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Dynamic properties of cross-laminated timber and timber truss building systems
Engineering Structures 186 (2019) 525–535
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Dynamic properties of cross-laminated timber and timber truss building systems
T
Ida Edskär , Helena Lidelöw ⁎
Timber Structures, Luleå University of Technology, 971 87 Luleå, Sweden
ARTICLE INFO
ABSTRACT
Keywords: Serviceability limit state Tall buildings Wind-induced vibration Mode shape Natural frequency Modal analysis Modal mass
Incorrect prediction of dynamic behaviour of tall buildings can lead to discomfort for humans. It is therefore important to understand the dynamic characteristics such as natural frequency, mode shape, damping and the influence they have on acceleration levels. The aim of this study is to compare two timber building types, cross laminated panels and post-and-beam with stabilising diagonals, through a parameter study applying modal analysis. Empirical formulae for predicting the natural frequency and mode shape are compared to measured and numerical results. Tall building assumptions such as ‘line-like’ behaviour and lumped mass at certain points were evaluated for both systems. The post-and-beam system showed a stiffer behaviour than the cross laminated system where more shear deformation occurred. Empirical formulae should be used with care until more data is collected. For the post-and-beam systems an assumption of linearity may be appropriate, but for cross laminated systems the approximation can give results on the unsafe side. Finally, the relationship between stiffness and mass for cross laminated timber systems and its effect on dynamic properties needs to be further investigated.
1. Introduction Comfort evaluation is an important aspect when designing taller buildings, especially timber buildings due to the low self-weight compared to concrete and steel buildings. The dynamic characteristics of a tall building are the natural frequencies, the mode shape, and the damping ratio. Incorrect prediction of these dynamic characteristics can lead to undesirable behaviour of the structure when completed or unnecessary construction costs [1]. ISO 10137 [2] can be used to evaluate comfort criteria. The natural frequencies of a building can be predicted by theoretical, empirical or numerical methods. Each natural frequency has a corresponding mode shape that can be generated from a modal analysis or estimated using formulas in Eurocode 1-4 [3,4]. The mode shape can either be a bending or a shear mode but is often a combination of the two. Both the natural frequency and the mode shape affect the acceleration levels. When evaluating tall buildings, assumptions such as the building behaving ‘line-like’ and masses being lumped at certain points are made [5]. These assumptions may not be suitable for high-rise timber buildings. The aim of this research is to elucidate the dynamic behaviour of two building types, cross-laminated timber panels and timber post-and-beam with trusses, through a parameter study applying modal analysis. Simulated results are compared with measurements of existing timber buildings. The empirical formula to
⁎
estimate the first natural frequency from Eurocode 1-4 [3] is challenged and the mode shape is analysed. Eurocode 1-4 [3] presents f = 46/ h as an empirical expression to estimate the first natural frequency of buildings. Depending on the geometry of the building, different mode shapes result. Shorter buildings deform predominantly in a shear mode, while taller buildings are slender enough to deform in bending. Eurocode 1-4 [3] recommends several expressions for engineers when estimating the mode shape. Through generating the mode shape from modal analysis, conclusions are drawn on the applicability of Eurocode 1-4 [3] formulas and tall building assumptions [5] are evaluated for high-rise timber buildings. 2. Theory In the serviceability limit state, large deformations and uncomfortable vibrations should be limited to design a functional building. Due to the low mass of timber buildings, wind induced vibrations may be an issue. Two of the tallest (2018) residential timber buildings are “Treet”, Norway with a height of 49 m and “Brock Commons”, Canada with a reported height of 53 m [6,7]. “Treet” has timber truss systems to stabilise the structure and additional concrete slabs to increase the mass [6]. “Brock Commons” is a hybrid system where two concrete cores are used to stabilise the structure, timber
Corresponding author. E-mail address: [email protected] (I. Edskär).
https://doi.org/10.1016/j.engstruct.2019.01.136 Received 6 July 2018; Received in revised form 23 December 2018; Accepted 30 January 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
Engineering Structures 186 (2019) 525–535
I. Edskär and H. Lidelöw
columns are used for the vertical load-carrying system and CLT slabs with concrete topping are used as floor elements [7]. “Treet” has a total mass of 135 kg/m3 [8], compared to tall concrete and steel buildings where the total mass is 300 kg/m3 and 160 kg/m3 respectively [9,10].
where M and K are the N × N matrices of mass and stiffness and u denotes the generalised displacement:
u(t ) = qn (t )
n is the deflected shape and qn (t ) can be described by a simple harmonic function, which gives the following algebraic equation also known as the eigenvalue problem:
2.1. Dynamic properties Tall buildings can be modelled as a vibrating uniform cantilever beam, fixed to the foundation [4,5]. To predict the natural frequency of a building theoretical models, empirical formulae, or numerical methods can be used.
fi =
2 L2
EI m
i = 1, 2, 3,
,n
i
G
2 L
i = 1, 2, 3,
,n
= (2i
2 n M]
det[K 2 n
n
The solution will give the eigenvalue of squared natural frequencies and the corresponding eigenvector n to each eigenvalue: 1
=
2
iz L
cos
iz L
i
sinh
iz L
sin
iz L
(10)
where L is the length of the beam/height of the building, z varies along the building height and for the first mode 1 = 1.875, 1 = 0.734 . Eq. (11) is for the first shear mode:
(1)
(2
(z ) =
1) z
(11)
L
These two equations represent pure bending mode and pure shear mode [11]. The eigenvector is normalised to make the amplitude unique, n . The mode can be scaled so the eigenvector ( N )n = 1 at the specified coordinate N , the mode can be scaled at the coordinate N where the mode has it maximum displacement, or the mode is scaled so the generalised mass or modal mass, Mn , has a specified value. The modal mass is defined by:
(2)
T n M n
(12)
Mn = 1 is often used and since the product of has the unit mass entails Mn = 1 kg . The generalised stiffness or modal stiffness for the nth mode is defined by: T n M n
(3)
T n K n
Kn =
(13)
For the nth mode and multiplied by T n K n
=
2 T n ( n M n)
T n
Eq. (7) can be written as: (14)
The squared natural frequencies can then be related to the modal mass and modal stiffness by: 2 n
=
Kn Mn
(15)
Each resonance mode have an associated modal mass and modal stiffness [5]. For a free vibration continuous system the modal mass is defined by:
(4)
where f f is the fundamental frequencies from the flexural beam model and fs from the shear beam model. Tall buildings consist of several elements and can be very complex in structure therefore finite element (FE) software is useful to solve the multiple degree of freedom system. Modal analysis is used to solve the free-vibration equation of motion [4], expressed as:
Mu¨ + Ku= 0
,N (9)
(z ) = cosh
and is the shear coefficient, G is the shear modulus, and is the mass density. The flexural beam model relates the frequency as proportional to 1/L2 and in the shear beam model to 1/L. The Timoshenko beam model considers, except from flexural and shear deformation also the effect of rotary inertia. The introduction of the shear deformation and rotary inertia reduces the natural frequencies compared to using only the flexural beam theory. However, the effect of rotary inertia is smaller than the effect of shear deformations [11]. There is no general closed form solution for the frequency when combining flexural and shear deformations, but Dunkerleýs approximation can be used which estimates a lower bound of the natural frequency [11] expressed as:
1 1 1 = 2+ 2+ f2 ff fs
n = 1, 2,
n is the number of modes. The eigenvector is also known as the natural mode and represents the mode shape. Depending on the structural system the mode shape will be a bending mode, a shear mode or a combination of bending and shear. The bending mode for a cantilever beam is presented in Eq. (10):
Mn = 1)
(8)
=0
N
where i
(7)
=0
Eq. (7) has a nontrivial solution if
where L is the length of the beam, E is the modulus of elasticity, I is the moment of inertia, m is the mass per unit length and subscript i represents the mode of vibration. For the first natural frequency 1 = 1.875 [11]. For slender beams the flexural deformations will dominate, but for less slender beams shear deformation may become important and needs to be considered. Eq. (2) estimates the natural frequencies of a fixed uniform cantilever beam, considering shear deformation only [11]:
fi =
2 n M] n
[K
2.1.1. Theoretical and numerical models For simple structures, a model based on the theory of an idealised cantilever beam with simple boundary conditions can be used. The dynamic system can be represented by either a lumped or a distributed mass system. Lumped masses are connected by elements (walls, columns) with a certain stiffness and damping. To solve the system and determine the natural frequencies, modal analysis can be used. The system can be scaled and the dynamic behaviour is described by second order differential equations and results in a finite number of natural frequencies. The classic Euler-Bernoulli beam model estimates the natural frequency of beams. The theory only considers flexural deformations of a slender beam with linear elastic, homogeneous material. Eq. (1) estimates the natural frequencies of a fixed uniform cantilever beam [11]: 2 i
(6)
n
h
Mn =
m (z ) 0
2 n dz
(16)
An assumption is made that the dimensions in X and Y directions, Fig. 5, are constant and do not vary along the building. If the building is irregular it needs to be considered in Eq. (16). The modal mass can be generated from the FE software directly or be calculated from the
(5) 526
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I. Edskär and H. Lidelöw
(z ) =
z h
(23)
= 0.6 for slender frame structures with non load-distributing walls or cladding. = 1.0 for buildings with a central core plus peripheral columns or larger columns plus shear bracings. = 1.5 for slender cantilever buildings and buildings supported by central reinforced concrete cores. Mode functions with different value are displayed in Fig. 1. = 1.5 corresponds to a bending mode shape, while = 1.0 is a linear relationship. When solving the eigenvalue problem numerically in FE software, the eigenvector for each natural frequency will be generated and thereby the mode shape is captured. 2.1.2. Empirical formulae Several empirical formulae to estimate the first natural frequency have been suggested by researchers [12–15]. In Eurocode 1-4 [3] the fundamental frequency f can be estimated by Eq. (24) for multi-storey buildings with a height larger than 50 m.
f=
eigenvector. Values of the eigenvector can be defined by taking points at certain levels of the building where the maximum amplitude is generated, N . The modal mass can then be calculated using:
1 2 N
(17)
When dealing with ‘tall’ buildings, the building is considered slender and some assumptions are often made. The building is considered to be ‘line-like’/linear and with a constant mass over the height, but often the mass decreases with height since smaller dimensions are needed to carry the weight. Another assumption is that the mass is lumped in certain points and a normalised amplitude is used. The mode shape can for tall buildings normally be represented by a straight line, Eq. (18):
(z ) =
z h
f=
Mn =
50 h
(25)
where h is the height of the structure. In [13] data was also analysed with respect to structural material, Eqs. (26)–(28):
f=
45 h
steel buildings
(26)
f=
55 h
reinforced concrete buildings
(27)
f=
57 h
hybrid buildings
(28)
(18)
The mode shape represented by a straight line is presented in Fig. 1 with = 1. If using the assumption for tall buildings and inserting Eq. (18) into Eq. (16) the modal mass will be:
m (z ) h3
(24)
where h is the height of the structure in meters. The formula is based on a collection of 163 rectangular plan buildings with different types of materials; concrete, steel, and mixed [12]. To estimate the first natural frequency, different predictors were studied and compared to the measured frequency. The height was a better predictor than including the width of the building in the formula. Eq. (24) has a correlation coefficient of R2 = 0.88. By looking at the frequency plotted against height in [12], one finds that the largest scatter of data exists for shorter buildings. However, there are more data points for shorter buildings than taller ones. From a collection of 185 different buildings (steel, reinforced concrete, mixed, pre-cast, masonry, unknown), the following equation to predict the natural first natural frequency was proposed [13]:
Fig. 1. Mode shape.
