Finite Element Modelling Results of High-Rise Timber Buildings: Dynamic Modal Analysis

Finite Element Modelling Results of High-Rise Timber Buildings

Dynamic Modal Analysis

Ida Edskär

Finite Element Modelling Results of High-Rise Timber Buildings

Dynamic Modal Analysis

Ida Edskär

Luleå University of Technology

ISSN 1402-1536

ISBN 978-91-7790-519-6 (pdf) Luleå 2019

www.ltu.se

Abstract

Tall timber buildings are built around the world and the development is moving forward. Tall timber buildings are sensitive to wind-induced vibration and can cause discomfort for

humans. In this report is the results from a dynamic analysis preformed on a cross-laminated buildings system presented. Size and placement of openings and floor plans have been investigated using finite element analysis and dynamic modal analyses. The results show that asymmetric placement of openings and asymmetric floor plans may affect the dynamic behaviour of tall timber buildings. Asymmetry can cause modes with a tendency to rotate and even diagonal modes. Timber buildings may have problem to fulfil the comfort criteria already at a slenderness of 1.5-2.1.

Contents

Abstract ... I

Introduction ... 1

Theory ... 1

Method ... 2

Part A ... 3

Part B ... 3

FE-model and material properties ... 4

Results – Part A ... 5

Case O ... 6

Case 1X ... 7

Case 2X ... 8

Case 1X1Y ... 9

Case 1X2Y ... 10

Case 2X2Y ... 11

Result – Part B ... 14

Timber ... 15

Hybrid I ... 16

Hybrid II ... 17

Concrete ... 18

Summary ... 23

References ... 25 Appendix A1 ... A1.1 Appendix A2 ... A2.1 Appendix B.1 ... B1.1

Introduction

Taller timber buildings are sensitive to wind-induced vibrations, which can cause discomfort for humans. Tall timber building can be stabilised by cross-laminated timber panels (CLT) and the placement of stabilising elements are generally not symmetric in plane. Several walls may have large openings for windows and doors. The geometry of the floor plan is different in buildings, where mid- and high-rise buildings often are organised around the elevator shaft.

The shaft can be a part of the stabilising system and can be placed in the central core or at the exterior part of the building. Placement of the shaft and openings may affect the dynamic behaviour of the building. The aim of this report is to present the results from a modal analysis of a CLT building where asymmetrical floor plans and openings have been studied.

Theory

The theory of vibrations of a uniform cantilever beam, fixed to the foundation can be used to model a tall building. Since tall buildings can be very complex and consist of several

elements, finite element (FE) software is useful to solve the multiple degree of freedom problem. In FE software the modal analysis is used to solve the free-vibration equation of motion (Chopra, 2012):

Mü + Ku = 0 (1)

Where M and K are the 𝑁𝑁 × 𝑁𝑁 matrices of mass and stiffness and u denotes the generalised displacement. Further derivation of the equation is referred to Chopra (2012). The dynamic properties such as the natural freqency, mode shape and modal mass can be determined by modal analaysis. For each resonance mode there is an associated modal mass and a modal stiffness (Jeary, 1997). The squared natural frequencies are related to the modal mass and modal stiffness by (Chopra, 2012):

𝜔𝜔𝑛𝑛2 =_𝑀𝑀^{𝐾𝐾�𝑛𝑛}

�𝑛𝑛 (2)

Where 𝐾𝐾� is the modal stiffness and 𝑀𝑀𝑛𝑛 � is the modal mass. From the squared natural 𝑛𝑛

frequencies, the natural frequencies can be written as:

𝑓𝑓𝑛𝑛 =^𝜔𝜔_2𝜋𝜋^𝑛𝑛 (3)

Since the modal mass is generated from the modal analysis the modal stiffness can be calculated by using Eq.(2) and (3). For a continuous system the modal mass is defined by (Jeary, 1997):

𝑀𝑀� = ∫ 𝑚𝑚(𝑧𝑧)𝜙𝜙𝑛𝑛 ₀^ℎ 𝑛𝑛2𝑑𝑑𝑧𝑧 (4)

Eq.(4) assume that the dimensions in the X- and Y-directions are constant and do not vary along the building, figure 1. Irregularities in X- and Y-directions need to be considered. The equation of modal mass for a 3D-system is by definition a volume integral, (Jeary, 1997).

The three fist modes for a building are normally two orthogonal translation modes and one torsional mode given that the building is symmetric. Asymmetry in the building can yield other mode combinations (Jeary, 1997).

