Variant three The third variant consists of a linear actuator attached to the hinge and the attachments of both profiles. When the linear actuator becomes longer, both the hinge as the attachments are pushed away, hereby rotating both profiles and thus the structure in that specific point. The variant is visualized in Figure 16.


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Figure 16: Conceptual design of variant three

The force in the actuator does not provide a moment directly as was the case for variant one and two. It provides for a resultant force in the attachments, which in combination with the distance from the line in action to the hinge (arm) results in a moment. When again setting the design angle β at 45° and the initial bar length l at 0.4 m, the force of the actuator is lower than in the other variants. Where variant two had a force in the actuator of 100.5 kN, this this variant only has a force of 83.2 kN. This is visualized in Figure 17a. However, this variant has a larger material use in the connection than the other since the attachment on itself is very long, namely 1.045 m. In order to make a complete comparison of material use between the variants, the attachment is adjusted to the same length as variant two, which resulted in the variant visualized in Figure 17b. Now the force in the actuator has increased largely to a value of 200.9 kN, which can be explained by the arm that is significantly reduced from 0.370 to 0.283 m. Additionally, the angle between the actuator and the attachment is smaller, so that the resultant force is larger in the actuator. These factors cause the large increase of force in the actuator.

Figure 17: Force distribution in node C of variant three for (a) β = 45° and l = 0.400 m; (b) an adjusted length of

the attachment

Concluding, variant three has a larger force in the actuator and still uses more material for the connection in comparison with variant two. Therefore variant two is preferred over variant three.

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3.1.4 Variant four Variant four consists of a linear actuator that is connected to a bar which is connected to two fixed attachments. A visualisation of this variant is provided by Figure 18.

Figure 18: Conceptual design of variant four

This variant is investigated to explore the possibilities for a continuous structure in the node, so that the moment could be transferred directly to the next profile. In the other variants the actuator was used to transfer moments and to rotate the structure, while this variant only uses the linear actuator for rotation. This principle is visualized in Figure 19a and b for variant four with and without linear actuator respectively. As can be seen from this Figure, the connection can transfer a moment without actuator. However, the forces in the bar and attachments are significantly decreased with actuator.

Figure 19: Variant four (a) with actuator; (b) without linear actuator; (c) forces to make an angle of 1°

Furthermore, the actuator needs to rotate the structure. When analysing this behaviour, it becomes clear from Figure 19c that the force in the actuator per rotational angle is much larger than for other variants. This is caused through the resistance of the material against deformation. Hence, there is no free rotation and the actuator needs to deliver an extra force to overcome the resistance of the material apart from the force caused by wind. Therefore this variant is not preferred over the other ones, making variant two still the most suitable one.


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3.1.5 Variant five The last variant consists of a rotary actuator (presented in Figure 10b), in contrast to the linear variants mentioned earlier. The conceptual design of the rotary actuator in the connection is visualized in Figure 20. The upper profile is connected to the rotary pin of the actuator, whereas the lower profile is connected to the fixed part of the actuator.

Figure 20: Conceptual design of variant five

Rotary actuators are integrated in line with the structure, which does not require a separate constructional part for actuation. This also implies that the input is directly transferred to the hinge, which gives better control of the required rotational angle. A large drawback of this variant compared to the other variants are its large dimensions. To increase the moment of the motor, a gearbox can be mounted on the rotating axis of the motor, also resulting in a decrease of rotary speed. A suitable ratio between moment and speed then has to be established. It is expected that technological improvements will lead to an increase in capacity and a decrease in dimensions of both linear and rotary actuators. The principle of linear actuators are based on force times distance, whereas the principle of rotary actuators are based on direct moment transfer. Linear actuators thus always need extra material to create an arm, in contrast to rotary actuators that really can be integrated in the structure. The key issue for this project is to use actuation in reducing material use of the structural profiles. Considering these technological improvements, rotary actuators then fit best for this purpose. Not only they use less material, they also can be integrated in the structure, which together with the smaller profile dimensions increase the transparency of the facade. This makes that variant five is preferred over variant two and thus forms the answer on sub-question three. Implementing variant five has consequences for both the numerical as the experimental model. For the numerical model the variant is modelled as a node without any attachments. Then the required rotation can be implemented by means of a distortion in radians. Only for a dynamic linear analysis a distortion cannot be inserted via a load curve, which is why for this analysis the active deformation is inserted static linearly. Consequently, the deformation by wind and the active deformation are combined via a combination case, resulting in the final deformation for a dynamic linear analysis. This is valid since both cases are linear and can thus be added. For the experimental model the choice is made for electric actuators, since these need few equipment than hydraulic or pneumatic actuators. Moreover, since it concerns a scale model, the forces are expected not to be that high. Therefore, variant five results in four rotational actuators of the type Parker Digiplan ML2340A, as visualized in Figure 21a. Each actuator is controlled by means of a BLHX30 servo regulator, which is visualized in Figure 21b.

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Figure 21 Parker Digiplan (a): ML2340A servo motor; (b): BLHX30 servo regulator


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3.2 Numerical test The main goal of the numerical test is to verify whether the deformations are controlled, hereby making use of the database as presented in the methodology. Chapter 3.2.1 presents the final deformations under different loads and by means of different analyses and verifies whether they are controlled. The influence of time response and mass is investigated in Chapters 3.2.2 and 3.2.3 respectively. The enlarged graphs can found in Appendix F. The input parameters for both the database as the numerical model are presented in Table 6. The mass of the facade is an estimation based on a reference project with glass facade panels attached to the millions (Hunter Douglas Contract, 2010). Aluminium is chosen as material, since the concept of the Structural Active Facade strengthens for materials that only have a low stiffness as main disadvantage. The viscous damping ratio of continuous metal structures (Irvine, 2014) is hereby used as damping ratio for aluminium and the 0.2 % proof stress as limit for elastic calculations (Soetens, Maljaars, van Hove, & Pawiroredjo, 2014, p. 39). As profile a square hollow section is used, since these are insensitive to lateral torsional buckling according to NEN-EN 1999-1-1, article 6.3.1.4 (1) (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011d). The design check of this profile can be found in Appendix D. The mass of one actuator of the type as visualized in Figure 10b is 505 kg. The maximum load that is based on wind measurements over time in Eindhoven from 1971-2015 (KNMI, 2013a), as was visualized in Figure 2. Using the extreme wind velocity of 34.5 m/s in that time period results in a maximum wind pressure of: 𝑞𝑝(𝑧) = 1

2⁄ ∙ 𝜌 ∙ 𝑣𝑏2 = 1

2⁄ ∙ 1.25 ∙ 34.52 = 𝟎. 𝟕𝟒𝟒 𝒌𝑵/𝒎𝟐 (35)

Consequently, this value is multiplied with the centre to centre distance of the millions to obtain the load qmax. When calculating the design wind pressure by means of the Eurocode (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011c) a maximum wind pressure is found of 0.94 kN/m2 (Appendix C), which indeed gives a safe upper bound of the maximum wind pressure that can occur. Since this research is concerned with the real wind velocities over time, the value of equation (35) is used for the database. Next, the maximum load is divided into five steps, with thus six values for every load. This results in a total of 7776 load combinations and an equal number of rows in the database. The precise equation used for the database can be found in Appendix E.

Table 6: Parameter values for the database and the numerical model

Type Parameter Unit Database Numerical model

Material ρ [kg/m3] - 2710

E [kN/m2] 7.00e10 7.00e10

Profile

h [m] 0.135 0.135

b [m] 0.135 0.135

t [m] 0.10 0.10

Structure

H [m] 15 15

c.t.c. [m] 2 2

mfacade [kg/m2] - 30

ζ [%] - 2

Load

q [kN/m] - 0 - 1.488

qmin [kN/m] 0 -

qmax [kN/m] 1.488 -

a [-] 5 -

b [m] 3 -

Actuation mactuator [kg] - 505

Tresponse [s] - 0; 0.3; 0.5; 1.0

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As mentioned in the methodology, the numerical model uses two types of load for the analysis. The first one is the maximum wind pressure as explained above, the second one is based on wind measurements during one minute at the Auditorium of the Technical University of Eindhoven (Sleddens, 2012). Subsequently, this wind spectrum is enlarged by duplicating the spectrum and scaled so that the maximum wind velocity corresponds to the extreme wind velocities of the past 35 years (Figure 2). The result is visualized in Figure 22, whereby every load q has a randomly selected starting time and a duration of one minute.

Figure 22: Wind spectrum of 120 seconds with spectra for q1-q5

With these input parameters the numerical model is analysed according the methodology (Table 5), which is presented again by means of Table 7. This chapter presents the most significant graphs and insights and finally answers sub-question four.

Table 7: Analysis of the numerical model concerning deformation control

Deformations

Analysis type Wind load Response time Mass

SL qmax 0.0 s mall

SNL qspectrum 0.3 s mall-act

DL 0.5 s m-

1.0 s

3.2.1 Final deformations The three types of analyses and two wind loads are used to verify whether the final deformations satisfy the deformation requirements, whereby the response time and mass are set on 0.0 s. and mall respectively. Figure 23 contains the results of a linear static analysis whereby the structural active facade is subjected to a maximum wind load. As can be seen from the figure, the deformations by wind are reduced from 1.072 m to 0.073 m. by actuation, which is a reduction of 93.2%. According the deformation requirements of equation (10), the maximum deformation should be 0.060 m. in the midst of the facade, which is almost satisfied. This is caused by the active deformation that deviates little from the theoretical calculation.


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Figure 23: Static linear analysis with maximum wind load (a) Deformations by wind; (b) Active deformations; (c)

Final deformations

Figure 24 visualizes the linear static analysis with a wind spectrum, whereby the deformations in the midst of the facade are presented against time. It can be seen that the deformations by wind are significantly reduced and that the final deformations clearly satisfy the deformation requirements.

Figure 24: Static linear deformations in the middle of the facade under wind load spectrum

The second type of analysis is nonlinear and is considered for a maximum wind load and wind spectrum again. Figure 25 presents the first load case, whereby it can be seen that the deformations are clearly larger in comparison to the linear analysis. This can be ascribed to the mass of the

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actuators (505 kg each) which enlarge the deformations further after initial deformation, known as the second order effect. From Figure 25c it can be seen that although the deformations are reduced (0.109m.) with 93.4%, the requirements (0.060 m.) are not met. This is because the limit of the database is reached, which originates from linear calculations and thus gives a too small range for the expected deformations and strains. For deformation control the actuation should have been larger.