Mn =
46 h
(19)
Two equations have been proposed by AIJ [16] in [14] for estimating the first natural frequency, one for reinforced concrete buildings and one for steel buildings:
(20)
f = 67/ h
reinforced concrete buildings
(29)
If m (z ) h is assumed to be the total mass of the building, Mtot the modal mass will be:
f = 50/ h
steel buildings
(30)
3h2
And for a pure shear mode Eq. (16) will be:
Mn =
m (z ) h 2
M Mn = tot 3
where h is the height of the structure. The equations are used for comfort assessment and have been derived from 68 reinforced concrete buildings and 137 steel buildings in Japan. In the National Building Code of Canada [17] in [15], several empirical formulae are given to calculate the first natural frequency. [15] stated that the following equations could be used to predict the natural frequencies for a timber building:
(21)
and for shear:
Mn =
Mtot 2
(22)
Depending on the mode shape the modal mass will vary. Eq. (21)) is often used as an approximation for tall buildings with a constant mass per unit height [5]. As seen in Eqs. (16), (19) and (20) the modal mass is dependent on the mass distribution over the height, z . In Eurocode 1-4 [3], an expression for the mode shape of different types of buildings is stated:
f = 40/ h f=
20 h3/4
braced frames
(31)
shear walls and other structures
(32)
Eq. (32) suggests a relationship between f and h that is not exactly inversely proportional. Measurements of timber buildings have been 527
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I. Edskär and H. Lidelöw
apeak =
(35)
x (z ) k p
The equivalent mass per unit length is given by Eq. (36): h
m (z )·
0
me =
h 0
me, app = collected in [8], and dynamic tests (ambient vibration tests) have been carried out on 11 timber buildings. Some of the buildings have the first floor in concrete and some are composite timber-concrete or timbersteel, but the main vertical and lateral load-carrying system is of timber, which includes cross-laminated timber, glulam and light timber frames. Based on the collected data a single relationship between height and natural frequency was found for timber buildings Eq. (33): (33)
2.2. Acceleration and comfort
=
According to Eurocode 1-4 [3], a maximum horizontal displacement and a characteristic standard deviation of acceleration along wind at building height z, x (z ),should be satisfied. Acceptable comfort levels may be found in the standard ISO 10137 [2]. To evaluate the comfort level, the acceleration and first natural frequency need to be calculated. ISO 10137 [2] is based on a 1-year return period of wind and peak acceleration. The return period can either be handled by using a 2-year return period for the wind velocity or apply the acceleration level as 0.72 times the acceleration level for a 5-year return period [18,19]. The results will be the same. Equations for the standard deviation of acceleration may be found in Eurocode 1-4 [3]. The equation used in this paper is from the national annex in Sweden EKS 10 [18] where the standard deviation of acceleration along wind is given by:
(z ) =
3· Iv (h)·R· qm (h)·b ·cf · me
1, x (z )
(36)
mtot h
(37)
Inserting Eqs. (18) and (21) into (36) will give this approximation. The choice of mode shape will not affect the equivalent mass. Depending on structural systems, materials, connections etc. the damping in the building will vary. The magnitude of motion will be reduced more quickly in a building with high damping. Damping of a certain building can be measured once the building is completed. It is difficult to predict the damping ratio before the building is built. There are two types of damping systems, passive and active. Passive damping systems has a fixed property related to chosen layout and structural materials, while active dampers modify the system properties and realise an active mechanism. Damping of the structural system itself is a passive damper [21]. The damping ratio can be non-linear depending on the amplitude of the motion [22] c.f. Results and Analysis. The damping properties are included in the resonance factor R in Eq. (34). Both structural damping, aerodynamic damping and damping from special advices are included. The total logarithmic decrement for damping is given by:
It should be noted that timber buildings have different stabilising systems, which may behave differently under wind-induced vibrations. In Fig. 2 Eqs. (24), (27), (29), (31)–(33) are plotted. Eqs. (29) and (31) constitute the lower and higher bounds. Studying Fig. 2, the different empirical formulae display the same tendency, but with considerable spread.
x
2 1 (z ) dz
h is the height of the structure, m (z ) is the mass per unit length, and 1is the first eigenmode. The numerator in Eq. (36) is the modal mass. During calculations in FE software the eigenvector is normalised as in Eq. (12). Two alternative methods to arrive at the modal mass exist: either the modal mass in Eq. (36) is set to 1 and the eigenvector is normalised, or the software can produce the value of the modal mass. The denominator is the integral of the mode shape over the height of the building where the amplitude has its maximum. The number of integration points can be set to the mesh size over the height. Integration intervals can alternatively be set to the storey height and intervals can be defined floor to floor or centre of storey to centre of storey. The chosen method will give slightly different equivalent masses in the end, which may affect the acceleration. The equivalent mass can approximately be taken as the total mass of the building divided by the total height:
Fig. 2. Empirical formulae.
f = 55/ h
2 1 (z ) dz
s
+
a
+
d
(38)
s is the logarithmic decrement of structural damping, a is the logarithmic decrement of aerodynamic damping and d is the logarithmic decrement due to special devices e.g. tuned mass dampers, sloshing thanks [3]. In the old Swedish handbook “Snow load and wind action” [23] values on the logarithmic decrement for timber structures with and without mechanical connections are presented. Structural damping is stated as s = 0.06 for a timber structure without mechanical connections and s = 0.09 for a timber structure with mechanical connections. The values correspond to about 1.0% and 1.5% critical damping. In Eurocode 1-4 [3] a value of s = 0.06 0.12 is tabulated for timber bridges, which corresponds to 1.0–1.9% critical damping. In Table 1, dynamic properties are presented for measured buildings and values from numerical analyses. Critical damping values are reported between 1.3 and 9.1% for buildings with cross laminated timber as the main load bearing structure, 1.4–2.4% for post-and-beam systems and 1.0–6.9% for hybrid structures. For steel and reinforced concrete structures, the critical damping, tabulated in Eurocode 1-4 [3], are 0.8% and 1.6% respectively. The presented values in Eurocode 1-4 [3] are not related to the amplitude or height of the building but from measured timber buildings the damping ratio varies with the amplitude and may have a non-linear behaviour [22]. The damping ratio tends to decrease with increased height [8].
(34)
Iv (h) is the turbulence intensity at height h , R is the square root of the resonant response, qm (h) is the mean pressure at height h , b is the width of the structure, cf is the force coefficient, 1, x (z ) is the mode shape value at height z , me is the equivalent mass per unit length, z is the height above the ground, and h is the height of the structure. The standard deviation of the acceleration is multiplied with a peak factor to get the peak acceleration [3,18,20]. 528
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Table 1 Summary of first natural frequencies and damping ratio for timber buildings. Number in Fig. 11
Building type
Height [m]
Measured
Simulated/estimated
f1 [Hz]
ξ1 [%]
CLT 1 2 3 3 4 5 6 6 7 8
CLT CLT, post-and-beam CLT CLT + 50 mm cement screed on the floor CLT, timber frame, first storey in concrete CLT, first storey concrete CLT core, post-and-beam CLT core, post-and-beam CLT core, post-and-beam CLT
9.1 12.6 21 21 25 27 29.5 29.5 74.8 31–68.2
3.9 3.2 2.7 2.45 2.28 2.26 1.1 1.6
5.4 1.3 3.20–5.60 5.20–9.10 2.3 1.9 3 2
Post-and-beam 9 10 11
Post-and-beam Post-and-beam, concrete slab every fourth storey Post-and-beam, concrete slab in the upper part
18.3 45 81
2.8 1.02–1.03
1.4 1.6–2.4
Hybrid 12 13 14 14 15 16
CLT, concrete core Post-and-beam with concrete Post-and-beam with concrete Post-and-beam with concrete Solid timber, concrete core Post-and-beam with concrete
15.6 21.8 22.1 22.1 23.9 53
4.03–4.08 2.1 2.7 2.8 2.33–2.34
4.9–6.9 1.5 1 4 1.82–2.90
a b c
core core and shear walls core and shear walls core, CLT
f1 [Hz]
Reference
ξ1 [%]
0.6 2.78–0.97
1.5 1.5
[15] [15]a [22]a [22] [24] [24] [25]a [25] [26]b [27]
0.75 0.33
1.9 1.9
[15] [8,6,28] []c
1.5
[29] [15]a [25]a [25] [30] [7]
2.58 2.05 1.94
0.5
Without non-structural elements (partition wall, finishing, etc.). Mass from non-structural material was not included. Magne Aanstad Bjertnæs, Sweco Norway AS, personal communication.
2.3. Dynamic properties of timber buildings The number of dynamically tested timber buildings is growing. In the following Table 1, a summary of measured, numerically calculated and/or estimated (theoretical or empirical models) timber buildings are listed. In Fig. 11, the values are plotted. The buildings have been divided in three types CLT, Post-and-beam, and Hybrid. For CLT the main stabilisation is through shear wall elements. For Post-and-beam, a bracing/truss system with diagonals in timber is used for stabilisation. In the Hybrid type, most of the buildings have concrete parts e.g. a core or shear walls in concrete for the main stabilising system and timber elements for vertical loads and load distribution. To measure dynamic properties ambient vibration tests (AVT) have been made in most cases, but also a few forced vibration tests (FVT) have been carried out [15,22,24,25,28–30]. For simulated cases, FE software and modal analysis have been used [7,26,27]. Flexural beam models have been used in theoretical estimations [22,24]. In [22,25] measurements have been taken during construction so the effect of non-structural elements such as partition walls, finishing etc. has been evaluated. The natural frequencies decreased when cement screed was added in [22] and for [25] the natural frequencies increased when partition walls, finishing etc. were added. The frequency of the measured buildings is 1.02–4.08 Hz with a variation in height of 9.1 m up to 53 m. Simulated values are between 0.33 Hz and 2.78 Hz with a height variation between 31 and 81 m.
Fig. 3. Effect of changing parameters.
also shown that the mode shape changes when parameters are varied. In [20] the shear stiffness was attributed too high a value. For a 10storey building, the natural frequency, equivalent mass and peak acceleration on top floors were calculated to f = 3.31 Hz, me = 27,579 kg/m and apeak = 0.0326 m/s2 respectively. With the correct shear stiffness and thicker wall and floor elements, calculations resulted in f = 2.40 Hz, me = 31,132 kg/m and apeak = 0.0443 m/s2. The mode shape changed from bending to bending/shear.
2.3.1. Previous study In a previous study [20], different parameters were studied to predict the behaviour of a CLT structure under wind-induced vibration, Fig. 3. By changing the stiffness, the first natural frequency increases and the acceleration levels decrease. The stiffness can be changed by changing the material properties, E- and G-modulus, or by changing for example the geometry of the building/elements, floor layout, placement of lateral stabilising system, or thickness of elements. Additional mass will decrease both the natural frequencies and the acceleration. An increased damping ratio will reduce the acceleration level. It was
3. Method In this paper the same floor layout has been used as in [20]. The 529
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Fig. 4. Mode shapes for (a) the post-and-beam structure and (b) the CLT structure.
Fig. 5. Mode shapes, the dashed lines shows the mode shape: (a) translation in Y-direction, (b) translation in X-direction, and (c) Torsional. (Units [–]).
floor layout is 20 × 20 m2 with a core placed in the middle for elevator and staircase. There are four apartments on each storey connected to the core. Two building types have been evaluated, one with cross
laminated timber and one with post-and-beam elements with diagonal bracing. In Figs. 4 and 5, the floor layout and the placement of structural elements is presented for the post-and-beam system. The building 530
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I. Edskär and H. Lidelöw
types have been evaluated in ultimate limit state on a 14-storey high (42 m) building to get a sense of the approximate dimensions on elements. The utilisation is between 65 and 90%. Connections and openings have not been checked in the model. The building types have been studied for a height of 18–66 m, which corresponds to 6–22 storeys. A step of 4 storeys was used.
element. A diagonal bracing system in the façade was used to stabilise the building type. The diagonals span over two storeys, Fig. 4. For the columns a dimension of 450 × 450 mm was used. For the beams 200 × 400 mm and for the diagonals 300 × 450 mm were used. Grade GL30h was assumed for all columns and beams. The post-and-beam building type was assumed to have the same insulation, gypsum, surface finishing etc. as the CLT building type. The difference between the systems is only the load bearing structure and the stabilising system. The posts and beams were modelled as beam elements with six degrees of freedom in each end. The columns are continuous through the building and fixed to the foundation. The diagonals and the beams are pinned to the columns.