To evaluate the comfort criteria, the acceleration level needs to be calculated. In the Swedish national annex, EKS 10 (2015) the following equation is stated for estimating the standard deviation of acceleration:

𝜎𝜎_𝑥𝑥(𝑧𝑧) = ^{3∙𝐼𝐼𝑣𝑣(ℎ)∙𝑅𝑅∙𝑞𝑞𝑚𝑚(ℎ)∙𝑏𝑏∙𝑐𝑐}_𝑚𝑚 ^{𝑓𝑓∙𝜙𝜙1,𝑥𝑥(𝑧𝑧)}

𝑒𝑒 (5)

𝐼𝐼_𝑣𝑣(ℎ) is the turbulence intensity at height ℎ, 𝑅𝑅 is the square root of the resonant response, 𝑞𝑞𝑚𝑚(ℎ) is the mean pressure at height ℎ, 𝑏𝑏 is the width of the structure, 𝑐𝑐𝑓𝑓 is the force coefficient, 𝜙𝜙1,𝑥𝑥(𝑧𝑧) is the mode shape value at height 𝑧𝑧, 𝑚𝑚𝑒𝑒 is the equivalent mass per unit length, 𝑧𝑧 is the height above the ground, and ℎ is the height of the structure. According to Eurocode 1-4 (2005) the equivalent mass per unit length is given by:

𝑚𝑚_𝑒𝑒 = ∫ 𝑚𝑚(𝑧𝑧)∙𝜙𝜙₀^ℎ 12(𝑧𝑧)𝑑𝑑𝑧𝑧

∫ 𝜙𝜙₀^ℎ ₁²(𝑧𝑧)𝑑𝑑𝑧𝑧 (6)

The equivalent mass is the modal mass presented per unit length and has the unit kg/m. The numerator is the modal mass and the denominator is the integral of the mode shape over the height of the building for a given mode. The equivalent mass can be approximate by average value of the mass of the upper third of the structure or the total mass of the building divided by the total height (Eurocode 1-4, 2005):

𝑚𝑚𝑒𝑒,𝑎𝑎𝑎𝑎𝑎𝑎 = ^𝑚𝑚_ℎ^{𝑡𝑡𝑡𝑡𝑡𝑡} (7)

From a modal analysis which is performed on a 3D model, the mode shape is visually

presented in 3D. The sum of modal masses in X- Y- and Z-directions represent the total mass for the associated vibration modes:

𝑀𝑀� = 𝑚𝑚𝑛𝑛 � + 𝑚𝑚𝑋𝑋 � + 𝑚𝑚𝑌𝑌 � 𝑍𝑍 (8)

Where 𝑚𝑚� , 𝑚𝑚_𝑋𝑋 � 𝑚𝑚_𝑌𝑌, � are the modal mass in each direction X- Y- and Z-direction for the _𝑍𝑍 current mode (Autodesk Inc., 2015)

Method

This study is divided in two parts, Part A and Part B. In part A, the effects of openings have been studied and in part B different floor plans and hybrid solutions have been studied. The

study is based on same fictitious building as presented in Edskär & Lidelöw (2017). The building represents a common floor plan and building type for mid- and high-rise timber buildings. The building footprint is 20x20m² and the structure consists of cross-laminated timber, stabilised through diaphragm action. The slenderness of the building, h/d where h is the height and d is the depth of the building, has been varied between 1.5 to 4.5 with a step of 0.6. To evaluate a very high building h/d = 9.0 was added.

Eq.(6) was used to calculate the equivalent mass from the generated modal mass and mode shape. Eq.(7) was used to calculated the approximate equivalent mass. From Eq.(2) and (3) the modal stiffness was calculated by using the natural frequency and the associated modal mass for each case.

Part A

The inner walls are neglected in Part A and only the outer walls and the floors were

considered. Opening sizes of (height x width) 1.5x1.0m² and 1.5x2.5 m² have been studied with 4 openings in each wall. 6 different configurations of opening placement have been studied, figure 1:

- In case O no openings are added,

- Case 1X has openings on one wall in the X-direction - Case 2X has openings on both walls in the X-direction

- Case 1X1Y has openings on one wall in X-direction and one wall in Y-direction - Case 1X2Y has openings on one wall in X-direction and on both walls in Y-direction - Case 2X2Y have openings on all walls.

Figure 1 Placement of openings, Z-direction is out-of-plane

Part B

In Part B openings for windows have been removed, but door openings are left. Four cases have been studied with different placement of the elevator shaft, figure 2. The placements of the shaft are denoted I, II and III, with an additional case III – Hybrid with the same floor plan

O 1X 2X 1X1Y 1X2Y 2X2Y

X Y

but an extra wall added opposite to the shaft. Different building system are analysed for each case:

- Timber, all structural elements are in CLT - Concrete, all structural elements are in concrete

- Hybrid I, all structural elements are in CLT except for the shaft which is in concrete - Hybrid II, the shaft is in concrete and a 7 m long concrete wall on the opposite side

while the rest of the structural elements are CLT Hybrid II has only been studied for floor plan III.

