Master Thesis

Numerical Simulation of Timber Connections with Slotted-in Steel Plates

Michael Nygaard Nonbo s052719

July 8th 2010

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Preface

This master thesis within the field of constructional engineering has been made in order to try and clarify some of the accuracy issues by designing timber connections in accordance with the conventional hand calculation methods.

The idea for the thesis was presented to me by my supervisor, Sigurdur Omarsson, and the logical issue with the approach of the conventional hand calculation method of distributing the fastener forces in a connection group from a geometric point of view and not in respect to the stiffness of the wood material around the fasteners, appealed to me and I took an interest in further investigating the problematic.

In that connection I would like to thank Sigurdur Omarsson for his great help through the entire process of this thesis. He has been very motivating and a great support by always taking time to answer questions.

Lyngby, July 2010

_____________________________________

Michael Nygaard Nonbo

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Summary

This master thesis describes the hand calculations and FEM-simulation of a loaded three hinged glulam frame with moment stiff corners with a view to clarify differences in the distribution and magnitude of the contact forces between the dowel and the timber in worst load corner connection. The hand calculations are carried out in accordance with Eurocode and an applied calculation method from (Larsen & Enjily 2009) under the assumption that the wood material behaves elastically. The dowel forces found by the hand calculation method are based on cross sectional forces by considering the frame as consisting of beam elements. In the numerical analysis of the frame the wood material is also assumed having elastic behaviour. However, the simulation is here carried out on the frame consisting of beam element but with the worst loaded corner fully 3D-modelled.

The analysis show that the dowel forces found by hand calculation in general are perpendicular to the direction to the respective connection groups and that the largest of the peripheral dowel forces (80.9 kN) is about twice as big as the smallest one (35.8 kN). With regards to the FEM-simulation the dowel force are pointing more in the fibre direction than those found by hand calculation and the largest of the peripheral dowel forces (95.9 kN) is here about 7 times bigger than the smallest one (13.6 kN). From the FEM-simulation it is shown that you cannot expect dowel loaded perpendicular to the fibre direction to take up nearly as much of the moment induced force in the connection as by the hand calculation method.

Moreover, a FEM-simulation is carried out where the frame corner in question is subjected to shrinkage corresponding to a reduction of 3% of the moisture content of the wood. This causes radial stresses exceeding the strength of timber and dowel forces exceeding the load carrying capacity of the dowel connection.

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Resume

Dette kandidatspeciale behandler hndberegninger p, og en FEM-simulering af en belastet tre-hngslet limtrsramme med momentstive hjrner med henblik p at klarlgge forskelle i fordelingen af, og retningen p kontaktkrfterne mellem dyvler og limtr i den vrst belastede hjrnesamling. Hndberegningerne er udfrt i overensstemmelse med Eurocode og en anvendt beregningsmetode fra (Larsen & Enjily 2009) under antagelse af at tret opfrer sig elastisk. Dyvelkrfterne fundet ved hndberegning bygger p snitkrfter fundet ved at se rammen som bestende af bjlkeelementer. I den numeriske analyse af rammen antages tret ligeledes at opfrer sig elastisk, dog er simuleringerne her foretaget p rammen bestende af bjlkeelementer, men hvor det vrst belastede rammehjrne er fuldt 3D-modelleret.

Analyserne viser at dyvelkrfterne fundet ved hndberegning generelt set er vinkelrette p retningen til centrum af de pgldende dyvelsamlinger og at den strste af de perifere dyvelkrfter (80.9 kN) er ca. dobbelt s stor som den mindste (35.8 kN). Hvad angr FEM-simuleringen peger dyvelkrfter her mere i fiberretningen end dem fundet ved hndberegning og den strste af de perifere dyvelkrfter (95.9 kN) er her ca. 7 gange s stor som den mindste (13.6 kN). Ud fra FEM-simulering er det vist, at man ikke kan forvente at dyvler lastet vinkelret p fiberretning tager nr s stor en del af de momentforrsagede krfter i samlingen som ved hndberegningsmetoden.

Ydermere er der foretaget en FEM-simulering, hvor det pgldende rammehjrne udsttes for svind svarende til en reduktion af trets fugtindhold p 3 %. Dette forrsager radiale spndinger, der overskrider styrken af tret og dyvelkrfter, der overstiger breevnen af dyvelsamlingerne.

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Content

Preface .............................................................................................................................. 1Summary ........................................................................................................................... 2Resume ............................................................................................................................. 3List of figures ................................................................................................................... 6List of tables ..................................................................................................................... 91 Introduction ............................................................................................................. 10

1.1 General ............................................................................................................. 101.2 Problem identification ...................................................................................... 101.3 Structure of the thesis ...................................................................................... 121.4 Literature review .............................................................................................. 12

2 Hypothesis ............................................................................................................... 133 Geometry ................................................................................................................. 14

3.1 Frame and overall corner dimensions .............................................................. 143.2 Dowel placing and numbering ......................................................................... 163.3 Dimension of the steel plates ........................................................................... 17

4 Materials .................................................................................................................. 184.1 Wood ................................................................................................................ 184.2 Steel ................................................................................................................. 19

5 Load combination .................................................................................................... 206 Hand calculations .................................................................................................... 23

6.1 Method ............................................................................................................. 236.2 Results .............................................................................................................. 27

6.2.1 Cross sectional forces ............................................................................... 276.2.2 Deformations ............................................................................................ 296.2.3 Dowel forces ............................................................................................. 306.2.4 Load carrying capacity of the dowel connections .................................... 32

7 FEM-analysis .......................................................................................................... 347.1 Structure of the model ...................................................................................... 347.2 Results .............................................................................................................. 42

7.2.1 Deformation .............................................................................................. 42

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7.2.2 Stresses ..................................................................................................... 437.2.3 Dowel forces ............................................................................................. 48

8 Comparing hand and FEM-calculations .................................................................. 509 Moisture load .......................................................................................................... 52

9.1 Applying moisture load ................................................................................... 539.2 Results .............................................................................................................. 54

9.2.1 Deformations ............................................................................................ 549.2.2 Stresses ..................................................................................................... 569.2.3 Dowel forces ............................................................................................. 59

9.3 Combining moisture with other load cases ...................................................... 609.3.1 Dead and moisture load combination ....................................................... 609.3.2 Design and moisture load combination .................................................... 62

10 Discussion and Conclusion .................................................................................. 6411 Future research .................................................................................................... 6512 References ........................................................................................................... 66Appendix A ...................................................................................................................... 1Appendix B ....................................................................................................................... 5Appendix C ..................................................................................................................... 15Appendix D .................................................................................................................... 16Appendix E ..................................................................................................................... 17Appendix F ..................................................................................................................... 18Appendix G .................................................................................................................... 22Appendix H .................................................................................................................... 25Appendix I ...................................................................................................................... 26Appendix J ...................................................................................................................... 27Appendix K .................................................................................................................... 28Appendix L ..................................................................................................................... 29Appendix M .................................................................................................................... 32Appendix N .................................................................................................................... 33Appendix O .................................................................................................................... 36Appendix P ..................................................................................................................... 37Appendix Q .................................................................................................................... 38