3.1. FE-Modelling To calculate the natural frequency and the mode shape of the structures commercial FE software (Autodesk ® Robot™ Structural Analysis) was used to perform the modal analysis. In the FE model, only the structural elements were modelled such as cross laminated timber walls, floors, columns, beams, and diagonals. All non-structural materials, studs, insulation, gypsum boards and so on, were modelled as masses. The masses were applied as uniformed loads (kN/m2 or kN/m) and converted to masses when running the modal analysis. The Z-direction for the mass was ignored in the analysis to avoid of local vibrations in floors; X and Y directions (transversal vibrations) were set to active, Figs. 4 and 5. The subspace iteration method with a consistent mass matrix was used in the analysis. A value of 1.5% critical damping was assumed.
3.2. Acceleration, mass and mode shape By using Eq. (34), the standard deviation of acceleration was calculated. The equation gives an r.m.s value of the acceleration. The wind velocity was set to 24 m/s and terrain type III was used [3,18]. The wind velocity was reformulated from a 50 year return period to a 5 year return period according to [18]. The r.m.s acceleration with a 5 year return period of the wind velocity was then multiplied with 0.72 to represent a 1 year return period. The peak acceleration was calculated by multiplying the r.m.s acceleration value with a peak-factor, Eq. (35), which can be found in [18]. By multiplying the peak acceleration level with the mode shape, acceleration levels for each floor is established. The peak acceleration on the top floor was used for evaluation. The equivalent mass was calculated according to Eq. (36) where the numerator, modal mass, was generated from the FE software. The denominator, the integral of the eigenvector, was calculated with values from the FE software where the amplitude has its maximum. For the CLT building type values of the amplitude has been collected in each node along the height with a distance of 0.4–0.75 m depending on the case. For the post-and-beam building type, values at each storey has been collected since the beam element extends the whole storey height of 3 m. Values of the eigenvector in the FE model were taken on each storey where the amplitude peaked.
3.1.1. Building type with a CLT structure The CLT building type consists of external and internal walls, shafts and floors of CLT elements as the load bearing structure. The building type is stabilised through shear walls and diaphragm action in the floor elements. Post-and-beams have been used due to large spans of the floor elements [20]. The material grade was GL30h [31]. The thickness of the wall is 200 mm and it consists of five layers. The floor element is made of seven layers where the outmost parts has two parallel layers to increase the stiffness in the stiff direction. The thickness of the floor is 280 mm. Both the wall and the floor is made of graded C24 in all layers. The density was taken as 400 kg/m3 for CLT and 430 kg/m3 for the post-and-beam system. The wall and floor elements are completed with insulation, gypsum, surface finish etc. In [20] a detailed description can be found of the wall and floor build-up. Calculation of the stiffness of the elements can be done either by using the supplier’s data or handbooks on CLT. The only difference is how the layers in the CLT are built up. In [20] material properties were used from a supplier of CLT elements. In this paper, the thickness of the walls and floors were changed, and the shear stiffness was corrected to enable proper comparison with results generated in this paper. The elements were modelled as fournode quadrilateral shell elements and the stiffness matrix was calculated and manually defined in the software. Since the stiffness matrix is defined manually, the orthotropic properties are handled properly. Connections between CLT elements were modelled as fixed except the connection between wall and floor elements that was assumed to be pinned. The walls were assumed to be continuous through the building and fixed to the ground. In [20] pinned connections between walls and foundations and between walls were tested to compare with fixed connections. No differences in global results such as mode shape or acceleration were observed in the modal analysis. In [6] the stiffness of connections in a truss system was varied. Rotational stiffness decrease did not have any effect on the mode shape, while an axial connection stiffness lower than 25% of the member stiffness affected the mode shape and increased the acceleration. [32] report axial connection stiffnesses of 67% of the axial member stiffness. Assuming fixed connections is thus deemed appropriate when evaluating global dynamic behaviour.
4. Results and analysis In Table 2 the height of the building, h, the first natural frequency, f1, the mode shape direction, the modal mass, M1, the equivalent mass, me, peak acceleration on the top floor, apeak and calculated mode shape factor are presented for cases A to J. The equivalent mass and peak acceleration have been calculated from the generated mode shape, using the approximation in Eq. (37) and using the calculated mode shape factor, . Cases A to E represent the CLT building type and cases F to J the post-and-beam building type with diagonals. Each storey in the CLT building type weighs m = 113,990 kg and the roof storey weighs 126,873 kg where 38% of the total mass originates from non-structural elements. For the post-and-beam building type each storey weighs m = 103,785 kg and the roof storey 116,668 kg where 41% is mass from non-structural elements. The CLT building type weighs around 10% more than the post-and-beam building type. The total mass/m3 is 96 kg/m3 which is almost 1/3 compared to the concrete building in [9]. For all simulated cases, the first two modes are orthogonal translation modes in either X or Y-direction and the third a rotational mode, Fig. 5. The first natural frequency varies between 0.91 and 4.14 Hz. The CLT and post-and-beam building types have natural frequencies in the same range. The first mode shape direction is in the Y-direction, Fig. 4, except for cases I and J where the first mode shape direction changes to the Xdirection. The change of direction may be due to the slight asymmetry of the building floor plan layout rendering the X-direction to become the weak direction when increasing the height. The approximation in Eq. (37) will give a higher equivalent mass me compared with the simulated results. For the CLT building type me,app is between 14 and 22% higher than the calculated me. The difference
3.1.2. Building type with a post-and-beam structure The post-and-beam building type has columns in the perimeter of the façade and in the core. Beams are connected between the columns. The same floor elements were used as in the CLT building type. Columns and beams were placed to reduce the span of the floor 531
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Table 2 Results from FE simulations. Generated mode
Approximation
Calculated mode shape
Case
Lateral resisting system
h [m]
mtot [kg]
f1 [Hz]
M1 [kg]
Mode shape direction
me [kg/m]
apeak [m/s2]
me,app [kg/m]
apeak,app [m/s2]
ζ
me,ζ [kg/m]
apeak,ζ [m/s2]
me,app/me [%]
A B C D E F G H I J
CLT CLT CLT CLT CLT Truss Truss Truss Truss Truss
18 30 42 54 66 18 30 42 54 66
696,826 1,152,788 1,608,750 2,064,711 2,520,673 635,594 1,050,735 1,465,876 1,881,017 2,296,158
4.11 2.40 1.65 1.22 0.94 4.14 2.47 1.67 1.21 0.91
245,699 385,801 530,730 674,808 805,404 305,452 461,267 593,491 699,549 799,741
Y Y Y Y Y Y Y Y X X
33,610 31,625 31,658 32,766 33,486 39,509 38,458 37,849 37,497 37,143
0.018 0.042 0.066 0.090 0.120 0.017 0.034 0.052 0.080 0.111
38,713 38,426 38,304 38,235 38,192 35,311 35,025 34,902 34,834 34,790
0.016 0.034 0.055 0.078 0.106 0.019 0.037 0.056 0.086 0.118
0.81 0.81 0.84 0.89 0.94 0.83 0.85 0.92 1.02 1.09
34,782 33,133 33,332 34,210 34,689 36,652 36,456 36,363 36,262 35,896
0.0181 0.0403 0.0641 0.0886 0.1157 0.0169 0.0351 0.0577 0.0839 0.1151
1.15 1.22 1.21 1.17 1.14 0.89 0.91 0.92 0.93 0.94
the difference is almost zero, which results in the building behaving linearly and Eq. (37) would be appropriate. Models where the beam elements in the post-and-beam building type have been divided into smaller elements have been evaluated. The distribution of the mass will then not be lumped to each storey and the equivalent mass will be affected and become larger than the approximation. This is because the stiffness of the post-and-beam system is not reduced as for the CLT system and the building behaves more like a cantilever beam with nonlinear distribution of the mass. However, the effect of non-linear distribution of the mass decreases with increased height and the building tends to be a continuous system with linearly distributed mass. In Figs. 7 and 9, the simulated mode shapes from the FE model are presented. The mode shapes are normalised and scaled relative to each other. The amplitude is larger for the CLT building type for all cases. The mode shape shows linear behaviour for about the first 15 m and continues with more curvature. The slope of the amplitude is steeper the first 3 m for the post-and-beam building type. In cases E and J the
Fig. 6. Acceleration levels for simulated mode shape and approximation for the CLT building type.
decreases with increased height. The equivalent mass affects the acceleration level, higher equivalent mass reduces the acceleration level. In Fig. 6, the peak acceleration on the top floor is compared to the comfort criteria in ISO 10137 [2] for cases A-E. The comfort limit for residences is represented by the solid line and for offices by the dashed line. Acceleration levels from a simulated mode shape are represented by a circle and from the approximation by a square. The approximation gives a lower acceleration compared with the simulated mode. The acceleration levels using approximated values can give an acceleration level that satisfies the comfort criterion, whilst the simulated results shows that the criterion is not satisfied. For the post-and-beam building type there are small differences between me and me,app. The concentrated mass contributes to a higher equivalent mass since the mass becomes lumped at each storey, Table 2, cases F-J. The difference between approximated and generated equivalent mass are between 6 and 11%, which decreases with increased height. The mass of the CLT building type will vary along the height of the building while for the post-and-beam building type the mass will be lumped on each floor. Openings for doors and windows in the CLT building type will reduce the stiffness and the mass of the building, which in turn affects the modal mass and amplitude. The openings affect the stiffness more than the mass. Buildings with a height of 78–114 m have been evaluated for the CLT building type. The difference between me and me,app decreases with height. At a height of 81 m
Fig. 7. Normalised mode shape, cases A and F. 532
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Fig. 10. Mode shape function, Eurocode and generated, cases E and J.
Fig. 8. Mode shape function, Eurocode and generated, cases A and F.
slightly above the curve for = 1.0 and case J follows this tendency throughout the envelope. For case E it changes to be between the curves for = 1.0 and = 0.6 . By using the method of least squares the exponent, , has been calculated for all cases, Table 2. For the CLT building type, the exponent varies from 0.81 to 0.94 and for the postand-beam building type from 0.83 to 1.09. Using the calculated exponent factor with the associated modal mass will give an equivalent mass, me,ζ, which is around 5% higher than the generated equivalent mass me. This will affect the acceleration level, but less than the approximated equivalent mass, me,app, see Table 2. The equivalent mass will be overestimated, and the acceleration levels will be underestimated, which can give uncertain results when evaluating the comfort criteria. The stiffness of the two types of stabilising system is different, which is shown in Figs. 7 and 9 where the post-and-beam type has a lower amplitude compared with the CLT building type. Depending on the structural system the mode shape will vary, which is displayed in Figs. 8 and 10. The building tends to have a shear mode for shorter specimens and when the height is increased the building starts to deform in bending lower down and shifts over to shear higher up. The post-andbeam type displays more bending behaviour than the CLT building type for taller building specimens. The mode shape is a combination of bending and shear, Fig. 9. In Eq. (23) the constant = 1.5 is set to represent a slender building that will deform in bending, which timber buildings at these heights do not, Figs. 8 and 10. The post-and-beam building type with a height of 66 m have bending tendencies. For the CLT building type, only the first 15 m shows bending behaviour. The CLT and post-and-beam building types do not display this behaviour lower down, Fig. 8, while higher up, Fig. 10, the bending mode shape fits better especially for the post-and-beam building type. The CLT building type tends to have more of the shear mode shape, especially higher up in the structure. Since the post-and-beam type is stiffer (amplitude is smaller in Fig. 8), the bending mode shape will appear for shorter building specimens than compared with the CLT building type. The bending/shear stiffness ratio is different in the two building systems and will change with increasing building height. The assumed mode shape function in Eurocode 1-4 [3] does not represent the behaviour of the building and the equivalent mass will therefore be over- or underestimated if they are used and the simulated mode shape not considered. The acceleration levels on each storey are dependent on the mode shape. The approximation of the equivalent mass is a linear
Fig. 9. Normalised mode shape, cases E and J.