I II

III III - Hybrid

Figure 2 Placement of the shaft, floor layout

FE-model and material properties

From a modal analysis, dynamic properties such as the natural frequencies, the mode shape, and the modal mass can be generated. To perform the modal analysis a commercial FE- software (Autodesk ® Robot^TM Structural Analysis) was used. The CLT walls are 200 mm thick and the floors are 280 mm. The wall element consist of five layers and the floor element consist of seven layer. Graded C24 are used for all layers with a E-modulus of 11,000 MPa, a G-modulus of 690 MPa, and a density taken as 400 kg/m³. Only dead-load was considered in the model. The element stiffness has been calculated from CLT handbook and manually

defined in the FE software so the orthotropic properties are handled properly (Svenskt Trä, 2017). Four-node quadrilateral shell elements were used for all panels. The mass in Z- direction was ignored to avoid local vibrations of floor elements. Only X- and Y-directions (transversal vibration) were set to be active in the analysis, figure 1.

Results – Part A

In appendix A1 and A2 all results from Part A are presented. In appendix A1 the results from opening sizes of 1.5x2.5 m² are presented and in appendix A2 the results from opening sizes of 1.5x1.0m² . In the following figure 3 to 8, the modal mass, modal stiffness, and natural frequency are presented for the two first modes for each case with size opening 1.5x2.5m². The two first modes represent the two first transversal modes. Some of the modes show a tendency to rotate and even diagonal modes occur, see appendix A1 and A2. The peak acceleration for selected cases are presented in the appendix A1 and A2. The peak

acceleration for cases with opening size of 1.5x2.5 m² and natural frequency are presented in figure 9 and 10 in relation to the comfort standard ISO 10137 (ISO 10137, 2008) for selected slenderness, h/d.

Case O

a b

c d

e f

Figure 3 Case O – size opening 1.5x2.5 m² a) slenderness vs natural frequency mode 1 b) slenderness vs natural frequency mode 2 c) slenderness vs modal mass mode 1 d) slenderness vs modal mass mode 2 e) slenderness vs modal stiffness mode 1 f) slenderness vs modal stiffness mode 2

Case 1X

a b

c d

e f

Figure 4 Case 1X - size opening 1.5x2.5 m² a) slenderness vs natural frequency mode 1 b) slenderness vs natural frequency mode 2 c) slenderness vs modal mass mode 1 d) slenderness vs modal mass mode 2 e) slenderness vs modal stiffness mode 1 f) slenderness vs modal stiffness mode 2

Case 2X

a b

c d

e f

Figure 5 Case 2X - size opening 1.5x2.5 m² a) slenderness vs natural frequency mode 1 b) slenderness vs natural frequency mode 2 c) slenderness vs modal mass mode 1 d) slenderness vs modal mass mode 2 e) slenderness vs modal stiffness mode 1 f) slenderness vs modal stiffness mode 2

Case 1X1Y

a b

c d

e f

Figure 6 Case 1X1Y - size opening 1.5x2.5 m² a) slenderness vs natural frequency mode 1 b) slenderness vs natural frequency mode 2 c) slenderness vs modal mass mode 1 d) slenderness vs modal mass mode 2 e) slenderness vs modal stiffness mode 1 f) slenderness vs modal stiffness mode 2

Case 1X2Y

a b

c d

e f

Figure 7 Case 1X2Y - size opening 1.5x2.5 m² a) slenderness vs natural frequency mode 1 b) slenderness vs natural frequency mode 2 c) slenderness vs modal mass mode 1 d) slenderness vs modal mass mode 2 e) slenderness vs modal stiffness mode 1 f) slenderness vs modal stiffness mode 2

Case 2X2Y

a b

c d

e f

Figure 8 Case 2X2Y - size opening 1.5x2.5 m² a) slenderness vs natural frequency mode 1 b) slenderness vs natural frequency mode 2 c) slenderness vs modal mass mode 1 d) slenderness vs modal mass mode 2 e) slenderness vs modal stiffness mode 1 f) slenderness vs modal stiffness mode 2

Acceleration

a

b c

d e

Figure 9 ISO 10137 with peak acceleration and natural frequency for a) Case O – mode 1 and 2 b) Case 1X – mode 1 c) Case 1X – mode 2 d) Case 2X – mode 1 e) Case 2X – mode 2

a b

c d

Figure 10 ISO 10137 with peak acceleration and natural frequency for a) Case 1X1Y – mode 1 b) Case e 1X1Y – mode 2 c) Case 1X2Y – mode 1 d) Case 1X2Y – mode 2 e) Case 2X2Y – mode 1 and 2

