IN
DEGREE PROJECT CIVIL ENGINEERING AND URBAN MANAGEMENT,
SECOND CYCLE, 30 CREDITS STOCKHOLM SWEDEN 2020,
Effect of birch in CLT elements
An investigation of how introducing birch effects the strength properties of CLT elements
RICHARD ERIKSSON
MARIA KARLSSON
Effect of birch in CLT elements
- An investigation of how introducing birch effects the strength properties of CLT elements
Richard Eriksson and Maria Karlsson
Master thesis, 2020 TRITA-ABE-MBT-20206
©Richard Eriksson and Maria Karlsson Royal Institute of Technology (KTH)
Department of Civil and Architectural Engineering Stockholm, Sweden, 2020
Abstract
In this thesis, material properties for a CLT element have been examined with regard to the structure of the element and its content of different types of wood. The focus has been on how Birch affects the properties, since its properties in shear are much better compared to the commonly used pine and spruce.
Calculations have been made in python script within the Rhino 6 software and its Grasshopper plugin.
Grasshopper basically works with coding but with a visual interface that is very user friendly. This pro- gram has been used because of its parametric structure which makes changes in the structure very simple and the whole model changes with the changes in parameters.
The basis for the survey has been the following parameters:
• Number of layers within the KL element
• Widths of lamellae
• Thicknesses of lamellae
• Material properties of Birch, Pine and Spruce
• Quotas between Birch, Pine and Spruce
The results show that the shear stiffness of the CLT-element increases with the amount of birch. The value of the G-moduli, however, does not only depend on the amount but the positioning as well. The shear stiffness is the greatest when the birch is placed closest to the middle of the element.
The conclusion made is that further investigation of the practical and economical aspects are required to make an informed decision about whether the use of birch or increased board width is the most efficient method of improving the shear stiffness of CLT.
Sammanfattning
I detta examensarbetet har materialegenskaper för ett KL-element undersökts med hänsyn till elementets uppbyggnad och dess innehåll av olika träslag. Fokus har legat på hur björk påverkar egenskaperna då dess egenskaper i skjuvning är betydligt bättre jämfört med de vanligtvis använda träslagen gran och furu.
Beräkningar har gjorts i pythonscript inom programvaran Rhino 6 och dess plugin Grasshopper. Grasshopper jobbar med kodning i grund och botten men med ett visuellt gränssnitt som är väldigt användarvänligt. Detta program har använts på grund av sin parametriska uppbyggnad vilket gör förändringar inom uppbyggnaden väldigt enkla och hela modellen ändras med förändringarna.
Basen för undersökningen har varit följande parametrar:
• Antal lager inom KL-elementet
• Bredder av lameller
• Tjocklekar av lameller
• Marerialegenskaper för Björk, Furu och Gran
• Kvoter mellan Björk, Furu och Gran
Resultaten visar att desto mer björk desto högre skjuvstyvhet. Det är dock inte bara mängden, utan placeringen som är avgörande. Skjuvstyvheten blir högst när störst mängd björk placeras i mitten av elementet.
Slutsatsen är att vidare undersökning av praktiska och ekonomiska faktorer krävs för att besluta om KL-element i björk eller med ökad brädbredd är effektivast för att förbättra skjuvstuvheten.
Preface
This master thesis concludes our civil engineering degree program at the Royal Institute of Technology, KTH, and comprises 30 higher education credits. It has been 5 long years that have finally taken us to this day and examination. The work has been carried out in collaboration with Tyrens AB where we, with the help of Karl Graah Hagelbäck, worked out the idea for the thesis.
We want to take this opportunity to thank everyone who helped us along the way, two special people in par- ticular.
Magnus Yngvesson, our supervisor at Tyrens, for his support and help with his technical knowledge, which has been invaluable in completing the work.
Then we would like to extend a special thanks to our supervisor and examiner at KTH, Bert Norlin. His vast knowledge of calculations and methods has made this work possible and incredibly educational.