amplitude is quite similar at the top of the building. Mode shape functions from Eurocode 1-4 [3] with associated factors = 0.6 , = 1.0 and = 1.5 are presented in Figs. 8 and 10 together with the generated mode shape from the FE model. Case A in Fig. 8 starts with a curvature as = 1.5 but changes almost immediately towards = 1.0 and finally ends up with the curve for = 0.6 . Case F lies between = 1.0 and = 0.6 in the beginning and passes through the curve for = 0.6 . Finally, in Fig. 10 cases E and J are presented. Both cases E and J start 533
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have a higher natural frequency after completion as compared to simulated results. One reason could be that the building is stiffer when finished than in the simulation. Calibrated models are required to capture the phenomenon. The equation in Eurocode 1-4 [3] might be applicable to shorter timber buildings e.g. height < 25 m since most of the shorter buildings have a measured natural frequency above 2 Hz. If the building is stiffer when finished compared with the simulations, the natural frequency will increase, and the acceptable acceleration will decrease. However, the mass plays an important role in this. If the mass is too low, the acceleration levels will increase. If the natural frequency is below 1 Hz it is appropriate to increase the mass to reduce the acceleration and for a natural frequency above 2 Hz an increase in stiffness is more appropriate. In cases where the natural frequencies are below 1 Hz, which will be the case for taller timber buildings, additional mass can be used to decrease the acceleration level. Since additional mass decreases the natural frequency, and as seen in [13] where different formulae was suggested depending on material, it might be appropriate to have different formulae for hybrid structures and pure timber structures for estimating the natural frequency. Some of the empirical formulae were categorised based on stabilising system [17,15], but in Fig. 11 there is no clear difference between stabilising systems. If the equivalent mass is not handled in a proper way, the acceleration levels can be on the unsafe side independent of the calculated natural frequency. Since there are not so many tall timber buildings (built and measured) > 50 m where Eq. (24) is valid, it is difficult to judge the appropriateness of Eq. (24). The calculated acceleration levels are dependent on the damping ratio. Fig. 12 shows peak acceleration values, Eq. (35) depending on damping ratio, which forms part of the resonance factor R in Eq. (34). The damping ratio has been varied between 0 and 10% for case C. The relationship between peak acceleration level and damping ratio is nonlinear in Fig. 12. By changing the damping ratio from 1.5% to 3.0% the acceleration level decreases around 30% for a 14-storey building. Table 1 displays a large variation in damping ratio, which would have to be evaluated in each individual application.
Fig. 11. Empirical formulation together with measured, simulated and estimated natural frequencies.
assumption, which does not capture the reduction of stiffness due to openings in the walls. Higher up in the building, the post-and-beam building type, case J, shows a linear behaviour, Fig. 10, and a linear approximation is appropriate for the mode shape. In Fig. 11, the empirical Eqs. (24) and (33) are presented together with measured, simulated and estimated values from Table 1. The letters a and b stand for measured and simulated/estimated values respectively. Buildings that have been measured in two steps are labelled with 1, for measurements taken before partition walls, finishing etc. was added to the building. Most of the measured, simulated and estimated values in Fig. 11 have higher natural frequencies than generated by Eq. (24). The measured values follow the same trend as the presented empirical formulation. Buildings 11 and 10 in Fig. 11 have concrete slabs, which add mass to the building and decrease the natural frequency. Building 15 is a hybrid structure with a concrete core, which contributes to decreasing the natural frequency. For building 6 the frequency increased when partition walls, finishing etc. were added which indicates a stiffening of the structure. The mass of non-structural materials was not added in building 7, which would decrease the natural frequency if added. Some of the hybrid structures have higher natural frequencies than Eq. (33), e.g. 12 and 14. The numerical values derived in this paper for the CLT and the postand-beam building types show higher natural frequencies than measured ones, Fig. 11. The same results occur for simulated values in [27], building 8. It is difficult to specifically determine the cause since there are several parameters that affect the result and are connected to each other. The adoption of stiff connections for example between walls and pinned connections which only consider free rotation and do not take account translation stiffness in the connections may contribute to a stiffer building and thereby higher natural frequencies. In ISO 10137 [2], Fig. 6, acceptable acceleration levels vary depending on the natural frequency. The toughest acceleration requirements are between 1 and 2 Hz with an acceleration level of 0.04 m/s2. 1–2 Hz, which corresponds to a building height of 46 and 23 m for Eq. (24) (higher structures have lower frequencies). Eq. (33) will give a higher natural frequency than Eurocode 1-4 [3]. Eq. (24) is more conservative than Eq. (33) since decreased frequency results in increased acceleration. In that aspect Eq. (31) is most conservative of all the empirical equations. From Table 1 and Fig. 11 the conclusion is drawn that buildings
5. Conclusions Two timber building types have been evaluated in a parameter study, CLT and post-and-beam, regarding their dynamic properties. The use of empirical formulae for estimating the first natural frequency has
Fig. 12. Peak acceleration vs damping ratio. 534
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been discussed. The assumptions for tall buildings have been evaluated in the context of high-rise timber buildings. The main conclusions are:
Technologies, Sydney; 2016. [2] SS-ISO 10137, Bases for design of structures - Serviceability of buildings and walkways againts vibration, Stockholm: SIS Förlag AB; 2008. [3] SS-EN 1991-1-4. Actions on structures - Part1-4: General actions - Wind actions, Stockholm: SIS Förlag AB; 2005. [4] Chopra AK. Dynamics of structures: theory and applications to earthquake engineering. 4th ed. red., Upper Saddle River, N.J.: Prentice Hall; 2012. [5] Jeary A. Designeŕs guide to the dynamic response of structures. Hong Kong: E & FN Spon; 1997. [6] Malo KA, Abrahamsen RB, Bjertnæs MA. Some structural design issues of the 14storey timber framed building “Treet” in Norway. Eur J Wood Wood Prod 2016;74(3):407–24. [7] Fast P, Gafner B, Jackson R, Li J. Case study: An 18 storey tall mass timber hybrid student residence at the University of British Columbia, Vancouver. In: World conference of timber engineering, Vienna; 2016. [8] Reynolds T, Feldmann A, Ramage M, Chang W-S, Harris RDP. Design parameters for lateral vibration of multi-storey timber buildings. In: International network on timber engineering research, Graz; 2016. [9] Yang JN, Agrawall AK, Samali B, Wu JC. Benchmark problem for response control of wind-excited tall buildings. J Eng Mech 2004;130(4):437–46. [10] Huang S, Li Q, Xu S. Numerical evaluation of wind effects on a tall steel building by CFD. J Constr Steel Res 2007;63(5):612–27. [11] Blevins RD. Formulas for natural frequency and mode shape. New York: Van Nostrand Reinhold, cop; 1979. [12] Ellis B. An assessment of the accuracy of predicting the fundamental natural frequencies of buildings and the implications concerning the dynamic analysis of structures. Proc Inst Civ Eng 1980;69(3):763–76. [13] Lagomarsion S. Forecast models for damping and vibration periods of buildings. J Wind Eng Ind Aerodyn 1993;48(2–3):221–39. [14] Kim Y, Kanda J. Wind response characteristics for habitability of tall buildings in Japan. Struct Des Tall Spec Build 2008;17:683–718. [15] Hu L, Omeranovic A, Gagnon S, Mohammad M. Wind-induced vibration of tall wood buildings–is it an issue?. In: Wolrd conference on timber engineering, Quebec City, Canada; 2014. [16] AIJ, Damping in Buildings, Tokyo: Architectural Institution of Japan; 2000. [17] National Research Council of Canada, National Building Code of Canada, Ottawa: National Research Council of Canada; 2010. [18] BFS 2015:6-EKS 10. Boverkets konstruktionsregler, EKS 10 (Application of the European construction standards, EKS 10), Karlskrona; 2016. [19] ISO 6897. Guidelines for the evaluation of the response of occupants of fixed structures, especially buildings and off-shore structures, to low-frequency horizontal motion (0,063 to 1 Hz), International Organization for Standardization; 1984. [20] Edskär I, Lidelöw H. Wind-induced vibrations in timber buildings-parameter study of cross-laminated timber residential structures. Struct Eng Int 2017;27(2):205–16. [21] Ali MM, Moon KS. Structural developments in tall buildings: current trends and future prospects. Archit Sci Rev 2007;350:205–23. [22] Reynolds T, Harris R, Chang W-S, Bregulla J, Bawcome J. Ambient vibration tests of a cross-laminated timber building. Instituion of Civil Engineers; 2014. [23] BSV 97. Boverkets handbok om snö- vindlast (Snow load and wind action), Karlskrona; 1997. [24] Reynolds T, Bolmsvik Å, Vessby J, Chang W-S, Harris R, Bawcombe J, et al. Ambient vibration testing and modal analysis of multi-storey cross-laminated timber buldings. In: World conference on timber engineering, Quebec City; 2014. [25] Hu L, Karch E, Gagnon S, Dagenais C, Ramzi R. Dynamic performance measured on two 6-storey building made from wood structures before and after their completion and occupancy. In: World conference on timber engineering, Vienna; 2016. [26] Johansson M, Linderholt A, Jarnerö K, Landel P. Tall timber buildings - a preliminary study of wind-induced vibration of a 22-storey building. In: World conference on timber engineering, Vienna; 2016. [27] TräCentrumNorr (TCN), Innovativa höga träbaserade flerfamiljshus, Luleå; 2016. [28] Fjeld Olsen M, Hansen O. Measuring vibrations and assessing dynamic properties of tall timber buildings. [Master Thesis], NTNU- Norwegian University of Science and Technology, Trondheim; 2016. [29] Reynolds T, Casagrande D, Tomasi R. Comparison of multi-storey cross-laminated timber and timber frame buildings by in situ modal analysis. Constr Build Mater 2016;102:1009–17. [30] Feldmann A, Huang H, Chang W-S, Harris R, Dietsch P, Gräfeoch M, Hein C. Dynamic properties of tall timber structures under wind-induced vibration. In: World conference on timber engineering, Vienna; 2016. [31] SS-EN 14080, Timber structures – Glued laminated timber and glued solid, SIS Förlag ABl 2013. [32] Landel P, Linderholt A, Johansson M. Dynamical properties of a large glulam truss for a tall timber building. In: World conference on timber engineering, Seoul; 2018.
– The CLT and post-and-beam building types have similar first natural frequencies. The post-and-beam type showed a stiffer behaviour and more bending tendency than the CLT type where more shear behaviour was present. – In comparison, the post-and-beam building type has less actual weight m than the CLT building type. However, due to higher equivalent mass me which results in reduced acceleration levels, the post-and-beam system will provisionally be more useful if the goal is to construct tall timber buildings. – More measurements are needed to evaluate the appropriateness of equation f = 46/h. The theory indicates that some division according to stabilising system and material, pure timber buildings or hybrid, may be appropriate. Overall, empirical equations should be used with care since they rely on data with high variation and buildings that are not made of timber. – The assumption the mass is evenly distributed in tall buildings when calculating equivalent mass should be used with care for timber buildings. For a post-and-beam system the approximation can be appropriate but for a CLT system the approximation of the equivalent mass is not reliable and can give acceleration levels on the unsafe side. This becomes a problem because timber buildings fall short of the comfort criteria for acceleration at shorter building heights as compared to buildings of concrete or steel. Since the conclusions drawn here are based on FE simulations and the results are based on one certain type of floor layout and dimension of buildings generalisation is limited. 6. Further research The first natural frequencies of a building depend on several parameters, structural material (stiffness and mass), geometry of the building, stabilising system, position of the stabilising system etc. It is too early to draw any conclusions of the use of Eq. (24), more measurements are needed. Since openings reduce the stiffness of CLT panels and affect the dynamic properties the relation between stiffness and mass for timber buildings should be further explored. Declarations of interest None. Acknowledgements Fruitful discussions with Thomas Hallgren, COWI AB, Sweden and Magne Aanstad Bjertnæs SWECO AS, Norway are gratefully acknowledged. The authors would like to gratefully acknowledge SWECO Structures AB, Sweden for their financial support and interest. Finally, TräCentrum Norr support by VINNOVA (Sweden’s Innovation Agency) is acknowledged for financial support. References [1] Moore AJ. Dynamic characteristics and wind-induced response of a tall building. [Doctoral Thesis], University of Sydney, Faculty of Engineering and Information
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Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Dynamic properties of cross-laminated timber and timber truss building systems
T
Ida Edskär , Helena Lidelöw ⁎
Timber Structures, Luleå University of Technology, 971 87 Luleå, Sweden
ARTICLE INFO
ABSTRACT
Keywords: Serviceability limit state Tall buildings Wind-induced vibration Mode shape Natural frequency Modal analysis Modal mass
Incorrect prediction of dynamic behaviour of tall buildings can lead to discomfort for humans. It is therefore important to understand the dynamic characteristics such as natural frequency, mode shape, damping and the influence they have on acceleration levels. The aim of this study is to compare two timber building types, cross laminated panels and post-and-beam with stabilising diagonals, through a parameter study applying modal analysis. Empirical formulae for predicting the natural frequency and mode shape are compared to measured and numerical results. Tall building assumptions such as ‘line-like’ behaviour and lumped mass at certain points were evaluated for both systems. The post-and-beam system showed a stiffer behaviour than the cross laminated system where more shear deformation occurred. Empirical formulae should be used with care until more data is collected. For the post-and-beam systems an assumption of linearity may be appropriate, but for cross laminated systems the approximation can give results on the unsafe side. Finally, the relationship between stiffness and mass for cross laminated timber systems and its effect on dynamic properties needs to be further investigated.