Result – Part B

In appendix B1 the results from Part B are presented. In figure, 11 mode directions for floor plan I (a-c) and III (d-f) are presented. Figure 11a and b show the two translational mode directions and figure 11c presents the torsion mode. Figure 11d presents a translational mode in X-direction with a tendency to rotate (X^*) for floor plan III and figure 11e presents the translational mode in Y-direction and figure 11f presents the torsional mode.

a Translation in X-direction b Translation in Y-direction c Torsion

d Translation in X-direction with a

tendency to rotate, X^* e Translation in Y-direction f Torsion Figure 11 Mode directions

Normalised mode shapes are presented in figure 12 to 15 for floor plan III and the two first modes for each case. The difference between mode shapes generated for floor plans I-III is small and only floor plan III is presented. The slenderness ratios of 1.5, 3.9 and 9.0 are presented. The peak acceleration for all cases are presented in the appendix B1. The peak acceleration and natural frequency are presented in figure 16 to 19 in relation to the comfort standard ISO 10137 (ISO 10137, 2008). In figure 19 the acceleration levels for slenderness of 1.5 and 2.7 are not presented due to the levels are outside of the comfort standard.

Timber

a b

c d

e f

Figure 12 Mode shape Timber floor plan III a) h/d=1.5 mode 1 b) h/d=1.5 mode 2 c) h/d=3.9 mode 1 d) h/d=3.9 mode 2 e) h/d=9.0 mode 1 f) h/d=9.0 mode 2

Hybrid I

a b

c d

e f

Figure 13 Mode shape Hybrid I floor plan III a) h/d=1.5 mode 1 b) h/d=1.5 mode 2 c) h/d=3.9 mode 1 d) h/d=3.9 mode 2 e) h/d=9.0 mode 1 f) h/d=9.0 mode 2

Hybrid II

a b

c d

e f

Figure 14 Mode shape Hybrid II floor plan III a) h/d=1.5 mode 1 b) h/d=1.5 mode 2 c) h/d=3.9 mode 1 d) h/d=3.9 mode 2 e) h/d=9.0 mode 1 f) h/d=9.0 mode 2

Concrete

a b

c d

e f

Figure 15 Mode shape Concrete floor plan III a) h/d=1.5 mode 1 b) h/d=1.5 mode 2 c) h/d=3.9 mode 1 d) h/d=3.9 mode 2 e) h/d=9.0 mode 1 f) h/d=9.0 mode 2

Acceleration

a b

c

Figure 16 ISO 10137 with peak acceleration and natural frequency for Timber a) floor plan I – mode 1 b) d floor plan I-III – mode 2 c) floor plan II mode 1 d) floor plan III mode 1

a b

c d

e f

Figure 17 ISO 10137 with peak acceleration and natural frequency for Hybrid I a) floor plan I – mode 1 b) floor plan I – mode 2 c) floor plan II – mode 1 d) floor plan II - mode 2 e) floor plan III – mode 1 f) floor plan III – mode 2

a b

Figure 18 ISO 10137 with peak acceleration and natural frequency for Hybrid II floor plan III a) mode 1 b) mode 2

a

b c

d e

Figure 19 ISO 10137 with peak acceleration and natural frequency for Concrete a) floor plan I mode 1 and 2 b) floor plan II mode 1 c) floor plan mode 2 d) floor plan III mode 1 e) floor plan III mode 2

Summary

In this report, size and placement of openings have been studied. Three different floor plans have been studied for four different cases with varying structural material: 1) cross-laminated timber, 2) a timber-concrete hybrid solution where the shaft is in concrete 3) a concrete shaft and a 7m long concrete wall, and 4) all structural elements in concrete. The slenderness has been varied between 1.5 to 4.5 with a step of 0.6 and an additional slenderness of 9.0.

Asymmetry in both placement of openings and floor plan affect the dynamic properties. The asymmetry create eccentricities between centre of mass and centre of rigidity resulting in some of the modes tends to rotate or even be diagonal. Depending on the floor plan, timber buildings with a slenderness of 1.5-2.1 may have problem to fulfil the comfort criteria.

Additional mass from non-structural elements has not been included, which may affect the acceleration level.

References

Autodesk Inc. (2015). Technical Documentation for Calculating the Modal Mass.

BFS 2015:6 EKS 10. (2015). Boverkets föreskrifter om ändring i verkets föreskrifter och allmänna råd (2011:10) om tillämpning av europeiska konstruktionsstandarder (eurokoder). Boverket. Karlskrona: In Swedish.

Chopra, A. K. (2012). Dynamics of structures : theory and applications to earthquake engineering (4th ed. ed.). Upper Saddle River, N.J.: Prentice Hall.

Edskär, I., & Lidelöw, H. (2017). Wind-induced vibrations in timber buildings-parameter study of cross-laminated timber residential structures. Structural Engineering International, 27(2), 205-216. doi:10.2749/101686617X14881932435619

Eurocode 1-4 SS-EN 1991-1-4. (2005). Actions on structures - Part1-4: General actions - Wind actions. Stockholm: SIS Förlag AB.