Stockholm, june, 2020
Richard Eriksson and Maria Karlsson
Authors
Richard Eriksson and Maria Karlsson Structural engineering
KTH Royal Institute of Technology
Place for thesis
Tyrens Stockholm, Sweden
Examiner and KTH supervisor
Bert Norlin
KTH Royal Institute of Technology
Supervisor
Magnus Yngvesson Structural engineer Tyrens
Contents
Abstract iv
Sammanfattning vi
Preface viii
List of figures xvii
List of tables xviii
Nomenclature xxi
1 Introduction 1
1.1 CLT . . . . 1
1.1.1 Enviroment . . . . 1
1.1.2 Manufacturing . . . . 2
1.1.3 Wood in CLT . . . . 4
1.2 Parametric design . . . . 5
1.3 Research question . . . . 6
2 Method 7 2.1 Structure of the analysis . . . . 7
2.2 Material properties . . . . 7
2.2.1 Assumptions and limitations . . . . 8
2.2.2 Cross-sectional properties . . . . 9
2.2.3 Elastic modulus . . . . 9
2.2.3.1 Input data . . . . 9
2.2.3.2 Geometric centre . . . . 9
2.2.3.3 Axial stiffness . . . . 10
2.2.3.4 Bending stiffness . . . . 10
2.2.3.5 Fictious elastic modulus . . . . 11
2.2.4 Shear modulus perpendicular to the plane . . . . 11
2.2.4.1 Input data . . . . 11
2.2.4.2 Transverse shear stiffness . . . . 11
2.2.4.3 Shear transformation factor . . . . 14
2.2.4.4 Fictious shear modulus . . . . 15
2.2.5 Shear modulus parallel to the plane . . . . 15
2.2.5.1 Assumptions and limitations . . . . 16
2.2.5.2 Input data . . . . 17
2.2.5.3 Derivation of governing equations . . . . 18
2.2.5.4 Fictious shear modulus . . . . 22
2.3 Setting up the Grasshopper and Karamba environment . . . . 22
2.3.1 Python script . . . . 28
2.3.1.1 Input values . . . . 28
2.3.1.2 Script . . . . 29
2.3.2 Verification of results . . . . 30
3 Result 31 3.1 Verification of results . . . . 31
3.2 Material properties . . . . 32
3.2.1 Wood type combinations . . . . 35
3.2.1.1 Tables . . . . 35
3.2.1.2 Graphs . . . . 35
3.2.2 Board thickness combinations . . . . 39
3.2.2.1 Tables . . . . 39
3.2.2.2 Graphs for board thickness combinations . . . . 39
3.2.3 Board width variation . . . . 42
3.2.3.1 Tables . . . . 42
3.2.3.2 Graph, board width variation . . . . 42
3.2.4 Wood type combination vs Board width variation . . . . 43
4 Discussion 47 4.1 Process . . . . 47
4.2 Method . . . . 47
4.3 Results . . . . 49
4.3.1 Analysis of the parametric study . . . . 49
4.3.1.1 Wood type combinations . . . . 49
4.3.1.2 Board thickness combinations . . . . 50
4.3.1.3 Board width variations . . . . 50
4.3.1.4 Wood type combinations vs Board width variations . . . . 51
4.3.2 Source of errors . . . . 51
4.3.2.1 Calculation . . . . 51
4.3.2.2 Model . . . . 51
4.4 Conclusion . . . . 52
4.5 Future development . . . . 52
Bibliography 53
Appendices 55 Python script . . . . 55 Result tables . . . . 61
List of Figures
1.1 Schematic manufacturing process from timber to construction . . . . 2
1.2 Finger-jointing . . . . 3
1.3 CNC-machine and a example of a CLT-element cut to its ordered dimensions [6] . . . . 4
1.4 Example of appearance classes . . . . 4
1.5 Pine, cell structure . . . . 5
1.6 Spruce, cell structure . . . . 5
1.7 Birch, cell structure . . . . 5
1.8 The chain model and the crypt of Colonia Güell . . . . 6
2.1 Shear stresses in a CLT panel [6] . . . . 7
2.2 Definition of the directions . . . . 8
2.3 Orientation of the axis, [6] . . . . 9
2.4 Structure and numbering of a symmetric cross-section, [6] . . . . 10
2.5 Shear mechanisms I and II . . . . 16
2.6 Shear deformation for lamellae 1 and 2 caused by mechanism I . . . . 18
2.7 Mechanism II seen from above, xz-plane . . . . 19
2.8 Mechanism II, side view, xy-plane . . . . 20
2.9 Lab setup for the beam load test [4] . . . . 22
2.10 Overview of the Grasshopper worksheet . . . . 23
2.11 Calculation of material properties . . . . 24
2.12 Geometry setup in the GH model . . . . 25
2.13 Defining the model mesh . . . . 25
2.14 Definition of model inputs . . . . 26
2.15 Assembly and analysis of the model . . . . 27
2.16 Visualization of the model . . . . 27
2.17 Displaying the results . . . . 28
2.18 Material properties input for calculation of the material properties . . . . 28
2.19 Geometric properties input for calculation of the material properties . . . . 29
2.20 Composition of spruce, pine and birch input for calculation of the material properties . . . . . 29
3.1 Graph from the lab test in Lund . . . . 31
3.2 The values for the E-moduli for wood type combinations of birch and pine or spruce for a 3-Layer CLT element . . . . 36
3.3 The values for the G-moduli for wood type combinations of birch and pine or spruce for a 3-Layer CLT element . . . . 36
3.4 The values for the E-moduli for wood type combinations of birch and pine or spruce for a 5-Layer CLT element . . . . 37
3.5 The values for the G-moduli for wood type combinations of birch and pine or spruce for a 5-Layer CLT element . . . . 37 3.6 The values for the E-moduli for wood type combinations of birch and pine or spruce for a 7-Layer
CLT element . . . . 38 3.7 The values for the G-moduli for wood type combinations of birch and pine or spruce for a 7-Layer