Figure 25: Static nonlinear analysis with maximum wind load (a) Deformations by wind; (b) Active deformations;

(c) Final deformations

Figure 26 considers the nonlinear static deformations in the midst of the facade for a wind load spectrum, whereby it can be seen that the final deformations never satisfy the requirements. At first sight it looks like the final deformations are the result from the summation of the deformations by wind and the active deformations, which is remarkable. Namely, this would imply superposition, a principle that does not hold for a nonlinear analysis. A closer consideration proofs that this summation and the final deformations are not exactly the same, which thus excludes the principle of superposition. However, it is still remarkable that these deformations are that close. The cause can be found in the way the actuation is inserted in the program. Since the aspect of time is not considered for a static analysis, both the beam load (wind) as the distortion (actuation) are implemented simultaneously. This results in a cambered beam subjected to a wind load, which behaves differently than a straight beam subjected to the same wind load. Namely, a cambered beam is more stiff, which is why the wind load introduces a smaller deformation then for a straight beam. Moreover, the wind load decreases the curvature and thus the second order effect, resulting in smaller deformations as well. This explains why the final deformations are smaller than the deformation by wind or actuation and are at the same side of the facade as the camber. However, no conclusion can be drawn from this nonlinear static analysis on deformation control since it concerns a different type of beam than intended. The aspect of time has to be taken into account, since then the deformation by wind and the actuation develop simultaneously.


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Figure 26: Static nonlinear deformations in the middle of the facade under wind load spectrum

The last type of analysis is dynamic and is visualized by Figure 27. This type of analysis takes also the influence of the preceding actions into account. There are two phenomena that need to be highlighted, namely the vibrations and the larger deformations in comparison with the linear static analysis. The vibrations in the graph for wind are the results of a slender and heavy structure that accelerates by a wind load (inertia effect) and decelerates by the structure’s strain energy, whereby an equilibrium is found between these two continuously. This strain energy results in larger structural deformations than for the linear static analysis. Since the active deformations originate from the measured strains of the deformations by wind, these deformations also contain vibrations. These vibrations are less fluently since the solving procedure contains some tolerances to fit the final deformations between the requirements.

Figure 27: Dynamic linear deformations in the middle of the facade under wind load spectrum

Interesting to see is that the final deformations also contain vibrations, which can be seen more clearly from the enlarged graph in Figure 28. As can be seen from the Figure, the final deformations are nearly always controlled, except for some small peaks. These peaks result from the inaccuracy of the

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database and will be better controlled for a database containing smaller step sizes. The period of the graph changes continuously because of the changing wind load, but is on average about 2.5 s. After calculation the eigenfrequency of the structure turns out to be 1.715 Hz, which results in an eigenperiod of 0.583 s. Since the eigenperiod of the structure does not corresponds to the continuously changing period of the structural behaviour, there is no resonance effect, which can be recognized in both graphs by no continuously increasing amplitude (Sleddens, 2012).

Figure 28: Dynamic linear final deformations in the middle of the facade under wind load spectrum

3.2.2 Influence of response time To understand the influence of response time of the actuators, the final deformations in the middle of the facade are plotted against time for different delay times. The linear analysis is presented by Figure 29, whereby it can be seen that the deformation line shifts to the right for an increase in response time. The peak at the beginning of the graph is the result of the discontinuity of the analysis and should therefore not be considered for deformation control. Considering the rest of the graph it can be concluded that the deformations are controlled for 0.0 and 0.3 s. and not controlled for 0.5 and 1.0 s. This sets the lower bound of the required response time of the actuators at 0.3 s. and the upper bound at 0.5 s. Moreover, this analysis points out that response time of the actuators plays a major role in deformation control, since a response time of 1.0 s. already can double the deformations.

Figure 29: Static linear final deformations in the middle of the facade under wind load spectrum for delay times

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δ[m

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T [s]

δfinal δallowed δallowed

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0.5

0 5 10 15 20 25 30 35 40 45 50 55 60

δ[m

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T [s]

δfinal;0.0 δfinal;0.3 δfinal;0.5 δfinal;1.0 δallowed δallowed


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Figure 30 visualizes the dynamic linear final deformations for different delay times. Hereby it can be seen that the deformations exceed the requirements already after 0.3 s. For a delay time of 1.0 s., the active deformations reach their maximum value at the moments that the deformation by wind is at its minimum. This results in final deformations being thirteen times larger than a structure without any delay time. So the response time of the actuators is even more important when the acceleration of the structure is taken into account.

Figure 30: Dynamic linear final deformations in the middle of the facade under wind load spectrum for delay times

3.2.3 Influence of the mass Chapter 3.2.1 already stated that for a nonlinear analysis no direct conclusions can be drawn on deformation control since it concerns a different type of beam. In spite of this limitation, still the nonlinear analysis for varying masses can provide for valuable results and conclusions. Figure 31 presents the nonlinear deformations for all the masses, all the masses without actuators and no mass at all. Only the deformations by wind are presented, since the active deformations and final deformations make no sense for the nonlinear analysis in this case. From the differences in deformations between the graphs it can be seen that the mass of the actuators influences the deformations significantly. Namely, the difference between the graphs for mall and mall-act is larger than the difference between the graphs for mall-act and m-. This can be explained by the large mass of one actuator, namely 505 kg, that increases the deformations through the second order effect. Although in the final state these deformations will not occur and the second order effect is less, it is still important to account for the mass of the actuators. Namely, when the actuators malfunction, the deformations can be significantly increased, hereby increasing the stresses in the million. Therefore it is important to account for the second order effect in structural design to provide the required level of safety and to minimize the masses of the actuators.

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0.8

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T [s]

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Figure 31: Static nonlinear deformations by wind in the middle of the facade under wind load spectrum for masses

For the dynamic linear analysis the mass directly affects the deformations by wind and indirectly affects the active and final deformations. Therefore, Figure 32 only presents the deformations by wind, whereby the last variation of the mass cannot be visualized since the dynamic calculation cannot be performed without any mass. From the Figure it can be seen that a decrease in mass results in smaller deformations, smaller vibrations and smaller periods, which can be explained by the inertia effect as mentioned earlier. Smaller deformations by wind result in smaller active deformations for deformation control, which sets thus less strict specifications for the actuators concerning the magnitude of the rotational angles. However, smaller vibrations and periods indicate that the structure is more sensitive, which requires fast actuators to steer the deformations for deformation control.

Figure 32: Dynamic linear deformations by wind in the midst of the facade under wind load spectrum for different

masses

So to answer sub-question four about numerical deformation control it can be concluded that the method as proposed in the methodology can be used to control the final deformations as long as the Structural Active Facade behaves linearly. The nonlinear analysis results in a different type of beam

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and although it cannot be used to verify deformation control, it still provides for valuable insights into the structural behaviour. Namely, it proves that the second order effect of the large masses of the actuators significantly increase the deformations. Apart from this effect, the inertia effect of a large mass enlarges the deformations as well. It is therefore important to minimize the masses, which also results in a more sensitive structure that requires faster actuators. A delay time in response can enlarge the deformations significantly, especially when the active deformations are at their maximum and the deformations by wind at their minimum. There is no resonance effect since the eigenperiod of the structure does not correspond to the real occurring period, which changes continuously.

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3.3 Experimental test Likewise the numerical test, the main goal of the experimental test is to verify whether the final deformations are controlled and to measure the response time of the system. Chapter 3.3.1 presents the final deformations for different loads and verifies whether the deformations are controlled. Chapter 3.3.2 investigates the influence of the response time of the system. Since the deformations and moments are quite smaller than in the numerical model, the units m and kN are changed to mm and N. First two secondary tests were performed to determine the maximum torque of the actuator and the bending stiffness and relation between moment and strain of the plate, as presented by Figure 33a and b respectively. The maximum torque was determined by multiplying the measured force at the load cell with the distance of the cantilever, hereby also taking into account its self-weight. The three-point-bending test measured the applied force and deflection, which resulted in the bending stiffness by means of the following derivation:

𝛿 = 𝛿𝐹𝑜𝑟𝑐𝑒 + 𝛿𝑆𝑒𝑙𝑓−𝑤𝑒𝑖𝑔ℎ𝑡 =1

48∙

𝐹 ∙ 𝐿3

𝐸𝐼+

5

384∙

𝑞 ∙ 𝐿4

𝐸𝐼 (36)

𝐸𝐼 =1

𝛿(

1

48∙ 𝐹 ∙ 𝐿3 +

5

384∙ 𝑞 ∙ 𝐿4 ) (37)

Apart from the applied force and deflection, also the strains were measured, which was done by means of a strain gauge. Since the moment can be derived from the applied load this resulted in the relation between moment and strains. The mean of two plates, tested at both sides, resulted in the values for EI and M-ε as presented in Table 8. The graphs of these secondary tests can be found in Appendix G. The enlarged graphs of the results can be found in Appendix H.

Figure 33: (a) maximum-torque test of the actuator ML2340A; (b) three-point-bending test of the plate

Table 8: Parameter values for the database and the experimental model

Type Parameter Unit Database Experimental model

Material EI [Nm2] 3.61 3.61

Profile M-ε [1/Nm] 0.000371 0.000371

Structure

H [m] 1.550 1.550

mactuator-frame [kg] - 3.250

mplate [kg] - 0.977

Load

q [N/m] - 0 – 196

qmin [N/m] 0 -

qmax [N/m] 392 -

a [-] 3 -

b [m] 0.012 -

Actuation mactuator [kg] - 2.100

Mactuator;max [Nm] 1.8 1.8


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The scale model of the Structural Active Facade is presented by Figure 34. The structure consists of two actuators and three plates of the types that were tested above. At one end the structure is supported by a hinge, at the other end by a roller hinge. A frame on top of the actuator connects two plates and one actuator and consists of a linear axis and a casing. The linear axis is fixed to both the axis of the actuator and one of the plates. The other plate is fixed to the casing, so that a continuous structure is formed. The maximum torque of 1.8 Nm per actuator limited the applied load to 0.240 kg. However, to visualize deformation control, the deformations needed to be large in relation to the height. Therefore, a steel plate of 0.8 mm was used. This small thickness was not sufficient to carry the self-weight of the actuators, which required a separation of the direction in which the loads were applied and the direction of the self-weight. Since the latter cannot be changed, the structure was placed horizontally, whereby the loads were applied in horizontal direction. Steel bended strips were welded to the plates in vertical direction to provide for torsional stiffness, without affecting the bending stiffness in the direction of the load. Initially the plan was to use four actuators in combination with five plates. However, the roller supports under the actuators resulted in too much friction to be overcome by the actuators. Removing these roller supports resulted in an unstable structure, since the mass of four actuators is too large to be carried by the plates. Therefore, the system was reduced to two actuators and three plates, as visualized by Figure 34. The low loads in combination with the large deformations resulted in slip of the load cable on the potentiometer. Therefore, a laser displacement sensor was used, which provided for contactless measurement of the deformations in the middle of the facade. In preparation for the experimental test, a database was constructed based on the values for EI and M-ε of the secondary tests. Further on, the maximum load was doubled, so that a larger occurring deformation than expected also results in a match during testing. The maximum load was divided into 19 steps, which resulted in a total of 8000 load combinations and rows in the database. The number of loads (a) is three and the width (b) of the load comes from the diameter of the washer (Table 8).