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List of figures

Figure 1.1: Collapsed connections of the Siemens Super Arena in Ballerup. ................ 10Figure 1.2: A three hinge frame with moment stiff corners where the left corner has been fully modelled for further analysis. ........................................................................ 11Figure 1.3: Principle construction of the corner section of the frame. ........................... 11Figure 2.1: Sketch showing, in principle, the difference in load distribution on the dowels between the hand calculation method and the numerical analysis. .................... 13Figure 3.1: Dimensions expressed by variables and key points on the left of the frame and the left corner. .......................................................................................................... 15Figure 3.2: Numbering of the dowels with connection group 1 in the beam part and connection group 2 in the column part and the required distances in accordance with the Eurocode 5. ..................................................................................................................... 16Figure 3.3: Key points on the steel plates and the distance from the timber edge to the steel plate edge. .............................................................................................................. 17Figure 4.1: Illustration of the three orthotropic material directions of a wooden beam longitudinal, l, radial, r, and tangential, t. ....................................................................... 19Figure 5.1: Illustration of the pressure coefficients for external pressure used on the frame. .............................................................................................................................. 21Figure 5.2: Dead, wind and snow load son the frame. ................................................... 22Figure 6.1: Illustration of the dowel group and the load action on the dowel group 1. . 23Figure 6.2: Failure modes for connection with two slotted-in steel plates. .................... 26Figure 6.3: The distribution of normal forces in N in the frame due to design load. ..... 28Figure 6.4: The distribution of shear forces in N in the frame due to design load. ........ 28Figure 6.5: The moment distribution in Nm on the frame due to design load. .............. 29Figure 6.6: Deformation of the frame due to design load. ............................................. 29Figure 6.7: Magnitude and direction of the hand calculated dowel forces..................... 32Figure 7.1: Part 1 Corner peace of the column. ........................................................... 35Figure 7.2: Part 2 Corner peace of the beam. .............................................................. 35Figure 7.3: Part 3 One of the steel plates slotted into the wooden column and beam. 35Figure 7.4: Part 4 One of the steel dowels. .................................................................. 35Figure 7.5: Part 5 One of the wooden ring parts which are placed in the holes of the larger wooden corner parts. ............................................................................................ 35Figure 7.6: Part 6 One of the steel rings which are placed in the holes of the steel plates. .............................................................................................................................. 35Figure 7.7: The interface to define the material properties of the wood material. ......... 36Figure 7.8: The assembling of the frame with numbering of Part 1, 2, 7, 8, 9 and 10 and the material orientations indicated by yellow coordinate axis. ...................................... 37Figure 7.9: Contact surface between dowel and hole where the two mesh grids do not fit perfectly together. ........................................................................................................... 39

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Figure 7.10: Meshing done in 20-node hexahedrons (black nodes at the left) with 27 Gauss points (red nodes at the right). ............................................................................. 40Figure 7.11: Meshing of Part 1. ...................................................................................... 41Figure 7.12: Meshing of Part 2. ...................................................................................... 41Figure 7.13: Meshing of Part 3. ...................................................................................... 41Figure 7.14: Meshing of Part 4. ...................................................................................... 41Figure 7.15: Meshing of Part 5. ...................................................................................... 41Figure 7.16: Meshing of Part 6. ...................................................................................... 41Figure 7.17: Deformation of the frame from the design load (magnitude 20 times). .... 42Figure 7.18: Deformation of the frame corner and the local deformation of the dowel holes 3 and 9 in CG 1 and 7 and 9 in CG2 magnified 20 times. .................................... 43Figure 7.19: Significant rise in value of longitudinal stresses at the edge of an element shown in principle. ......................................................................................................... 44Figure 7.20: The longitudinal stresses in the corner of the frame with tension and compression indicated by red and blue colours, respectively. ....................................... 45Figure 7.21: Longitudinal stresses in the timber around dowel 3 and 9 in CG1 and 7 and 9 in CG2. ........................................................................................................................ 45Figure 7.22: Radial stresses in the frame corner. ........................................................... 46Figure 7.23: The radial stress distribution on connection group 1 and 2. ...................... 46Figure 7.24: Shear stresses in the timber in the corner connection. ............................... 47Figure 7.25: The dowel forces acting on the wood material .......................................... 49Figure 8.1: The dowel forces from both the hand calculation and FEM-analysis plotted as black and red arrows, respectively, along with some of the angle differences. ......... 51Figure 9.1: Boundary conditions indicated by red arrows on the frame corner subjected to moisture load. ............................................................................................................. 54Figure 9.2: The moisture load is applied as an additional material behaviour to the wood material. .......................................................................................................................... 54Figure 9.3: Deformations of the frame corner and dowel 3 and 9 in CG1 and 7 and 9 in CG2 (magnified 20 times). ............................................................................................. 55Figure 9.4: Deformation of the frame corner around CG2 seen from the outer edge of the frame (magnified 20 times). ..................................................................................... 56Figure 9.5: The longitudinal stresses due to decrease in the timber of 3%. ................... 57Figure 9.6: Radial stresses in the frame corner and in the timber around dowel 3 and 9 in CG1 and 7 and 9 in CG2 due to moisture loading. .................................................... 57Figure 9.7: Shear stresses in the timber in the corner connection. ................................. 58Figure 9.8: Sketch showing magnitude and direction of dowel forces caused by moisture load (3% shrinkage), with magnitude and angle of the five dowel forces which exceed the load carrying capacities. ........................................................................................... 59Figure 9.9: The dowels forces caused by the combination of dead and moisture load (3% shrinkage) shown by purple arrows. The magnitude and angle of those dowel forces exceeding the load carrying capacity are also shown. ......................................... 62

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Figure 9.10: The dowels forces caused by the combination of design and moisture load (3% shrinkage) shown by purple arrows. The magnitude and angle of those dowel forces exceeding the load carrying capacity are also shown. ......................................... 63

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List of tables

Table 4.1: Characteristic and design strength parameters of GL32 [MPa]. ................... 18Table 4.2: Stiffness properties of the wood material. ..................................................... 18Table 6.1: Displacement of the frame top and corners in mm from hand calculation due to design load. ................................................................................................................. 29Table 6.2: The x- and y-components and magnitude of the moment contribution to the resulting dowel force. ..................................................................................................... 30Table 6.3: Hand calculated magnitudes and angles of the dowel forces in relation to the fibre direction due to design load. .................................................................................. 31Table 6.4: Magnitude and angle of dowel force, load carrying capacity and load carrying capacity check of each of the 18 dowels. ......................................................... 33Table 7.1: Displacement of the frame top and corners in mm from the numerical analysis due to the design load. ...................................................................................... 42Table 7.2: Magnitude and angle of the dowel forces in relation to the fibre direction due to design load. (Abaqus) ................................................................................................. 48Table 7.3: Magnitude and angle of dowel force from the design load, load carrying capacity and load carrying capacity check of each of the 18 dowels. ............................ 49Table 8.1: Comparison of reaction force found by Abaqus and hand calculation. ........ 50Table 9.1: Magnitude and angle of the dowel forces in relation to the fibre direction due to moisture load. ............................................................................................................. 59Table 9.2: Magnitude and angle of dowel forces due to 3% decrease in timber moisture content, load carrying capacity and load carrying capacity check of each of the 18 dowels. ............................................................................................................................ 60Table 9.3: x-, y- and z-components and magnitudes of the dowel forces cause by the dead load of the frame. ................................................................................................... 61Table 9.4: Magnitude and angle of dowel forces due to a combination of dead and moisture load, load carrying capacity and load carrying capacity check of each of the 18 dowels. ............................................................................................................................ 61Table 9.5: Magnitude and angle of dowel forces due to a combination of design and moisture load, load carrying capacity and load carrying capacity check of each of the 18 dowels. ............................................................................................................................ 63

Introduction

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1 Introduction

1.1 General Through the ages wood has been one of the most important construction materials and it is still widely used, for instance in small or large roof or frames constructions as the ones used in the Siemens Super Arena in Ballerup. With large timber constructions comes heavy loaded connections and in these kinds of constructions the laterally loaded connections often are quite complex. The connections often involve two or more slotted-in steel plates which are connected to the timber members with steel dowels or bolts. The forces are typically transferred through the connections by mechanical contact or friction between the dowel/bolts and the timber. For complex connections with unknown line of force actions or connection exposed to climate changes (e.g. moisture content variations) the stresses in the connection and around the dowels can be difficult to determine (Omarsson, Dahlblom & Nygaard 2010). Incorrect design or miscalculations may lead to critical under dimensioning with fatal consequences like to collapse of the Siemens Super Arena.

Figure 1.1: Collapsed connections of the Siemens Super Arena in Ballerup.

1.2 Problem identification The way timber connections are calculated by hand is based on isotropic elastic behaviour of the timber and a geometric distribution of the dowel/bolt forces. Thereby dowels loaded perpendicular to the fibre direction should be able to obtain the same forces as those loaded in the fibre direction and that is not realistic since stiffness of the wood is highly dependent on the fibre direction.