1. Introduction Comfort evaluation is an important aspect when designing taller buildings, especially timber buildings due to the low self-weight compared to concrete and steel buildings. The dynamic characteristics of a tall building are the natural frequencies, the mode shape, and the damping ratio. Incorrect prediction of these dynamic characteristics can lead to undesirable behaviour of the structure when completed or unnecessary construction costs [1]. ISO 10137 [2] can be used to evaluate comfort criteria. The natural frequencies of a building can be predicted by theoretical, empirical or numerical methods. Each natural frequency has a corresponding mode shape that can be generated from a modal analysis or estimated using formulas in Eurocode 1-4 [3,4]. The mode shape can either be a bending or a shear mode but is often a combination of the two. Both the natural frequency and the mode shape affect the acceleration levels. When evaluating tall buildings, assumptions such as the building behaving ‘line-like’ and masses being lumped at certain points are made [5]. These assumptions may not be suitable for high-rise timber buildings. The aim of this research is to elucidate the dynamic behaviour of two building types, cross-laminated timber panels and timber post-and-beam with trusses, through a parameter study applying modal analysis. Simulated results are compared with measurements of existing timber buildings. The empirical formula to
⁎
estimate the first natural frequency from Eurocode 1-4 [3] is challenged and the mode shape is analysed. Eurocode 1-4 [3] presents f = 46/ h as an empirical expression to estimate the first natural frequency of buildings. Depending on the geometry of the building, different mode shapes result. Shorter buildings deform predominantly in a shear mode, while taller buildings are slender enough to deform in bending. Eurocode 1-4 [3] recommends several expressions for engineers when estimating the mode shape. Through generating the mode shape from modal analysis, conclusions are drawn on the applicability of Eurocode 1-4 [3] formulas and tall building assumptions [5] are evaluated for high-rise timber buildings. 2. Theory In the serviceability limit state, large deformations and uncomfortable vibrations should be limited to design a functional building. Due to the low mass of timber buildings, wind induced vibrations may be an issue. Two of the tallest (2018) residential timber buildings are “Treet”, Norway with a height of 49 m and “Brock Commons”, Canada with a reported height of 53 m [6,7]. “Treet” has timber truss systems to stabilise the structure and additional concrete slabs to increase the mass [6]. “Brock Commons” is a hybrid system where two concrete cores are used to stabilise the structure, timber
Corresponding author. E-mail address: [email protected] (I. Edskär).
https://doi.org/10.1016/j.engstruct.2019.01.136 Received 6 July 2018; Received in revised form 23 December 2018; Accepted 30 January 2019 0141-0296/ © 2019 Elsevier Ltd. All rights reserved.
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I. Edskär and H. Lidelöw
columns are used for the vertical load-carrying system and CLT slabs with concrete topping are used as floor elements [7]. “Treet” has a total mass of 135 kg/m3 [8], compared to tall concrete and steel buildings where the total mass is 300 kg/m3 and 160 kg/m3 respectively [9,10].
where M and K are the N × N matrices of mass and stiffness and u denotes the generalised displacement:
u(t ) = qn (t )
n is the deflected shape and qn (t ) can be described by a simple harmonic function, which gives the following algebraic equation also known as the eigenvalue problem:
2.1. Dynamic properties Tall buildings can be modelled as a vibrating uniform cantilever beam, fixed to the foundation [4,5]. To predict the natural frequency of a building theoretical models, empirical formulae, or numerical methods can be used.
fi =
2 L2
EI m
i = 1, 2, 3,
,n
i
G
2 L
i = 1, 2, 3,
,n
= (2i
2 n M]
det[K 2 n
n
The solution will give the eigenvalue of squared natural frequencies and the corresponding eigenvector n to each eigenvalue: 1
=
2
iz L
cos
iz L
i
sinh
iz L
sin
iz L
(10)
where L is the length of the beam/height of the building, z varies along the building height and for the first mode 1 = 1.875, 1 = 0.734 . Eq. (11) is for the first shear mode:
(1)
(2
(z ) =
1) z
(11)
L
These two equations represent pure bending mode and pure shear mode [11]. The eigenvector is normalised to make the amplitude unique, n . The mode can be scaled so the eigenvector ( N )n = 1 at the specified coordinate N , the mode can be scaled at the coordinate N where the mode has it maximum displacement, or the mode is scaled so the generalised mass or modal mass, Mn , has a specified value. The modal mass is defined by:
(2)
T n M n
(12)
Mn = 1 is often used and since the product of has the unit mass entails Mn = 1 kg . The generalised stiffness or modal stiffness for the nth mode is defined by: T n M n
(3)
T n K n
Kn =
(13)
For the nth mode and multiplied by T n K n
=
2 T n ( n M n)
T n
Eq. (7) can be written as: (14)
The squared natural frequencies can then be related to the modal mass and modal stiffness by: 2 n
=
Kn Mn
(15)
Each resonance mode have an associated modal mass and modal stiffness [5]. For a free vibration continuous system the modal mass is defined by:
(4)
where f f is the fundamental frequencies from the flexural beam model and fs from the shear beam model. Tall buildings consist of several elements and can be very complex in structure therefore finite element (FE) software is useful to solve the multiple degree of freedom system. Modal analysis is used to solve the free-vibration equation of motion [4], expressed as:
Mu¨ + Ku= 0
,N (9)
(z ) = cosh
and is the shear coefficient, G is the shear modulus, and is the mass density. The flexural beam model relates the frequency as proportional to 1/L2 and in the shear beam model to 1/L. The Timoshenko beam model considers, except from flexural and shear deformation also the effect of rotary inertia. The introduction of the shear deformation and rotary inertia reduces the natural frequencies compared to using only the flexural beam theory. However, the effect of rotary inertia is smaller than the effect of shear deformations [11]. There is no general closed form solution for the frequency when combining flexural and shear deformations, but Dunkerleýs approximation can be used which estimates a lower bound of the natural frequency [11] expressed as:
1 1 1 = 2+ 2+ f2 ff fs
n = 1, 2,
n is the number of modes. The eigenvector is also known as the natural mode and represents the mode shape. Depending on the structural system the mode shape will be a bending mode, a shear mode or a combination of bending and shear. The bending mode for a cantilever beam is presented in Eq. (10):
Mn = 1)
(8)
=0
N
where i
(7)
=0
Eq. (7) has a nontrivial solution if
where L is the length of the beam, E is the modulus of elasticity, I is the moment of inertia, m is the mass per unit length and subscript i represents the mode of vibration. For the first natural frequency 1 = 1.875 [11]. For slender beams the flexural deformations will dominate, but for less slender beams shear deformation may become important and needs to be considered. Eq. (2) estimates the natural frequencies of a fixed uniform cantilever beam, considering shear deformation only [11]:
fi =
2 n M] n
[K
2.1.1. Theoretical and numerical models For simple structures, a model based on the theory of an idealised cantilever beam with simple boundary conditions can be used. The dynamic system can be represented by either a lumped or a distributed mass system. Lumped masses are connected by elements (walls, columns) with a certain stiffness and damping. To solve the system and determine the natural frequencies, modal analysis can be used. The system can be scaled and the dynamic behaviour is described by second order differential equations and results in a finite number of natural frequencies. The classic Euler-Bernoulli beam model estimates the natural frequency of beams. The theory only considers flexural deformations of a slender beam with linear elastic, homogeneous material. Eq. (1) estimates the natural frequencies of a fixed uniform cantilever beam [11]: 2 i
(6)
n
h
Mn =
m (z ) 0
2 n dz
(16)
An assumption is made that the dimensions in X and Y directions, Fig. 5, are constant and do not vary along the building. If the building is irregular it needs to be considered in Eq. (16). The modal mass can be generated from the FE software directly or be calculated from the
(5) 526
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(z ) =
z h
(23)
= 0.6 for slender frame structures with non load-distributing walls or cladding. = 1.0 for buildings with a central core plus peripheral columns or larger columns plus shear bracings. = 1.5 for slender cantilever buildings and buildings supported by central reinforced concrete cores. Mode functions with different value are displayed in Fig. 1. = 1.5 corresponds to a bending mode shape, while = 1.0 is a linear relationship. When solving the eigenvalue problem numerically in FE software, the eigenvector for each natural frequency will be generated and thereby the mode shape is captured. 2.1.2. Empirical formulae Several empirical formulae to estimate the first natural frequency have been suggested by researchers [12–15]. In Eurocode 1-4 [3] the fundamental frequency f can be estimated by Eq. (24) for multi-storey buildings with a height larger than 50 m.
f=
eigenvector. Values of the eigenvector can be defined by taking points at certain levels of the building where the maximum amplitude is generated, N . The modal mass can then be calculated using:
1 2 N
(17)
When dealing with ‘tall’ buildings, the building is considered slender and some assumptions are often made. The building is considered to be ‘line-like’/linear and with a constant mass over the height, but often the mass decreases with height since smaller dimensions are needed to carry the weight. Another assumption is that the mass is lumped in certain points and a normalised amplitude is used. The mode shape can for tall buildings normally be represented by a straight line, Eq. (18):
(z ) =
z h
f=
Mn =
50 h
(25)
where h is the height of the structure. In [13] data was also analysed with respect to structural material, Eqs. (26)–(28):
f=
45 h
steel buildings
(26)
f=
55 h
reinforced concrete buildings
(27)
f=
57 h
hybrid buildings
(28)
(18)
The mode shape represented by a straight line is presented in Fig. 1 with = 1. If using the assumption for tall buildings and inserting Eq. (18) into Eq. (16) the modal mass will be:
m (z ) h3
(24)
where h is the height of the structure in meters. The formula is based on a collection of 163 rectangular plan buildings with different types of materials; concrete, steel, and mixed [12]. To estimate the first natural frequency, different predictors were studied and compared to the measured frequency. The height was a better predictor than including the width of the building in the formula. Eq. (24) has a correlation coefficient of R2 = 0.88. By looking at the frequency plotted against height in [12], one finds that the largest scatter of data exists for shorter buildings. However, there are more data points for shorter buildings than taller ones. From a collection of 185 different buildings (steel, reinforced concrete, mixed, pre-cast, masonry, unknown), the following equation to predict the natural first natural frequency was proposed [13]:
Fig. 1. Mode shape.