CLT element . . . . 38 3.8 The values for the E-moduli for board thickness combinations for a 3-Layer CLT element . . . 39 3.9 The values for the G-moduli for board thickness combinations for a 3-Layer CLT element . . . 40 3.10 The values for the E-moduli for board thickness combinations for a 5-Layer CLT element . . . 40 3.11 The values for the G-moduli for board thickness combinations for a 5-Layer CLT element . . . 41 3.12 The values for the E-moduli for board thickness combinations for a 7-Layer CLT element . . . 41 3.13 The values for the G-moduli for board thickness combinations for a 7-Layer CLT element . . . 42 3.14 The values of G12 for board width variations, for 3-, 5- and 7-Layer CLT elements, with material
properties of pine or spruce. . . . 43 3.15 The values of G12 for wood type combinations (lines) and board width variations (bars), for a
3-Layer CLT element, with material properties of pine and spruce. . . . 44 3.16 The values of G12 for wood type combinations (lines) and board width variations (bars), for a
5-Layer CLT element, with material properties of pine and spruce. . . . 44 3.17 The values of G12 for wood type combinations (lines) and board width variations (bars), for a
7-Layer CLT element, with material properties of pine and spruce. . . . 45 4.1 The Rhino model and the associated Grasshopper work sheet . . . . 48 4.2 Components used in Karamba . . . . 48
List of Tables
1.1 Properties of lamellae . . . . 2
1.2 Mechanical properties of wood in use for CLT-element [12][3] . . . . 5
3.1 Comparison of deflections measured in [mm] . . . . 31
3.2 Wood type combinations for a 3-layer CLT element . . . . 32
3.3 Wood type combinations for a 5-layer CLT element . . . . 32
3.4 Wood type combinations for a 7-layer CLT element . . . . 33
3.5 Board thickness combinations for a 3-layer CLT element . . . . 33
3.6 Board thickness combinations for a 5-layer CLT element . . . . 34
3.7 Board thickness combinations for a 7-layer CLT element . . . . 34
3.8 Wood type combination of birch and pine for a 3-Layer CLT element . . . . 35
3.9 Results for board thickness combinations according to Table 3.3, with the material properties of pine for a 3-Layer CLT element and no birch . . . . 39
3.10 Board width variation with the material properties of pine for a 3-Layer CLT element . . . . . 42
4.1 The best performing combinations of the wood type combinations with birch and pine. . . . 49
4.2 The best performing combinations of the wood type combinations with birch and spruce. . . . 49
4.3 The best performing combinations of the board thickness combinations with pine, b = 140 mm and no birch. . . . 50
4.4 The best performing combinations of the board thickness combinations with spruce, b = 140 mm and no birch. . . . 50
4.5 The required board width for a CLT element with pine or spruce to match the value of G12 for a CLT element of birch with board width 140mm and ti = 40 mm . . . . 51
Nomenclature
γ Shear displacement angle κ Shear transformation factor
µ Transformation factor to account for the variation of the elastic moduli of the lamina σ Shear strain
τ Shear stress
τav Average shear stress υ Shear flow
ε Shear longitudinal displacement Agross The gross cross-sectional area bx Width in the x-direction by Width in the y-direction
E0 Young’s modulus parallel to grain
Ef ict,l/E1 Fictious elastic modulus for the entire CLT element in the longitudinal direction Ef ict,t/E2 Fictious elastic modulus for the entire CLT element in the transverse direction Eli Elastic modulus of lamina i in the longitudinal direction
Eref Chosen reference value for modulus of elasticity Eti Elastic modulus of lamina i in the transverse direction fs Shape factor for the cross-section
G0 Shear modulus parallel to grain G90 Shear modulus perpendicular to grain
Gli Shear modulus of the lamina in the longitudinal direction Gti Shear modulus of the lamina in the transverse direction hCLT Total thickness of the CLT element
ML Bending moment caused by external work MU Bending moment caused by internal virtual work S First moment of inertia
ti Thickness of lamina
Uint Internal virtual work/internal strain energy VL Shear force caused by external work VU Shear force caused by internal virtual work Wext External work
z Distance between the centre of each layer and the CLT elements center axis I Second moment of inertia
1. Introduction
1.1 CLT
CLT is the abbreviation of cross laminated timber and refers to the way the elements are manufactured. Sheets made up by laminae is placed in layers with a 90 degrees rotation to the adjacent layer. CLT-elements can be used both as walls and slabs in the load bearing framework of a building due to its versatility. Swedish woods CLT-handbook [6] lists 8 features that make the CLT profile unique and a valuable building material.
• The flexibility of CLT makes a valuable contribution to the development of construction.
• High strength in relation to the self-weight of the material.
• Small manufacturing tolerances and good dimensional stability.
• Good load-bearing capacity in fire.
• Good thermal insulation capacity.
• Low self-weight, which means lower transport and assembly costs, as well as lower foundation costs.
• Good capacity to tolerate chemically aggressive environments.
• Flexible production that even allows the manufacture of curved surfaces.