Since wood takes up water from the surrounding air it is sensitive to climate changes and the change of moisture content causes swelling/shrinkage of the timber. The dowel/bolts in the connection are then retaining the deformations of the timber resulting

Introduction

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in additional forces in the fasteners. This poses a risk that the resulting forces in the fasteners are larger or having more critical angles to the fibre direction the forces from the mechanical loads the connection is design for. Moreover, the milling of grooves for the slotted-in steel plates reduces the timber cross section but increases the surface area and thereby increasing the sensitivity to climate changes.

A fastener force comparison of FEM-simulation and the hand calculation method will be carried out on a three hinge frame with moment stiff corners as the one shown in Figure 1.1. The hand calculation will be carried out on a beam model of the frame and the FEM-simulation on a beam model of the frame with a fully solid modelled corner (see Figure 1.1 and Figure 1.2).

The main emphasis of the analysis carried out in this thesis is on finding the resulting dowel force found by hand calculation and FEM-simulation and comparing the results. Moreover, assessing the effects of moisture content differences on the dowel forces and the stress distributions.

Figure 1.2: A three hinge frame with moment stiff corners where the left corner has been fully modelled

for further analysis.

Figure 1.3: Principle construction of the corner section of the frame.

Introduction

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1.3 Structure of the thesis The thesis is build so the subjects common for both analysis methods are described and defined first. This relates to the material properties, the geometry and load combination on the frame. Afterwards the hand calculations will be gone through with a description of the method and a presentation of the results, with the main emphasis on the dowel forces in the left corner connection. The FEM-analysis is then gone through with a description of the modelling process and a presentation of the results, again with the main emphasis on the dowel forces and here the stresses in the left corner connection as well. The results from the two analysis methods are then compared and discussed. Lastly the results of a FEM-analysis of the frame corner exposed to drying is presented and discussed.

1.4 Literature review FEM-analysis of timber connections has earlier been done, e.g. has the collapse of the abutment connections been analysed by Jessen and Mougaard in (Jessen & Mougaard 2003), (Jessen & Mougaard 2003) and (Jessen & Mougaard 2004). They here implement beam elements instead of dowels and applies some spring effect in the connection with the perimeter of the holes. In (Hafsteinsson 2009) nailed bracket connection has been analysed in Abaqus to find nail to timber stresses.

Hypothesis

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2 Hypothesis

When performing a FEM-analysis on the corner connection in question a different outcome than that for the hand calculations must be expected, since the load distribution on the dowels are quite different from the two methods. In the hand calculation method used in this thesis ((Johansen 1949) and (Larsen & Enjily 2009)) the loads are distributed between the dowels in relation to the geometry regardless of the material stiffness in the respective load direction. I.e. a dowel loaded perpendicular to the fibre direction (where the stiffness of the timber is very low) will be subjected to the same force as a dowel loaded parallel to fibre direction (where the stiffness of the timber is high) if it has the same distance to the centre of the connection group. However, this is not realistic and it must be expected, for the analysis of the corner connections, that the angles of the dowel forces will tend to point more in the fibre direction and that the dowels loaded in the fibre direction will take up more force than the ones loaded perpendicular to the fibre direction of the timber. The difference in the load distributions in the dowels between the hand calculation method and the numerical analysis is principally sketched in the figure below. Two pieces of timber are here connected by a slotted-in steel plate and four dowels in each of the two connection groups. The dowel forces are distributed evenly in the four dowels for the hand calculation method due to the double symmetric connection groups, but in the numerical analysis the dowels in the top and bottom of the connection groups will take up much more load than those in between because they are loaded in the fibre direction of the timber.

Figure 2.1: Sketch showing, in principle, the difference in load distribution on the dowels between the hand calculation method and the numerical analysis.

Geometry

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3 Geometry

The frame and corner in question is build up as shown in Figure 1.2 and the dimensions of the frame and the left corner, where the dowel forces are to be found, will obviously have to be determined before any calculation can be made. For the hand calculation only a beam frame, to determine the cross sectional forces at the left corner, and the dimensions of the connection groups, to determine the force distribution in the dowels, are necessary but for the numerical analysis the frame and corner have to be modelled, in some extent, as solid elements. However, if the numerical analysis should be carried out on a fully modelled frame it would demand a large amount of calculation power and time. Therefore, since only the dowel forces and the stresses at the connection groups are of particular interest only some of the left corner will be modelled as solid elements and the rest of the frame will be modelled as beam elements with material and profile properties matching the frame in question. The frame and corner is defined by various key points with variable coordinates, which makes it easy to changes the geometry of the frame or corner. The geometry is divided into three parts; first the overall dimensions of the frame (length and height) and the perimeter dimensions of the left corner, secondly the dimensions and placing of the dowels and thirdly the dimensions of the steel plates.

3.1 Frame and overall corner dimensions In Figure 3.1 the dimensions of half a frame and the numbering of key points on it and the corner is shown. Here the red dashed lines (L1, L4 and L5) are indicating the dimensions of the part of the frame that has been replaced by beam elements, and the beam elements shown as single continues black lines (L2 and L6) are indicating the system lines for the frame which intersect in point P8. The system lines/beam elements are perpendicular to the cut-off faces of the beam and column parts. g and k4 indicates the original height and bottom point of the beam part at the corner before the making cut-off at the angle of the top of the column part. The height of the beam then varies from g at the corner to f at the top and the width of the column varies from e in the top to d in the bottom. As it appear from the figure the centre of left support is located at , 0 and the centre of the top at

, . The remaining coordinates to the point

and the expressions for the lines are given in Appendix D. The further hand calculations and numerical analysis will be carried out on a frame with the values of the variables in Figure 3.1 to Figure 3.3 listed below.

Geometry

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Figure 3.1: Dimensions expressed by variables and key points on the left of the frame and the left corner.

a = 3.5 m Inside column height b = 6.0 m Frame height at system line c = 7.0 m Half frame width d = 0.40 m Column width at support e = 0.69 m Column width at frame corner f = 0.40 m Beam height at frame top g = 0.69 m Beam height at frame corner

bt = 0.20 m Thickness of frame st = 10 mm Thickness of steel plates v = 10 mm Distance from timber to steel plate edge

dd = 28 mm Diameter of dowels t1 = 45 mm Thickness of outer member of corner t2 = 90 mm Thickness of middle member of corner

a1 = 5dd = 140 mm Minimum distance between fasteners in

the fibre direction a2 = 4dd = 112 mm Minimum distance between fasteners

perpendicular to the fibre direction a3 = max(7dd , 80) = 196 mm Minimum distance from timber end to

the fastener a4 = 4dd = 112 mm Minimum distance from timber edge to

the fastener

Geometry

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3.2 Dowel placing and numbering The dowels are divided into two connection groups (CG 1 and CG 2); number 1 in the beam part and number 2 in the column part. They are placed with the required distances to each other and to the ends and edges of the timber in accordance with (EC-5-1-1 2007). Here a1 and a2 are the required minimum distances between the dowels in and perpendicular to the fibre direction, respectively, and a3 and a4 are the required minimum distances to the edges and ends. The values of a3 and a4 are determined, conservatively, as the minimum distances to loaded ends and edges and with the most unfavourable angle to the fibre direction. The numbering of the dowels is shown in Figure 3.2.

Figure 3.2: Numbering of the dowels with connection group 1 in the beam part and connection group 2 in the column part and the required distances in accordance with the Eurocode 5.

Geometry

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3.3 Dimension of the steel plates The perimeters of the steel plates are defined by six points numbered from S1 to S6 (see Figure 3.3) and the plates are not entirely filling out the milled grooves. The edges of the plates are parallel to the edges of the timber but with a distance of v between the two material edges. As well as for the coordinates of the dowels the coordinates to the steel plate perimeter point are defined in Appendix D. The distance from the timber edge to the milled grooves, t1, and the distance, t2, between them are shown in Figure 3.3 and indicate the three-dimensional location of the steel plates. The plates are located of the glulam thickness from the edge which means that t2 = 2t1.

Figure 3.3: Key points on the steel plates and the distance from the timber edge to the steel plate edge.

Materials

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4 Materials

The frame is done in two types of materials; wood and steel. In this section the material properties of these materials will be described.