Figure 34: Experimental model of the Structural Active Facade

Actuator B Actuator C

δ-sensor

Roller hinge support Control system

Hinge support

Plate 1 Plate 2 Plate 3

qwind;1 qwind;2

qwind;3

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3.3.1 Final deformations Figure 35 presents the final deformations in the middle of the facade, whereby it can be seen that the deformations are not controlled. Three tests were performed wherein the facade was subjected to a load spectrum of three steps: from 0 to 0.120, from 0.120 to 0.240 and from 0.240 to 0 kg again. The load introduction for the first two steps starts at the second plate, followed by the first plate and ending with the third plate. The load removal in the last step is in reversed sequence. The load attachments within each step can be recognized in the Figure from the sudden increases in deformation. After each step the actuators rotated in response to the measured strains, which is at 72, 137 and 187 s. for test one, at 68, 125 and 182 s. for test two and 88, 144 and 194 s. for test three. These moments of actuation can be recognized in the Figure from small decreases or even increases in deformation, which are far from the allowed deformation line. The last heavy vibrations can be ascribed to the sudden release of the final load attachment and after the last moment of actuation the actuators were turned off.

Figure 35: Final deformations in the middle of the facade under load spectrum for test one to three

Although Figure 35 answers sub-question five, it does not explain why the deformations are not controlled. The core of the methodology lies within the relation between the strains and the rotations. Therefore, both the strains that are measured in the midst of every plate and the rotations by the actuators are presented by Figure 36 and Figure 37 respectively. Hereby test two and three are not presented, since the pattern for all tests is the same and adding six more graphs would only obscure both Figures. The measured strains of Figure 36 are used to find the best match in the database, which provides for the rotations of Figure 37.

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Figure 36: Strains in the three plates for test one

Figure 37: Time-specific (t) and constant (c) rotations for actuators B and C for test one

Figure 37 contains two types of rotations, namely time-specific (t) and constant (c) rotations. Time-specific rotations were actually send to the actuators in the laboratory, which is done specifically at the instants of actuation. Constant rotations are afterwards computed rotations based on every measured strain, which could be seen as a real time operating system. As can be seen from the Figure, these two types of rotations match at the specific real instants of actuation. This confirms that the methodology of coupling the strains to the rotations was performed correctly. A further investigation into the strains resulted in Figure 38, whereby the expected values of the strains based on the real measured deformations are added to the graph of Figure 36.

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Figure 38: Strains in the three plates for test one, including the expected values based on the real measured

deformations

As can be from Figure 38, the expected strains are almost always higher than the real measured strains, especially for the real measured strains of plate one in the first 100 s. and plate three in-between 135 and 165 s. This indicates that the system behaves stiffer than expected, which thus results in an underestimation of the deformations and too small rotations for the actuators. The positive strains of plate one are remarkable, since the strains were attached at the compression side of the facade. This can be ascribed to the initial position of the actuators. Namely, since the actuators hang on the plates, their position is quite instable and it was observed that this instability can result in a difference of even 200 μm/m. Apart from an underestimation of the deformations, there is another reason why the deformations are not controlled. As can be seen from Figure 36, the strains increased for every load introduction and decreased for every load removal, which is in accordance with the deformations of Figure 35. However, the strains also increased for actuation when there is a load applied and decreased for actuation when no load was applied, which was not expected. Namely, according the theory used in the methodology, the strains would not be affected by actuation. The measured increase in strains indicates that the plates are further bended in the load direction, which requires energy. Since there is no other external parameter than the actuators that change the system at the instant of actuation, it is expected that this energy is delivered by the actuators. Therefore, the produced moment for each actuator is visualized by Figure 39.

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Figure 39: Produced moment per actuator for test one

As can be seen from the Figure, the moments increased for every load introduction, which is in accordance with the theory. However, the moments increased as well at the instants of actuation when there is a load applied to the structure (at 72 and 137 s.). Together with the observation that the strains increased at that same instant, it can be concluded that the structure rather bends further than that it moves. Apparently, the roller friction of the end support is still too large to be overcome by the actuators, despite the precise and well thought-out made design in the laboratory. The decrease of produced moment and the decrease in strains after load removal and the last actuation (at 187 s.) both indicate that the system relaxes. This is in accordance with the explanation given above, since at those instances the structure rotates in the opposite direction (Figure 37) and thus decreases the bending and energy in the system. At last, it can be seen from Figure 36 that the strains do not reach zero for load removal and that they decrease largely when the actuators are turned off. This is again proof that the plates are bended by the actuators and that the system contains a lot of energy that is released as soon as the actuators no longer give any resistance. So due to an underestimation of the deformations based on the strains and a too small stiffness of the structure, the deformations are not controlled. To investigate the principle of deformation control further, the founded rotations are multiplied with a factor of six. Also, each instant of actuation is performed twice, so that it is more likely that the required rotational position of the actuator is reached. Theoretically this iterative approach does not influence the rotations, since only the difference between the real and required rotational position of the actuator is implemented. Again, three tests were performed according the same procedure as the previous tests, of which the final deformations in the midst of the facade are visualized by Figure 40.

0

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0.2

0.3

0.4

0.5

0 25 50 75 100 125 150 175 200 225

M [

Nm

]

T [s]

MB MC

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Figure 40: Final deformations in the middle of the facade under load spectrum for test four to six

It can be seen from this Figure that the deformations now are significantly reduced by actuation, namely at 72, 76, 136, 139, 177 and 181 s. for test four, 59, 63, 123, 126, 168 and 172 s. for test five and 62, 68, 122, 125, 163 and 167 s. for test six. Due to the large rotations the system even moves to the opposite side of the load introduction, whereby the point of turnover can be recognized by the heavy vibrations around 140 s. for test four and 130 s. for tests five and six. The measured strains are more than doubled in comparison with the first three tests and the actuators reach their maximum moments at 1.8 Nm without break down. This thus confirms that for even larger rotations the energy in the structure is increased. The principle of deformation control at the moment just before and just after actuation is visualized by Figure 41 and Figure 42 respectively for test four. The Figures really clarify the reduction in deformations through actuation and the oblique position of the structure after turning over.

Figure 41: Principle of deformation control for test four just before actuation at t=130 s. (a) top view; (b) front view

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Figure 42: Principle of deformation control for test four just after actuation at t=146 s. (a) top view; (b) front view

So to answer sub-question five about experimental deformation control it can be concluded that the deformations are not controlled due to an underestimation of the deformations based on the strains and a too small stiffness of the structure. The small stiffness of the structure is just a limitation of this research, which would be solved by stronger actuators and stiffer plates. However, an underestimation of the deformations based on the strains is fundamental and indicates that the accuracy of the model as proposed by the methodology is extremely important.

3.3.2 Influence of time response Initially the plan was to measure the time response of the system and its influence on the deformations. However, since this system dated back to 1997, it already became clear before testing that this system would not be fast enough to be used in a real Structural Active Facade. Therefore it is decided not to measure the time response of the system.

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Conclusion and discussion

49

4. CONCLUSION AND DISCUSSION

This research is based on the following main question:

How can the deformations of a structural facade be actively controlled in response to its changing loading environment over time?

whereby the method as described in Chapter 2 is proposed as answer. First the conclusion considers whether this method is sufficient for deformation control, after which the discussion interprets the results, discusses the limitations and gives recommendations for further research. For the conclusion and discussion the method is divided into the following three translations that are used as guideline:

Strains to deformations

Deformations to rotations

Rotations to deformations

4.1 Conclusion Starting with the last translation, the results of the variant study show that rotating actuators have the largest potential in deformation control for the Structural Active Facade. The experimental tests four to six prove that electric rotary actuators can be used to steer the deformations (Figure 40 to Figure 42), hereby reducing the final deformations significantly. So translating rotations to deformations works well via the method of this research. The translation from deformations to rotations is formed by the solver function and its targets as explained by Chapter 2.1.5 and the control algorithm that is used to find the best match in the database. A visualization of this solver function and its targets show that the final deformation line is within the allowed deformation lines for every step. The results of experimental test one (Figure 37) demonstrate that the control algorithm succeeds in finding the best match based on the measured strains. These results prove that the second translation via this research’ method also succeeds as long as the boundaries set by the solver function and its targets are met. At last, the translation from strains to deformations as explained in Chapter 2.1.1 is considered. The results of the numerical static and dynamic linear tests (Figure 24 and Figure 27) show that the deformations are controlled and that also this last translation seems to work fine. However, the experimental tests one to three (Figure 35) demonstrate that the deformations are not controlled. Since this research’ method originates from linearity, it explains why this translation works fine for the numerical tests which are linear, and does not work for the experimental tests, which do not confine to a specific type such as linearity. At first sight it seems like a research model which accounts for both nonlinearity and dynamics will correct the relation between strains and deformations. With a correct state of deformation as input for the other two translations the deformations then can be controlled. However, not only requires such a model large computation times and high modelling skills, it also does not account for imperfections in the system. Namely, the results of all the experimental tests show that strains are extremely sensitive to imperfections as an oblique initial position of the experimental model already results in a difference of 200 μm/m. Further on, the results of the numerical model on the influence of response time (Figure 29 and Figure 30) show that a response time of only one second can enlarge the deformations with a factor of thirteen, which also emphasizes the importance of the correct translation between strains and deformations. Moreover, the system of this research’s method does not get any feedback on its control parameter (δm). It only measures the strains, which are not affected by actuation (for a stiffer and stronger system than the experimental model). Since the system cannot measure whether it approaches or distances from its control target, it cannot correct its deformation when the real state deviates from the expected (computed) state. So to conclude, the method as proposed in this research is not sufficient for deformation control of the Structural Active Facade. The main reason lies within the reconstruction of the deformations from the strains, which is too sensitive to imperfections and makes the system highly liable to errors. Determining the required rotations from the deformations works well via this method, however within the boundaries of the accuracy of the model. Finally, rotating actuators can be used to implement the rotations and actively steer the deformations of the Structural Active Facade.


S.T van Ruitenbeek

50

4.2 Discussion The discussion follows again the same translations as the conclusion and positions the results within the scope of the main question.

