Master’s Thesis in Structural Engineering
Parameterized and Adaptive Modelling of Mechanical Connections in Timber Frame Structures
Authors: Joan Gikonyo
Pierre Modig
Supervisor, LNU: Sigurdur Ormarsson Examiner, LNU: Torbj¨orn Ekevid Course Code: 4BY35E
Semester: Spring 2018, 15.0 credits
Abstract
This study investigates the global stiffness of a timer frame structure under wind loading using the finite element method by creating parameterized script files. Of key interest was the accuracy of the global stiffness determined from an adaptive 3D beam model in comparison to a 2D beam model and, the stiffness of a 3D beam model when subjected to different types of bracing in the presence of internal bracing provided by a lift shaft structure. Investigation of contact forces on the surfaces between the fastener and the timber at the connection was carried out and a design check for the specified bolts shear capacity done with respect to Eurocode 5.
A 3D adaptive connection was created for a 2D frame model and the stiffness of the structure was studied. A comparison of the maximum displacement of the structure in the x direction, under the same wind loading, spring stiffness and boundary conditions, with a 2D beam structure without the adaptive connection initially showed a difference in the displacement. This implied that the rotational stiffness in the beam model was greater than that of the adaptive connection created. Therefore after altering the rotational stiff- ness of the beam model to achieve similar displacement as in the adaptive model, the rotational stiffness of the created connection was found to be 33.4 · 106Nm. The study also determined the contact forces generated at the surfaces between the fasteners and the timber using the finite element method to integrate over the surfaces and calculate the forces. The results were generated using the History Output in the step module. The only disadvantage of acquiring the contact forces was that, the contact surface simulation caused larger run times for the model to complete the time step. For the adaptive model it took 18 hours to complete each step.
Further investigation into the stiffness of a 3D frame structure was conducted. The model of the 3D structure was created by a parameterized script which makes it easy to change input variables such as number of internal walls, geometry in x-z-plane, number of storeys, cross-sectional dimensions, material properties number of diagonals and location of diagonals. A variety of models with different conditions was analyzed. This showed that stiffness has a major impact on the magnitude of reaction forces and displacements.
Keywords: Timber frame structure, Parameterized modelling, Python script, Adaptive connection, ABAQUS.
Acknowledgment
We would like to thank our supervisor, Professor Sigurdur Ormarsson at the department of Building Technology at Linnaeus University in V¨axj¨o, for guiding us through this mas- ters thesis. Furthermore, we sincerely appreciate his patience and the time he took to respond to all our queries.
In a special way we would like to also thank Le Kuai for his previous work in creat- ing a parameterized timber structure model that we used to carry out this study.
Joan Gikonyo & Pierre Modig V¨axj¨o 13th of July 2018
Contents
1 Introduction . . . 1
1.1 Background and problem description . . . 2
1.2 Aim and purpose . . . 3
1.3 Hypothesis and limitations . . . 3
1.4 Reliability, validity and objectivity . . . 4
2 Literature review . . . 5
3 Theory . . . 12
3.1 Wood . . . 12
3.1.1 Structure of wood . . . 12
3.1.2 Natural defects in wood . . . 13
3.1.3 Mechanical properties of wood . . . 14
3.1.4 Linear elasticity . . . 15
3.1.5 Glulam . . . 17
3.2 Connection Design . . . 18
3.2.1 Functional requirements of a connection . . . 18
3.2.1.1 Strength. . . 18
3.2.1.2 Stiffness. . . 18
3.2.1.3 Failure Modes. . . 18
3.2.1.4 Load Reversal. . . 19
3.2.1.5 Ease of Construction. . . 19
3.2.1.6 Aesthetics. . . 19
3.2.1.7 Other functional requirements of timber connec- tions. . . 19
3.2.2 Johansen Theory of Timber Connections . . . 20
3.2.2.1 Embedding Strength. . . 20
3.2.2.2 Friction. . . 21
3.2.2.3 Toothed Washer. . . 21
3.2.2.4 Bolt Connections. . . 21
3.2.3 Bolted Slotted-in Steel Plate Connection . . . 25
3.2.3.1 Why Slotted-in-Steel Plates? . . . 25
3.2.4 Shear Capacity of steel-to-wood connections . . . 26
3.2.5 Yield moment and characteristic embedment strength for dowelled and bolted connections . . . 27
3.2.6 Bolt or dowel spacing requirements . . . 28
3.3 Adaptive design . . . 30
3.4 The Finite Element Method . . . 30
3.4.1 Beam theory . . . 31
3.4.2 Plate theory . . . 33
3.5 Wind load . . . 35
4 Methods . . . 38
4.1 Parameterized Python script . . . 38
4.2 Models . . . 38
4.2.1 3D structure . . . 38
4.2.2 2D structure . . . 38
4.3 ABAQUS . . . 39
4.3.1 Beams and columns in 3D-structure . . . 39
4.3.2 Walls and floors in 3D-structure . . . 40
4.3.3 Beams and columns in 2D-structure . . . 40
4.3.4 3D connection in 2D-structure . . . 40
5 Implementation . . . 41
5.1 Modelling of 2D fame structure with 3D connection . . . 41
5.1.1 Part module . . . 41
5.1.1.1 Solid Timber Beam and Column. . . 41
5.1.1.2 Steel plate. . . 42
5.1.1.3 Timber Peripheral and Steel Peripheral. . . 42
5.1.1.4 Fastener. . . 43
5.1.1.5 Wire Beam Elements representing the Frame columns and Beams. . . 43
5.1.2 Property module . . . 44
5.1.3 Assembly module . . . 44
5.1.4 Interaction module . . . 45
5.1.4.1 Tie Constraints . . . 46
5.1.4.2 Surface to Surface Contact . . . 46
5.1.4.3 Coupling Constraints . . . 47
5.1.4.4 Small Beams specially tied to fasteners . . . 47
5.1.4.5 Spring interactions in the 2-dimensional frame structure . . . 48
5.1.5 Step module . . . 48
5.1.6 Load module . . . 49
5.1.7 Mesh module . . . 50
5.1.8 Job module . . . 50
5.1.9 Visualization module . . . 50
5.2 Modelling of 3D structure . . . 51
6 Results . . . 52
6.1 2D structure . . . 52
6.1.1 Influence of the 3D Adaptive solid element connection on the structural behaviour of a timber frame structure. . . . 52
6.1.2 Discussion of the results . . . 53
6.1.3 Contact Forces Generated at Adaptive Connection . . . . 54
6.1.4 Determination of Bolt Capacity . . . 57
6.2 3D structure . . . 58
6.3 Influence of diagonals and number of storeys . . . 59
6.3.1 Different number of storeys without diagonals . . . 59
6.3.2 Different number of storeys with diagonals . . . 61
6.4 Influence of number of diagonals . . . 64
6.5 Structure loaded with external forces . . . 66
7 Analysis . . . 69
7.1 Analysis of 2D structure with 3D connection . . . 69
7.1.1 Influence of the Adaptive Connection in Beam Model . . . 69
7.1.2 Contact Forces on the Timber . . . 69
7.2 Analysis of 3D structure . . . 69
7.2.1 Influence of diagonals and number of storeys . . . 69