4.1 Wood The wood or timber is an orthotropic material with different material properties in three perpendicular directions longitudinal, l, radial, r, and tangential, t as sketched in Figure 4.1. The frame analysed in this thesis is constructed in glulam (glued laminated timber) consisting of thin wood lamellas glued together. The top and bottom lamellas are placed with pith outwards to avoid sapwood splitting and the rest of the lamellas are placed with their pith in the same direction in order to reduce moisture stresses. When it comes to moisture stability, glulam is considered more stable than regular structural timber because it is delivered dry with an approximate moisture content of 12%, and the increased dimensions of glulam members in relation to timber members slows down the moisture change (Larsen & Enjily 2009). When calculating the load carrying capacities of the dowels the strength parameters listed in are based on glulam in strength class GL32 with a density of k = 550 kg/m3. The design material strength parameters are calculated from equation (4.1) (Larsen & Enjily 2009), where mk is the characteristic value of the material parameter, kmod is a modification factor and M partial factor. Due to the orthotropic of the wood the stiffness parameters shown in Table 4.2 varies from each of the three directions. The modulus of elasticity, E, Poissons ratio, , and shear modulus, G, are based on values from (Omarsson 1999).

(4.1)

Table 4.1: Characteristic and design strength parameters of GL32 [MPa]. Strength Characteristic Design Bending: fm 32.0 23.04 Tension, parallel: ft,0 22.5 16.20 Compression, parallel: fc,0 29.0 20.88 Tension, perpendicular: ft,90 0.5 0.36 Compression, perpendicular: fc,90 3.3 2.376 Shear: fv 3.8 2.736

Table 4.2: Stiffness properties of the wood material. El = 10.00 GPa Er = 500.0 MPa Et = 400.0 MPa lr = 0.35 lt = 0.55 rt = 0.23 Glr = 800.0 MPa Glt = 600.0 MPa Grt = 100.0 MPa

Materials

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Figure 4.1: Illustration of the three orthotropic material directions of a wooden beam longitudinal, l, radial, r, and tangential, t.

For the shrinkage/swelling coefficients of the glulam used in this analysis the following values are used based on (Omarsson 1999):

l = 0.007 (4.2) r = 0.19 (4.3) t = 0.35 (4.4)

Here l, r and t indicate the longitudinal, radial and tangential strains, respectively. The shrinkage/swelling strains are found as:

i = i (4.5) , where i is either l, r or t and is the change in moisture content of the timber. 4.2 Steel Opposite the wood material the steel is considered an isotropic material which means that is has the same material properties in all directions. The steel used for the slotted-in plates and the dowels in this analysis has a module of elasticity of Es = 210 GPa, a Poisson ratio of s = 0.3 and a characteristic yield strength of fuk = 400 MPa.

Load combination

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5 Load combination

The frame will be designed to withstand a combination of permanent and variable loads in accordance with the relevant Eurocode. The permanent load consists of the self weight of the structure and the variable load includes the snow and wind loads. Based on (EC-0 2007) the design load combination is determined. It is assessed that the worst load case will be obtained from a situation with a dominating snow load. Therefore, the chosen load combination for this frame is given as follows:

(5.1), where g is the dead load, s is the snow load, w is the wind load, G = 1.35 is the partial coefficient on the permanent loads, Q = 1.50 (according to table A1.2 (B) in (EC-0 2007)) is the partial coefficient on the variable loads and 0 = 0.7 (according to A1.1 in (EC-0 2007)).

Dead load The dead load of both the roof of the frame and the column are assumed to be uniformly distributed based on average cross sectional dimensions. For the column the surface area in the x-y-plane is , and the volume is then , where bt is the thickness of the frame. For a timber density of w = 550 kg/m3 this gives the following uniformly distributed dead load on the column:

/ 0.794 kN/m (5.2)

The surface area of the roof/beam of half a frame is , and the volume is and the uniform dead load on the beam is then:

2 0.794 kN/m (5.3)

Snow load The design snow load, s, is given by (5.1) in (EC-1-1-3 2007) (see eq. (5.5)). Here, the snow load shape, i, is 0.8 for roof slopes smaller than or equal to 30 (the roof slopes is 17 for the frame in question), the exposure coefficient, Ce, is 1.0 which applicable for the normal topography. The thermal coefficient, Ct, is also 1.0, since the factor, otherwise, only reduces the snow load for roofs with high thermal transmittance. The characteristic value for snow on the ground, sk, can, according to table C.1 in (EC-1-1-3 2007), be calculated from the following expression:

0.264 0.002 1 256

(5.4)

Load combination

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, here Z is the zone number (varies from 3 to 4.5 for Denmark) and A is the altitude above sea level. However, according to The Danish National Annex for (EC-1-1-3 2007) sk can be set to 0.9 kN/m2.

720 kN m2 (5.5)When assuming that the distance between the frames in the building they should be placed in is 4 m the uniform snow load is then:

4.32 kN/m (5.6)Wind load The wind load on the frame is determined from the following equation from (EC-1-1-4 2007).

(5.7), where qp(ze) is the peak velocity pressure at the reference height ze and cpe is the pressure coefficient for external pressure.

720 kN/m2 (5.8), where the exposure factor ce(z) = 2.0 for a terrain category II and a height of z = 7.5 m, 360 kN/m2.

It is determined that the combination of pressure coefficients, cpe, shown in Figure 5.1 generates the largest cross sectional forces in the left corner and, thereby, the worst load case.

Figure 5.1: Illustration of the pressure coefficients for external pressure used on the frame.

In Figure 5.2 the total load case is illustrated and here it is seen that the wind generates a pressure on the right beam and column of 1.800 kN and 2.088 kN, respectively. While the left beam is not affected by the wind the left column is subjected to a suction of

Load combination

22

1.296 kN. This load case results in the reaction forces at the two supports listed below and shown in Figure 5.2.

RA,h = 32.59 kN RA,v = 46.04 kN RB,h = 16.12 kN RB,v = 46.07 kN

Figure 5.2: Dead, wind and snow load son the frame.

Hand calculations

23

6 Hand calculations

6.1 Method The hand calculations are performed in accordance with (EC-5-1-1 2007) and consist of a load carrying capacity control of the frame itself and an analysis of the load carrying capacity of the dowel connections. When checking the load carrying capacity of the frame the shear and compression strength and a combination of bending and shear compression is checked. Moreover, the load carrying capacity in respect to the compressive stresses on the tapered edge is checked and the general load carrying capacity control of the cross sectional dimensions of the frame is carried out simultaneously as the calculation of dowel force in Appendix B with the use of the material strength properties of GL24 in order to make sure it is the corner connection that dictates the load carrying capacity of the frame. The frame is calculated as acting elastically which means that the connection is considered failing if one dowel fails.

Figure 6.1: Illustration of the dowel group and the load action on the dowel group 1.

In order to transfer the cross sectional forces, shown in Figure 6.1, from the beam part of the frame to the column part the dowels are subjected to different forces different in magnitude and direction. The distribution of internal forces between the dowels is found by considering the frame corner connection as an eccentrically loaded connection with an unknown line of force action. It is an approach also used in (Larsen & Enjily 2009)

Hand calculations

24

and the resulting force in each dowel consists of a contribution from the normal and shear forces and one from the moment.

The dowel forces from the normal and shear forces are evenly distributed between the dowels in the two connection groups:

,, = (6.1)

,, =

(6.2)

,, = (6.3)

,, =

(6.4)

, here i is the dowel number in the connection group, i.e. i = [1, 2, ... , 9], n1 and n2 are

the number of dowels in connection group 1 and 2, respectively. cos sin and sincos are unit vectors pointing in the opposite direction of the normal and shear forces.

The moment contribution to the dowel forces depends on the dowels position in relation to the centre of gravity of the connection group.

,, =

, , (6.5)

,, =

, , (6.6)

, here , , , , is the polar moment of inertia, , , and , , are the distances shown in Figure 6.1. ri,1 and ri,2 are the radii from the centre of the connection groups to the dowels and ,

,

,, and ,

, ,, are unit vectors perpendicular to a vector from the

centre of the connection group to the dowel.