4.2.1 Interpretation results The translation from strains to deformations is used in preceding research (Neuhäuser et al., 2013; van Bommel, 2016) to determine the correct state of the system. The advantage of using strains is that apart from deformations, also forces and vibrations can be measured. This gives large freedom in optimizing structures by means of actuation. However, as is proven by the results of this research, the strains are also highly sensitive to imperfections and their translation requires an extremely accurate model in advance to simulate the different states. For this model, it is expected that the frame of the actuators would hardly affect the assumed stiffness of the total structure, since it accounts for only 4% of the total height. However, it appears that despite its experimental determination, the bending stiffness of the plate gives an underestimation of the bending stiffness of the total structure, hereby causing errors in the translation from strains to deformations. Remarkable for this translation is that first the simulated deformations need to be translated to strains, which subsequently are compared with the measured strains to finally control the deformations again. It would be better to skip this translation entirely, which not only would decrease computation time and complexity of the model, it also would fasten the response of the system and reduce its liability to errors. The database translates the deformations to rotations and is in analogy with preceding research (van Bommel, 2016). Where preceding research used a static geometric nonlinear numerical model to set up the database, this research uses the principle of superposition (equation (20)). Both researches show that as long as the structural behaves within the boundaries of the solver function and its targets, the deformations were controlled (this research) or the peak stresses minimized (preceding research). What these researches have in common is that they both try to predict the structural behaviour based on existing theory and assumptions. However, since it concerns a new aspect of structural engineering, it would be better to work the other way around and start with an as good as identical numerical and experimental model. Analysing and comparing their behaviour carefully by means of theory and preceding outcomes could then provide for a relation between the influence parameter (rotations of the actuators) and the control parameter (δm or σmax). The structure is then able to respond directly without the use of any database. This again not only provides for a decrease in computation and response time, it also gives the system the ability to correct for small errors, hereby preventing error accumulation. In the scope of the last translation, this research proves that rotary actuators can be used as well to control deformations, which is an addition to literature where only linear types were used (Sobek, Teuffel, Weilandt, & Lemaitre, 2006; Neuhäuser et al., 2013; Sobek & Teuffel, 2001; Fest, Shea, & Smith, 2004; Kmet, Pltako, & Mojdis, 2012; Noack, Ruth, & Müller, 2006; Senatore, 2015). It also makes clear that an active connection is a fifth way of static shape control, what was not yet investigated by literature. The variant study shows that rotary actuators provide direct load transfer and better control of the required rotation then linear actuators. Namely, linear actuators require a separate constructional part for actuation, hereby implementing more variable parameters that can influence actuation. Since rotary actuators do not have this extra constructional part, material is saved and in the picture of technological improvements, rotary actuators can actually be integrated in the structure. Together with the smaller profile dimensions that are within reach through actuation, the facade’s transparency is increased.

4.2.2 Limitations Firstly, the static nonlinear analysis by the numerical test limits this research. Namely, instead of a straight beam, a cambered beam was inserted in the program, which behaves more stiff and differently than intended. This makes that only the deformations by wind can be analysed nonlinearly. However, when the deformations by wind and the active deformations were implemented simultaneously, the influence of the 2nd order effect on the final deformations would have become more clear. Moreover, it could have resulted in deformation control, since then the following two assumptions for linearity are met for the final deformations:

material stress-strain relationship is linear (material Young's Modulus is constant)

displacement and strain relationship is linear (small displacement problem)

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Conclusion and discussion

51

Now it has not become clear whether the method as proposed in this research would have been sufficient for nonlinearity. What became clear is that for a real good understanding, not only a nonlinear analysis is necessary, also the aspect of time needs to be taken into account. Secondly, the analyses in this research are mainly static, with the exception of the deformations by wind. Namely, the static active deformations are superimposed upon the dynamic deformations by wind, which makes that the final deformations agree for a delay time of zero. A delay in response time could thus have more consequences on deformation control through dynamical effects, which makes that the analysis into the influence of response time provides for an first boundary of the response time of the system. A detailed dynamical analysis would be required to analyse this boundary further. At last, the actuators with its control system form a limitation to the experimental tests of this research. Namely, due to the low torque capacity of the actuators it was quite a puzzle to construct an experimental model that on the one hand needed to visualize deformation control and on the other hand might not collapse under its self-weight. Eventually, an appropriate construction was found and carefully constructed by means of experienced laboratory staff and accurate equipment. However, the structure was still unstable around its initial position, hereby influencing the strains and thus deformation control. Moreover, the required small plate thickness resulted in a too low stiffness of the total structure to really move the system, which was more likely to bend instead. So to remove this limitation in further research, it requires stronger and faster rotary actuators to construct a stiffer and more stable structure to control the deformations.

4.2.3 Recommendations As a result of interpretation of the results and the limitations of this research, the following recommendations are proposed for further research and design of the Structural Active Facade. Now that the variant study and the experimental model crystallized the concept of the Structural Active Facade, a numerical and experimental model can be constructed. Hereby the numerical model needs to be geometric nonlinear and dynamic and must form a detailed representation of the experimental model. The next step is to determine for which rotations the deformations are controlled in both models, which can be done through a searching algorithm that rotates the actuators until the deformations are controlled. This algorithm can be optimized, as was done in this research (Chapter 2.1.5). Then, by carefully comparing and analysing the results of both models by means of theory, literature and preceding results, a relation can be found between the required rotations and the final deformations. This relation forms the base of the control algorithm for deformation control and can be tested via the numerical and experimental model again, hereby replacing the searching algorithm. Since the numerical model can be subdivided into different types of analyses, as was already done for this research, the influence of the 2nd order and inertia effect on deformation control can be clarified. Also, an iterative approach can be used to steer the deformations, which means that the system is able to correct itself for (small) errors in deformation, hereby preventing error accumulation. To get feedback on deformation control, it is important that the input parameter corresponds to the control parameter. For this case this means that the deformations both need to be controlled and measured. In real, this would result in (laser) sensors at the top and bottom of the facade, since there is no possibility to measure the deformations horizontally in for example a large foyer. Further on, the connection that is formed by the rotating actuator can be optimized further. As resulted from this research, the mass of the actuators is very important in deformation control. Decreasing this mass would also result in a more sensitive and dynamical structure, which requires faster actuators. On the other hand, the forces in the structure due to wind are still significant, which requires strong actuators. A study into the newest developments of actuators that takes into account the reversed correlation between force and speed could optimize the connection further. Hereby, also the number of actuators, the allowed curvature and practical implementation could be investigated. Additionally, the actuators could be placed at the supports instead of integrated in the facade, so that also the forces can be influenced within the structure. The plastic strength of the material can be used, hereby extending the material savings further. Moreover, it is recommended to numerically simulate and experimentally test the wind behaviour on the Structural Active Facade. And at last, a strong recommendation is to involve other disciplines within this project, since it not only concerns Structural Design, it also concerns the aspects of Mechanical Engineering, Electrical Engineering and Systems and Control.


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References

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5. REFERENCES

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KNMI. (2013b, October). KNMI Hydra Project. Retrieved from http://www.knmi.nl/samenw/hydra/cgi-

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tec.co.uk/linear-actuators.php

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earthquake and wind loading control response – a review. Intelligent Buildings International,

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Neuhäuser, S., Weickgenannt, M., Witte, M., Haase, C., Sawodny, O., & Sobek, W. (2013). Stuttgart

SmartShell – A Full Scale Prototype of an Adaptive Shell Structure (Vol. 54, pp. 259–270).

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Noack, T., Ruth, J., & Müller, U. (2006). Adaptive hybrid structures. In International conference on

adaptable building structures, Eindhoven, Netherlands. Retrieved from

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Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’. (2011a). Eurocode 0:

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gewicht en opgelegde belastingen voor gebouwen (NEN-EN 1991). Delft: Nederlands

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Appendices

57

6. APPENDICES

Appendix A: Indication material savings Normally, the required dimensions for a normal beam are based on stiffness, resulting in a required second moment of area of:

𝐼𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 =5

384∙

𝑞 ∙ 𝐻4

𝐸 ∙ 𝛿𝑎𝑙𝑙𝑜𝑤𝑒𝑑

(38)

𝐼𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 =5

384∙

1.5 ∙ 154

2.1 ∙ 108 ∙ 0.06= 7.85 ∙ 10−5𝑚4 (39)

Which gives a common profile of HEA260 with a surface of 0.008682 m2. An active beam only has to be based on strength, resulting in a section modulus of:

𝑊𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 =1

8∙

𝑞 ∙ 𝐻2

𝜎𝑦

(40)

𝑊𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 =1

8∙

1.5 ∙ 152

3.55 ∙ 105= 1.19 ∙ 10−4𝑚3 (41)

This gives a common profile of HEA140 with a surface of 0.003142 m2, which is a reduction of 63.9%.

Appendix B: Derivation ∆l variants

Variant 1 From the figure of the geometrical representation it can be derived that:

sin(𝛽

2) =

12

𝑙

𝑏=

𝑙

2𝑏

(42)

sin(𝛽

2−

𝜑

2) =

12

𝑙 −12

∆𝑙

𝑏=

𝑙 − ∆𝑙

2𝑏

(43)

Combining equation (42) and (43) gives:

sin(𝛽

2−

𝜑

2) =

𝑙 − ∆𝑙

𝑙∙ sin(

𝛽

2) (44)

Difference formula for sin:

sin(𝛽

2−

𝜑

2) = sin(

1

2𝛽) cos(

𝜑

2) − cos(

1

2𝛽) sin(

𝜑

2) (45)

Combining equations (44) and (45) gives: sin(

𝛽2

) cos(𝜑2

) − cos(𝛽2

) sin(𝜑2

)

sin(𝛽2

)=

𝑙 − ∆𝑙

𝑙 (46)

cos(𝜑

2) −

sin(𝜑2

)

tan(𝛽2

)=

𝑙 − ∆𝑙

𝑙 (47)

The change in length is then given by:


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∆𝑙𝑎𝑐𝑡;𝑣1 = 𝑙(1 − cos (𝜑

2) +

sin (𝜑2

)

tan(𝛽2

)) (48)

Variant 2 The first length to be determined are: 𝑐 = 𝑎 ∙ sin(𝛽 − 2 ∙

𝜑

2) (49)

𝑑 = 𝑎 ∙ cos(𝛽 − 2 ∙

𝜑

2) (50)

𝑏 =𝑙

sin(𝛽) (51)

𝑒 = (𝑏 − 𝑑) =𝑙 − 𝑎 ∙ cos(𝛽 − 𝜑) ∙ sin(𝛽)

sin(𝛽) (52)

𝑎 =𝑙

tan(𝛽) (53)

Combining equation (49), (52) and (53) gives:

𝑙 − ∆𝑙 = √𝑙2 ∙ (cos2(𝛽) + 1 − 2 ∙ cos(𝛽 − 𝜑) ∙ cos(𝛽))

sin2(𝛽) (54)

The change in length ∆l is then given by:

∆𝑙𝑎𝑐𝑡;𝑣2 = l ∙ (1 −√1 − 2 ∙ cos(𝛽 − 𝜑) ∙ cos(𝛽) + cos2(𝛽)

sin(𝛽)) (55)

Appendix C: Maximum wind pressure Eurocode Starting with the fundamental value of the basic wind velocity: 𝑣𝑏,0 = 24.5 𝑚/𝑠 (56)

According to the code, the basic wind velocity can be computed with: 𝑣𝑏 = 𝑐𝑝𝑟𝑜𝑏 ∙ 𝑐𝑑𝑖𝑟 ∙ 𝑐𝑠𝑒𝑎𝑠𝑜𝑛 ∙ 𝑣𝑏,0 (57)