7.2.2 Influence of number of diagonals . . . 70
7.2.3 Displacement caused by external forces . . . 70
8 Conclusions and discussion . . . 71
8.1 Conclusions for 2D structure with 3D connection . . . 71
8.2 Conclusions for 3D structure . . . 71
References . . . 72
List of Figures
1 Dowel forces from both the hand calculations and FEM-Analysis plotted
as black and red arrows respectively, along with the angle differences [1]. . 5
2 3D model showing modelled spring stiffness at connections [2]. . . 6
3 Comparison of reaction forces from Model with rotational stiffness to one without rotational stiffness for multi-storey timber structure with pre- scribed displacement at the top [2]. . . 7
4 Relationship between connection stiffness and reaction force for multi- storey timber structure with prescribed displacement at the top [2]. . . 8
5 Typical dowel-type timber connection with a single dowel; photo of a spec- imen during conduction of the experiment (left), sketch (right) [3]. . . 9
6 Multiple Shear Plane Connection Procedure [4]. . . 10
7 Wood structure. . . 12
8 Knot. . . 13
9 Spiral grain angle. . . 13
10 Reaction wood. . . 14
11 Wood cell for mature wood. . . 15
12 Linear elastic constitutive relation. . . 16
13 Local coordinate system for wood. . . 16
14 Glulam beam. . . 17
15 Yield curves for embedding test [5]. . . 20
16 Relation between slip g and force P [5]. . . 21
17 Relation between slip g and force P for a Toothed Washer [5]. . . 22
18 Single shear connection with a dowel. The left side shows a dowel with sufficient Capacity, the right side shows a dowel with Insufficient Capacity. 22 19 Sufficient Dowel Capacity in a Double shear Joint. . . 23
20 Double shear connection with a dowel with insufficient Capacity Dowel. . . 24
21 Slotted-in steel plates. . . 25
22 Failure modes for steel to timber connections [6]. . . 26
23 Definition of edge and ends distances and spacing between fasteners [6]. . . 29
24 Euler-Bernoulli beam and Timoshenko beam. . . 31
25 Wind zones for walls of rectangular buildings. . . 37
26 Meshed geometry of the solid beam and column parts. . . 42
27 Meshed geometry of the steel plate. . . 42
28 Timber peripheral and steel peripheral. . . 43
29 Fastener. . . 43
30 Assembled connection. . . 44
31 Assembled structure. . . 45
32 Location of Tie Constraint. . . 46
33 Location of surface to surface contact. . . 47
34 Coupled interaction. . . 47
35 Spring locations in the structure. . . 48
36 Loaded structure. . . 49
37 Deformed shape in the x direction for the 2D frame with the Adaptive Connection. . . 52
38 Deformed shape in the x direction for the 2D frame without the Adaptive Connection. . . 53
39 Displacement in the x direction for the 2D frame set to the Rotational stiffness of the Adaptive model . . . 54
40 Adaptive 3D Solid Element Connection. . . 55
41 Location for bolts in adaptive connection. . . 55
42 Resultant bolt forces acting on the timber . . . 57
43 Structure without diagonals and prescribed displacements. . . 59
44 Two examples of structures with different number of storeys. . . 60
45 Variation in reaction forces for structures without diagonals. . . 61
46 Structure with diagonals and prescribed displacements. . . 62
47 Two examples of structures with different number of storeys. . . 63
48 Variation in reaction forces for structures with diagonals. . . 63
49 Frame structures with different number of diagonals. . . 64
50 Values for different number of diagonals. . . 65
51 Structure with prescribed forces. . . 66
52 Deformed structure. . . 67
53 Displecements due to prescribed forces. . . 68
List of Tables
1 Spacing and end/edge distances for bolts. . . 29
2 Spacing and end/edge distances for dowels. . . 30
3 Terrain categories and terrain parameters . . . 36
4 Pressure coefficients for walls of rectangular buildings . . . 37
5 Unit system in ABAQUS/CAE. . . 39
6 Bolt Contact Forces . . . 56
7 Values for structures without diagonals. . . 61
8 Values for structures with diagonals . . . 62
9 Reaction forces for structure with different number diagonals. . . 65
10 Values for structure with different forces. . . 68
1 Introduction
Today, countries the world over, are working together to combat climate change which if not retarded may lead to devastating environmental consequences for future genera- tions. Sweden in its bid to contribute to the global effort, to fight climate change, has embarked on seeking more eco-friendly solutions to its day-to-day problems. One of these eco-friendly solutions is an increase in timber construction to reduce carbon dioxide emis- sions.
Sweden’s forest cover is about 27,5 million hectares, which is about 61 % of the entire country. This makes timber a readily available construction material in Sweden. Accord- ing to Swedish Wood, 78 percent of the forest coverage is productive, while the remaining percentage is set aside for conservation and unproductive areas [7]. A report by Swedish Forest Industries Federation, observes that wood is the only renewable construction ma- terial [8]. For this reason, an increase in the proportion of timber used in construction and reduction in the use of non-renewable materials such as concrete, brick and steel will reduce carbon dioxide emissions. Wood provides an eco-friendly construction option.
In addition to its environmental advantages, timber construction is more economically feasible as compared to construction of similar buildings using steel and/or concrete [9].
In recent years construction of timber frame structures has increased due to technical and environmental benefits of using wood in construction [8].
With both economic and environmental advantages illustrated above, the Swedish Forest Industries Federation acknowledges the need for further research into the behaviour of timber as a construction material [8]. The interest in the behaviour of timber is due to its reduced stiffness as compared to commonly used construction materials such as steel and reinforced concrete. Currently there are several wood products used in construction in Sweden, these are Structural Timber, Glued Laminated Timber (glulam), Laminated Veneer Lumber (LVL), Fibre Boards, Particleboards, Oriented Strand Boards (OSB), Cross Laminated Timber (CLT) and Plywood. All these products are used for different purposes in the construction process. This study will focus on the use of glulam which is preferred in the construction of multi-storey buildings due to its formability and worka- bility as compared to other wood products.
One of the notable timber frame structures is Treet. The high-rise is located in Bergen, Norway. It was completed in 2015, and is currently the tallest timber frame structure in Europe with 14 storeys. It consists of box units that make up the individual rooms.