1.1.1 Enviroment
Is building with CLT environmentally friendly? The construction industry is today the source for 36 % of the global CO2 emissions and 40 % of the energy usage [8]. There is still much to research when it comes to the environmental impact of CLT but some numbers show that it may very well be the way to go. Compared to steel and concrete frames, timber frames use 30 % energy [7] and 60 cubic meters can store over 45 tons of CO2 over its life span [7].
The greatest uncertainty lies in the forestry and the size of its carbon footprint. Today the market for CLT is still in its infancy and there is no way of knowing what the development looks like if massive timber takes over the construction market completely. The manufacturing of the elements is easier to evaluate. Sweden today has got a sustainable forestry, which means that the outtake is lower than the growth of forest, and the manufacturing process is very energy efficient [7] Biproducts from the process such as sawdust and splints are reused as a heat source for the kilns, this makes the process even more energy efficient [6].
Compared to other construction wood products, like MDF and Plywood, the adhesive is a contributor to emissions. But while adhesives in MDF and Plywood stands for 10 % weight it is only 1 % in CLT elements [10]. Construction with CLT can reduce the need of concrete, due to its low weight. For example, a timber building weighs about 20 % of a concrete building. When using a lighter material, the foundation does not need to be as massive and that reduces the use of concrete [10].
1.1.2 Manufacturing
Figure 1.1 [6] shows the process from timber felled in the forest to CLT-elements ready to use in construction sites. CLT-elements are produced mainly according to the standard of SS-EN 16351, but also to meet the properties set by the manufacturer in its European Technical Approval [6].
Figure 1.1: Schematic manufacturing process from timber to construction
A brief overview of the stages in the process follows:
• Production of boards
Timber is sawed into boards with different dimensions depending on the configuration of the element. In Sweden, the most commonly used sort of wood is pine or spruce. Table ?? presents available properties for boards.
Table 1.1: Properties of lamellae Thickness, t 20-60 mm
Width, b 40-300 mm
Strength class C14-C30 Common ratio, b/t 4:1
• Drying
The board is dried to meet the required level of 12 % ± 2 % [13] before going through to the grading stage.
• Strength grading
Grading of the wood is done according to to the standard SS-EN 14081-1 [6].
• Finger-jointing
This step is usually the first one at the factory for making CLT-elements when the drying and classification often is made at the sawmill. To be able to manufacture elements as long as up to 30 meters [6] the boards have to be finger-jointed in the longitudinal direction. Example of different finger-joints are shown in figure 1.2. It is important that the moisture level does not differ more than 5 % to minimize the risk of cracking.
Figure 1.2: Finger-jointing
• Planing
It is in this stage the board gets its precise thickness.
• Application of glue
Boards are placed in sheets and adhesive is applied. The adhesive used is often melamine-urea-formaldehyde, single component polyurethane or emulsion polymer isocyanate [13]. The recommendation is that the adhesive presents the same mechanical properties as the basic material.
• Gluing under pressure
This is the stage where the cross section is constructed for the element. To utilize the strength of the boards in the best way possible, wood with a higher strength class is used in the surface layers, oriented in the main load direction.
Technical regulations states that the maximum spacing between boards is 6 mm . If the boards are placed without spacing it is possible to apply glue along the narrow side. This is however not proved to contribute to the overall strength and is recommended to be neglected or at least limited to the internal layers of the element [13].
When the arrangement is done and the adhesive applied, the element is set under pressure during the hardening of the adhesive. This pressure is applied using hydraulic jacks, vacuum jacks or by using bolts, clamps and nails.
• Finishing
When it comes to finishing the element, it is all about what the customer has ordered. The element is cut in the right dimension and holes are made for windows and doors. Milling channels for installations and drilling holes to fit for example fixtures can also be made. All this is made in a CNC machine, is a milling, drilling and cutting device that runs under Computer Numerical Control .
(a) CNC machine (b) CLT element
Figure 1.3: CNC-machine and a example of a CLT-element cut to its ordered dimensions [6]
Depending of the usages for the element there is different classes of appearance. In figure 1.4 taken from the CLT-handbook given by Swedish wood shows a couple of examples of classes and classification.
Figure 1.4: Example of appearance classes
• Packaging
• Construction
1.1.3 Wood in CLT
There are different types of woods and with that different properties that can be utilized. Woods are typically divided into soft- and hardwood depending on what kind of trees the wood originates from. Softwood comes from gymnosperm trees and hardwood comes from angiosperm trees [14]. The most common woods used in CLT-production in Sweden today are the softwoods pine and spruce [6]. A Swedish hardwood possible to use in CLT-elements is birch. The material properties for pine and spruce are obtained from [12] and the properties for birch from [3]. These are the values used for this report, but in general, these values can differ from region to region and from sub-species to sub-species. The values given throughout this report are given in Table 1.2.
Table 1.2: Mechanical properties of wood in use for CLT-element [12][3]
Property Unit Scots Pine Spruce Yellow Birch
E0 MPa 16 300 10 700 13 900
G0 MPa 680 620 945
G90 MPa 70 30 236
Density kg/m3 470 440 630
As shown in table 1.2 most material properties are higher for birch then for pine and spruce. This comes from the general connection between a higher density and greater strength value [9]. The density can be traced to the structure of the tree where conifers have a different cell-structure depending on the season while the deciduous have the same over the year. This difference is shown in figures 1.5 to 1.7.