Mn =
46 h
(19)
Two equations have been proposed by AIJ [16] in [14] for estimating the first natural frequency, one for reinforced concrete buildings and one for steel buildings:
(20)
f = 67/ h
reinforced concrete buildings
(29)
If m (z ) h is assumed to be the total mass of the building, Mtot the modal mass will be:
f = 50/ h
steel buildings
(30)
3h2
And for a pure shear mode Eq. (16) will be:
Mn =
m (z ) h 2
M Mn = tot 3
where h is the height of the structure. The equations are used for comfort assessment and have been derived from 68 reinforced concrete buildings and 137 steel buildings in Japan. In the National Building Code of Canada [17] in [15], several empirical formulae are given to calculate the first natural frequency. [15] stated that the following equations could be used to predict the natural frequencies for a timber building:
(21)
and for shear:
Mn =
Mtot 2
(22)
Depending on the mode shape the modal mass will vary. Eq. (21)) is often used as an approximation for tall buildings with a constant mass per unit height [5]. As seen in Eqs. (16), (19) and (20) the modal mass is dependent on the mass distribution over the height, z . In Eurocode 1-4 [3], an expression for the mode shape of different types of buildings is stated:
f = 40/ h f=
20 h3/4
braced frames
(31)
shear walls and other structures
(32)
Eq. (32) suggests a relationship between f and h that is not exactly inversely proportional. Measurements of timber buildings have been 527
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apeak =
(35)
x (z ) k p
The equivalent mass per unit length is given by Eq. (36): h
m (z )·
0
me =
h 0
me, app = collected in [8], and dynamic tests (ambient vibration tests) have been carried out on 11 timber buildings. Some of the buildings have the first floor in concrete and some are composite timber-concrete or timbersteel, but the main vertical and lateral load-carrying system is of timber, which includes cross-laminated timber, glulam and light timber frames. Based on the collected data a single relationship between height and natural frequency was found for timber buildings Eq. (33): (33)
2.2. Acceleration and comfort
=
According to Eurocode 1-4 [3], a maximum horizontal displacement and a characteristic standard deviation of acceleration along wind at building height z, x (z ),should be satisfied. Acceptable comfort levels may be found in the standard ISO 10137 [2]. To evaluate the comfort level, the acceleration and first natural frequency need to be calculated. ISO 10137 [2] is based on a 1-year return period of wind and peak acceleration. The return period can either be handled by using a 2-year return period for the wind velocity or apply the acceleration level as 0.72 times the acceleration level for a 5-year return period [18,19]. The results will be the same. Equations for the standard deviation of acceleration may be found in Eurocode 1-4 [3]. The equation used in this paper is from the national annex in Sweden EKS 10 [18] where the standard deviation of acceleration along wind is given by:
(z ) =
3· Iv (h)·R· qm (h)·b ·cf · me
1, x (z )
(36)
mtot h
(37)
Inserting Eqs. (18) and (21) into (36) will give this approximation. The choice of mode shape will not affect the equivalent mass. Depending on structural systems, materials, connections etc. the damping in the building will vary. The magnitude of motion will be reduced more quickly in a building with high damping. Damping of a certain building can be measured once the building is completed. It is difficult to predict the damping ratio before the building is built. There are two types of damping systems, passive and active. Passive damping systems has a fixed property related to chosen layout and structural materials, while active dampers modify the system properties and realise an active mechanism. Damping of the structural system itself is a passive damper [21]. The damping ratio can be non-linear depending on the amplitude of the motion [22] c.f. Results and Analysis. The damping properties are included in the resonance factor R in Eq. (34). Both structural damping, aerodynamic damping and damping from special advices are included. The total logarithmic decrement for damping is given by:
It should be noted that timber buildings have different stabilising systems, which may behave differently under wind-induced vibrations. In Fig. 2 Eqs. (24), (27), (29), (31)–(33) are plotted. Eqs. (29) and (31) constitute the lower and higher bounds. Studying Fig. 2, the different empirical formulae display the same tendency, but with considerable spread.
x
2 1 (z ) dz
h is the height of the structure, m (z ) is the mass per unit length, and 1is the first eigenmode. The numerator in Eq. (36) is the modal mass. During calculations in FE software the eigenvector is normalised as in Eq. (12). Two alternative methods to arrive at the modal mass exist: either the modal mass in Eq. (36) is set to 1 and the eigenvector is normalised, or the software can produce the value of the modal mass. The denominator is the integral of the mode shape over the height of the building where the amplitude has its maximum. The number of integration points can be set to the mesh size over the height. Integration intervals can alternatively be set to the storey height and intervals can be defined floor to floor or centre of storey to centre of storey. The chosen method will give slightly different equivalent masses in the end, which may affect the acceleration. The equivalent mass can approximately be taken as the total mass of the building divided by the total height:
Fig. 2. Empirical formulae.
f = 55/ h
2 1 (z ) dz
s
+
a
+
d
(38)
s is the logarithmic decrement of structural damping, a is the logarithmic decrement of aerodynamic damping and d is the logarithmic decrement due to special devices e.g. tuned mass dampers, sloshing thanks [3]. In the old Swedish handbook “Snow load and wind action” [23] values on the logarithmic decrement for timber structures with and without mechanical connections are presented. Structural damping is stated as s = 0.06 for a timber structure without mechanical connections and s = 0.09 for a timber structure with mechanical connections. The values correspond to about 1.0% and 1.5% critical damping. In Eurocode 1-4 [3] a value of s = 0.06 0.12 is tabulated for timber bridges, which corresponds to 1.0–1.9% critical damping. In Table 1, dynamic properties are presented for measured buildings and values from numerical analyses. Critical damping values are reported between 1.3 and 9.1% for buildings with cross laminated timber as the main load bearing structure, 1.4–2.4% for post-and-beam systems and 1.0–6.9% for hybrid structures. For steel and reinforced concrete structures, the critical damping, tabulated in Eurocode 1-4 [3], are 0.8% and 1.6% respectively. The presented values in Eurocode 1-4 [3] are not related to the amplitude or height of the building but from measured timber buildings the damping ratio varies with the amplitude and may have a non-linear behaviour [22]. The damping ratio tends to decrease with increased height [8].
(34)
Iv (h) is the turbulence intensity at height h , R is the square root of the resonant response, qm (h) is the mean pressure at height h , b is the width of the structure, cf is the force coefficient, 1, x (z ) is the mode shape value at height z , me is the equivalent mass per unit length, z is the height above the ground, and h is the height of the structure. The standard deviation of the acceleration is multiplied with a peak factor to get the peak acceleration [3,18,20]. 528
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Table 1 Summary of first natural frequencies and damping ratio for timber buildings. Number in Fig. 11
Building type
Height [m]
Measured
Simulated/estimated
f1 [Hz]
ξ1 [%]
CLT 1 2 3 3 4 5 6 6 7 8
CLT CLT, post-and-beam CLT CLT + 50 mm cement screed on the floor CLT, timber frame, first storey in concrete CLT, first storey concrete CLT core, post-and-beam CLT core, post-and-beam CLT core, post-and-beam CLT
9.1 12.6 21 21 25 27 29.5 29.5 74.8 31–68.2
3.9 3.2 2.7 2.45 2.28 2.26 1.1 1.6
5.4 1.3 3.20–5.60 5.20–9.10 2.3 1.9 3 2
Post-and-beam 9 10 11
Post-and-beam Post-and-beam, concrete slab every fourth storey Post-and-beam, concrete slab in the upper part
18.3 45 81
2.8 1.02–1.03
1.4 1.6–2.4
Hybrid 12 13 14 14 15 16
CLT, concrete core Post-and-beam with concrete Post-and-beam with concrete Post-and-beam with concrete Solid timber, concrete core Post-and-beam with concrete
15.6 21.8 22.1 22.1 23.9 53
4.03–4.08 2.1 2.7 2.8 2.33–2.34
4.9–6.9 1.5 1 4 1.82–2.90
a b c
core core and shear walls core and shear walls core, CLT
f1 [Hz]
Reference
ξ1 [%]
0.6 2.78–0.97
1.5 1.5
[15] [15]a [22]a [22] [24] [24] [25]a [25] [26]b [27]
0.75 0.33
1.9 1.9
[15] [8,6,28] []c
1.5
[29] [15]a [25]a [25] [30] [7]
2.58 2.05 1.94
0.5
Without non-structural elements (partition wall, finishing, etc.). Mass from non-structural material was not included. Magne Aanstad Bjertnæs, Sweco Norway AS, personal communication.
2.3. Dynamic properties of timber buildings The number of dynamically tested timber buildings is growing. In the following Table 1, a summary of measured, numerically calculated and/or estimated (theoretical or empirical models) timber buildings are listed. In Fig. 11, the values are plotted. The buildings have been divided in three types CLT, Post-and-beam, and Hybrid. For CLT the main stabilisation is through shear wall elements. For Post-and-beam, a bracing/truss system with diagonals in timber is used for stabilisation. In the Hybrid type, most of the buildings have concrete parts e.g. a core or shear walls in concrete for the main stabilising system and timber elements for vertical loads and load distribution. To measure dynamic properties ambient vibration tests (AVT) have been made in most cases, but also a few forced vibration tests (FVT) have been carried out [15,22,24,25,28–30]. For simulated cases, FE software and modal analysis have been used [7,26,27]. Flexural beam models have been used in theoretical estimations [22,24]. In [22,25] measurements have been taken during construction so the effect of non-structural elements such as partition walls, finishing etc. has been evaluated. The natural frequencies decreased when cement screed was added in [22] and for [25] the natural frequencies increased when partition walls, finishing etc. were added. The frequency of the measured buildings is 1.02–4.08 Hz with a variation in height of 9.1 m up to 53 m. Simulated values are between 0.33 Hz and 2.78 Hz with a height variation between 31 and 81 m.
Fig. 3. Effect of changing parameters.
also shown that the mode shape changes when parameters are varied. In [20] the shear stiffness was attributed too high a value. For a 10storey building, the natural frequency, equivalent mass and peak acceleration on top floors were calculated to f = 3.31 Hz, me = 27,579 kg/m and apeak = 0.0326 m/s2 respectively. With the correct shear stiffness and thicker wall and floor elements, calculations resulted in f = 2.40 Hz, me = 31,132 kg/m and apeak = 0.0443 m/s2. The mode shape changed from bending to bending/shear.
2.3.1. Previous study In a previous study [20], different parameters were studied to predict the behaviour of a CLT structure under wind-induced vibration, Fig. 3. By changing the stiffness, the first natural frequency increases and the acceleration levels decrease. The stiffness can be changed by changing the material properties, E- and G-modulus, or by changing for example the geometry of the building/elements, floor layout, placement of lateral stabilising system, or thickness of elements. Additional mass will decrease both the natural frequencies and the acceleration. An increased damping ratio will reduce the acceleration level. It was
3. Method In this paper the same floor layout has been used as in [20]. The 529
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Fig. 4. Mode shapes for (a) the post-and-beam structure and (b) the CLT structure.
Fig. 5. Mode shapes, the dashed lines shows the mode shape: (a) translation in Y-direction, (b) translation in X-direction, and (c) Torsional. (Units [–]).
floor layout is 20 × 20 m2 with a core placed in the middle for elevator and staircase. There are four apartments on each storey connected to the core. Two building types have been evaluated, one with cross
laminated timber and one with post-and-beam elements with diagonal bracing. In Figs. 4 and 5, the floor layout and the placement of structural elements is presented for the post-and-beam system. The building 530
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types have been evaluated in ultimate limit state on a 14-storey high (42 m) building to get a sense of the approximate dimensions on elements. The utilisation is between 65 and 90%. Connections and openings have not been checked in the model. The building types have been studied for a height of 18–66 m, which corresponds to 6–22 storeys. A step of 4 storeys was used.
element. A diagonal bracing system in the façade was used to stabilise the building type. The diagonals span over two storeys, Fig. 4. For the columns a dimension of 450 × 450 mm was used. For the beams 200 × 400 mm and for the diagonals 300 × 450 mm were used. Grade GL30h was assumed for all columns and beams. The post-and-beam building type was assumed to have the same insulation, gypsum, surface finishing etc. as the CLT building type. The difference between the systems is only the load bearing structure and the stabilising system. The posts and beams were modelled as beam elements with six degrees of freedom in each end. The columns are continuous through the building and fixed to the foundation. The diagonals and the beams are pinned to the columns.