ISO 10137. (2008). Bases for design of structures - Serviceability of buildings and walkways againts vibration. Stockholm: SIS Förlag AB.

Jeary, A. (1997). Designer´s guide to the dynamic response of structures. Hong Kong: E &

FN Spon.

Svenskt Trä. (2017). KL-trähandbok (CLT handbook). (E. Borgström, & J. Fröbel, Eds.) Stockholm: Skogsindutrierna, Svenskt Trä (In Swedish).

Appendix A1

d – depth of the building w – width

h - height h/d – slenderness

XCM – Centre of mass, X-coordinate YCM – Centre of mass, Y-coordinate

XCR – Centre of rigdity, X-coordinate YCR – Centre of rigidity, Y-coordinate f – frequency

mtot – total mass 𝑀𝑀� – modal mass

𝑚𝑚� – modal mass X-direction _𝑋𝑋

𝑚𝑚� – modal mass Y-direction 𝑌𝑌

𝐾𝐾� – modal stiffness me – equivalent mass

me,app – equivalent mass approximation apeak – peak acceleration at the top

* the mode has a tendency to rotate Case Size opening / mode Mode

direction d [m] w

[m] h

[m] h/d XCM

[m] YCM

[m] XCR

[m] YCR

[m] f

[Hz] mtot

[kg] 𝑴𝑴�

[kg] 𝒎𝒎� 𝑿𝑿

[kg] 𝒎𝒎� 𝒀𝒀

[kg] 𝑲𝑲�

[kNm] me

[kg/m] me,app

[kg/m] me,app / me

apeak

[m/s²]

O - m1 X 20 20 30 1.5 0 0 0 0 3.14 636,303 291,111 291,110 1 113,312 22,414 21,210 0.946 0.044

O - m2 Y 20 20 30 1.5 0 0 0 0 3.16 636,303 289,896 2 289,894 114,282 22,397 21,210 0.947 0.043

O - m1 X 20 20 42 2.1 0 0 0 0 2.18 890,824 386,908 386,908 0 72,591 22,246 21,210 0.953 0.069

O - m2 Y 20 20 42 2.1 0 0 0 0 2.19 890,824 385,591 1 385,590 73,009 22,253 21,210 0.953 0.069

O - m1 X 20 20 54 2.7 0 0 0 0 1.62 1,145,345 470,496 470,496 0 48,747 22,125 21,210 0.959 0.010

O - m2 Y 20 20 54 2.7 0 0 0 0 1.63 1,145,345 469,058 0 469,058 49,200 22,129 21,210 0.958 0.099

O - m1 X 20 20 66 3.3 0 0 0 0 1.26 1,399,866 544,089 544,088 0 34,101 22,034 21,210 0.963 0.134

O - m2 Y 20 20 66 3.3 0 0 0 0 1.27 1,399,866 542,526 0 542,526 34,545 22,037 21,210 0.962 0.132

O - m1 X 20 20 78 3.9 0 0 0 0 1.01 1,654,388 610,446 610,446 0 24,584 21,963 21,210 0.966 0.170

O - m2 Y 20 20 78 3.9 0 0 0 0 1.02 1,654,388 608,785 0 608,785 25,005 21,966 21,210 0.966 0.168

O - m1 X 20 20 90 4.5 0 0 0 0 0.83 1,908,909 672,079 672,079 0 18,278 21,905 21,210 0.968 0.208

O - m2 Y 20 20 90 4.5 0 0 0 0 0.83 1,908,909 670,353 0 670,353 18,231 21,907 21,210 0.968 0.208

O - m1 X 20 20 180 9 0 0 0 0 0.27 3,817,818 1,100,494 1,100,494 0 3,167 21,647 21,210 0.980 0.526

O - m2 Y 20 20 180 9 0 0 0 0 0.27 3,817,818 1,098,987 0 1,098,987 3,163 21,644 21,210 0.980 0.526