Figure 1.5: Pine, cell structure
Figure 1.6: Spruce, cell structure Figure 1.7: Birch, cell structure
In figures 1.5 to 1.6 we can clearly see the annual rings of the wood. These rings appear due to difference in growth. During the spring/early summer, under the right conditions with temperature and precipitations, the tree forms a light and brighter wood. When the late summer/ autumn comes and the growth process decreases the wood gets darker and more dense [2].
Deciduous trees, which birch belongs to, has no clear rings because it produces similar wood the same through out the year [2]. As we can see in figure 1.7 the wood has vessels uniformly distributed over the cross section.
Most probably, it is this distribution of vessels that is the reason for the bending and tension strength and sometimes also the E-modulus to be higher than for pine and spruce [9].
1.2 Parametric design
From the world of mathematics comes the term parametric equations that uses changeable inputs (parameters) that defines the output, in our case the shape of a model. This is a relatively new approach for architects in their work with geometries, but the foundation of this way of working can be traced at least 100 years back.
The famous architect Antonio Gaudi used a modelling technique that uses the same thinking as parametric modelling today.
When modelling the crypt for the church Colonia Güell, Gaudi used chains hanging from the ceiling to find the shape of the arches. By hanging the model upside down he let the gravity shape the arches in its catenary curve. Gravity acts like a uniform load on every particle having the same mass. When flipping the model vertically, the load changes from tension to compression, after which the curved shapes use arch action to withstand the load.
(a) Chain model (b) Crypt
Figure 1.8: The chain model and the crypt of Colonia Güell
1.3 Research question
How does the elastic and shear moduli change in a CLT-element when introducing birch? How does the elastic and shear moduli change in a CLT-element when the width and thickness of the boards in the CLT-element differ? How does these variations of the moduli compare? Is it more efficient to increase the
dimensions of the boards or adding birch to increase the stiffness of a CLT-element?
2. Method
2.1 Structure of the analysis
The goal of the analysis is to perform a parametric study of the material properties due to variations in geometry, composition of hard wood and soft wood, positioning and proportions.
The calculation of the structural properties is done in a python script, the geometry is defined in grasshopper and the verification of the calculations is done with a finite element analysis in Karamba and compared to test results.
2.2 Material properties
The distribution of stress in CLT is a complicated matter. Firstly, wood is an anisotropic material, which means that the material properties differ in the longitudinal, tangential and radial directions. In a CLT element, the orientation of the boards vary with 90 degrees for each lamina. Then there is the matter of the interaction between the laminae and the interactions between the boards in the lamina. If there is no edge bonding between the boards within each lamina there is no shear stress at the longitudinal edges of the board, only transverse shear.
Assuming bonding between the lamina, the crossing surface between boards will be subjected to torsional shear due to the transverse shear of the differently oriented boards, which will result in rolling shear in the boards.
In addition to this, the CLT element will, for this project, contain boards of different material properties in each lamina.
(a) Shear stresses in relation to layer thick- ness in CLT panel
(b) Shear stress between the layers of a CLT panel
Figure 2.1: Shear stresses in a CLT panel [6]
The material properties, used as input for the finite element analysis in Karamba, are coded in a python script.
Karamba can only account for isotropic or orthotropic properties. Due to this restrictions of the software,
fictitious values for the elastic and shear moduli are calculated for the entire element. As if the element was a massive beam with orthotropic material properties.
Figure 2.2: Definition of the directions
The input values for Karamba are the elastic modulus for the longitudinal and transverse directions, E1 and E2, transverse shear modulus for the longitudinal and transverse directions, G31 and G32, and the in-plane shear modulus, G12. The directions 1, 2 and 3 are defined in figure [?].
In these calculations regular beam theories are used with orthotropic plate properties .
The parameters that can vary for the parametric study are, number of lamina, board thickness of each lamina, board width (same for all the boards in the element), the composition of hard and soft wood in each lamina.
2.2.1 Assumptions and limitations
• Since the CLT element analysed in this project is a wall, the reinforcing boards of birch will be evenly distributed and for these calculations the mean value of the material properties for each lamina will be calculated:
G0,layer1= G0,birch· %birch+ G0,pine· %pine+ G0,spruce· %spruce (2.1) G90,layer1= G90,birch· %birch+ G90,pine· %pine+ G90,spruce· %spruce (2.2) E0,layer1= E0,birch· %birch+ E0,pine· %pine+ E0,spruce· %spruce (2.3)
• The modulus of elasticity perpendicular to the grain, E90, is only 3-6% of the modulus of elasticity parallel to the grain, E0, and is therefor neglectable, especially since the open interface between boards will reduce this percentage towards zero.
E90= 0 (2.4)
• The number of lamina can be 3, 5 or 7.
• The material composition can consist of Scots pine, Spruce and Yellow birch.