The box units are laid on top of each other supported by the glulam timber frame ele- ments. Key design details of this project are the joint connections, which are composed of slotted-in steel plates and dowels [10]. This type of connection will be the basis of this study.
Another illustration of existing timber frame structures is, The Cube. It is located in London and was completed in 2015. The ten storey timber frame structure, whose frame is composed of three main materials, cross laminated timber (CLT), steel and concrete.
The architecture of the building is unique with a twisted character, which was been made possible by the hybrid concept. The building is stabilized by a reinforced concrete core and integrated steel elements, which allows several cantilevered overhangs. The wood- based material used in the structure largely consists of CLT [11].
In both illustrations above the connection design was critical. McLain [12] observes that
”a structure is a constructed assembly of joints separated by members”. The capacity of the joints in the frame of the structure to withstand the applied load determines the members used. The study observed that the stiffness of the joints will influence the global stability of the structure [12]. It is for this reason that the focus of this study is on the design of critical joint connection of a timber frame structure.
1.1 Background and problem description
In the past, timber structures were limited to one or two storeys in Sweden. Timber construction has grown tremendously in the recent years as sited by Swedish Forest In- dustries Federation [8]. According to the Swedish Forest Industries Federation, this could be attributed to Swedens effort to reduce on carbon emissions. It further sites that ad- vancing knowledge relating to timber construction and properties of timber will increase the use of wood in construction [8]. In this study by focusing on the design of the critical connection in a multi-storey timber frame structure, there will be further insight into the behaviour of timber as a construction material.
This study seeks to further the works of Kuai, Parameterized Finite Element Simula- tion of Multi-Storey Timber Structures [2] and Nonbo, Numerical Simulation of Timber Connections with Slotted in Steel Plates [1]. As such this study will aim on developing an adaptive three dimensional solid element model of the critical moment stiff connection in multi-storey timber structure.
Kuai [2] research focused on the relationship between local connection stiffness and the global stiffness of the timber structure. The research revealed a nonlinear relationship be- tween the increase in connection stiffness and the global stiffness of the timber structure.
From the study, it was observed that at first the increase in the connection stiffness led to a large increase in the global stiffness of the timber structure, however as the connection stiffness continued to increase, the increase in global stiffness increased but the effect was smaller to a point where an increase in the connection stiffness did not have any effect on the global stiffness.
On the other hand, Nonbo [1] investigated the resulting design forces at the critical connection of a timber frame structure by comparing both hand calculations and 3D solid element calculations. It was clear at the end of the study that there was a notable dif- ference in the forces. This was attributed to the fact that hand calculations only take
into consideration material properties in the fibre direction, while three dimensional solid finite element modeling takes into consideration orthotropic material properties. From this study it was found that design forces at the connection obtained using the adaptive 3D solid finite element model were more accurate as compared to those obtained by hand calculations.
Due to the increased demand for multi-storey timber based structures in Europe, many construction companies are investing in research into the stability of taller timber struc- tures. For instance, The Moelven Group [13] that deals with timber based construction has invested in the research on the stability of tall timber buildings. The company hopes to provide safe structures that can withstand all types of loading. In cases where these tall structures are subjected to large wind loads, they are prone to deflecting. The stability of the structure to withstand the deflection is thus critical and hence the interest in the global stability of the structure. Even though past studies have shown that the global stability of a timber structure is dependent on the connection stiffness, the research has not focused on the design of the connections. This study will purpose to design the critical connection and with the results from the design provide further knowledge on the effect of connection stiffness on the global geometric behaviour of the structure.
1.2 Aim and purpose
Based on the works of Kuai [2] and Nonbo [1], this investigation will:
i. Firstly, develop a more realistic parameterized three dimensional beam based finite element model by inserting internal walls into the existing model created by Kuai as the existing model only provides for external walling.
ii. Secondly, create a three dimensional solid element adaptive connection model of the critical moment stiff connection between the column and primary beam supporting the floor element.
Since this study seeks to further existing works, its purpose is to improve the accuracy of the results from the three dimensional solid finite element model. As a result of the analysis, determine the design forces at the critical connections and subsequently design the connection. The results from this study are expected to give a deeper insight into the effect of connection stiffness on the global stiffness of the entire timber structure.
1.3 Hypothesis and limitations
As this study aims to design a critical connection in a multi-storey timber structure, the research will focus on whether adaptive design gives better insight into the overall global geometric behaviour of a timber structure.
This study will be subject to the following limitations:
i. Though computer models are instrumental in capturing real life structures, they are never completely representative of the structure. Numerical models are never fully representative of the real structure, as such results from numerical models should be carried out repeatedly to determine their validity [14].
ii. The geometry of the model created is quite simple and it may not adequately repre- sent the more complex real life geometry. However; albeit the simple geometry used for the study, the developed script design code of parameterized frame structure will allow for future users to determine the geometry of the structure to be analyzed.
iii. This study will only carry out the connection design for one model therefore, the study will only provide conclusive results for the particular structure.As such,generalization of the results may be inaccurate.
iv. The effect of moisture on the connection will not be considered. Therefore the results may not be fully representative of the actual behaviour in reality.
1.4 Reliability, validity and objectivity
This study will utilize the finite element method. It is a numerical method used for solving engineering problems, by which general differential equations can be solved in an approximate manner. The region under analysis is divided into smaller elements referred to as finite elements, and an approximation is carried out over each element [15]. Abaqus which is a finite element analysis tool will be used in this study to create the model and determine the stresses. The parameterized model will be created using python script.
Therefore, one could edit the script to suit the structure under analysis. The connection design will be checked against existing European Standard Manuals for timber based con- struction.
Furthermore, it is best to continue the works using the finite element method as it was used in previous works by Kuai and Nonbo, which this study will focus on advancing.
This will allow for accurate comparisons to be made with respect to whether the model created is an improvement of the original version.
2 Literature review
This study aims to further existing research by Kuai [2] by focusing on the study carried out by Nonbo [1] regarding the numerical simulation of a connection. These two studies are essential to this study. Nonbo [1] focused on the difference in hand calculated and 3D numerically simulated design forces at a connection. The study found that there was a clear pattern in the direction and magnitude differences of the calculated forces. Dowel forces calculated using the finite element method (FEM) were found to lean more towards the fibre direction than those from hand calculations. According to Nonbo, dowel forces found by the hand calculation method were based on cross sectional forces. In the numer- ical analysis of the frame the wood material was assumed to have elastic behaviour but the critical moment stiff connection was modelled as a 3D adaptive connection while the rest of the structural frame was modelled using beam elements. The analysis showed that the dowel forces found by hand calculation in general are perpendicular to the direction of the respective connection groups. In addition FEM-simulation the dowel forces pointed more in the fibre direction than those found by hand calculation. Based on the results of the Study, Nonbo [1] showed that dowels loaded perpendicularly to the fibre direction take up nearly as much of the moment induced force in the adaptive connection as by the hand calculation method.