Filling in the factors for probability, wind direction and season: 𝑣𝑏 = 1,0 ∙ 1,0 ∙ 1,0 ∙ 24.5 = 24.5 m/s (58) The mean wind speed is calculated with: 𝑣𝑚 = 𝑐𝑟(𝑧) ∙ 𝑐0(𝑧) ∙ 𝑣𝑏 (59) Herein the roughness factor is given by: 𝑐𝑟(𝑧) = 𝑘𝑟 ∙ ln (

𝑧

𝑧0

) 𝑧𝑚𝑖𝑛 ≤ 𝑧 ≤ 𝑧𝑚𝑎𝑥 (60)

𝑐𝑟(𝑧) = 𝑐𝑟(𝑧𝑚𝑖𝑛) 𝑧 ≤ 𝑧𝑚𝑖𝑛 (61)

And the terrain factor:

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Appendices

59

𝑘𝑟 = 0.19 ∙ (

𝑧0

0.05)

0.07

(62)

To compare the calculation of the NEN-EN 1991-1-4 NB with the wind pressure of equation (35) by means of wind measurements, the height z is set at 10 m and the roughness length z0 at a value of 0.03 m. Together with the values of minimal height from table 4.1 of NEN-EN 1991-1-4 NB, equation (62) becomes:

𝑘𝑟 = 0.19 ∙ (0.03

0.05)

0.07

= 0.183 (63)

And the roughness factor:

𝑐𝑟(𝑧) = 0.183 ∙ ln (10

0.03) = 1.065 (64)

The mean wind speed is then: 𝑣𝑚 = 1.065 ∙ 1.0 ∙ 24.5 = 26.1 𝑚/𝑠 (65) The standard deviation of the turbulence can be computed with: 𝜎𝑣 = 𝑘𝑟 ∙ 𝑣𝑏 ∙ 𝑘𝑙 (66) 𝜎𝑣 = 0.183 ∙ 24.5 ∙ 1.0 = 4.491 𝑚/𝑠 (67) The turbulence intensity is the ratio of the standard deviation to the mean value: 𝐼𝑣(𝑧) =

𝜎𝑣

𝑣𝑚 (𝑧) (68)

𝐼𝑣(𝑧) =4.491

26.1= 0.172 (69)

Finally the wind pressure can be calculated: 𝑞𝑝(𝑧) = (1 + 7𝐼𝑣(𝑧)) ∙ 1

2⁄ ∙ 𝜌 ∙ 𝑣𝑚2(𝑧) (70)

𝑞𝑝(𝑧) = (1 + 7 ∙ 0.172) ∙ 1

2⁄ ∙ 1.25 ∙ 4.4912 = 𝟎. 𝟗𝟒 𝒌𝑵/𝒎𝟐 (71)

Appendix D: Design check strength

Check bending According to NEN-EN 1999-1-1, article 6.2.5.1 (2) (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011d) the design value of the elastic section modulus is determined by equation 6.25 for this case, which can be rearranged into:

𝑊𝑒𝑙 =𝑀𝐸𝑑 ∙ 𝛾𝑀1

𝛼 ∙ 𝑓0

(72)

With: 𝑀𝐸𝑑 = maximum occurring bending moment

𝛾𝑀1 = partial safety factor which accounts for the resistance against instability with a value of 1.10

𝛼 = form factor, the minimum value recommended by the Eurocode is 1.0 𝑓0 = characteristic value of the 0.2% proof stress, table 3.2b NEN-EN 1999-1-1 for type 6082-T6 Using the parameters as explained in Chapter 3.2, the required elastic section modulus becomes:

𝑊𝑒𝑙 =(𝑞𝑝 ∙ 𝑐. 𝑡. 𝑐. ) ∙ 𝐻2 ∙ 𝛾𝑀1

8 ∙ 𝑓0

=(0.744 ∙ 2) ∙ 152 ∙ 1.1

8 ∙ 250 ∙ 103= 1.84 ∙ 10−4𝑚3 (73)


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60

A square hollow tube profile of 0.132x0.132x0.010 m in aluminium satisfies for this requirement.

Check shear NEN-EN 1999-1-1, article 6.2.8 (2) states the influence of shear on bending can be neglected if (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011d):

𝑉𝐸𝑑 ≤1

2𝑉𝑝𝑙,𝑅𝑑 (74)

When the facade is subjected to the maximum wind load, the maximum occurring shear forces become 11.16 kN at the supports. First it must be determined whether it concerns a slender or non-slender section: ℎ𝑤

𝑡𝑤

< 39휀 (75)

With

휀 = √250

𝑓0

(76)

2 ∙ (0.132 − 2 ∙ 0.010)

0.010< 39 ∙ √

250

250 (77)

The maximum plastic shear strength is defined as:

𝑉𝑝𝑙,𝑅𝑑 = 𝐴𝑣 ∙ 𝑓0

√3 ∙ 𝛾𝑀1

(78)

Filling in equation (74) and assuming that the shear stresses are uniformly distributed makes that:

11.16 𝑘𝑁 ≤1

2∙

0.00488 ∙ 250 ∙ 103

√3 ∙ 1.1= 320.17 𝑘𝑁 (79)

Which makes it clear that shear is of no influence on bending.

Check normal force Since the actuator has a large weight, its necessary to check the influences of the normal forces as well. According to NEN-EN 1999-1-1, article 6.2.9.1 (1) the following criteria needs to be satisfied (Normcommissie 351 001 ‘Technische Grondslagen voor Bouwconstructies’, 2011a):

(𝑁𝐸𝑑

𝜔0 ∙ 𝑁𝑅𝑑

)𝜉0

+𝑀𝑦,𝐸𝑑

𝜔0 ∙ 𝑀𝑦,𝑅𝑑

≤ 1.00 (80)

(𝑁𝐸𝑑

𝜔0 ∙ 𝑁𝑅𝑑

)𝜂0

+ (𝑀𝑦,𝐸𝑑

𝜔0 ∙ 𝑀𝑦,𝑅𝑑

)

𝛾0

+ (𝑀𝑧,𝐸𝑑

𝜔0 ∙ 𝑀𝑧,𝑅𝑑

)

𝛾0

≤ 1.00 (81)

With: 𝑁𝐸𝑑 = maximum occurring normal force 𝑁𝑅𝑑 = normal strength

𝑀𝑦,𝐸𝑑 = maximum occurring bending moment around the y-axis

𝑀𝑦,𝑅𝑑 = bending strength around the y-axis

𝑀𝑦,𝐸𝑑 = maximum occurring bending moment around the z-axis

𝑀𝑦,𝑅𝑑 = bending strength around the z-axis

𝜔0, 𝜂0, 𝜉0, 𝛾0 = 1.0

Graduation project

Appendices

61

Assuming that the actuators can be integrated in the structure with hardly any offset, the moment in z-direction can be neglected. Then, by filling in the factors, the check on normal force becomes: 𝑁𝐸𝑑

𝐴𝑒𝑓𝑓 ∙ 𝑓0,𝑑

+𝑀𝑦,𝐸𝑑 ∙ 𝛾𝑀1

𝑊𝑒𝑙 ∙ 𝑓0

≤ 1.00 (82)

The weight of one actuator is about 505 kg (Parker Pneumatic, 2015). The lowest profile in the facade carries all the four actuators together with the weight of the other profiles. Although the moment will be lower in this profile, this profile still is checked with the maximum bending moment. After filling in equation (82): (505 ∙ 4 + 2700 ∙ 0.00488 ∙ 15) ∙

9.811000

0.00488 ∙ 227+

0.744 ∙ 28

∙ 152 ∙ 1.10

1.84 ∙ 10−4 ∙ 250= 1.02 ≥ 1.00

(83)

So the criterion is not satisfied. Therefore the profile is changed into 0.135x0.135x0.010 m, resulting in a profile that now satisfies on bending, shear and normal force.

Appendix E: Formula’s deformations by the environment

Numerical test The following equations (string format) are used in Chapter 2.1.1) to construct the database for the numerical model: q(z) = q[1]*Heaviside(z-1/5*H)-q[1]*Heaviside(z)+q[2]*Heaviside(z-2/5*H)-q[2]*Heaviside(z-

1/5*H)+q[3]*Heaviside(z-3/5*H)-q[3]*Heaviside(z-2/5*H)+q[4]*Heaviside(z-4/5*H)-q[4]*Heaviside(z-3/5*H)+q[5]*Heaviside(z-H)-q[5]*Heaviside(z-4/5*H)

V(z) = -q[1]*Heaviside(z-1/5*H)*z+1/5*q[1]*Heaviside(z-1/5*H)*H+q[1]*Heaviside(z)*z-

q[2]*Heaviside(z-2/5*H)*z+2/5*q[2]*Heaviside(z-2/5*H)*H+q[2]*Heaviside(z-1/5*H)*z-1/5*q[2]*Heaviside(z-1/5*H)*H-q[3]*Heaviside(z-3/5*H)*z+3/5*q[3]*Heaviside(z-3/5*H)*H+q[3]*Heaviside(z-2/5*H)*z-2/5*q[3]*Heaviside(z-2/5*H)*H-q[4]*Heaviside(z-4/5*H)*z+4/5*q[4]*Heaviside(z-4/5*H)*H+q[4]*Heaviside(z-3/5*H)*z-3/5*q[4]*Heaviside(z-3/5*H)*H-q[5]*Heaviside(z-H)*z+q[5]*Heaviside(z-H)*H+q[5]*Heaviside(z-4/5*H)*z-4/5*q[5]*Heaviside(z-4/5*H)*H-7/50*q[4]*H-9/50*q[5]*H-1/50*q[1]*H-3/50*q[2]*H-1/10*q[3]*H-4/25*H*q[1]*Heaviside(H)-2/25*H*q[2]*Heaviside(H)+2/25*H*q[4]*Heaviside(H)+4/25*H*q[5]*Heaviside(H)

M(z) = -q[1]*Heaviside(z-1/5*H)*z+1/5*q[1]*Heaviside(z-1/5*H)*H+q[1]*Heaviside(z)*z-

q[2]*Heaviside(z-2/5*H)*z+2/5*q[2]*Heaviside(z-2/5*H)*H+q[2]*Heaviside(z-1/5*H)*z-1/5*q[2]*Heaviside(z-1/5*H)*H-q[3]*Heaviside(z-3/5*H)*z+3/5*q[3]*Heaviside(z-3/5*H)*H+q[3]*Heaviside(z-2/5*H)*z-2/5*q[3]*Heaviside(z-2/5*H)*H-q[4]*Heaviside(z-4/5*H)*z+4/5*q[4]*Heaviside(z-4/5*H)*H+q[4]*Heaviside(z-3/5*H)*z-3/5*q[4]*Heaviside(z-3/5*H)*H-q[5]*Heaviside(z-H)*z+q[5]*Heaviside(z-H)*H+q[5]*Heaviside(z-4/5*H)*z-4/5*q[5]*Heaviside(z-4/5*H)*H-7/50*q[4]*H-9/50*q[5]*H-1/50*q[1]*H-3/50*q[2]*H-1/10*q[3]*H-4/25*H*q[1]*Heaviside(H)-2/25*H*q[2]*Heaviside(H)+2/25*H*q[4]*Heaviside(H)+4/25*H*q[5]*Heaviside(H)