• Ordinary beam bending theory is assumed to be valid, i.e. small displacements and linear elastic materials apply.
• Symmetry about the centre layer is assumed, both regarding lamina thickness and the composition of soft and hard wood.
2.2.2 Cross-sectional properties
In the following calculations, the orientation of the axes, the dimensions and numbering are as shown in figure 2.3, taken from the CLT handbook [6].
Figure 2.3: Orientation of the axis, [6]
Example of a 5-lamina element:
t = (t1, t2, t3, t4, t5)T (2.5)
hCLT = t1+ t2+ t3+ t4+ t5 (2.6)
2.2.3 Elastic modulus
2.2.3.1 Input data
Moduli of elasticity for direct stress are assembled in vectors for the longitudinal and transverse directions, as both directions are investigated. Index l for the longitudinal direction and t for the transverse direction.
Example of longitudinal and transvers modulus of elasticity vectors for a CLT element with 5 lamina:
El= (E0,lam1, 0, E0,lam3, 0, E0,lam5)T (2.7)
Et= (0, E0,lam2, 0, E0,lam4, 0)T (2.8)
2.2.3.2 Geometric centre
Since symmetry, both concerning geometry and material properties, about the geometric center axis is assumed, the distances from it can be calculated for a symmetric cross-section.
Values of ai shown in figure 2.4 are denoted zi for this project. For a 5-lamina CLT element they are calculated as follows.
z =
t1
2 + t2+t23
t2 2 +t23 0
−t23 +t24
−t23 + t4+t25
(2.9)
Figure 2.4: Structure and numbering of a symmetric cross-section, [6]
2.2.3.3 Axial stiffness
The axial stiffness is calculated for two principal directions, the longitudinal and transverse. There are two limiting cases for the evaluation, full contact or no contact between the boards within the lamina. If there is no contact the axial stiffness for the lamina is equal to zero. If there is full contact, the axial stiffness can be calculated for uniform compression or tension. In reality the contact will not be perfect. Glue will be pushed between the boards, but the distribution is never exact and cracks will develop over time. Here no contact is assumed, which is a valid assumption for most practical purposes.
EAl=X
i
(Eliti) EAt=X
i
(Etiti) (2.10)
Fictious elastic moduli
From the axial stiffness the for tension or compression fictious modulus of elasticity is derived:
Ec,t,l= E1= EAl
hCLT
Ec,t,t= E2= EAt
hCLT (2.11)
2.2.3.4 Bending stiffness
The bending stiffness is calculated for bending about both the transverse and longitudinal axes:
EIl=
n
X
i=1
El,i(ti)3
12 + El,iti(zi)2
EIt=
n
X
i=1
Et,i(ti)3
12 + Et,iti(zi)2
(2.12)
Fictious elastic moduli for pure bending
From the bending stiffness the fictious modulus of elasticity is derived:
Em,l= E1= 12EIl
t3 Em,t= E2= 12EIt
t3 (2.13)
2.2.3.5 Fictious elastic modulus
If the fictious moduli derived from axial or bending stiffness should be used as input for E1 and E2 depends on the load case for the model in question. If the response is dominated by in-plane tension and compression forces, then it is best to use values according to (2.11), but (2.13) is better if plate bending is dominating the scene. Expression (2.11) is used throughout this report, as it is only covering for in-plane actions. As Karamba only has one modulus of elasticity for each of the in-plane directions, it is not possible to give two values as input (one for tension and/or compression and one for bending). In a more refined FE-program it would be possible to directly give the two axial stiffnesses and the two bending stiffnesses as direct input, meaning that both (2.11) and (2.13) can be used simultaneously, which is obviously the preferred procedure if the studied plate is subjected to in-plane and bending actions at the same time.
Ef ict,l= E1 (2.14)
Ef ict,t= E2 (2.15)
2.2.4 Shear modulus perpendicular to the plane
2.2.4.1 Input data
The shear moduli perpendicular to the plane have 2 different values, i.e. in the transverse and longitudinal directions. The shear moduli vectors of individual layers are either shear along the grain or rolling shear of each lamina. Example of the shear moduli for a CLT element with 5 laminae:
Gl= (G0,lam1, G90,lam2, G0,lam3, G90,lam4, G0,lam5)T (2.16) Gt = (G90,lam1, G0,lam2, G90,lam3, G0,lam4, G90,lam5)T (2.17)
2.2.4.2 Transverse shear stiffness
When deriving the fictious shear modulus the aim is to find the shear stiffness, GA, that gives the same total shear angle as the combined effect of all the individual boards would do. According to ordinary beam theory, the shear stiffness for a beam with a rectangular cross section will be GA/1.2, where 1.2 is the shape factor, fs, for a rectangular cross section. The gross cross section for the CLT element is kind of rectangular, but the shear modulus and the modulus of elasticity vary from lamina to lamina and that must be taken into account.
When deriving the shape factor for the CLT element the assumption is made that the plane sections remain plane, inferring that the stresses themselves do not deform the material, resulting in a slightly lower shear stiffness.