The resulting dowel is then the sum of the three contributions:

, = ,,,, = ,, ,, ,, (6.7)

, = ,,,, = ,, ,, ,, (6.8)

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It should be noticed that the shear and normal forces are creating moments opposite to each other, and then the sum of these two moments can be optimised to be zero with the right combination of eV,1 and eN,1.

When the dowel forces are set up as vectors the angle between the forces and the fibre direction can be found as the angle between the force vector and a vector parallel to the fibres. The fibre vector in the beam part is pointing towards the top of the frame and the fibre vector in the column part is pointing towards the corner of the frame. The vectors can be written as follows:

, = 1tan (6.9)

, = 01 (6.10)

In order to find the signed angle and not absolute angle the MatLab function atan2(y,x) is used (see Appendix B) and it returns the angle from the x-axis to the vector (x,y) within the closed interval [-;], where the positive direction is counter clockwise from the x-axis. I.e. when subtracting the atan2 product of the fibre vector from the force vector the angle between the dowel forces and fibre direction can be written as:

, = atan2,,, ,, atan2,,, ,, (6.11)

, here j is the number of the connection group.

When it comes to the load carrying capacity of the dowel connections the theory presented is based on Johansens theory from 1949 and describes the failure of dowel and dowel connections in wooden materials. The different failure modes of a dowel connection are illustrated in Figure 6.2 and they are based on symmetric timber-to-steel connections with a single fastener and two slotted-in steel plates. The failure mode depends on the hypothesis that the dowel remains straight during yielding, and the yield moment of the dowel is reached at several points (Sawata, Sasaki & Kanetaka 2006). According to (Johansen 1949) the strength of the connections depends partly on the embedding strength of the timber and partly on the resistance of the dowel to bending and the embedding of the timber and the bending of the dowels are both assumed to be rigid-plastic. The characteristic embedding strength is given as:

,, = ,, (6.12)

, where ,, 0.0821 0.01 is the characteristic embedding strength in the fibre direction, qk is the characteristic wood density, 1.35 0.015 for softwood and is the angle between the force direction and the fibre direction.

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The embedding strength is not a material but a system property depending on the properties of both the timber and dowel materials. In accordance to (Thelanderson & Larsen 2003) some of the most important parameters influencing on the embedding strength is: density (embedding strength varies linear with density), fastener and hole diameter (embedding strength decreases when fastener diameter increases), angle between load and grain/fibre direction (It is seen that the embedding strength is highly influenced by the angle and varies from fh,0,k to 0.56 fh,0,k when varies from 0 to 90 for a dowel diameter of 28 mm.), friction between dowel and the surrounding timber (increased friction between dowel and timber surface increases embedding strength) and moisture content (a decrease in moisture content of timber increases bending and compression strength but shrinkage reduces cross sectional area all parameters affecting the embedding strength).

Figure 6.2: Failure modes for connection with two slotted-in steel plates.

In order to check the characteristic load carrying capacity of the connections the failure modes are set up as follows in accordance with (Sawata, Sasaki & Kanetaka 2006).

Failure mode I is a situation where the bending strength of the dowel exceeds the embedding strength of the timber and the dowel therefore moves in timber;

,, = 2 1 (6.13)

Failure mode II is a situation where both the dowel and the timber fail. Two yield hinges occurs in the dowel at the outer surfaces of the steel plates but the dowel remains straight between them where the timber fails;

,, = 2 2

1 1 (6.14)

Failure mode III is a situation similar to that in failure mode II but here with four yield hinges in the dowel;

,, = 1 (6.15)

Common for failure mode I to III is that the embedding strength of the centre member exceeds the yield strength of the dowel. Failure mode IV is similar to failure mode I but here the dowel has three yield hinges one at the middle and one at each inner surface of the steel plates;

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27

,, = 2 1 (6.16)

Failure mode V is a combination of failure mode II and IV. Here the dowel has five yield hinges one at the middle, one at each inner surface of the steel plates and one at each outer surface of the steel plates;

,, = 2 2

1 1

(6.17)

Failure mode VI is a combination of failure mode III and IV. Here the dowel has seven yield hinges one at the middle, one at each inner surface of the steel plates, one at each outer surface of the steel plates and one somewhere in each of the outer members of the timber;

,, = (6.18)

, where t1 is the smaller of the thickness of the glulam side member, t2 is thickness of the glulam middle member, ns is the number of steel plates, ,,, ,,,, d is the dowel diameter, fu,k is the characteristic strength of the dowels, fe,,k is the characteristic embedding strength of the timber the direction of the dowel force action.

6.2 Results

6.2.1 Cross sectional forces The program created by Niels Holck, attached as Appendix A, is made up from several script files and is run by a main file (fe_frame.m) containing the dimensions and strength and stiffness parameters of the frame, and the load on the frame. When running the main file an interface will provide the possibility to get the desired outputs and the program is used to find the deformations of, the normal and shear forces and the moments in the frame. The moment, normal and shear force distribution is shown in Figure 6.5, Figure 6.3 and Figure 6.4, where the x- and y-axis indicates the dimensions of the frame. From the distribution figures the cross sectional forces just above the left corner are as listed below. Notice that the normal force and moment are listed positive due to their directions in the principally illustration in Figure 6.1.

N = 39.96 kN V = 32.10 kN M = 105.35 kNm

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Figure 6.3: The distribution of normal forces in N in the frame due to design load.

Figure 6.4: The distribution of shear forces in N in the frame due to design load.

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29

Figure 6.5: The moment distribution in Nm on the frame due to design load.

6.2.2 Deformations The load case described in section 5 with a combination of an uneven distributed wind load and an even distributed snow load causes the frame to deform as shown in Figure 6.6. The displacements in the global x- and y-direction of the left and right corner and of the top are here as shown in the table below.

Table 6.1: Displacement of the frame top and corners in mm from hand calculation due to design load.

Left corner Top Right corner x-direction -26.8 -13.3 -0.07 y-direction 0.96 -40.5 -0.16

Figure 6.6: Deformation of the frame due to design load.

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30

It is seen that large displacements occur in the top and the left corner; at the top primarily as a large horizontal displacement and at the left corner primarily as a large vertical one. Even though the displacement of the right corner is very small the corner still deforms due to a large corner moment. The moment in the left corner is even larger hence a more distinctive deformation of this particular corner.

6.2.3 Dowel forces When using the equations (6.1) to (6.6) the contribution to the dowel force from the moment, normal and shear force can be calculated. With an angle of = 18 of the roof/beam part of the frame and n1 = n2 = 9 dowels in each connection group the force contribution from the normal and shear force is:

,,1 ,,2 4.2121.403 kN

,,1 ,,2 1.1283.384 kN

For Ip1 = 345.1103 mm2, Ip2 = 404.2103 mm2, eN = 1.188 m, eV = 1.128 m (see Appendix B) and the direction vector mi,j as listed below the moment contribution to the dowel forces can be calculated (see Table 6.2).

,1 ,1 ,1 ,1 ,1 ,1 ,1 ,1 ,1 x-component -0.153 -0.195 -0.237 0.042 0.0 -0.042 0.237 0.195 0.153 y-component -0.134 0.0 0.134 -0.134 0.0 0.134 -0.134 0.0 0.134

,2 ,2 ,2 ,2 ,2 ,2 ,2 ,2 ,2 x-component -0.085 -0.140 -0.195 0.055 0.0 -0.055 0.195 0.140 0.085 y-component -0.212 0.0 0.212 -0.212 0.0 0.212 -0.212 0.0 0.212

Table 6.2: The x- and y-components and magnitude of the moment contribution to the resulting dowel force.

Dowel no.