φ(z) = 1/15000*(6000*q[4]*H*Heaviside(z-4/5*H)*z^2+2700*q[4]*H^2*Heaviside(z-3/5*H)*z-

4500*q[4]*H*Heaviside(z-3/5*H)*z^2-7500*q[5]*H^2*Heaviside(z-H)*z+7500*q[5]*H*Heaviside(z-H)*z^2+4800*q[5]*H^2*Heaviside(z-4/5*H)*z-6000*q[5]*H*Heaviside(z-4/5*H)*z^2-1200*H*z^2*q[1]*Heaviside(H)-600*H*z^2*q[2]*Heaviside(H)+600*H*z^2*q[4]*Heaviside(H)+1200*H*z^2*q[5]*Heaviside(H)-300*q[1]*H^2*Heaviside(z-1/5*H)*z+1500*q[1]*H*Heaviside(z-1/5*H)*z^2-1200*q[2]*H^2*Heaviside(z-2/5*H)*z+3000*q[2]*H*Heaviside(z-2/5*H)*z^2+300*q[2]*H^2*Heaviside(z-1/5*H)*z-1500*q[2]*H*Heaviside(z-1/5*H)*z^2-2700*q[3]*H^2*Heaviside(z-3/5*H)*z+4500*q[3]*H*Heaviside(z-3/5*H)*z^2+1200*q[3]*H^2*Heaviside(z-2/5*H)*z-3000*q[3]*H*Heaviside(z-2/5*H)*z^2-4800*q[4]*H^2*Heaviside(z-4/5*H)*z+490*H^3*q[2]*Heaviside(H)+750*H^3*q[3]*Heaviside(H)+1010*H^3*q[4]*Heaviside(H)+1318*H^3*q[5]*Heaviside(H)+182*H^3*q[1]*Heaviside(H)-1269*q[5]*H^3-101*q[1]*H^3-


S.T van Ruitenbeek

62

565*H^3*q[3]-875*H^3*q[4]-315*q[2]*H^3-450*q[2]*H*z^2+1500*q[3]*H^2*z-300*q[1]*H^2*z*Heaviside(H)-900*q[2]*H^2*z*Heaviside(H)-1500*q[3]*H^2*z*Heaviside(H)-2100*q[4]*H^2*z*Heaviside(H)-2700*q[5]*H^2*z*Heaviside(H)+300*q[1]*H^2*z+900*q[2]*H^2*z-750*q[3]*H*z^2+2100*q[4]*H^2*z-1050*q[4]*H*z^2+2700*q[5]*H^2*z-1350*q[5]*H*z^2-150*H*z^2*q[1]-20*q[2]*Heaviside(z-1/5*H)*H^3-2500*q[3]*Heaviside(z-3/5*H)*z^3+540*q[3]*Heaviside(z-3/5*H)*H^3+2500*q[3]*Heaviside(z-2/5*H)*z^3-160*q[3]*Heaviside(z-2/5*H)*H^3-2500*q[4]*Heaviside(z-4/5*H)*z^3+1280*q[4]*Heaviside(z-4/5*H)*H^3+2500*q[4]*Heaviside(z-3/5*H)*z^3-540*q[4]*Heaviside(z-3/5*H)*H^3-2500*q[5]*Heaviside(z-H)*z^3+2500*q[5]*Heaviside(z-H)*H^3+2500*q[5]*Heaviside(z-4/5*H)*z^3-1280*q[5]*Heaviside(z-4/5*H)*H^3-2500*q[1]*Heaviside(z-1/5*H)*z^3+20*q[1]*Heaviside(z-1/5*H)*H^3+2500*q[1]*Heaviside(z)*z^3-2500*q[2]*Heaviside(z-2/5*H)*z^3+160*q[2]*Heaviside(z-2/5*H)*H^3+2500*q[2]*Heaviside(z-1/5*H)*z^3)/EI

δ(z) = -

1/15000*(175*q[4]*Heaviside(H)*H^4+369*q[5]*Heaviside(H)*H^4+q[1]*Heaviside(H)*H^4+15*q[2]*Heaviside(H)*H^4+65*q[3]*Heaviside(H)*H^4-625*q[1]*Heaviside(z)*z^4+625*q[2]*Heaviside(z-2/5*H)*z^4+16*q[2]*Heaviside(z-2/5*H)*H^4-625*q[2]*Heaviside(z-1/5*H)*z^4-q[2]*Heaviside(z-1/5*H)*H^4+625*q[3]*Heaviside(z-3/5*H)*z^4+81*q[3]*Heaviside(z-3/5*H)*H^4-625*q[3]*Heaviside(z-2/5*H)*z^4-16*q[3]*Heaviside(z-2/5*H)*H^4+625*q[4]*Heaviside(z-4/5*H)*z^4+256*q[4]*Heaviside(z-4/5*H)*H^4-625*q[4]*Heaviside(z-3/5*H)*z^4-81*q[4]*Heaviside(z-3/5*H)*H^4+625*q[5]*Heaviside(z-H)*z^4+625*q[5]*Heaviside(z-H)*H^4-625*q[5]*Heaviside(z-4/5*H)*z^4-256*q[5]*Heaviside(z-4/5*H)*H^4+625*q[1]*Heaviside(z-1/5*H)*z^4+q[1]*Heaviside(z-1/5*H)*H^4-175*q[4]*H^4-H^4*q[1]-15*H^4*q[2]-65*H^4*q[3]-369*H^4*q[5]-20*q[1]*H^3*Heaviside(z-1/5*H)*z+150*q[1]*H^2*Heaviside(z-1/5*H)*z^2-500*q[1]*H*Heaviside(z-1/5*H)*z^3-160*q[2]*H^3*Heaviside(z-2/5*H)*z+600*q[2]*H^2*Heaviside(z-2/5*H)*z^2-1000*q[2]*H*Heaviside(z-2/5*H)*z^3+20*q[2]*H^3*Heaviside(z-1/5*H)*z-150*q[2]*H^2*Heaviside(z-1/5*H)*z^2+500*q[2]*H*Heaviside(z-1/5*H)*z^3-540*q[3]*H^3*Heaviside(z-3/5*H)*z+1350*q[3]*H^2*Heaviside(z-3/5*H)*z^2-1500*q[3]*H*Heaviside(z-3/5*H)*z^3+160*q[3]*H^3*Heaviside(z-2/5*H)*z-600*q[3]*H^2*Heaviside(z-2/5*H)*z^2+1000*q[3]*H*Heaviside(z-2/5*H)*z^3-1280*q[4]*H^3*Heaviside(z-4/5*H)*z+2400*q[4]*H^2*Heaviside(z-4/5*H)*z^2-2000*q[4]*H*Heaviside(z-4/5*H)*z^3+540*q[4]*H^3*Heaviside(z-3/5*H)*z-1350*q[4]*H^2*Heaviside(z-3/5*H)*z^2+1500*q[4]*H*Heaviside(z-3/5*H)*z^3-2500*q[5]*H^3*Heaviside(z-H)*z+3750*q[5]*H^2*Heaviside(z-H)*z^2+450*q[5]*H*z^3+350*q[4]*H*z^3+1269*q[5]*H^3*z-1350*q[5]*H^2*z^2-182*H^3*z*q[1]*Heaviside(H)-490*H^3*z*q[2]*Heaviside(H)-750*H^3*z*q[3]*Heaviside(H)-1010*H^3*z*q[4]*Heaviside(H)-1318*H^3*z*q[5]*Heaviside(H)+450*q[2]*H^2*z^2*Heaviside(H)+750*q[3]*H^2*z^2*Heaviside(H)+1050*q[4]*H^2*z^2*Heaviside(H)+1350*q[5]*H^2*z^2*Heaviside(H)+150*q[1]*H^2*z^2*Heaviside(H)-2500*q[5]*H*Heaviside(z-H)*z^3+1280*q[5]*H^3*Heaviside(z-4/5*H)*z-2400*q[5]*H^2*Heaviside(z-4/5*H)*z^2+2000*q[5]*H*Heaviside(z-4/5*H)*z^3+400*H*z^3*q[1]*Heaviside(H)+200*H*z^3*q[2]*Heaviside(H)-200*H*z^3*q[4]*Heaviside(H)-400*H*z^3*q[5]*Heaviside(H)-150*q[1]*H^2*z^2-450*q[2]*H^2*z^2-750*q[3]*H^2*z^2-1050*q[4]*H^2*z^2+101*H^3*z*q[1]+315*H^3*z*q[2]+565*H^3*z*q[3]+875*H^3*z*q[4]+50*H*z^3*q[1]+150*H*z^3*q[2]+250*H*z^3*q[3])/EI

Experimental test The following equations (string format) are used in Chapter 2.1.1) to construct the database for the experimental model: q(z) = q[1]*Heaviside(z-.1705376344*H)-1.*q[1]*Heaviside(z-.1627956989*H)+q[2]*Heaviside(z-

.5038709678*H)-1.*q[2]*Heaviside(z-.4961290324*H)+q[3]*Heaviside(z-.8372043009*H)-1.*q[3]*Heaviside(z-.8294623659*H)

V(z) = -.64516128e-2*q[3]*H-.12903226e-2*q[1]*H-.38709677e-2*q[2]*H-

.4961290323*q[2]*H*Heaviside(z-.4961290324*H)-1.*q[3]*z*Heaviside(z-

.8372043009*H)+.8372043011*q[3]*H*Heaviside(z-.8372043009*H)+1.*q[3]*z*Heaviside(z-

Graduation project

Appendices

63

.8294623659*H)-.8294623656*q[3]*H*Heaviside(z-.8294623659*H)-1.*q[2]*z*Heaviside(z-

.5038709678*H)+.5038709677*q[2]*H*Heaviside(z-.5038709678*H)+1.*q[2]*z*Heaviside(z-

.4961290324*H)-1.*q[1]*Heaviside(z-.1705376344*H)*z+.1705376344*q[1]*Heaviside(z-

.1705376344*H)*H+1.*q[1]*Heaviside(z-.1627956989*H)*z-.1627956989*q[1]*Heaviside(z-

.1627956989*H)*H-.5161290380e-2*H*q[1]*Heaviside(H)+.5161290300e-2*H*q[3]*Heaviside(H)

M(z) = -.5000000000*q[3]*z^2*Heaviside(z-.8372043009*H)-.3504555209*q[3]*H^2*Heaviside(z-

.8372043009*H)+.5000000000*q[3]*z^2*Heaviside(z-

.8294623659*H)+.3440039079*q[3]*H^2*Heaviside(z-.8294623659*H)-

.5000000000*q[2]*z^2*Heaviside(z-.5038709678*H)-.1269429761*q[2]*H^2*Heaviside(z-