1X 1.5x2.5 m² - m1 X* 20 20 30 1.5 0 0.191 0 2.333 2.54 624,372 205,687 201,737 3,950 52,388 15,716 20,812 1.324 0.057 1X 1.5x2.5 m² - m2 Y 20 20 30 1.5 0 0.191 0 2.333 3.15 624,372 280,329 2 280,328 109,812 22,077 20,812 0.943 0.044 1X 1.5x2.5 m² - m1 X* 20 20 42 2.1 0 0.191 0 2.333 1.77 874,121 279,896 274,915 4,981 34,618 15,569 20,812 1.337 0.086 1X 1.5x2.5 m² - m2 Y 20 20 42 2.1 0 0.191 0 2.333 2.16 874,121 369,156 2 369,155 67,995 21,906 20,812 0.950 0.070 1X 1.5x2.5 m² - m1 X* 20 20 54 2.7 0 0.191 0 2.333 1.35 1,123,870 354,365 349,060 5,305 25,496 15,784 20,812 1.319 1X 1.5x2.5 m² - m2 Y 20 20 54 2.7 0 0.191 0 2.333 1.59 1,123,870 445,964 0 445,964 44,510 21,780 20,812 0.956 1X 1.5x2.5 m² - m1 X* 20 20 66 3.3 0 0.191 0 2.333 1.06 1,373,619 423,568 418,092 5,476 18,789 15,990 20,812 1.302 1X 1.5x2.5 m² - m2 Y 20 20 66 3.3 0 0.191 0 2.333 1.23 1,373,619 513,794 0 513,793 30,687 21,685 20,812 0.960 1X 1.5x2.5 m² - m1 X* 20 20 78 3.9 0 0.191 0 2.333 0.87 1,623,368 497,029 492,254 4,775 14,852 16,520 20,812 1.260 1X 1.5x2.5 m² - m2 Y 20 20 78 3.9 0 0.191 0 2.333 0.98 1,623,368 576,162 0 576,161 21,845 21,608 20,812 0.963 1X 1.5x2.5 m² - m1 X* 20 20 90 4.5 0 0.191 0 2.333 0.72 1,873,117 562,735 558,288 4,447 11,517 16,834 20,812 1.236 1X 1.5x2.5 m² - m2 Y 20 20 90 4.5 0 0.191 0 2.333 0.79 1,873,117 634,720 0 634,720 15,639 21,544 20,812 0.966 1X 1.5x2.5 m² - m1 Y 20 20 180 9 0 0.191 0 2.333 0.25 3,746,233 1,057,506 0 1,057,506 2,609 21,261 20,812 0.979 1X 1.5x2.5 m² - m2 X 20 20 180 9 0 0.191 0 2.333 0.25 3,746,233 1,025,699 1,023,961 1,738 2,531 18,847 20,812 1.104