Connection group 1 Connection group 2 x-

component [kN]

y-component

[kN] Magnitude

[kN]

x-component

[kN]

y-component

[kN] Magnitude

[kN] 1 -51.98 -41.32 66.40 15.95 55.81 58.042 -64.12 -2.531 64.17 31.00 -2.531 31.103 -76.26 36.26 84.44 46.04 -60.87 76.324 4.545 -41.32 41.57 -22.64 55.81 60.235 -7.596 -2.531 8.01 -7.596 -2.531 8.016 -19.74 36.26 41.29 7.445 -60.87 61.327 61.07 -41.32 73.74 -61.23 55.81 82.858 48.93 -2.531 49.00 -46.19 -2.531 46.269 36.79 36.26 51.66 -31.15 -60.87 68.38

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31

The sum of the contributions in the x- and y-direction of the normal and shear force and the moment is then the components of the resulting dowel force acting on the wood material. In Appendix C the resulting x- and y-force components and the force magnitudes are listed and it is seen that the sum of the y-components is -43.09 kN for each of the two connection groups which shows that there is equilibrium between them. As shown in the previous section the dead load on the columns is 0.794 kN/m and from the geometry it is found that the centre of CG2 is located 3.709 m above the support. The sum of the column dead load and the sum of y-components in one connection group is -46.03 kN which is assessed to be sufficiently close to the vertical reaction force of 46.04 kN.

In Table 6.3 the resulting forces are listed with magnitudes and angles in relation to the fibre direction of the timber in the respective connection group. It should be noticed that the forces in dowel no. 3 and 7 in both connection groups are of large magnitudes and that there angles to the fibre directions are rather steep. This could indicate that for the hand calculation these two dowels will be decisive factors to the design load. The dowels in question are also those with the greatest distance to the centres of the connection groups (the no. 5 dowels) and according to the hand calculation method from (Larsen & Enjily 2009) they will obtain the most of the forces caused by the moment as it appears from Table 6.2. The resulting magnitudes of the dowel forces and their angles from Table 6.3 and they are illustrated in Figure 6.7.

It is seen that all the peripheral dowel forces (i.e. all but the one in the centre) are more or less perpendicular to their direction to the centre of the respective connection groups. It is also seen that the largest of these forces in each of the connection groups (79.40 kN and 80.87 kN) are about twice as big as the smallest ones (37.25 kN and 35.83 kN).

Table 6.3: Hand calculated magnitudes and angles of the dowel forces in relation to the fibre direction due to design load.

Dowel no.

Connection group 1 Connection group 2 Force [kN] Angle [] Force [kN] Angle []

1 64.44 -155.87 68.51 -22.88 2 59.80 166.17 41.95 -83.45 3 79.40 136.22 78.00 -133.35 4 44.51 -96.68 64.25 10.73 5 5.70 -141.22 5.70 -32.79 6 37.25 95.70 56.53 -161.30 7 78.74 -52.03 80.87 38.69 8 53.65 -23.55 35.83 82.32 9 53.49 21.04 57.33 159.08

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Figure 6.7: Magnitude and direction of the hand calculated dowel forces

6.2.4 Load carrying capacity of the dowel connections The design load carrying capacities of the dowels are used to assess the dowel forces. This value is found by multiplying the values from equation (6.13) to (6.18) by the modification factor kmod = 0.9, which concerns glulam subjected to subjected to short-term actions (snow and wind), and the partial factor for material properties which is M = 1.25 for glulam. The design load carrying capacities for each of the 18 dowels in each of the six failure modes are listed in Appendix C and from this it is seen that the lowest load carrying capacity is for failure mode I in all of the dowels which indicates that it is the glulam that fails before the dowels. In Table 6.4 the magnitude and angle of each of the dowel forces are listed (as in Table 6.3) and the load carrying capacity of the hole/dowel in the particular direction of the dowel force. Lastly a control is made dividing the dowel force by the load carrying capacity and from this it is seen that the dowel connection in the corner of the frame can carry the loads and that the dowels closest to failure are number 3 and 7 in both connection groups. These are also the

Hand calculations

33

dowels which the moment in the corner has the largest influence on due their longer distance to the centre of the respective dowel groups.

Table 6.4: Magnitude and angle of dowel force, load carrying capacity and load carrying capacity check of each of the 18 dowels.

Dowel no.

Connection group 1 Connection group 2 Fdowel [kN]

Angle []

FRd [kN]

Fdowel/FRd Fdowel [kN]

Angle []

FRd [kN]

Fdowel/FRd

1 64.44 -155.87 104.40 0.617 68.51 -22.88 105.55 0.6492 59.80 166.17 112.87 0.530 41.95 -83.45 66.95 0.6273 79.40 136.22 86.10 0.922 78.00 -133.35 83.74 0.9314 44.51 -96.68 66.97 0.665 64.25 10.73 114.77 0.5605 5.70 -141.22 90.50 0.063 5.70 -32.79 96.12 0.0596 37.25 95.70 66.86 0.557 56.53 -161.30 109.19 0.5187 78.74 -52.03 79.70 0.988 80.87 38.69 90.58 0.8938 53.65 -23.55 104.94 0.511 35.83 82.32 67.09 0.5349 53.49 21.04 107.20 0.499 57.33 159.08 107.30 0.534

FEM-analysis

34

7 FEM-analysis

7.1 Structure of the model The corner for the frame has been modelled in the finite element program Abaqus/CAE. As mentioned in section 3 the model geometry is made from variables which provides a certain amount of freedom within the choice of size of the frame. The values of these variables including the load and strength parameters are chosen in the beginning of the script (see in Appendix D). In Abaqus the modelling process is done graphically and divided into a number of modules. In the following the structure of the model and there by the script file (py-file) will be described in relation to these modules with an indication of in which lines the modules are defined.

Part (Line 235 to 722) The part module is where the different parts of the model is constructed. The frame modelled for this analysis consists of 10 different parts. Part 1 to 6 are solid elements creating the fully modelled left frame corner and Part 7 to 10 are beam elements with no modelled cross sectional dimensions, i.e. 1D elements. Each element is first constructed in the x-y-plane using the key point coordinates described in section 3, and then the planes are extruded in the z-direction to their respective thicknesses. In Part 1 and 2, which are the column and beam parts shown in Figure 7.1 and Figure 7.2, cuts/milled grooves for the steel plates have been made as well as holes for the wooden ring twice the diameter of the dowels. The wooden rings shown in Figure 7.5 are constructed from a cylinder fitting to the size of the holes and only one is made and then later duplicated and named in accordance with the number and numbering of dowels. A hole through the cylinder with the diameter of the dowels is made and two cuts for the steel plates are cutting the cylinder into three rings so they fit into the three timber members in Part 1 and 2. The steel plates (see Figure 7.3) are denoted as Part 3 and, as for the wooden cylinder, only one is modelled and then later duplicated. Holes twice the diameter of the dowels are also made in the steel plate(s) for the steel rings shown in Figure 7.6. The rings are made as discs in the x-y-plane and then extracted in the z-direction to the thickness of the steel plates. A hole is made in the centre to fit the dowels.

The beam elements are, as mentioned earlier, 1D-elements with beam properties. These elements are following the system lines of the frame. Part 7 is the beam element corresponding to the cut part of the left column. Part 8 is the beam element corresponding to the cut part of the left beam. Part 9 and 10 are the beam elements corresponding to the right column and beam, respectively.

FEM-analysis

35

Figure 7.1: Part 1 Corner peace of the column.

Figure 7.2: Part 2 Corner peace of the beam.

Figure 7.3: Part 3 One of the steel plates slotted into the wooden column and beam.

Figure 7.4: Part 4 One of the steel dowels.

Figure 7.5: Part 5 One of the wooden ring parts which are placed in the holes of the

larger wooden corner parts.

Figure 7.6: Part 6 One of the steel rings which are placed in the holes of the steel plates.

FEM-analysis

36

Properties (Line 723 to 1554) The property module is divided into a material and section part. First the properties of the different materials used are defined (see Figure 7.7) then, by sectioning, these properties are assigned to the respective parts and the material orientation for each part is assigned. The material properties follows the ones defined section 4 and the Type is chosen as Engineering Constants in order to apply the orthotropic elastic properties of the wood material.

Figure 7.7: The interface to define the material properties of the wood material.

The material properties and section assignment can be edited later in the modelling process but the partitioning of the parts remains the same throughout the rest of the modelling process at has great influence on the geometry. Abaqus number the points, edges and faces in an apparently random pattern and therefore, it is of great importance that the partitioning is done right before continuing the modelling, otherwise, the user risks having to change assembling of the parts, the user defined surfaces, areas subjected to load etc.