.5038709678*H)+.5000000000*q[2]*z^2*Heaviside(z-

.4961290324*H)+.1230720084*q[2]*H^2*Heaviside(z-.4961290324*H)+.6451612800e-2*q[3]*H^2+.38709677e-2*q[2]*H^2+.129032259e-2*q[1]*H^2-.5000000000*q[1]*Heaviside(z-.1705376344*H)*z^2-.1454154238e-1*q[1]*Heaviside(z-.1705376344*H)*H^2+.5000000000*q[1]*Heaviside(z-.1627956989*H)*z^2+.1325121979e-1*q[1]*Heaviside(z-.1627956989*H)*H^2+.5038709677*q[2]*H*z*Heaviside(z-.5038709678*H)-.4961290323*q[2]*H*z*Heaviside(z-.4961290324*H)+.8372043011*q[3]*H*z*Heaviside(z-.8372043009*H)-.8294623656*q[3]*H*z*Heaviside(z-.8294623659*H)-.64516128e-2*q[3]*H*z-.12903226e-2*q[1]*H*z-.3870967700e-2*q[2]*H*z+.1705376344*q[1]*H*Heaviside(z-.1705376344*H)*z-.1627956989*q[1]*H*Heaviside(z-.1627956989*H)*z-.5161290380e-2*H*z*q[1]*Heaviside(H)+.5161290300e-2*H*z*q[3]*Heaviside(H)-.1290322590e-2*q[1]*Heaviside(H)*H^2-.3870967700e-2*q[2]*Heaviside(H)*H^2-.6451612800e-2*q[3]*Heaviside(H)*H^2

φ(z) = .2000000000e-12*(-2180422340.*q[1]*H^3-7258112850.*q[2]*H^3-

.1448634145e11*q[3]*H^3+6451612950.*q[1]*H^2*z-3225806450.*q[1]*H*z^2+.1935483850e11*q[2]*H^2*z-9677419000.*q[2]*H*z^2+.3225806400e11*q[3]*H^2*z-.1612903200e11*q[3]*H*z^2+.1066048006e12*H^3*q[2]*Heaviside(z-.5038709678*H)+.8333333335e12*q[2]*z^3*Heaviside(z-.4961290324*H)-.1017659942e12*H^3*q[2]*Heaviside(z-.4961290324*H)-.8333333335e12*q[3]*z^3*Heaviside(z-.8372043009*H)+.4890047825e12*H^3*q[3]*Heaviside(z-.8372043009*H)+.8333333335e12*q[3]*z^3*Heaviside(z-.8294623659*H)-.4755638250e12*H^3*q[3]*Heaviside(z-.8294623659*H)-.8333333335e12*q[2]*z^3*Heaviside(z-.5038709678*H)+4133133740.*q[1]*Heaviside(z-.1705376344*H)*H^3-.8333333335e12*q[1]*Heaviside(z-.1705376344*H)*z^3+.8333333335e12*q[1]*Heaviside(z-.1627956989*H)*z^3-3595402645.*q[1]*Heaviside(z-.1627956989*H)*H^3+3823113583.*H^3*q[1]*Heaviside(H)+9677419380.*H^3*q[2]*Heaviside(H)+.1553172555e11*H^3*q[3]*Heaviside(H)+.4263440860e12*q[1]*H*Heaviside(z-.1705376344*H)*z^2+.6625609895e11*q[1]*H^2*Heaviside(z-.1627956989*H)*z-.4069892472e12*q[1]*H*Heaviside(z-.1627956989*H)*z^2-.1240322581e13*H*z^2*q[2]*Heaviside(z-.4961290324*H)-.1752277604e13*q[3]*H^2*z*Heaviside(z-.8372043009*H)-.6347148805e12*q[2]*H^2*z*Heaviside(z-.5038709678*H)+.1259677419e13*H*z^2*q[2]*Heaviside(z-.5038709678*H)+.2093010753e13*q[3]*H*z^2*Heaviside(z-.8372043009*H)+.1720019540e13*q[3]*H^2*z*Heaviside(z-.8294623659*H)-.2073655914e13*q[3]*H*z^2*Heaviside(z-.8294623659*H)+.6153600420e12*q[2]*H^2*z*Heaviside(z-.4961290324*H)-.7270771190e11*q[1]*H^2*Heaviside(z-.1705376344*H)*z-6451612950.*q[1]*H^2*z*Heaviside(H)-.1935483850e11*q[2]*H^2*z*Heaviside(H)-.3225806400e11*q[3]*H^2*z*Heaviside(H)-.1290322595e11*H*z^2*q[1]*Heaviside(H)+.1290322575e11*H*z^2*q[3]*Heaviside(H))/EI

δ(z) = .2000000000e-12*(-2180422340.*q[1]*H^3-7258112850.*q[2]*H^3-

.1448634145e11*q[3]*H^3+6451612950.*q[1]*H^2*z-


S.T van Ruitenbeek

64

3225806450.*q[1]*H*z^2+.1935483850e11*q[2]*H^2*z-9677419000.*q[2]*H*z^2+.3225806400e11*q[3]*H^2*z-.1612903200e11*q[3]*H*z^2+.1066048006e12*H^3*q[2]*Heaviside(z-.5038709678*H)+.8333333335e12*q[2]*z^3*Heaviside(z-.4961290324*H)-.1017659942e12*H^3*q[2]*Heaviside(z-.4961290324*H)-.8333333335e12*q[3]*z^3*Heaviside(z-.8372043009*H)+.4890047825e12*H^3*q[3]*Heaviside(z-.8372043009*H)+.8333333335e12*q[3]*z^3*Heaviside(z-.8294623659*H)-.4755638250e12*H^3*q[3]*Heaviside(z-.8294623659*H)-.8333333335e12*q[2]*z^3*Heaviside(z-.5038709678*H)+4133133740.*q[1]*Heaviside(z-.1705376344*H)*H^3-.8333333335e12*q[1]*Heaviside(z-.1705376344*H)*z^3+.8333333335e12*q[1]*Heaviside(z-.1627956989*H)*z^3-3595402645.*q[1]*Heaviside(z-.1627956989*H)*H^3+3823113583.*H^3*q[1]*Heaviside(H)+9677419380.*H^3*q[2]*Heaviside(H)+.1553172555e11*H^3*q[3]*Heaviside(H)+.4263440860e12*q[1]*H*Heaviside(z-.1705376344*H)*z^2+.6625609895e11*q[1]*H^2*Heaviside(z-.1627956989*H)*z-.4069892472e12*q[1]*H*Heaviside(z-.1627956989*H)*z^2-.1240322581e13*H*z^2*q[2]*Heaviside(z-.4961290324*H)-.1752277604e13*q[3]*H^2*z*Heaviside(z-.8372043009*H)-.6347148805e12*q[2]*H^2*z*Heaviside(z-.5038709678*H)+.1259677419e13*H*z^2*q[2]*Heaviside(z-.5038709678*H)+.2093010753e13*q[3]*H*z^2*Heaviside(z-.8372043009*H)+.1720019540e13*q[3]*H^2*z*Heaviside(z-.8294623659*H)-.2073655914e13*q[3]*H*z^2*Heaviside(z-.8294623659*H)+.6153600420e12*q[2]*H^2*z*Heaviside(z-.4961290324*H)-.7270771190e11*q[1]*H^2*Heaviside(z-.1705376344*H)*z-6451612950.*q[1]*H^2*z*Heaviside(H)-.1935483850e11*q[2]*H^2*z*Heaviside(H)-.3225806400e11*q[3]*H^2*z*Heaviside(H)-.1290322595e11*H*z^2*q[1]*Heaviside(H)+.1290322575e11*H*z^2*q[3]*Heaviside(H))/EI

Gra

duation p

roje

ct

App

end

ices

65

Ap

pen

dix

F:

Nu

meri

cal te

st

Fig

ure

F.1

: S

tatic lin

ea

r d

efo

rma

tio

ns in

the

mid

dle

of th

e fa

cad

e u

nd

er

win

d lo

ad

sp

ectr

um

-1.5-1

-0.50

0.51

1.5

05

10

15

20

25

30

35

40

45

50

55

60

δ[m]

T [s

]

δw

ind

δac

tuat

ion

δfi

nal

δal

low

edδ

allo

wed

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

66

Fig

ure

F.2

: S

tatic n

on

line

ar

de

form

atio

ns in t

he m

idd

le o

f th

e fa

cad

e u

nd

er

win

d lo

ad s

pe

ctr

um

-1.5-1

-0.50

0.51

1.5

05

10

15

20

25

30

35

40

45

50

55

60

δ[m]

T [s

]

δw

ind

δac

tuat

ion

δfi

nal

δal

low

edδ

allo

wed

Gra

duation p

roje

ct

App

end

ices

67

Fig

ure

F.3

: D

yn

am

ic lin

ea

r d

efo

rma

tio

ns in t

he m

idd

le o

f th

e fa

cad

e u

nd

er

win

d lo

ad s

pe

ctr

um

-1.5-1

-0.50

0.51

1.5

05

10

15

20

25

30

35

40

45

50

55

60

δ[m]

T [s

]

δw

ind

δac

tuat

ion

δfi

nal

δal

low

edδ

allo

wed

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

68

Fig

ure

F.4

: S

tatic lin

ea

r fina

l de

form

ation

s in

th

e m

iddle

of

the

fa

ca

de

un

de

r w

ind loa

d s

pectr

um

fo

r d

iffe

ren

t d

ela

y t

imes

-0.2

50

0.2

5

0.5

05

10

15

20

25

30

35

40

45

50

55

60

δ[m]

T [s

]

δfi

nal

;0.0

δfi

nal

;0.3

δfi

nal

;0.5

δfi

nal

;1.0

δal

low

edδ

allo

wed

Gra

duation p

roje

ct

App

end

ices

69

Fig

ure

F.5

: D

yn

am

ic lin

ea

r fin

al d

efo

rma

tio

ns in

the

mid

dle

of th

e f

aca

de

und

er

win

d lo

ad

spe

ctr

um

fo

r d

iffe

rent

dela

y t

imes

-0.8

-0.40

0.4

0.8

05

10

15

20

25

30

35

40

45

50

55

60

δ[m]

T [s

]

δfi

nal

;0.0

δfi

nal

;0.3

δfi

nal

;0.5

δfi

nal

;1.0

δal

low

edδ

allo

wed

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

70

Fig

ure

F.6

: S

tatic n

on

line

ar

de

form

atio

ns b

y w

ind

in

th

e m

idst o

f th

e f

aca

de u

nde

r w

ind

loa

d s

pectr

um

fo

r d

iffe

ren

t m

asse

s

0

0.51

05

10

15

20

25

30

35

40

45

50

55

60

δ[m]

T [s

]