The foundation for the derivation of the shape factor is the procedure given in section 10.9 in Mechanics of Materials [5].
The method of choice is the unit load method, which is based on the principal of virtual work. The virtual displacement is taken to be the "real" displacements caused by the "real" external loading producing (ML, VL), while the forces (Mu, Vu) moving through these "real" displacements are instead assumed to be caused by the unit load only. Respectively, M and V denote bending moment and shear force. Then the internally stored strain energy in some small material particle must be the force caused by the unit load multiplied by the displacement caused by the real load. The total internal work, Uint, resulting from the assumed virtual displacement field can be found by integration over the total volume of the structure. The external work, Wext,
produced by the unit load is simply the unit load times the displacement, DELTA, under the unit load itself.
As external and internal work must be equal, it all boils down to
Wext= 1∆ = Uint (2.18)
The internal strain energy in (2.18) can be determined by integration over the beam volume.
A normal stress, σ, caused by the "unit load" gives a force (σdy dz) that moves through the displacement (εdx) in which εis from the "real" load. Similarly, τgives a shear force (τdy dz) that moves through the shear displacement (γdx) caused by the "real" load.
The equation below shows the expression for the stored strain energy, which is simply the product of force and displacement.
dUint= (σdydz)(εdx) + (τ dydz)(γdx) (2.19)
Remember that σand τare caused by MUand VU, respectively. While εand γare corresponding to MLand VL. The assumption is made that the cross-section of the beam is rectangular. To account for the asymmetry the method of a transformed (or fictitious) cross-section is used. The width of the studied cross-section is set to b
= 1000mm, a fictious value only for analytical purpose.
A reference value is, for the modulus of elasticity, set to calculate the transformation factors. It can be the value for any of the lamellae in the element.
Eref = E0,lam1 (2.20)
The transformation factors are the ratio between the elastic modulus of a lamina and the reference value.
µl,i= El,i
Eref (2.21)
µt,i= Et,i Eref
(2.22)
Where i = 1, 2, ..., n and n is the number of layers.
Transformed width:
bl,i= µl,i∗ bl (2.23)
bt,i = µt,i∗ bt (2.24)
By using the transformed width in the calculations, the influence of different material properties is accounted for.
The second area moment of inertia of the transformed cross-section:
Il=
n
X
i=1
bl,i(ti)3
12 + bl,iti(zi)2
(2.25)
It=
n
X
i=1
bt,i(ti)3
12 + bt,iti(zi)2
(2.26)
Equation (2.19) for dUint, can be further expanded using elementary beam theory and Hook’s Law.
σ = Eε (2.27)
τ = Gγ (2.28)
From here on the calculations for actions in the longitudinal direction are presented. The same principal goes for the transverse direction.
Below, σ, ε,τand γare expressed for the ith layer for xy-shear and bending about the transverse axis:
Stress from the unit load
σ(i, z) =MU
Il
zµli (2.29)
Strain from the real load
ε(i, z) = ML
EiIl
zµli (2.30)
Shear stress from the unit load
τ (i, z) = VUSl(i, z)
Ilb (2.31)
Shear strain from the real load
γ(i, z) = VUSl(i, z)
GiIlb (2.32)
Substituting the equations for σ, ε, τ and γ into equation (2.19) gives
dUint=MUML
ErIl2 µliz2dxdydz +VUVLSl(i, z)2
Gi(bIl)2 dxdydz (2.33)
The integration of the equation is set up as one integration over the length of the beam (dx) and one over the cross-section (dydz = dA).
Uint=
Z MUML
ErIl2
Z
Eiz2dA
dx +
Z VUVL
(bIl)2
Z S(i, z)2 Gi dA
dx (2.34)
The first term in this equation is strain energy from bending and the second is due to shear. In this case the fictious shear modulus are to be evaluated and therefore it is only the second term than is relevant. The bending
term was just included for demonstration purposes, as engineers are usually more familiar with bending than shear. The goal is to express the equation for the internal shear strain energy as follows (i.e. to replace the real properties,that are different from lamina to lamina, with one that is uniform over the whole cross-sectional area):
Uint.shear =
Z fsVUVL
GrAgross
dx =
Z VUVL
(bIl)2
Z S(i, z)2 Gi
dA
dx (2.35)
fs: The so called "shape factor" that takes the real shape and material properties of the section into account.
VUVL is set to 1 for this evaluation, as this product must be the same on both sides of the equality sign.
2.2.4.3 Shear transformation factor
The shape factor can be derived from equality (2.35). The shape factor is a constant, unique for each cross- section. The reference value for the shear modulus, Gr, and the gross cross-sectional area, Agross, are also a constant and will not vary along x. The shear modulus will not vary within a lamina. Only the first moment of inertia, S(i,z), will vary within each lamina and will therefore be calculated by integration over the lamina thickness, ti, and zi is the centre of the lamina in the z-direction. With this in mind, equality (2.35) can be rewritten as follows:
fs
Grbt = 1 (bIl)2
n
X
i=1
1 Gli
Z zi+ti2 zi−ti2
S(i, z)2bdz
!