Assembly (Line 1555 to 3320) As the name indicated the Assembly module is used, among other things, to assemble the model by creating so called Instances. Each part is implemented in the numbers wanted and then constrained to each other. The solid parts forming the left corner of the frame (Part 1 to 6) all contains holes or at least curved surfaces and therefore, the Coaxial constraint is used to make sure the inside curved surface of a hole aligns properly with the corresponding curved outer surface of the part that should fit into the

FEM-analysis

37

respective hole. The coaxial constraint only gives the position in the x-y-plane and to make sure the parts are not rotating in relation to each other the straight surfaces of the parts in the x-y-plane and in the x-z-plane are constraint to each other with the Face to Face constraint. The Face to Face-constraint constrains two surfaces so they are placed in the same plane with the ability to define a clearance or distance between these planes (default is 0.0 m). I.e. Part 1 and 2 are Face to Face constraint on the contact surfaces, surfaces in the x-y-plane and surfaces in the y-z-planes. The steel plates are Coaxial constraint in one hole with the corresponding one in Part 1, Face to Face constraint on surfaces in the x-y-plane and Face to Face constraint on their bottom surfaces with the corresponding bottom surfaces of the milled grooves in Part 1. The wooden rings are Coaxial constraint to the holes in the wooden column and beam part (Part 1 and 2) and the straight surfaces in the x-y-plane are Face to Face constraint with the corresponding surfaces in the x-y-plane of Part 1 and 2. This means that they in principle are able to rotate about the centre of the hole but this is overcome by defining a fixed rotation so the material orientation of the wooden rings are matching the material orientation of the wooden part they are constrained to. The same goes for the steel rings, even though the alignment of the material orientations is of no importance. The dowels are Coaxial constraint to the holes in the steel plates and then Face to Face constraint with the outermost surface in the x-y-plane of Part 1but here with a clearance of , where bl is the extra length of the dowels in relation to the width, bt, of the frame.

When it comes to assembling the beam elements, they are connected to the corner by Coincident Points at the corresponding angle of the system lines. I.e. the left beam column element, Part 7, is fixed to the centre point of the bottom cut-off surface of Part 1 and Part 8 to the centre of the cut-off surface of Part 2. Part 8 and 10 are fixed at the top point of the frame and Part 9 and 10 at the right corner of the frame. The total assembled frame is shown in Figure 7.8 with the material orientations indicated numbers on Part 1, 2 and 7 to 10.

Figure 7.8: The assembling of the frame with numbering of Part 1, 2, 7, 8, 9 and 10 and the material orientations indicated by yellow coordinate axis.

FEM-analysis

38

In the Assemble module different surfaces are also defined. These surfaces are made to ease the process of defining which surfaces that are interacting with each other.

Step (Line 3321 to 3331) Creating multiple steps provides the possibility to capture changes in the loading and boundary conditions (Hibbitt, Karlsson & Srensen 2008). The boundary conditions are defined in the Initial step and the frame is then subjected to the loads in Step 1. The step module is also where the History output is chosen and for the analysis the CFN is chosen. CFN is the total forces and moments due to contact pressure and the output consists of four values per contact surface: CFN1 contact force on the master surface of the interacting surfaces in the global x-direction, CFN2 contact force on the master surface in the global y-direction, CFN3 contact force on the master surface in the global z-direction and CFNM magnitude of resulting contact force.

Interaction (Line 3332 to 4268) In the Interaction module the contact behaviour between the surfaces are defined. In this analysis two types of contact behaviour are used Constraint and Interaction. When choosing two surfaces for either Constraint or Interaction the first one chosen is the slave surface and the second one is the master surface. This means that every node on the slave surface will have the same values for its degrees of freedom as the point on the master surface.

The type of constraint used is primarily Tie, and as the name says it ties two separate surfaces together so that there is no relative motion between them. This type of constraint allows you to fuse together two regions even though the meshes created on the surfaces of the regions may be dissimilar (Hibbitt, Karlsson & Srensen 2008). Tie is chosen for the contact between the wooden rings and Part 1 and 2, the steel rings and the steel plates and between the steel rings and the dowels. The rings are obviously tie constrained to the column and beam part of the corner and the steel plates because they are only made in order to apply a finer mesh around the hole. The dowels are tie constraint to the steel rings and there is not defined any interaction property between the bottom surface of the milled grooves and the steel plate edges which allows the steel plates to merge together with the beam and column parts without creating any stresses. Thereby, all the load transfer through the frame corner will happen through the dowels and the results becomes as comparable to those from the hand calculation as possible.

The contact between the dowels and the timber is defined as an Interaction and the type of interaction is chosen as Surface-to-surface contact with a Degree of smoothing of the master surface of 0.2. When applying a smoothing to contacting surfaces the inaccuracies in contact pressures caused by mesh discretization on curved geometries is reduced (Hibbitt, Karlsson & Srensen 2008). The frictional behaviour for the dowel-to-

FEM-analysis

39

timber interaction is defined, via the contact properties for these interactions, as mechanical, tangential behaviour with an assumed uniform friction coefficient of 0.4 in the contact area. The Penalty friction formulation is used which allows some relative motion between the interacting surfaces a so called elastic slip. For instance, if a situation occurs where the edge of the meshed surface of a dowel is in contact with a face of the meshed hole surface, as illustrated in Figure 7.9, this would cause infinite stresses because all should be transferred through a line without an area. The penalty interaction and the smoothing prevent this by allows the surfaces to, in some extend, merge together creating an even surface the forces can be transferred through.

Figure 7.9: Contact surface between dowel and hole where the two mesh grids do not fit perfectly together.

Lastly, constraints are used between beam elements at the top of the frame, at the right corner and between the column and beam parts and the beam elements. Here the constraint is denoted as a Coupling and it is used to constrain the motion of a surface to the motion a single point.

Load (Line 4269 to 4438) The load module you have the opportunity top apply the loads in different steps. In this analysis the boundary conditions are applied in the Initial step and the load actions in Step-1. The frame is, as for the hand calculation, assumed simply supported and therefore, no movements in the x-, y- and z-direction is allowed. However, the two supports are not retained from rotating about any of the three axis. Moreover, the frame is retained from movement in the z-direction in the top, at the right corner, at the tip of every dowel and on a surface of both Part 1 and 2.

The dead, snow and wind loads are applied as line loads on the beam elements with the same directions and magnitudes as in the hand calculation and as uniform distributed loads on Part 1 and 2.

FEM-analysis

40

Mesh (Line 4439 to 4735) The meshing divides the model into a finite number of elements. Before meshing the parts the type of elements are chosen and the solid elements (Part 1 to 6) are 20-node hexahedrons with 27 Gauss points, as sketched in Figure 7.10. The meshing of the parts is done with the objective that each field (denoted Faces in Abaqus) should be approximately square (in the x-y-plane). The seeding is done as Edge By Number. Here it chosen how many seeding point there should be on the lines (denoted edges in Abaqus) created by the partitioning. The greater the number the finer the mesh becomes. This is only done for those edges where the user wants a certain number of seeds, for the rest of the edge Abaqus chooses a seeding number in order to mesh the whole part. In Figure 7.11 to Figure 7.16 the meshing of Part 1 to 6 is shown. The beam elements (Part 7 to 10) are divided into 12, 36, 15 and 41 seeds, respectively.

Figure 7.10: Meshing done in 20-node hexahedrons (black nodes at the left) with 27 Gauss points (red nodes at the right).

Job (Line 4736 to 4746) The job module is used to create a job and then initiate the analysis of the model created and the job can be run on the PC/MAC you are working at. For this particular analysis the job module was used to write an input file which then was run on a separate server.

Visualization The visualization module is where you can view the results from the analysis. This can be done by, e.g., viewing plots of the stress distributions or writing result data to report files.

FEM-analysis

41

Figure 7.11: Meshing of Part 1. Figure 7.12: Meshing of Part 2.

Figure 7.13: Meshing of Part 3. Figure 7.14: Meshing of Part 4.

Figure 7.15: Meshing of Part 5. Figure 7.16: Meshing of Part 6.