Case Size opening / mode Mode direction d

[m] w [m] h

[m] h/d XCM

[m] YCM

[m] XCR

[m] YCR

[m] f

[Hz] mtot

[kg] 𝑴𝑴�

[kg] 𝒎𝒎� 𝑿𝑿

[kg] 𝒎𝒎� 𝒀𝒀

[kg] 𝑲𝑲�

[kNm] me

[kg/m] me,app

[kg/m] me,app / me

apeak

[m/s²] 2X 1.5x2.5 m² - m1 X 20 20 30 1.5 0 0 0 0 2.08 612,442 286,332 286,332 0 48,905 21,970 20,415 0.929 0.073 2X 1.5x2.5 m² - m2 Y 20 20 30 1.5 0 0 0 0 3.13 612,442 270,990 2 270,988 104,810 21,721 20,415 0.940 0.044 2X 1.5x2.5 m² - m1 X 20 20 42 2.1 0 0 0 0 1.47 857,418 393,774 393,774 0 33,592 21,564 20,415 0.947 0.109 2X 1.5x2.5 m² - m2 Y 20 20 42 2.1 0 0 0 0 2.13 857,418 353,213 1 353,212 63,264 21,582 20,415 0.946 0.071 2X 1.5x2.5 m² - m1 X 20 20 54 2.7 0 0 0 0 1.14 1,102,395 491,874 491,874 0 25,236 21,345 20,415 0.956 2X 1.5x2.5 m² - m2 Y 20 20 54 2.7 0 0 0 0 1.56 1,102,395 424,114 0 424,114 40,747 21,422 20,415 0.953 2X 1.5x2.5 m² - m1 X 20 20 66 3.3 0 0 0 0 0.9 1,347,371 580,866 580,866 0 18,575 21,209 20,415 0.963 2X 1.5x2.5 m² - m2 Y 20 20 66 3.3 0 0 0 0 1.19 1,347,371 487,172 0 487,171 27,236 21,328 20,415 0.957 2X 1.5x2.5 m² - m1 X 20 20 78 3.9 0 0 0 0 0.76 1,592,348 657,066 657,066 0 14,983 21,120 20,415 0.967 2X 1.5x2.5 m² - m2 Y 20 20 78 3.9 0 0 0 0 0.94 1,592,348 546,443 0 546,443 19,062 21,242 20,415 0.961 2X 1.5x2.5 m² - m1 X 20 20 90 4.5 0 0 0 0 0.63 1,837,325 729,011 729,011 0 11,423 21,053 20,415 0.970 2X 1.5x2.5 m² - m2 Y 20 20 90 4.5 0 0 0 0 0.75 1,837,325 602,712 0 602,712 13,384 21,173 20,415 0.964 2X 1.5x2.5 m² - m1 Y 20 20 180 9 0 0 0 0 0.23 3,674,649 1,020,637 0 1,020,637 2,132 20,870 20,415 0.978 2X 1.5x2.5 m² - m2 X 20 20 180 9 0 0 0 0 0.24 3,674,649 1,160,846 1,160,846 0 2,640 20,818 20,415 0.981 1X1Y 1.5x2.5 m² - m1 XY* 20 20 30 1.5 0.195 0.195 2.333 2.333 2.39 612,442 259,501 141,597 117,904 58,519 19,896 20,415 1.026 0.061 1X1Y 1.5x2.5 m² - m2 XY 20 20 30 1.5 0.195 0.195 2.333 2.333 2.66 612,442 497,206 225,803 271,403 138,886 38,862 20,415 0.525 0.055 1X1Y 1.5x2.5 m² - m1 XY* 20 20 42 2.1 0.195 0.195 2.333 2.333 1.67 857,418 360,383 201,138 159,245 39,679 19,811 20,415 1.030 0.094 1X1Y 1.5x2.5 m² - m2 XY 20 20 42 2.1 0.195 0.195 2.333 2.333 1.83 857,418 637,884 281,672 356,212 84,334 36,964 20,415 0.552 0.086 1X1Y 1.5x2.5 m² - m1 XY* 20 20 54 2.7 0.195 0.195 2.333 2.333 1.28 1,102,395 465,496 265,547 199,949 30,109 20,374 20,415 1.002 1X1Y 1.5x2.5 m² - m2 XY 20 20 54 2.7 0.195 0.195 2.333 2.333 1.37 1,102,395 757,023 324,847 432,177 56,093 35,707 20,415 0.572 1X1Y 1.5x2.5 m² - m1 XY* 20 20 66 3.3 0.195 0.195 2.333 2.333 1.01 1,347,371 561,194 327,740 233,454 22,600 20,778 20,415 0.983 1X1Y 1.5x2.5 m² - m2 XY 20 20 66 3.3 0.195 0.195 2.333 2.333 1.06 1,347,371 854,668 355,150 499,519 37,911 34,536 20,415 0.591 1X1Y 1.5x2.5 m² - m1 XY* 20 20 78 3.9 0.195 0.195 2.333 2.333 0.83 1,592,348 652,356 401,431 250,925 17,742 21,417 20,415 0.953 1X1Y 1.5x2.5 m² - m2 XY 20 20 78 3.9 0.195 0.195 2.333 2.333 0.86 1,592,348 907,654 348,768 558,887 26,502 32,410 20,415 0.630 1X1Y 1.5x2.5 m² - m1 XY* 20 20 90 4.5 0.195 0.195 2.333 2.333 0.69 1,837,325 721,810 468,239 253,571 13,567 21,425 20,415 0.953 1X1Y 1.5x2.5 m² - m2 XY 20 20 90 4.5 0.195 0.195 2.333 2.333 0.7 1,837,325 945,512 331,820 613,693 18,290 30,409 20,415 0.671 1X1Y 1.5x2.5 m² - m1 XY 20 20 180 9 0.195 0.195 2.333 2.333 0.23 3,674,649 1,735,922 1,048,842 687,079 3,625 33,642 20,415 0.607 1X1Y 1.5x2.5 m² - m2 XY* 20 20 180 9 0.195 0.195 2.333 2.333 0.24 3,674,649 1,538,637 608,849 929,788 3,499 28,428 20,415 0.718