Figure 1: Dowel forces from both the hand calculations and FEM-Analysis plotted as black and red arrows respectively, along with the angle differences [1].
This conclusion was based on the fact that the hand calculations were based on the as- sumption of elastic isotropic timber and dowel behaviour described by Larsen and Enjily [16] to determine the dowel forces whereas, the load carrying capacity calculation method was based on Johansens theory from 1949.
This theory was based on the assumption that the timber and dowel materials behave as rigid-plastic material. That meant that orthotropic timber material properties were not taken into account when determining the dowel forces at the connection even though the angle between the line of action of the force and the fibre direction is critical in deter- mining the load bearing capacity of the connection. This showed that hand calculated forces for the connection were not accurate as they did not take into consideration the orthotropic material properties of timber.
With the conclusion from Nonbo [1] study, this research will focus on adaptive 3D design of the critical connection in a multi-storey timber frame structure. This will be important in order to determine accurate details regarding the relationship between the connection stiffness and its effect on the global stiffness of a timber structure.
Kuai [2] research focused on the global stiffness of a timber frame structure. The re- search created a parameterized numerical model of a multi-storey timber frame structure and investigated the effect of bracing and connection stiffness to the global stiffness be- haviour. The study found that the provision of bracing on the structure improved the global stiffness. This could be attributed to improved transmission of wind load to the foundation through the bracing. The comparison between symmetric and asymmetrical bracing found larger deformations were noted in an asymmetrically braced structure. The study also investigated the relationship between connection stiffness and the global stiff- ness. The connections between the timber elements were simulated by springs between the elements as shown in Figure 2 below.
Figure 2: 3D model showing modelled spring stiffness at connections [2].
Each spring element consisted of six degrees of freedom, three for slip and three for rotational degrees of freedom. The symmetric structure was subjected to prescribed dis- placement of 0.5m at the top of the structure for two types of connections, pinned and fully moment stiff connection. Thereafter, the total reaction forces at the foundation level were determined for each of the two cases. The test was conducted for a structure without any rotational stiffness and for another with rotational stiffness. The result showed that the total reaction was much larger in the structure with rotational stiffness as compared to the one without as shown in the Figure 3 below.
Figure 3: Comparison of reaction forces from Model with rotational stiffness to one without rotational stiffness for multi-storey timber structure with prescribed displacement at the top [2].
From the results it was clear that the rotational stiffness of the connections should be a significant factor in structural design of a multi-storey timber building. The study also investigated the relationship between the connection stiffness and the horizontal reaction force at the top of the structure. Figure 4 shows the relationship between rotational spring stiffness and the horizontal reaction force at the top of the structure caused by prescribed constant horizontal displacement at the top.
Figure 4: Relationship between connection stiffness and reaction force for multi-storey timber structure with prescribed displacement at the top [2].
The results showed that initially a small increase in connection stiffness caused a large increase in the reaction force at the foundation but further increase in stiffness did not result in a steep increase in the reaction force, as was observed initially. Eventually, in- crease in rotational stiffness did not result in an increase in the reaction.
Kuai [2] research modelled the connections as spring elements in 3D beam element model, which according to the work by Nonbo [1], do not generate accurate forces as the analysis only takes into consideration material properties in the fibre direction. Therefore the results from Kuai could be improved by conducting the 3D adaptive design.
Michael Dorn et al [3] study focused on experiments on dowel-type timber connections.
Johansen theory which is the basis of the mechanical failure of the connections does not account for potential brittle failure modes. The theory is the base of Eurocode 5 [6] yield model. This model lacks in stringent mechanical basis. According to the theory, when determining the stiffness of the connection, the thickness of the timber is not considered.
Slim as well as thicker connections are considered to have the same stiffness which is not a realistic assumption. The design of multi-dowel connections is based on the behaviour of a single dowel. The ultimate load and connection stiffness are both derived by scal- ing up by an effective number of acting dowels, depending on the distances between the dowels. The aim of the work was to study the load-carrying behaviour of the connection in detail, including all loading stages, from the initial contact between dowel and wood up to the ultimate load and failure. The researchers carried out 64 steel-to-timber dowel type connections that were loaded parallel to the fibre direction. The experiments were carried out as shown in the Figure 5 below.
Figure 5: Typical dowel-type timber connection with a single dowel; photo of a specimen during conduction of the experiment (left), sketch (right) [3].
The wood sections selected for the experiment varied in wood density and geometric properties. The study also investigated the effects of dowel roughness and lateral re- inforcement. The results from the study showed that connection of higher density had significantly higher ultimate loads as compared to those of lower density. Further to this, denser connections were found to be more prone to brittle failure than connections in light wood. With regard to dowel roughness, the study found that increase in rough- ness occasioned an increased in ultimate capacity of the connection. In summary, the study showed that the comparison of the experiments with the current state of design in structural timber engineering in Eurocode 5 [6] is partially in agreement of the design values from the test results. The distinction between the failure modes with the absence or creation of plastic hinge(s) is in accordance with Eurocode 5 [6].
The three studies discussed in this section thus far form a foundation this research.
Michael Dorn et al [3], and Nonbo [1] studies outline key design aspects that need to be taken into consideration during the analysis and design of the connection. The findings from these two researches will be critical in the adaptive design of the critical connection in Kuai [2] study which this study seeks to further. With regard to numerical simulation of the connection, the study below by Porteous et al [4] will illustrate the basis of the connection simulation as a series of three member symmetrical connections.
According to Porteous et al [4], where a connection may have multiple shear planes and the connected members are at varying angles to each other, the resistance of each shear plane is derived on the assumption that, the shear plane in the connection forms part of a series of three member connections and the connection strength is determined by combining the strength values of those failure modes that are compatible with each other.
Eurocode 5 requires that failure modes (a), (b), (g) and (h) in Table 10.2 or modes (c), (f) and (j/l) in Table 10.3 must not be combined with other failure modes.
The research focused on carrying out simulation of a five member single bolt connection that was subjected to design axial forces acting in the directions shown in Figure 6 (a).
Figure 6: Multiple Shear Plane Connection Procedure [4].
Members 1, 2 and 3 were all specified to have different properties. From the application of statics the value of the design force and its direction in each shear plane was derived and illustrated in Figures 6 (f) - (g). The fastener in this case was assumed to be rigid.