3.1. FE-Modelling To calculate the natural frequency and the mode shape of the structures commercial FE software (Autodesk ® Robot™ Structural Analysis) was used to perform the modal analysis. In the FE model, only the structural elements were modelled such as cross laminated timber walls, floors, columns, beams, and diagonals. All non-structural materials, studs, insulation, gypsum boards and so on, were modelled as masses. The masses were applied as uniformed loads (kN/m2 or kN/m) and converted to masses when running the modal analysis. The Z-direction for the mass was ignored in the analysis to avoid of local vibrations in floors; X and Y directions (transversal vibrations) were set to active, Figs. 4 and 5. The subspace iteration method with a consistent mass matrix was used in the analysis. A value of 1.5% critical damping was assumed.
3.2. Acceleration, mass and mode shape By using Eq. (34), the standard deviation of acceleration was calculated. The equation gives an r.m.s value of the acceleration. The wind velocity was set to 24 m/s and terrain type III was used [3,18]. The wind velocity was reformulated from a 50 year return period to a 5 year return period according to [18]. The r.m.s acceleration with a 5 year return period of the wind velocity was then multiplied with 0.72 to represent a 1 year return period. The peak acceleration was calculated by multiplying the r.m.s acceleration value with a peak-factor, Eq. (35), which can be found in [18]. By multiplying the peak acceleration level with the mode shape, acceleration levels for each floor is established. The peak acceleration on the top floor was used for evaluation. The equivalent mass was calculated according to Eq. (36) where the numerator, modal mass, was generated from the FE software. The denominator, the integral of the eigenvector, was calculated with values from the FE software where the amplitude has its maximum. For the CLT building type values of the amplitude has been collected in each node along the height with a distance of 0.4–0.75 m depending on the case. For the post-and-beam building type, values at each storey has been collected since the beam element extends the whole storey height of 3 m. Values of the eigenvector in the FE model were taken on each storey where the amplitude peaked.
3.1.1. Building type with a CLT structure The CLT building type consists of external and internal walls, shafts and floors of CLT elements as the load bearing structure. The building type is stabilised through shear walls and diaphragm action in the floor elements. Post-and-beams have been used due to large spans of the floor elements [20]. The material grade was GL30h [31]. The thickness of the wall is 200 mm and it consists of five layers. The floor element is made of seven layers where the outmost parts has two parallel layers to increase the stiffness in the stiff direction. The thickness of the floor is 280 mm. Both the wall and the floor is made of graded C24 in all layers. The density was taken as 400 kg/m3 for CLT and 430 kg/m3 for the post-and-beam system. The wall and floor elements are completed with insulation, gypsum, surface finish etc. In [20] a detailed description can be found of the wall and floor build-up. Calculation of the stiffness of the elements can be done either by using the supplier’s data or handbooks on CLT. The only difference is how the layers in the CLT are built up. In [20] material properties were used from a supplier of CLT elements. In this paper, the thickness of the walls and floors were changed, and the shear stiffness was corrected to enable proper comparison with results generated in this paper. The elements were modelled as fournode quadrilateral shell elements and the stiffness matrix was calculated and manually defined in the software. Since the stiffness matrix is defined manually, the orthotropic properties are handled properly. Connections between CLT elements were modelled as fixed except the connection between wall and floor elements that was assumed to be pinned. The walls were assumed to be continuous through the building and fixed to the ground. In [20] pinned connections between walls and foundations and between walls were tested to compare with fixed connections. No differences in global results such as mode shape or acceleration were observed in the modal analysis. In [6] the stiffness of connections in a truss system was varied. Rotational stiffness decrease did not have any effect on the mode shape, while an axial connection stiffness lower than 25% of the member stiffness affected the mode shape and increased the acceleration. [32] report axial connection stiffnesses of 67% of the axial member stiffness. Assuming fixed connections is thus deemed appropriate when evaluating global dynamic behaviour.
4. Results and analysis In Table 2 the height of the building, h, the first natural frequency, f1, the mode shape direction, the modal mass, M1, the equivalent mass, me, peak acceleration on the top floor, apeak and calculated mode shape factor are presented for cases A to J. The equivalent mass and peak acceleration have been calculated from the generated mode shape, using the approximation in Eq. (37) and using the calculated mode shape factor, . Cases A to E represent the CLT building type and cases F to J the post-and-beam building type with diagonals. Each storey in the CLT building type weighs m = 113,990 kg and the roof storey weighs 126,873 kg where 38% of the total mass originates from non-structural elements. For the post-and-beam building type each storey weighs m = 103,785 kg and the roof storey 116,668 kg where 41% is mass from non-structural elements. The CLT building type weighs around 10% more than the post-and-beam building type. The total mass/m3 is 96 kg/m3 which is almost 1/3 compared to the concrete building in [9]. For all simulated cases, the first two modes are orthogonal translation modes in either X or Y-direction and the third a rotational mode, Fig. 5. The first natural frequency varies between 0.91 and 4.14 Hz. The CLT and post-and-beam building types have natural frequencies in the same range. The first mode shape direction is in the Y-direction, Fig. 4, except for cases I and J where the first mode shape direction changes to the Xdirection. The change of direction may be due to the slight asymmetry of the building floor plan layout rendering the X-direction to become the weak direction when increasing the height. The approximation in Eq. (37) will give a higher equivalent mass me compared with the simulated results. For the CLT building type me,app is between 14 and 22% higher than the calculated me. The difference
3.1.2. Building type with a post-and-beam structure The post-and-beam building type has columns in the perimeter of the façade and in the core. Beams are connected between the columns. The same floor elements were used as in the CLT building type. Columns and beams were placed to reduce the span of the floor 531
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Table 2 Results from FE simulations. Generated mode
Approximation
Calculated mode shape
Case
Lateral resisting system
h [m]
mtot [kg]
f1 [Hz]
M1 [kg]
Mode shape direction
me [kg/m]
apeak [m/s2]
me,app [kg/m]
apeak,app [m/s2]
ζ
me,ζ [kg/m]
apeak,ζ [m/s2]
me,app/me [%]
A B C D E F G H I J
CLT CLT CLT CLT CLT Truss Truss Truss Truss Truss
18 30 42 54 66 18 30 42 54 66
696,826 1,152,788 1,608,750 2,064,711 2,520,673 635,594 1,050,735 1,465,876 1,881,017 2,296,158
4.11 2.40 1.65 1.22 0.94 4.14 2.47 1.67 1.21 0.91
245,699 385,801 530,730 674,808 805,404 305,452 461,267 593,491 699,549 799,741
Y Y Y Y Y Y Y Y X X
33,610 31,625 31,658 32,766 33,486 39,509 38,458 37,849 37,497 37,143
0.018 0.042 0.066 0.090 0.120 0.017 0.034 0.052 0.080 0.111
38,713 38,426 38,304 38,235 38,192 35,311 35,025 34,902 34,834 34,790
0.016 0.034 0.055 0.078 0.106 0.019 0.037 0.056 0.086 0.118
0.81 0.81 0.84 0.89 0.94 0.83 0.85 0.92 1.02 1.09
34,782 33,133 33,332 34,210 34,689 36,652 36,456 36,363 36,262 35,896
0.0181 0.0403 0.0641 0.0886 0.1157 0.0169 0.0351 0.0577 0.0839 0.1151
1.15 1.22 1.21 1.17 1.14 0.89 0.91 0.92 0.93 0.94
the difference is almost zero, which results in the building behaving linearly and Eq. (37) would be appropriate. Models where the beam elements in the post-and-beam building type have been divided into smaller elements have been evaluated. The distribution of the mass will then not be lumped to each storey and the equivalent mass will be affected and become larger than the approximation. This is because the stiffness of the post-and-beam system is not reduced as for the CLT system and the building behaves more like a cantilever beam with nonlinear distribution of the mass. However, the effect of non-linear distribution of the mass decreases with increased height and the building tends to be a continuous system with linearly distributed mass. In Figs. 7 and 9, the simulated mode shapes from the FE model are presented. The mode shapes are normalised and scaled relative to each other. The amplitude is larger for the CLT building type for all cases. The mode shape shows linear behaviour for about the first 15 m and continues with more curvature. The slope of the amplitude is steeper the first 3 m for the post-and-beam building type. In cases E and J the
Fig. 6. Acceleration levels for simulated mode shape and approximation for the CLT building type.
decreases with increased height. The equivalent mass affects the acceleration level, higher equivalent mass reduces the acceleration level. In Fig. 6, the peak acceleration on the top floor is compared to the comfort criteria in ISO 10137 [2] for cases A-E. The comfort limit for residences is represented by the solid line and for offices by the dashed line. Acceleration levels from a simulated mode shape are represented by a circle and from the approximation by a square. The approximation gives a lower acceleration compared with the simulated mode. The acceleration levels using approximated values can give an acceleration level that satisfies the comfort criterion, whilst the simulated results shows that the criterion is not satisfied. For the post-and-beam building type there are small differences between me and me,app. The concentrated mass contributes to a higher equivalent mass since the mass becomes lumped at each storey, Table 2, cases F-J. The difference between approximated and generated equivalent mass are between 6 and 11%, which decreases with increased height. The mass of the CLT building type will vary along the height of the building while for the post-and-beam building type the mass will be lumped on each floor. Openings for doors and windows in the CLT building type will reduce the stiffness and the mass of the building, which in turn affects the modal mass and amplitude. The openings affect the stiffness more than the mass. Buildings with a height of 78–114 m have been evaluated for the CLT building type. The difference between me and me,app decreases with height. At a height of 81 m
Fig. 7. Normalised mode shape, cases A and F. 532
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Fig. 10. Mode shape function, Eurocode and generated, cases E and J.
Fig. 8. Mode shape function, Eurocode and generated, cases A and F.
slightly above the curve for = 1.0 and case J follows this tendency throughout the envelope. For case E it changes to be between the curves for = 1.0 and = 0.6 . By using the method of least squares the exponent, , has been calculated for all cases, Table 2. For the CLT building type, the exponent varies from 0.81 to 0.94 and for the postand-beam building type from 0.83 to 1.09. Using the calculated exponent factor with the associated modal mass will give an equivalent mass, me,ζ, which is around 5% higher than the generated equivalent mass me. This will affect the acceleration level, but less than the approximated equivalent mass, me,app, see Table 2. The equivalent mass will be overestimated, and the acceleration levels will be underestimated, which can give uncertain results when evaluating the comfort criteria. The stiffness of the two types of stabilising system is different, which is shown in Figs. 7 and 9 where the post-and-beam type has a lower amplitude compared with the CLT building type. Depending on the structural system the mode shape will vary, which is displayed in Figs. 8 and 10. The building tends to have a shear mode for shorter specimens and when the height is increased the building starts to deform in bending lower down and shifts over to shear higher up. The post-andbeam type displays more bending behaviour than the CLT building type for taller building specimens. The mode shape is a combination of bending and shear, Fig. 9. In Eq. (23) the constant = 1.5 is set to represent a slender building that will deform in bending, which timber buildings at these heights do not, Figs. 8 and 10. The post-and-beam building type with a height of 66 m have bending tendencies. For the CLT building type, only the first 15 m shows bending behaviour. The CLT and post-and-beam building types do not display this behaviour lower down, Fig. 8, while higher up, Fig. 10, the bending mode shape fits better especially for the post-and-beam building type. The CLT building type tends to have more of the shear mode shape, especially higher up in the structure. Since the post-and-beam type is stiffer (amplitude is smaller in Fig. 8), the bending mode shape will appear for shorter building specimens than compared with the CLT building type. The bending/shear stiffness ratio is different in the two building systems and will change with increasing building height. The assumed mode shape function in Eurocode 1-4 [3] does not represent the behaviour of the building and the equivalent mass will therefore be over- or underestimated if they are used and the simulated mode shape not considered. The acceleration levels on each storey are dependent on the mode shape. The approximation of the equivalent mass is a linear
Fig. 9. Normalised mode shape, cases E and J.