Case Size opening / mode Mode direction d

[m] w [m] h

[m] h/d XCM

[m] YCM

[m] XCR

[m] YCR

[m] f

[Hz] mtot

[kg] 𝑴𝑴�

[kg] 𝒎𝒎� 𝑿𝑿

[kg] 𝒎𝒎� 𝒀𝒀

[kg] 𝑲𝑲�

[kNm] me

[kg/m] me,app

[kg/m] me,app / me

apeak

[m/s²] 1X2Y 1.5x2.5 m² - m1 Y 20 20 30 1.5 0 0.199 0 2.333 2.12 600,511 278,437 1 278,437 49,404 21,606 20,017 0.926 0.071 1X2Y 1.5x2.5 m² - m2 X* 20 20 30 1.5 0 0.199 0 2.333 2.46 600,511 167,992 160,279 7,713 40,135 13,057 20,017 1.533 0.058 1X2Y 1.5x2.5 m² - m1 Y 20 20 42 2.1 0 0.199 0 2.333 1.49 840,715 379,081 1 379,080 33,225 21,203 20,017 0.944 0.107 1X2Y 1.5x2.5 m² - m2 X* 20 20 42 2.1 0 0.199 0 2.333 1.7 840,715 235,003 226,402 8,601 26,812 13,291 20,017 1.506 0.090 1X2Y 1.5x2.5 m² - m1 Y 20 20 54 2.7 0 0.199 0 2.333 1.14 1,080,920 469,034 0 469,034 24,064 20,975 20,017 0.954 1X2Y 1.5x2.5 m² - m2 X* 20 20 54 2.7 0 0.199 0 2.333 1.28 1,080,920 304,998 297,062 7,936 19,728 13,933 20,017 1.437 1X2Y 1.5x2.5 m² - m1 Y 20 20 66 3.3 0 0.199 0 2.333 0.90 1,321,124 549,632 0 549,632 17,576 20,847 20,017 0.960 1X2Y 1.5x2.5 m² - m2 X* 20 20 66 3.3 0 0.199 0 2.333 1.00 1,321,124 370,504 363,258 7,246 14,627 14,477 20,017 1.383 1X2Y 1.5x2.5 m² - m1 Y 20 20 78 3.9 0 0.199 0 2.333 0.73 1,561,328 622,052 0 622,052 13,087 20,750 20,017 0.965 1X2Y 1.5x2.5 m² - m2 X* 20 20 78 3.9 0 0.199 0 2.333 0.8 1,561,328 435,032 428,751 6,281 10,992 15,058 20,017 1.329 1X2Y 1.5x2.5 m² - m1 Y 20 20 90 4.5 0 0.199 0 2.333 0.61 1,801,533 687,894 0 687,893 10,105 20,684 20,017 0.968 1X2Y 1.5x2.5 m² - m2 X* 20 20 90 4.5 0 0.199 0 2.333 0.66 1,801,533 498,494 493,213 5,281 8,572 15,628 20,017 1.281 1X2Y 1.5x2.5 m² - m1 X* 20 20 180 9 0 0.199 0 2.333 0.22 3,603,065 953,745 952,444 1,301 1,822 18,351 20,017 1.091 1X2Y 1.5x2.5 m² - m2 Y 20 20 180 9 0 0.199 0 2.333 0.22 3,603,065 1,102,000 19 1,101,982 2,106 21,554 20,017 0.929 2X2Y 1.5x2.5 m² - m1 X 20 20 30 1.5 0 0 0 0 2.08 588,580 269,983 269,983 0 46,113 21,236 19,619 0.924 0.073 2X2Y 1.5x2.5 m² - m2 Y 20 20 30 1.5 0 0 0 0 2.11 588,580 270,197 1 270,197 47,490 21,235 19,619 0.924 0.071 2X2Y 1.5x2.5 m² - m1 X 20 20 42 2.1 0 0 0 0 1.45 824,012 365,331 365,331 0 30,324 20,831 19,619 0.942 0.110 2X2Y 1.5x2.5 m² - m2 Y 20 20 42 2.1 0 0 0 0 1.48 824,012 364,839 0 364,839 31,549 20,846 19,619 0.941 0.108 2X2Y 1.5x2.5 m² - m1 X 20 20 54 2.7 0 0 0 0 1.11 1,059,444 449,656 449,656 0 21,872 20,614 19,619 0.952 2X2Y 1.5x2.5 m² - m2 Y 20 20 54 2.7 0 0 0 0 1.12 1,059,444 448,236 0 448,236 22,197 20,614 19,619 0.952 2X2Y 1.5x2.5 m² - m1 X 20 20 66 3.3 0 0 0 0 0.87 1,294,876 524,706 524,706 0 15,679 20,479 19,619 0.958 2X2Y 1.5x2.5 m² - m2 Y 20 20 66 3.3 0 0 0 0 0.88 1,294,876 522,432 0 522,432 15,972 20,483 19,619 0.958 2X2Y 1.5x2.5 m² - m1 X 20 20 78 3.9 0 0 0 0 0.72 1,530,309 588,208 588,208 0 12,038 20,388 19,619 0.962 2X2Y 1.5x2.5 m² - m2 Y 20 20 78 3.9 0 0 0 0 0.73 1,530,309 585,276 0 585,276 12,313 20,391 19,619 0.962 2X2Y 1.5x2.5 m² - m1 X 20 20 90 4.5 0 0 0 0 0.59 1,765,741 648,990 648,990 0 8,919 20,317 19,619 0.966 2X2Y 1.5x2.5 m² - m2 Y 20 20 90 4.5 0 0 0 0 0.6 1,765,741 645,517 0 645,517 9,174 20,321 19,619 0.965 2X2Y 1.5x2.5 m² - m1 X 20 20 180 9 0 0 0 0 0.2 3,531,481 1,047,724 1,047,724 0 1,654 20,051 19,619 0.978 2X2Y 1.5x2.5 m² - m2 Y 20 20 180 9 0 0 0 0 0.2 3,531,481 1,043,546 0 1,043,546 1,648 20,048 19,619 0.979