Using this approach to determine the strength of each shear plane, the three member connections were formed. The set up was such that the central member of the series of three member symmetrical connections was the actual joint member and the outer mem- bers were its adjacent members. For instance to form the three member double shear connection between member 1 and 2, member 1 was added to the other side as shown in Figure 6 (c). The other members were ignored. With this new arrangement the strength of shear plane x was calculated using the strength equations Eurocode 5 [6] tables 10.2 and 10.3 ensuring that the embedment strength of each connected member takes into consideration the angle of the shear plane load relative to the direction of the grain.
For shear plane y between members 2 and 3, the three member symmetrical connection was formed around member 3 using member 2 as the outer member as shown in Figure 6 (d) and the other members were ignored. The strength of the shear joint was derived using Eurocode 5 [6] as was done for the shear plane x. The results showed that the resultant force in shear plane y was half the value of the force in the central member and acted along the direction of the central axis of that member. To ensure that all failure modes were considered, a double shear connection with member 2 as the central member was set up as shown in Figure 6 (e).
For this study the simulation procedure by Porteous et al [4] will be utilized in order for the connection design to be carried out with respect to Eurocode 5. The FEM analysis will be done by setting up a series of three-member symmetrical connections as illustrated.
3 Theory
This chapter will discuss wood as a construction material and its properties along with the science behind the design of the steel-to-wood connection.
3.1 Wood
Wood comes from trees and the structure of the wood is determined by the circumstances to which the tree is exposed to during its life span before it is harvested. The tree stem grows in annular rings that strengthen the tree against wind loads. Trees also undergo internal pre-stressing to minimize the risk of crushing the wood at high compressive stresses. The fibre structure and mechanical properties of wood material are critical for timber used in construction.
3.1.1 Structure of wood
Figure 7: Wood structure.
Wood is made up of five main layers, pith, heartwood, sapwood, cambium and bark, as illustrated to the left in Figure 7. The bark is the outer most layer of the tree. It is made up of layers of dead cells that form the hard outer covering of the tree that protects it against damage from parasites, herbivorous animals and diseases, as well as dehydration and fire. The cambium which is the layer between the sapwood and the bark form new cells through cell division. The wood cells, shown to the right in Figure 7, transport mineral solutions and nutrition between cells through small openings (pits) on the sides
that connect them. The pits in the wood cells govern the moisture permeability of different species of trees. The heartwood is composed of older, harder non-living central wood that is usually darker, denser, less permeable, and more durable than the surrounding sapwood.
Heartwood is the main source timber. The sapwood transports liquids to the cells. These cells in the sapwood store substances that protect the tree against diseases. The pith is composed of soft, spongy cells, which store and transport nutrients throughout the tree [17].
3.1.2 Natural defects in wood
Wood has natural defects that are caused due various conditions that occur during its formation. Some of the key defects in wood are; knots, spiral grain angle, juvenile wood, and reaction wood.
Figure 8: Knot.
Knots occur when the fibres in the wood are no longer continuous in the main fibre direction. A typical knot is shown in Figure 8. The size, shape and location of the knots affect the mechanical properties of timber.
Figure 9: Spiral grain angle.
Spiral grain angle, as in Figure 9, is the angle between the fibre direction and the direction of the pith. The fibres of a tree tend to grow in a spiral around the truck of the tree.
During growth the spiral grain develops to a maximum value of between 3◦ - 4◦ within the first ten annual ring growths. The strength of the wood is reduced when the grain angle exceeds 3◦.
Figure 10: Reaction wood.
Reaction wood refers to abnormal wood tissues that are generated in a tree trunk due to subjection to compression or tension during their growth. This compression or tension is mostly caused by wind load on the tree. Reaction wood can result in abnormal wood tissue as in Figure 10.
Juvenile wood as the name suggests is not mature, according to studies [17], the cells in mature wood may be three to four times longer than those in juvenile wood. As such, juvenile wood has less desirable mechanical properties. It is also reported that juvenile wood shrinks along the grains nearly ten times as much as the mature wood.
3.1.3 Mechanical properties of wood
In most cases wood specimens are divided into small clear specimens and large specimens that have the anomalies discussed in chapter 3.1.2 above. Clear specimens have straight wood fibres that are clear of anomalies. The mechanical properties of the wood determine the size of forces that the timber members can withstand.
As shown in Figure 11, the structure of wood is a tube shaped cell which results in different properties in different fibre directions. This means that wood is essentially anisotropic.
It is therefore expected that compression strength parallel to the fibre direction is much higher than compression strength perpendicular to the fibre direction due to the tubular shape of the cells. With this in mind, it is clear that wood has different properties in different directions thus it is very important to keep track of fibre direction with regard to the stresses in the wood.
Figure 11: Wood cell for mature wood.
Wood is considered as an largely orthotropic material. This means that wood is consid- ered to have different properties in various orthotropic material directions, which means longitudinal l, radial r and tangential t directions referring to the fibre direction and the annual rings of the tree system. The radial and tangetial directions are often reduced to one direction considered perpendicular to the fibre direction. Figure 13 shows the or- thotropic material directions. With respect to this, there are five main properties that are important in the determination of the mechanical capabilities of timber:
• Tension parallel to the fibre direction, σt,0,d.
• Tension perpendicular to the fibre direction, σt,90,d.
• Compression parallel to the fibre direction, σc,0,d.
• Compression perpendicular to the fibre direction, σc,90,d.
• Shear stress to the fibre direction, σv,d. 3.1.4 Linear elasticity
Linear elasticity is described as the relation between stresses and strains, which is often referred to as the constitutive relation. For one dimensional linear elastic material behav- ior, this could be described as increased stresses σ will result in proportional increased strains and also decreased stresses σ will result in decreased strains , as in Figure 12.
The same principle applies for two and three dimensional linear elastic materials [15].
Figure 12: Linear elastic constitutive relation.
As previously mentioned wood is an orthotropic material and the three dimensional elastic behavior is defined by Hooks law for the local coordinate system, showed in Figure 13. The three directions in the local coordinate system are defined by the longitudinal direction l, the tangential direction t and the radial direction r. The linear relation between local stresses and local strains is described in equation (3.1), which can be rewritten as the local elastic strain in equation (3.2) and expanded to equation (3.3). The including variables in equation (3.1) and (3.2) consists of the local stress matrix ¯σ, the local constitutive matrix D, the local compliance matrix ¯¯ C and the local elastic strain matrix ¯e. [18].
¯
σ = ¯C−1¯e= ¯D ¯e (3.1)
¯
e = ¯C ¯σ (3.2)
Wood has different mechanical properties in various directions according to equation (3.3) and Figure 14. The shear modulus E represent the orthotropic plane lr, lt and rt. Youngs modulus G represent the orthotropic directions l, r and t. Poissons ratio v varies depending on the directions of the orthotropic material lr, rt, lt, tl, tr and rt [18].