(2.36)
Solving for the shape factor gives:
fs= Grt Il2
n
X
i=1
1 Gli
Z zi+ti2 zi−ti2
S(i, z)2dz
!
(2.37)
The known values in the equation for the shape factor are the second moment of inertia, Il, It, and the shear modulus for each lamina Gl, Gt.
Dot-product
Gr ·t is defined in the CLT-handbook as the dot-product:
Grlt =
n
X
i=1
(Gliti) (2.38)
First moment of inertia
The integral calculating the first moment of inertia is symbolically evaluated as follows:
Z zi+ti2 zi−ti2
S(i, z)2dz = Z zi+ti2
zi−ti2
Si+ bl,i(ti− 2z + 2zi)(2z + ti+ 2zi) 8
2
dz (2.39)
= (Si)2ti+Si(ti)3bl,i
6 + Si(ti)2zibl,i+(ti)5(bl,i)2
120 +(ti)4zi(bl,i)2
12 +(ti)3(zi)2bl,i)2
3 (2.40)
The quantity Si in this equation is the first moment of inertia for all lamellae except the current lamina and the values can be determined in advance by the scheme:
j = 2, ..., n (2.41)
S1= 0 (2.42)
Slj = Slj−1+ blj−1tj−1zj−1 (2.43) Shape factor
Now everything is known and the shape factor can be calculated:
fsl = Pn
i=1(Gl,iti) Il2
" n X
i=1
1 Gli
(Si)2ti+Si(ti)3bl,i
6 + Si(ti)2zibl,i (2.44) +(ti)5(bl,i)2
120 +(ti)4zi(bl,i)2
12 +(ti)3(zi)2bl,i)2 3
#
(2.45)
The shape factor for the longitudinal direction is now in place, the same procedure is used to evaluate the shape factor for the transverse direction, fst.
Shear transformation factor
The shear transformation factor is defined in the CLT-handbook and can be used to determine the shear modulus perpendicular to plane for the entire element. When evaluating the shape factor it becomes clear that the shear transformation factor is the inverse of the shape factor:
κlt = fs,lt−1 (2.46)
2.2.4.4 Fictious shear modulus
The fictious shear modulus can therefore be evaluated as the mean shear modulus for the element times the shear transformation factor:
Gl,f ict = G31= κl
Gr,l
hCLT (2.47)
Gt,f ict= G31= κt Gr,t
hCLT (2.48)
2.2.5 Shear modulus parallel to the plane
The evaluation of the shear modulus parallel to the plane is based on the theories presented by Thomas Bogensperger, Thomas Moosbrugger, Gregor Silly in their article "Verification of CLT-plates under loads in plane" [11]. The main principle is the same for the calculation of the in-plane shear modulus as for the transverse shear modulus: to derivea fictitious value, Gfict, that gives the same total shear displacement angle as the combined effect of all the individual board properties would. For the derivation of the fictious in plane shear modulus, we calculate the total shear angle caused by the shear mechanisms, γ.
Gf ict= υ
tγ (2.49)
The in-plane shear deformation is caused by several different mechanisms for a CLT-element. The two mech- anisms considered here are mechanism I and mechanism II. Mechanism I is the shear deformation caused by uniform ordinary shear, with no relative rotation between the neighbouring boards. Mechanism II is the shear deformation caused torsion, the relative rotation between the neighbouring boards. The other mechanisms, not covered here, are longitudinal and transversal slip between adjacent lamellae, which may reduce the effective shear modulus slightly more.
(a) Shear deformation from mechanism I (b) Shear deformation from mechanism II Figure 2.5: Shear mechanisms I and II
To simplify the calculations some additional assumptions are made.
2.2.5.1 Assumptions and limitations
• No edge bonding between the longitudinal edges of the boards. The edges are assumed not to be glued, or if glued dry cracks are assumed to run along the edges. Hence, all these edges are stress free.
• Each board can only transfer shear in the x- or z-directions when passing across an edge-to-edge unbonded interface (described above).
• The transverse shear flow within the boards is assumed to have a resultant acting at mid-thickness of this board. This can only be exactly true for internal,symmetrically positioned boards having the same material properties, but it is an acceptable assumption for most practical cases.
• The extreme rotation angle for mechanism II occurs at the mid-thickness of each board. This can only be exactly true for the same conditions as above.
• The total shear angle, γ, is assumed to be the same for the entire element, continuity requirement.
• The slip modulus of the crossing areas is controlled by the mean values of the shear moduli, parallel-to- grain and for rolling shear.
• Moment equilibrium for a unit cell across the total thickness is automatically fulfilled.
• The shear angle γ = γI + γII, results from two components. Where γI is the ordinary shear angle produced by the shear flow affecting any two neighbouring boards, while γIIis the relative rotation angle measured between the mid-thickness points of the two layers involved.
• Symmetry about the centre layer is assumed. Therefore, only half of the thickness of the element is modelled, except that the real thickness is used of the centre layer in all calculations.