δw

ind

;all

δw

ind

;all-

act

δw

ind

;-

Gra

duation p

roje

ct

App

end

ices

71

Fig

ure

F.7

: D

yn

am

ic lin

ea

r d

efo

rma

tio

ns b

y w

ind

in

th

e m

idst o

f th

e f

aca

de u

nde

r w

ind

loa

d s

pectr

um

fo

r d

iffe

ren

t m

asse

s

0

0.51

05

10

15

20

25

30

35

40

45

50

55

60

δ[m]

T [s

]

δw

ind

;all

δw

ind

;all-

act

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

72

Ap

pen

dix

G:

Se

co

nd

ary

te

sts

Maxim

um

to

rqu

e a

ctu

ato

r

Fig

ure

G.1

: Lo

ad

-tim

e g

raph

to

de

term

ine

th

e m

axim

um

to

rqu

e o

f th

e M

L2

34

0A

actu

ato

r

0

10

20

30

40

50

60

70

80

90

10

0

05

10

15

20

25

30

35

40

45

50

F [N]

T [s

]

Gra

duation p

roje

ct

App

end

ices

73

Ben

din

g s

tiff

ness p

late

s

Fig

ure

G.2

: Lo

ad-d

isp

lace

me

nt

gra

ph o

f th

e p

late

witho

ut

ribb

on

s (

Pla

te 0

) an

d th

e to

p (

t) a

nd

bo

tto

m (

b)

of

the p

late

s w

ith

rib

bo

ns (

Pla

tes 1

and

2)

0123456789

10

02

46

81

01

2

F [N]

δm

[mm

]

Pla

te 0

Pla

te 1

;bo

tto

mP

late

1;t

op

Pla

te 2

;bo

tto

mP

late

2;t

op

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

74

Rela

tio

n m

om

en

t an

d s

tra

ins

F

igu

re G

.3:

Mo

me

nt-

str

ain

gra

ph

of

the

to

p (

t) a

nd b

otto

m (

b)

of

the p

late

s w

ith

rib

bo

ns (

Pla

tes 1

an

d 2

)

0

0.2

0.4

0.6

0.81

1.2

-35

00

-25

00

-15

00

-50

05

00

15

00

25

00

35

00

M [Nm]

ε[μ

m/m

]

Pla

te 1

;bo

tto

mP

late

1;t

op

Pla

te 2

;bo

tto

mP

late

2;t

op

Gra

duation p

roje

ct

App

end

ices

75

Ap

pen

dix

H:

Ex

pe

rim

en

tal te

st

Fig

ure

H.1

: F

inal d

efo

rma

tion

s in

the

mid

dle

of th

e f

aca

de u

nd

er

loa

d s

pectr

um

fo

r te

st

one

to

th

ree

-50

-40

-30

-20

-100

10

02

55

07

51

00

12

51

50

17

52

00

22

5

δ[mm]

T [s

]

δm

;1δ

m;2

δm

;3δ

allo

wed

δal

low

ed

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

76

Test

1

Fig

ure

H.2

: S

tra

ins in

th

e t

hre

e p

late

s fo

r te

st

on

e

-30

0

-25

0

-20

0

-15

0

-10

0

-500

50

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3

Gra

duation p

roje

ct

App

end

ices

77

Fig

ure

H.3

: S

tra

ins in

th

e t

hre

e p

late

s fo

r te

st

on

e, in

clu

din

g th

e e

xp

ecte

d v

alu

es b

ase

d o

n th

e r

eal m

ea

su

red

de

form

atio

ns

-30

0

-25

0

-20

0

-15

0

-10

0

-500

50

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3εe

xp;1

εexp

;2εe

xp;3

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

78

F

igu

re H

.4:

Tim

e-s

pecific

(t)

an

d c

on

sta

nt

(c)

rota

tio

ns f

or

actu

ato

rs B

an

d C

fo

r te

st

one

-0.1

1

-0.0

9

-0.0

7

-0.0

5

-0.0

3

-0.0

1

0.0

1

02

55

07

51

00

12

51

50

17

52

00

22

5

ϕ[rad]

T [s

]

ϕt;

Bϕ

t;C

ϕc;

Bϕ

c;C

Gra

duation p

roje

ct

App

end

ices

79

Fig

ure

H.5

: P

rod

uce

d m

om

en

t fo

r actu

ato

r B

an

d C

fo

r te

st o

ne

-1.5-1

-0.50

0.51

1.5

02

55

07

51

00

12

51

50

17

52

00

22

5

M [Nm]

T [s

]

MB

MC

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

80

Test

2

Fig

ure

H.6

: S

tra

ins in

th

e t

hre

e p

late

s fo

r te

st

two

-30

0

-25

0

-20

0

-15

0

-10

0

-500

50

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3

Gra

duation p

roje

ct

App

end

ices

81

Fig

ure

H.7

: S

tra

ins in

th

e t

hre

e p

late

s fo

r te

st

two,

inclu

din

g th

e e

xp

ecte

d v

alu

es b

ase

d o

n th

e r

eal m

ea

su

red

de

form

atio

ns

-30

0

-25

0

-20

0

-15

0

-10

0

-500

50

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3εe

xp;1

εexp

;2εe

xp;3

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

82

Fig

ure

H.8

: T

ime-s

pecific

(t)

an

d c

on

sta

nt

(c)

rota

tio

ns f

or

actu

ato

rs B

an

d C

fo

r te

st

two

-0.1

1

-0.0

9

-0.0

7

-0.0

5

-0.0

3

-0.0

1

0.0

1

02

55

07

51

00

12

51

50

17

52

00

22

5

φ[rad]

T [s

]

ϕt;

Bϕ

t;C

ϕc;

Bϕ

c;C

Gra

duation p

roje

ct

App

end

ices

83

Fig

ure

H.9

: P

rod

uce

d m

om

en

t fo

r actu

ato

r B

an

d C

fo

r te

st

two

-1.5-1

-0.50

0.51

1.5

02

55

07

51

00

12

51

50

17

52

00

22

5

M [Nm]

T [s

]

MB

MC

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

84

Test

3

Fig

ure

H.1

0:

Str

ain

s in

the

th

ree

pla

tes fo

r te

st th

ree

-30

0

-25

0

-20

0

-15

0

-10

0

-500

50

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3

Gra

duation p

roje

ct

App

end

ices

85

Fig

ure

H.1

1:

Str

ain

s in t

he

th

ree

pla

tes fo

r te

st th

ree,

inclu

din

g t

he

expe

cte

d v

alu

es b

ase

d o

n t

he

re

al m

ea

su

red

de

form

atio

ns

-30

0

-25

0

-20

0

-15

0

-10

0

-500

50

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3εe

xp;1

εexp

;2εe

xp;3

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

86

Fig

ure

H.1

2:

Tim

e-s

pe

cific

(t)

an

d c

onsta

nt (c

) ro

tatio

ns f

or

actu

ato

rs B

an

d C

fo

r te

st th

ree

-0.1

1

-0.0

9

-0.0

7

-0.0

5

-0.0

3

-0.0

1

0.0

1

02

55

07

51

00

12

51

50

17

52

00

22

5

φ[rad]

T [s

]

Gra

duation p

roje

ct

App

end

ices

87

Fig

ure

H.1

3:

Pro

duce

d m

om

en

t fo

r a

ctu

ato

r B

an

d C

fo

r te

st

thre

e

-1.5-1

-0.50

0.51

1.5

02

55

07

51

00

12

51

50

17

52

00

22

5

M [Nm]

T [s

]

MB

MC

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

88

Fig

ure

H.1

4: F

ina

l de

form

atio

ns in

th

e m

idd

le o

f th

e f

acad

e u

nd

er

loa

d s

pectr

um

fo

r te

st

four

to s

ix

-50

-250

25

50

75

10

0

02

55

07

51

00

12

51

50

17

52

00

δ[mm]

T [s

]

δm

;4δ

m;5

δm

;6δ

allo

wed

δal

low

ed

Gra

duation p

roje

ct

App

end

ices

89

Test

4

Fig

ure

H.1

5:

Str

ain

s in

the

th

ree

pla

tes fo

r te

st fo

ur

-60

0

-50

0

-40

0

-30

0

-20

0

-10

00

10

0

20

0

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

90

Fig

ure

H.1

6:

Ro

tatio

ns f

or

actu

ato

rs B

an

d C

fo

r te

st

fou

r

-0.5

5

-0.4

5

-0.3

5

-0.2

5

-0.1

5

-0.0

5

0.0

5

02

55

07

51

00

12

51

50

17

52

00

22

5

φ[rad]

T [s

]

ϕB

ϕC

Gra

duation p

roje

ct

App

end

ices

91

Fig

ure

H.1

7:

Pro

duce

d m

om

en

t fo

r a

ctu

ato

r B

an

d C

fo

r te

st

fou

r

0

0.51

1.52

02

55

07

51

00

12

51

50

17

52

00

22

5

M [Nm]

T [s

]

MB

MC

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

92

Test

5

Fig

ure

H.1

8:

Str

ain

s in

the

th

ree

pla

tes fo

r te

st five

-60

0

-50

0

-40

0

-30

0

-20

0

-10

00

10

0

20

0

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3

Gra

duation p

roje

ct

App

end

ices

93

Fig

ure

H.1

9:

Ro

tatio

ns f

or

actu

ato

rs B

an

d C

fo

r te

st

five

-0.5

5

-0.4

5

-0.3

5

-0.2

5

-0.1

5

-0.0

5

0.0

5

02

55

07

51

00

12

51

50

17

52

00

22

5

φ[rad]

T [s

]

ϕB

ϕC

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

94

F

igu

re H

.20

: P

rod

uce

d m

om

en

t fo

r a

ctu

ato

r B

an

d C

fo

r te

st

five

0

0.51

1.52

02

55

07

51

00

12

51

50

17

52

00

22

5

M [Nm]

T [s

]

MB

MC

Gra

duation p

roje

ct

App

end

ices

95

Test

6

Fig

ure

H.2

1:

Str

ain

s in

the

th

ree

pla

tes fo

r te

st six

-60

0

-50

0

-40

0

-30

0

-20

0

-10

00

10

0

20

0

02

55

07

51

00

12

51

50

17

52

00

22

5

ε[μm/m]

T [s

]

ε1ε2

ε3

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

96

Fig

ure

H.2

2:

Ro

tatio

ns f

or

actu

ato

rs B

an

d C

fo

r te

st

six

-0.5

5

-0.4

5

-0.3

5

-0.2

5

-0.1

5

-0.0

5

0.0

5

02

55

07

51

00

12

51

50

17

52

00

22

5

φ[rad]

T [s

]

ϕB

ϕC

Gra

duation p

roje

ct

App

end

ices

97

Fig

ure

H.2

3:

Pro

duce

d m

om

en

t fo

r a

ctu

ato

r B

an

d C

fo

r te

st

six

0

0.51

1.52

02

55

07

51

00

12

51

50

17

52

00

22

5

M [Nm]

T [s

]

MB

MC

Str

uctu

ral A

ctive F

acade

S.T

van R

uite

nbe

ek

98

structural active facade · research proposes a structural active facade as adaptive structure,...

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