Figure 13: Local coordinate system for wood.
ell
err
ett
elr
elt
ert
=
1 El
−vrl Er
−vtl
Et 0 0 0
−vlr El
1 Er
−vtr Et
0 0 0
−vlt El
−vrt Er
1
Et 0 0 0
0 0 0 1
Glr 0 0
0 0 0 0 1
Glt
0
0 0 0 0 0 1
Grt
σll σrr σtt σlr σlt σrt
(3.3)
3.1.5 Glulam
According to Eurocode 5 [17], glulam is one of the oldest wood products that has been used in construction over the years. It consists of at least four laminations, usually finger jointed sawn boards bonded together using adhesives. Glulam is preferred in timber construction as the beams can be produced to required depths. Glulam can be homogeneous or designed to match the stress regions in the beam as shown in Figure 14. Normal thickness for glulam lamination is 45 mm. The maximum width of each lamina is 215 mm. Glulam beams are especially prone to tensile failure parallel to the grain when subjected to bending. Shear failure is also a common cause of failure of glulam beams, Eurocode [17]. For curved and tapered glulam beams, high tensile stresses perpendicular to the grain are critical in the design. In this study we will focus on the high tensile stresses on the peripheral of the bolt holes.
Figure 14: Glulam beam.
3.2 Connection Design
The strength of a structure is dependent on the strength of its members as much as the strength of its connections. The connection should fulfill the following requirements to be adequate.
3.2.1 Functional requirements of a connection
To design an adequate connection the design calculations are aimed that ensuring that the connection can adequately transfer axial forces, moments and shear forces from one structural member to another with acceptable deflections, rotations and adequate safety at a reasonable cost. However according to Madsen [19], this does not cover real require- ments for a connection to be adequate.
As such, the key functional requirements for a connection to be adequate are discussed in depth below.
3.2.1.1 Strength. The maximum forces acting in individual members should be ad- equately transmitted from one member to the next. It is imperative that the designer understand the path through which the forces in the structure are to be transmitted.
Ideally the connection should withstand the maximum moment developed in the adjacent members. In cases where holes have to be drilled in the main member to allow for con- nections, stresses develop around the periphery of these holes leading to localized stresses that reduce the mechanical capacity and stiffness of the main member. It is important to take into account these local stresses developed in order to carry out a realistic design. In this case an FE model that indicates exact locations of the localized stresses will deter- mine these critical stress areas. It will also allow for prediction of expected failure modes, Borg et al [19].
3.2.1.2 Stiffness. Slip which is the local deformation experienced at a connection affects the stiffness of the connection [20]. This local deformation which occurs between the structural members in the connection is what is referred to as slip. In this study we will refer to the connection stiffness also as the local stiffness of the timber structure.
This study will focus on the local stiffness of the timber structure as the deformations that take place at the connection have been found to somewhat contribute to the global stiffness of the structure, Kuai [2].
3.2.1.3 Failure Modes. Generally in design, brittle fracturing of members and con- nections is avoided. Desirable design allows for ductile failure as this mode of failure allows for potential energy to be absorbed in form of the displacements in the event of earthquakes or wind resonance. During the excitation by these excessive loads, the struc- ture may be damaged severely but it may not necessarily collapse. This makes the ductile failure mode safer as it allows for occupants to escape. On the other hand brittle failure mode does not allow for storage of energy and failure of members and connections is almost instantaneous once the structure is exposed to excessive loading, Borg et al [19].
3.2.1.4 Load Reversal. In cases where a structure is exposed to earthquake or wind load excitation, the structure should withstand the loading and at the same time revert back to initial state once the excitation has terminated, Borg et al [19]. It is expected that an adequate connection design will allow for some elasticity in the connection and members.
3.2.1.5 Ease of Construction. In this study we will focus on using slotted-in steel plates in the connection. The connection should be such that only an adequate size of holes is provided for installation, Borg et al [19]. Tolerances on the sizes of holes should be within tolerances. Oversized holes in timber especially may lead to development of unnecessary forces that may have to be transmitted by non-structural elements such as plaster, wall paper or glass. This could be very dangerous and may lead to failure.
Prefabricating of member and even a test fit in the workshop may help resolve this.
3.2.1.6 Aesthetics. Connections should be aesthetically satisfying. In this study, the timber structure under consideration is a residential building, whose aesthetics play a large role in the success of the project. The size of the connections should be adequate to withstand the forces but not so large that they protrude.
3.2.1.7 Other functional requirements of timber connections. Though equally important to the design of an adequate connection, the following list of requirements will not be investigated in this study.
• Load transferred through directed contact. This refers to connections that allow for the member to carry a portion of the force to be transmitted. However in this study the load transfer is not in direct contact with the members.
• Ease of manufacturing: Connections should be easily manufactured and installa- tion, except in the case where there is a special requirement. For instance, in cases where timber structures are exposed to moisture during transportation caus- ing fitting problems during installation, the connections are designed taking into consideration shrinkage once the members are dry.
• Cost: An adequately designed joint will allow for a smaller member to be used in the design this in turn has an overall effect on the cost of the structure. An adequate connection design may lead to a more economical structure.
3.2.2 Johansen Theory of Timber Connections
According to Johansen [5] timber connections might consist of a bolt with a toothed washer between the faces of the joint to prevent slipping of the timber on either side. The function of the connection according to the study was summarized into:
• The dowel effect of the bolt, which depends upon its resistance to bending and the resistance of the wood to crushing.
• The tensional effect of the bolt, which depends upon its resistance to tension and friction between the abutting surfaces.
• The effect of the washer, which depends on its form and strength together with the resistance of the wood to crushing.
The elements of the joint were considered as:
• Resistance of the wood to crushing under a dowel.
• Friction.
• Effect of the washer.
The three elements were studied as detailed below:
3.2.2.1 Embedding Strength. The experiment set up as shown in Figure 15 was used to investigate the yield curves for the various types of wood. The yield curves were found to end in straight lines as shown in Figure 15. Where P is the load per unit length of the dowel, the pressure on the bearing surface of the hole was defined as, SH = P/d where d is the diameter of the dowel.
Figure 15: Yield curves for embedding test [5].
3.2.2.2 Friction. Three pieces of timber were clamped together as shown in Figure 16 and the axial force N measured by tensometers. The center timber was loaded with the force P and its slip g with relation to the outer timber members was measured. The P g curves and the variation in force N were plotted as shown in Figure 16 below. It was found that the coefficient of friction varied between 0.4 and 1.4 with an average value of 0.67.
Figure 16: Relation between slip g and force P [5].
3.2.2.3 Toothed Washer. The experiments were carried out with a clamp instead of a bolt but with the addition of the toothed washers between the abutting surfaces. At the beginning of the experiment the clamps were lightly tensioned and the force N at the joints was applied. The resultant curves in Figure 17 from the experiment show that N begins to grow where the curve begins to bend. The increase in load capacity due to the washer with a coefficient of friction of 2/3 was found to be expressed as in equation (3.4).