amplitude is quite similar at the top of the building. Mode shape functions from Eurocode 1-4 [3] with associated factors = 0.6 , = 1.0 and = 1.5 are presented in Figs. 8 and 10 together with the generated mode shape from the FE model. Case A in Fig. 8 starts with a curvature as = 1.5 but changes almost immediately towards = 1.0 and finally ends up with the curve for = 0.6 . Case F lies between = 1.0 and = 0.6 in the beginning and passes through the curve for = 0.6 . Finally, in Fig. 10 cases E and J are presented. Both cases E and J start 533
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have a higher natural frequency after completion as compared to simulated results. One reason could be that the building is stiffer when finished than in the simulation. Calibrated models are required to capture the phenomenon. The equation in Eurocode 1-4 [3] might be applicable to shorter timber buildings e.g. height < 25 m since most of the shorter buildings have a measured natural frequency above 2 Hz. If the building is stiffer when finished compared with the simulations, the natural frequency will increase, and the acceptable acceleration will decrease. However, the mass plays an important role in this. If the mass is too low, the acceleration levels will increase. If the natural frequency is below 1 Hz it is appropriate to increase the mass to reduce the acceleration and for a natural frequency above 2 Hz an increase in stiffness is more appropriate. In cases where the natural frequencies are below 1 Hz, which will be the case for taller timber buildings, additional mass can be used to decrease the acceleration level. Since additional mass decreases the natural frequency, and as seen in [13] where different formulae was suggested depending on material, it might be appropriate to have different formulae for hybrid structures and pure timber structures for estimating the natural frequency. Some of the empirical formulae were categorised based on stabilising system [17,15], but in Fig. 11 there is no clear difference between stabilising systems. If the equivalent mass is not handled in a proper way, the acceleration levels can be on the unsafe side independent of the calculated natural frequency. Since there are not so many tall timber buildings (built and measured) > 50 m where Eq. (24) is valid, it is difficult to judge the appropriateness of Eq. (24). The calculated acceleration levels are dependent on the damping ratio. Fig. 12 shows peak acceleration values, Eq. (35) depending on damping ratio, which forms part of the resonance factor R in Eq. (34). The damping ratio has been varied between 0 and 10% for case C. The relationship between peak acceleration level and damping ratio is nonlinear in Fig. 12. By changing the damping ratio from 1.5% to 3.0% the acceleration level decreases around 30% for a 14-storey building. Table 1 displays a large variation in damping ratio, which would have to be evaluated in each individual application.
Fig. 11. Empirical formulation together with measured, simulated and estimated natural frequencies.
assumption, which does not capture the reduction of stiffness due to openings in the walls. Higher up in the building, the post-and-beam building type, case J, shows a linear behaviour, Fig. 10, and a linear approximation is appropriate for the mode shape. In Fig. 11, the empirical Eqs. (24) and (33) are presented together with measured, simulated and estimated values from Table 1. The letters a and b stand for measured and simulated/estimated values respectively. Buildings that have been measured in two steps are labelled with 1, for measurements taken before partition walls, finishing etc. was added to the building. Most of the measured, simulated and estimated values in Fig. 11 have higher natural frequencies than generated by Eq. (24). The measured values follow the same trend as the presented empirical formulation. Buildings 11 and 10 in Fig. 11 have concrete slabs, which add mass to the building and decrease the natural frequency. Building 15 is a hybrid structure with a concrete core, which contributes to decreasing the natural frequency. For building 6 the frequency increased when partition walls, finishing etc. were added which indicates a stiffening of the structure. The mass of non-structural materials was not added in building 7, which would decrease the natural frequency if added. Some of the hybrid structures have higher natural frequencies than Eq. (33), e.g. 12 and 14. The numerical values derived in this paper for the CLT and the postand-beam building types show higher natural frequencies than measured ones, Fig. 11. The same results occur for simulated values in [27], building 8. It is difficult to specifically determine the cause since there are several parameters that affect the result and are connected to each other. The adoption of stiff connections for example between walls and pinned connections which only consider free rotation and do not take account translation stiffness in the connections may contribute to a stiffer building and thereby higher natural frequencies. In ISO 10137 [2], Fig. 6, acceptable acceleration levels vary depending on the natural frequency. The toughest acceleration requirements are between 1 and 2 Hz with an acceleration level of 0.04 m/s2. 1–2 Hz, which corresponds to a building height of 46 and 23 m for Eq. (24) (higher structures have lower frequencies). Eq. (33) will give a higher natural frequency than Eurocode 1-4 [3]. Eq. (24) is more conservative than Eq. (33) since decreased frequency results in increased acceleration. In that aspect Eq. (31) is most conservative of all the empirical equations. From Table 1 and Fig. 11 the conclusion is drawn that buildings
5. Conclusions Two timber building types have been evaluated in a parameter study, CLT and post-and-beam, regarding their dynamic properties. The use of empirical formulae for estimating the first natural frequency has
Fig. 12. Peak acceleration vs damping ratio. 534
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been discussed. The assumptions for tall buildings have been evaluated in the context of high-rise timber buildings. The main conclusions are:
Technologies, Sydney; 2016. [2] SS-ISO 10137, Bases for design of structures - Serviceability of buildings and walkways againts vibration, Stockholm: SIS Förlag AB; 2008. [3] SS-EN 1991-1-4. Actions on structures - Part1-4: General actions - Wind actions, Stockholm: SIS Förlag AB; 2005. [4] Chopra AK. Dynamics of structures: theory and applications to earthquake engineering. 4th ed. red., Upper Saddle River, N.J.: Prentice Hall; 2012. [5] Jeary A. Designeŕs guide to the dynamic response of structures. Hong Kong: E & FN Spon; 1997. [6] Malo KA, Abrahamsen RB, Bjertnæs MA. Some structural design issues of the 14storey timber framed building “Treet” in Norway. Eur J Wood Wood Prod 2016;74(3):407–24. [7] Fast P, Gafner B, Jackson R, Li J. Case study: An 18 storey tall mass timber hybrid student residence at the University of British Columbia, Vancouver. In: World conference of timber engineering, Vienna; 2016. [8] Reynolds T, Feldmann A, Ramage M, Chang W-S, Harris RDP. Design parameters for lateral vibration of multi-storey timber buildings. In: International network on timber engineering research, Graz; 2016. [9] Yang JN, Agrawall AK, Samali B, Wu JC. Benchmark problem for response control of wind-excited tall buildings. J Eng Mech 2004;130(4):437–46. [10] Huang S, Li Q, Xu S. Numerical evaluation of wind effects on a tall steel building by CFD. J Constr Steel Res 2007;63(5):612–27. [11] Blevins RD. Formulas for natural frequency and mode shape. New York: Van Nostrand Reinhold, cop; 1979. [12] Ellis B. An assessment of the accuracy of predicting the fundamental natural frequencies of buildings and the implications concerning the dynamic analysis of structures. Proc Inst Civ Eng 1980;69(3):763–76. [13] Lagomarsion S. Forecast models for damping and vibration periods of buildings. J Wind Eng Ind Aerodyn 1993;48(2–3):221–39. [14] Kim Y, Kanda J. Wind response characteristics for habitability of tall buildings in Japan. Struct Des Tall Spec Build 2008;17:683–718. [15] Hu L, Omeranovic A, Gagnon S, Mohammad M. Wind-induced vibration of tall wood buildings–is it an issue?. In: Wolrd conference on timber engineering, Quebec City, Canada; 2014. [16] AIJ, Damping in Buildings, Tokyo: Architectural Institution of Japan; 2000. [17] National Research Council of Canada, National Building Code of Canada, Ottawa: National Research Council of Canada; 2010. [18] BFS 2015:6-EKS 10. Boverkets konstruktionsregler, EKS 10 (Application of the European construction standards, EKS 10), Karlskrona; 2016. [19] ISO 6897. Guidelines for the evaluation of the response of occupants of fixed structures, especially buildings and off-shore structures, to low-frequency horizontal motion (0,063 to 1 Hz), International Organization for Standardization; 1984. [20] Edskär I, Lidelöw H. Wind-induced vibrations in timber buildings-parameter study of cross-laminated timber residential structures. Struct Eng Int 2017;27(2):205–16. [21] Ali MM, Moon KS. Structural developments in tall buildings: current trends and future prospects. Archit Sci Rev 2007;350:205–23. [22] Reynolds T, Harris R, Chang W-S, Bregulla J, Bawcome J. Ambient vibration tests of a cross-laminated timber building. Instituion of Civil Engineers; 2014. [23] BSV 97. Boverkets handbok om snö- vindlast (Snow load and wind action), Karlskrona; 1997. [24] Reynolds T, Bolmsvik Å, Vessby J, Chang W-S, Harris R, Bawcombe J, et al. Ambient vibration testing and modal analysis of multi-storey cross-laminated timber buldings. In: World conference on timber engineering, Quebec City; 2014. [25] Hu L, Karch E, Gagnon S, Dagenais C, Ramzi R. Dynamic performance measured on two 6-storey building made from wood structures before and after their completion and occupancy. In: World conference on timber engineering, Vienna; 2016. [26] Johansson M, Linderholt A, Jarnerö K, Landel P. Tall timber buildings - a preliminary study of wind-induced vibration of a 22-storey building. In: World conference on timber engineering, Vienna; 2016. [27] TräCentrumNorr (TCN), Innovativa höga träbaserade flerfamiljshus, Luleå; 2016. [28] Fjeld Olsen M, Hansen O. Measuring vibrations and assessing dynamic properties of tall timber buildings. [Master Thesis], NTNU- Norwegian University of Science and Technology, Trondheim; 2016. [29] Reynolds T, Casagrande D, Tomasi R. Comparison of multi-storey cross-laminated timber and timber frame buildings by in situ modal analysis. Constr Build Mater 2016;102:1009–17. [30] Feldmann A, Huang H, Chang W-S, Harris R, Dietsch P, Gräfeoch M, Hein C. Dynamic properties of tall timber structures under wind-induced vibration. In: World conference on timber engineering, Vienna; 2016. [31] SS-EN 14080, Timber structures – Glued laminated timber and glued solid, SIS Förlag ABl 2013. [32] Landel P, Linderholt A, Johansson M. Dynamical properties of a large glulam truss for a tall timber building. In: World conference on timber engineering, Seoul; 2018.
– The CLT and post-and-beam building types have similar first natural frequencies. The post-and-beam type showed a stiffer behaviour and more bending tendency than the CLT type where more shear behaviour was present. – In comparison, the post-and-beam building type has less actual weight m than the CLT building type. However, due to higher equivalent mass me which results in reduced acceleration levels, the post-and-beam system will provisionally be more useful if the goal is to construct tall timber buildings. – More measurements are needed to evaluate the appropriateness of equation f = 46/h. The theory indicates that some division according to stabilising system and material, pure timber buildings or hybrid, may be appropriate. Overall, empirical equations should be used with care since they rely on data with high variation and buildings that are not made of timber. – The assumption the mass is evenly distributed in tall buildings when calculating equivalent mass should be used with care for timber buildings. For a post-and-beam system the approximation can be appropriate but for a CLT system the approximation of the equivalent mass is not reliable and can give acceleration levels on the unsafe side. This becomes a problem because timber buildings fall short of the comfort criteria for acceleration at shorter building heights as compared to buildings of concrete or steel. Since the conclusions drawn here are based on FE simulations and the results are based on one certain type of floor layout and dimension of buildings generalisation is limited. 6. Further research The first natural frequencies of a building depend on several parameters, structural material (stiffness and mass), geometry of the building, stabilising system, position of the stabilising system etc. It is too early to draw any conclusions of the use of Eq. (24), more measurements are needed. Since openings reduce the stiffness of CLT panels and affect the dynamic properties the relation between stiffness and mass for timber buildings should be further explored. Declarations of interest None. Acknowledgements Fruitful discussions with Thomas Hallgren, COWI AB, Sweden and Magne Aanstad Bjertnæs SWECO AS, Norway are gratefully acknowledged. The authors would like to gratefully acknowledge SWECO Structures AB, Sweden for their financial support and interest. Finally, TräCentrum Norr support by VINNOVA (Sweden’s Innovation Agency) is acknowledged for financial support. References [1] Moore AJ. Dynamic characteristics and wind-induced response of a tall building. [Doctoral Thesis], University of Sydney, Faculty of Engineering and Information
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