G = Pmax− 2 ·2
3N (3.4)
The values found for G divided by the embedded area, A, of the teeth alone give sH. The effective embedded area of these toothed washers, was therefore considered the effective are of the teeth.
3.2.2.4 Bolt Connections. The strength of a dowel connection was found to be partly dependent upon the resistance of the wood to the embedding of the dowel, i.e SH
and partly on the resistance of the dowel to bending i.e SB. As both the bending of the dowel and the embedding are plastic, the load capacity of a dowel connection can be formulated on this basis.
Figure 17: Relation between slip g and force P for a Toothed Washer [5].
In a single shear connection with a dowel of sufficient stiffness as shown in Figure 18 on the left. In case the dowel has adequate capacity, the bending moment of the dowel being zero, the equation of equilibrium for the dowel will give the expressions in equation (3.5).
Figure 18: Single shear connection with a dowel. The left side shows a dowel with sufficient Capacity, the right side shows a dowel with Insufficient Capacity.
P = SHdz (3.5)
Mmax = SHdx2 = 1
2SHdz2 (3.6)
Where,
x = z r1
2; l = z + 2x = z(1 +√
2) (3.7)
z = l(√
2 − 1) = 0.414l; x = 0.293l (3.8)
P = 0.414SHld (3.9)
In a single shear connection with a dowel of insufficient stiffness as shown in Figure 18 on the right, the dowel is not sufficiently stiff in bending. The plastic yield moment for a dowel with circular cross section is π/32 sBd3 and the transverse load is zero. The maximum moment in the dowel therefore can be considered to be Mmax and the transverse forces Q = 0 therefore,
P = SHdz (3.10)
Mmax = 1
2P z −1 2
P2 SHd = π
32SBd3 (3.11)
Therefore,
P = 0.44p
SBSH · d2 (3.12)
For a double shear connection with a stiff dowel in Figure 19 below the resulting formulas are:
P = SHdm when m ≤ 2l (3.13)
P = 2SHdl when m ≥ 2l (3.14)
Figure 19: Sufficient Dowel Capacity in a Double shear Joint.
A dowel of lesser stiffness bends in the centre as shown in Figure 20 on the left and yields at two points and outside these the state in the timber must be plastic as the movement of the rupture is due to the turning of the dowel.
P = 1
4SHl2+3 5SBd2
!rSH
SB (3.15)
Figure 20: Double shear connection with a dowel with insufficient Capacity Dowel.
However for a thinner dowel as shown in Figure 20 on the right, in the region between the yield hinges where the dowel has turned, the state of the wood must be plastic and the pressure against the dowel SHd, therefore
P = 2SHdz (3.16)
Further, the moments is given by,
Mmax+ Mmin = 2 π
32SBd3 = 1
2P z = SHdz2 (3.17) z = 0.442r SH
SB · d (3.18)
P = 0.885p
SBSH · d2 (3.19)
The conclusion was that the yield shear for a single shear connection is given by P = 0.44p
SBSH · d2 (3.20)
In summary according to Johansen theory by assuming plasticity in the wood and in the dowels or bolts, formulae are derived for the yield point and the ultimate load of connections with dowels, with bolts and with bolts and toothed washers.
3.2.3 Bolted Slotted-in Steel Plate Connection
Figure 21: Slotted-in steel plates.
Steel to wood connections are the most commonly used connections in glulam structures.A slotted-in steel connection is illustrated in Figure 21. This could be attributed to the improved mechanical properties of the connection. As sited by Uhre et al [21], the use of steel connection plates increases the moment carrying capacity at the connection. This study will focus on the critical stresses at the connection. According to Breyer et al [22], connections are classified by the direction of loading, for instance, shear connection exists where the load is applied perpendicular to the direction of the fastener. This type of connection can further be classified as either double or single shear connection. The later classification is dependent on the number of shear planes in the connection. A connection can have multiple shear planes and in such cases the design force experienced by the weakest shear plane is multiplied by the number of shear planes in the connection.
3.2.3.1 Why Slotted-in-Steel Plates? As discussed in section 3.1.5, glulam is prone to failure due to shear stress and high tensile stresses perpendicular to the grain. At any connection these stresses are critical and the connection design is aimed at ensuring that the forces caused by these stresses are below the yield strength of the materials at the connection. Since steel has tremendous tensile strength, especially compared to the traditional wood connectors, it offers a solution to the problem. The steel plates are used to transfer load between members. Aside from improved mechanical properties at the connection, slotted in steel plates have an additional advantage, that the steel plates are protected from fire. They are also aesthetically pleasing as they do not protrude from the main members.
3.2.4 Shear Capacity of steel-to-wood connections
The design capacity of a connection is given by fv,Rk, according to equation (3.21). The effects on strength due to load duration and service class is represented by kmod. Char- acteristic value for the shear is given by fv,Rk and partial coefficient for the material is represented by γM.
Fv,Rd = kmod· fv,Rk
γM (3.21)
The characteristic shear capacity for nails, dowels, bolts and screws per shear plane and per fastener is given by the minimum value for Fv,Rk. The characteristic capacity per fastener and share plane is given by Fv,Rk. Thicknesses of timber members is given by ti. The characteristic embeddment strength for the timber is represented by fh,i,k. The fastener diameter is given by d and the characteristic yield moment by My,Rk according to equation (3.8). Ratio between embedded strengths is represented by the character- istic strength of the middle timber member divided by the side member and it can be expressed as β = fh,2,k/fh,1,k. The characteristic withdrawal capacity of the fastener is given by Fax,Rk. The following equations (3.22), (3.23) and (3.34) correspond to the dif- ferent modes of failure in Figure 22 from Eurocode 5 for Steel to wood connections.
Figure 22: Failure modes for steel to timber connections [6].
Characteristic shear capacity for a double steel to wood connections with any thickness of the steel plate is given by equation (3.22). This equation corresponding to failure mode f, g, h in Figure 22.
Fv,Rk = min
fh,1,kt1d
fh,1,kt1d
"s
2 + 4My,Rk
fh,1,kdt21 − 1
#
+Fax,Rk
4
2.3pMy,Rkfh,1,kd +Fax,Rk
4
(3.22)
Characteristic shear capacity for a double steel to wood connection for thin steel plates with thickness that is equal or less then half fastener diameter, thickness of steel plate
≤ 0.5d is given by equation (3.23). The corresponding failure modes is j and k in accord- ing to Figure 22.
Fv,Rk = min
0.5fh,2,kt2d
1.15p2My,Rkfh,2,kd + Fax,Rk 4
(3.23)
Characteristic shear capacity for a double steel to wood connection for steel plates that is equal or thicker than the fasteners diameter, thickness of steel plate ≥ d is given by equation (3.24). The corresponding failure modes is l and m in according to Figure 22.
Fv,Rk = min
0.5fh,2,kt2d
2.3p2My,Rkfh,2,kd +Fax,Rk 4
(3.24)
3.2.5 Yield moment and characteristic embedment strength for dowelled and bolted connections
Characteristic value for yield moment for bolts and dowels is given by My,Rk in equation (3.25). The characteristic tensile strength for the fastener is given by fu in N/mm2 and the diameter of the fastener is represented by d in mm.
My,Rk = 0.3fud2.6 (3.25)
The characteristic embedment strength for d≤ 30 mm at angle α to the grain is given by fh,α,k, where fh,0,k is the embedment strength parallel to the fibre direction. The correction factor, depending wood properties is given by k90. Characteristic timber density is represented by ρk in kg/m3 and d is the fasteners diameter in mm.
fh,α,k = fh,0,k
k90sin2α + cos2α (3.26)
fh,0,k = 0.082(1 − 0.01d)ρk (3.27)
k90=
1.35 + 0.015d for softwood 1.30 + 0.015d for LVL 0.90 + 0.015d for hardwood
(3.28)
3.2.6 Bolt or dowel spacing requirements
For load carrying capacity of bolts or dowel parallel to the grain direction is the effective number of fasteners nef, according to equation (3.29). The number of fasteners in a row is given by n. Spacings between fasteners is represented by a1 in mm, according to Table 1 and Table 2. The diameter of the fastener is given by d in mm. Definition of edge and ends distances and spacing between fasteners are showed in Figure 23.
nef = min
n
n0.9 a1 13d
!0.25 (3.29)
For load perpendicular to the grain, nef = n for 0◦ ≤ α ≤ 90◦.
Figure 23: Definition of edge and ends distances and spacing between fasteners [6].
Table 1: Spacing and end/edge distances for bolts.
Spacing and end/edge distances Angle Minimum spacing or distance a1 (parallel to grain) 0◦ ≤ α ≤ 360◦ (4 + cos α )d
a2 (perpendicular to grain) 0◦ ≤ α ≤ 360◦ 4d
a3,t (loaded end) −90◦ ≤ α ≤ 90◦ max(7d; 80 mm) a3,c (unloaded end) 90◦ ≤ α ≤ 150◦ (1 + 6 sin α)d
150◦ ≤ α ≤ 210◦ 4d
210◦ ≤ α ≤ 270◦ (1 + 6 sin α )d a4,t (loaded edge) 0◦ ≤ α ≤ 180◦ max[(2 + 2 sin α)d; 3d]
a4,c (unloaded edge) 180◦ ≤ α ≤ 360◦ 3d
The tables above indicate the required spacing for bolted or dowelled connections.
Table 2: Spacing and end/edge distances for dowels.
Spacing and end/edge distances Angle Minimum spacing or distance a1 (parallel to grain) 0◦ ≤ α ≤ 360◦ (3 + 2 cos α )d
a2 (perpendicular to grain) 0◦ ≤ α ≤ 360◦ 3d
a3,t (loaded end) −90◦ ≤ α ≤ 90◦ max(7d; 80 mm) a3,c (unloaded end) 90◦ ≤ α ≤ 150◦ a3,t sin α
150◦ ≤ α ≤ 210◦ max(3.5d; 40 mm) 210◦ ≤ α ≤ 270◦ a3,t sin α
a4,t (loaded edge) 0◦ ≤ α ≤ 180◦ max[(2 + 2 sin α )d; 3d]
a4,c (unloaded edge) 180◦ ≤ α ≤ 360◦ 3d
3.3 Adaptive design
The finite element model of the timber structure created is quite large, therefore, mod- elling the entire structure as a 3D solid element model would result in very large run-time durations. Since the study is focused on the design of the critical connection, the study will focus on creating a 3D solid element model at the connection. The connection will have the orthotropic material properties of timber, therefore giving a more realistic view of the stresses. This technique has been used in previous studies such as the one carried out by Nonbo [1].
Adaptive design allows for the study to focus on the critical connection and ultimately yield the most realistic stresses at the connection which are of great interest to this study.
The stiffness of this 3D Solid element connection will be compared to that of a beam element connection. This information will provide better understanding on the effect of connection stiffness on the global structural behaviour of a timber frame structure.
3.4 The Finite Element Method
The Finite Element Method (FEM) is a numerical method for solving physical problems, often described with differential equations, in a approximate manner. This method can be used in several different fields, such as determine heat flows, stress distribution or predict the weather [15].
The course of action for the finite element method could be described in two main steps.
The first step is to formulate the physical problem using differential equations and thereby obtain a mathematical model. The second step is to use the mathematical model to con- duct a numerical analysis, which gives an approximate result for the physical problem.
In more detail, it can be described as follows; The mathematical model created from the physical problem consists of differential equations within a region enclosed by a boundary.
The differential equation gives a mathematical approach to the physical problem and the region describes the physical object, such as the area och the volume. Furthermore, the boundary describes the circumstances in the outskirts of the region. Before the numerical
analysis are conducted, the region are divided into several smaller parts called finite ele- ments, which consists of a region enclosed by a boundary. When all of the finite elements are clustered, they are called finite element mesh. The numerical analysis takes each individual finite element and the relation between them into account, which resulting in an approximate solution to the physical problem [15].
In addition to connections, beams and shells will be of great importance for this study.
Thereby is it important to understand how analysis of beams and plats is carried out ac- cording to the finite element method. By using the two previously mentioned main steps for the finite element method, the approximate solution is given. The first step contains establishment of strong formulation in order to derive the weak formulation. The second step is to get the approximate solution on terms of a FE-formulation from the weak for- mulation. Below follows two usual methods to calculate beams and plates according to the finite element method.
3.4.1 Beam theory
A beam are defined as a three dimensional element with a length that is significantly greater than the width and height. Furthermore, it can withstand twisting, axial and lat- eral loading. Although beams are three dimensional in reality, they are often considered to be one dimensional, due to dominating extension in axial direction, in order to simplify the calculations [15], [23].
If the deformations are assumed to be small for the considered beam element, the plane section will be normal to the longitudinal axis after deformation and thereby could the shear deformations be neglected. This method is called Euler-Bernoulli beam theory and are illustrated on the left side in Figure 24. For high beams will the plane section rotate and therefor not be a normal to the longitudinal axis after deformation. For this case are the shear deformations not neglected and the Timoshenko beam theory, illustrated on the right side in Figure 24 is used.
Figure 24: Euler-Bernoulli beam and Timoshenko beam.