FEM-analysis

42

7.2 Results

7.2.1 Deformation The deformation of the frame is shown in Figure 7.17 and the deformation shape is quite similar that found by the hand calculations. The displacements of the right corner, the left corner (at the system line intersection) and the top of the frame are listed in Table 7.1 but these values are not as similar to those found by hand. This could be because of the modelled frame corner in the FEM-analysis.

Table 7.1: Displacement of the frame top and corners in mm from the numerical analysis due to the design load.

Left corner Top Right corner x-direction -20.2 -9.32 -1.34 y-direction 0.66 -31.0 -0.10

Figure 7.17: Deformation of the frame from the design load (magnitude 20 times).

When subjected to the large negative moment and normal and shear force design load case described in Section 3 the frame corner deforms as shown in Figure 7.18. The figure also shows the deformation of the holes exposed to the largest dowel forces and it is easily seen that the holes are shaped elliptical due to the pressure from the dowels. The local deformation of the wood material around the dowels results in gaps between timber and steel opposite the force direction of the dowel. The four largest hole deformations are the ones in hole 3 and 9 in connection group 1 and hole 7 and 9 in connection group 2. As the Figure 7.18 also shows the column part and the beam part merge together at inner contact surface. This is because there are no interaction properties defined for contact between the top surface of the column and the bottom surface of the beam, which also is the case for the surfaces of the steel plates and the surfaces at the end of the milled grooves. As mentioned in section 7.1 the lack of

FEM-analysis

43

interaction between these abutting surfaces is made purposely in order to make the results from the FEM-analysis and the hand calculations as comparable as possible.

Figure 7.18: Deformation of the frame corner and the local deformation of the dowel holes 3 and 9 in CG 1 and 7 and 9 in CG2 magnified 20 times.

7.2.2 Stresses The stresses that will be investigated is only those in the wood material since these are the critical ones as it is shown from the hand calculation that timber fails before any of the dowels.

When assessing the stress plots of the frame corner it is import to keep the extrapolation that Abaqus performs in mind. For instance, when looking at a cut, A-A, in a timber ring element the longitudinal stresses might be distributed as sketched in Figure 7.19. The plots of the stresses are based on the stress value in the Gauss points (marked as black dots in the figure) and therefore, there is no nodal value of the stress at the edge of an element (in this case the inner perimeter of the hole). To overcome this issue Abaqus performs an extrapolation of the change in value from the second closest to the closest

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44

Gauss point. If the difference of some reasons is big between these two Gauss points the magnitude of the extrapolated values at the edge will be too big.

Figure 7.19: Significant rise in value of longitudinal stresses at the edge of an element shown in principle.

Longitudinal stresses, 11 In Figure 7.20 the stresses in the left frame corner are shown where the red areas are in tension and the blue ones are in compression. From the areas around the top right dowel of the uppermost connection group (dowel PF1,3) and around the bottom left dowel of the lowest connection group (dowel PF2,7) it is seen that outer edges of the frame is in tension at the corner which is due to the large negative moment present. The large moment obviously causes large stresses and since the corner connection is designed just to have the sufficient load carrying capacity for the hand calculations (referring to dowel PF1,7 with a dowel force to load carrying capacity ratio of 0.988 shown in Table 6.4), the timber around the dowels in the perimeter of the connection groups are, as expected, subjected to stresses near or above the design strength values listed in section 4. The largest stresses in the longitudinal direction of the timber are found around dowel PF1,3, PF1,9, PF2,7 and PF2,9 and they are shown in Figure 7.21. When looking at the stresses around the dowels it is seen that some areas are grey and others black. In the gray areas the tension exceeds the tension strength of ft,0,d = 16.2 MPa and in the black areas the compression exceeds the compression strength of fc,o,d = 20.88 MPa. The compression stresses are caused by the rotation of the slotted-in steel plates via the contact with the dowels and the tension stresses are caused by deformation of the dowel holes and the timber around dowel PF1,3 is especially exposed to both large tension and compression stresses. The largest stresses in the frame corner are c,0 = 39.7 MPa and t,0 = 56.8 MPa and both significantly exceeds the strength of the timber. It should be kept in mind that the analysis is carried out under the assumption that the timber has elastic material behaviours

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Figure 7.20: The longitudinal stresses in the corner of the frame with tension and compression indicated

by red and blue colours, respectively.

Figure 7.21: Longitudinal stresses in the timber around dowel 3 and 9 in CG1 and 7 and 9 in CG2.

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46

Radial stresses, 22 The stresses in the radial direction are shown in Figure 7.22 and Figure 7.23 and it is seen that large tensile stresses originates from the number 1 dowel of both connection groups and the timber in these areas are in great risk of splitting. As the gray colour indicates the tensile strength of ft,90 = 0.36 MPa perpendicular to the fibre direction has been exceeded in several areas and when looking closer at, for instance, dowel 3 and 9 in connection group 1 and dowel 7 and 9 in connection group 2 it is seen that also the compressive strength, fc,90 = 2.376 MPa, perpendicular to the fibre direction has been exceeded by the compression from the dowel contacts.

Figure 7.22: Radial stresses in the frame corner.

Figure 7.23: The radial stress distribution on connection group 1 and 2.

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47

Shear stresses, 12 The shear stresses shown in Figure 7.24 are concentrated along the dowels near the edges and they are largest at connection group 1. Here it is also seen that the shear stresses at the some of the surfaces in contact with the dowels are exceeding the strength of fv = 2.736 MPa and the largest shear stresses in the frame corner is 12 = 14.36 MPa. Besides from the extended sides of the holes, the shear stresses occur at the dowel-to-timber contact surfaces because of the friction between the timber and the dowels.

Figure 7.24: Shear stresses in the timber in the corner connection.

Even though the largest longitudinal, radial and shear stresses are exceeding the corresponding strength values it is difficult to comment on the failure of the corner connection because stresses should be assessed as a whole, and when combining shear and compression the strength can be increased. None the less, the Norris criteria is for instance obviously not obeyed because of this strength exceeding and it will therefore not be investigated further.

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7.2.3 Dowel forces The history output produced in Abaqus gives the components of each contact surface between the dowels and the wooden rings. However, these force components are listed with all the x-components first, after that the y-components, then the z-components and lastly the magnitude of the contact force in the raw data file abaqus.design.dowel.forces.rpt (see Appendix E). The MatLab-script abq_cfn_design.m in Appendix F lists the force components in order of dowel number and in Appendix G the history output is then shown as the three vectors per dowel (one for each of the three timber ring surfaces in contact with the respective dowel) containing the x-, y- and z-direction and magnitude of the contact force from the respective interacting surfaces, i.e. 54 vectors in total. Besides from sorting the history output data the script abq_cfn_design.m also puts out the summed x-, y- and z-components and force magnitude in each dowel (see Appendix H). When having the direction components of the forces in each dowel the angle between the dowel forces and fibre direction is then calculated as in the hand calculation method. From the numerical analysis in Abaqus the resulting forces and their angles in relation to the relevant fibre directions in each of the 18 dowels are shown in Table 7.2. The magnitude and direction of the forces are also illustrated in Figure 7.25.

It is seen that the peripheral dowel forces have a tendency lean towards the fibre direction and it is seen that the largest of these forces in each of the connection groups (95.89 kN and 67.42 kN) are about 7 times bigger than the smallest ones (13.58 kN and 9.54 kN).

Table 7.2: Magnitude and angle of the dowel forces in relation to the fibre direction due to design load. (Abaqus)

Dowel no.

Connection group 1 Connection group 2 Force [kN] Angle [] Force [kN] Angle []

1 49.82 -170.27 43.45 -9.19 2 53.95 172.25 9.54 -120.41 3 82.12 153.85 50.81 -166.97 4 13.58 -95.43 42.06 5.17 5 0.32 68.35 4.62 -177.83 6 21.34 74.80 48.13 179.39 7 60.77 -26.34 58.22 21.77 8 62.24 -5.80 15.32 120.73 9 95.89 9.17 67.42 169.82

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Table 7.3: Magnitude and angle of dowel force from the design load, load carrying capacity and load carrying capacity check of each of the 18 dowels.

Dowel no.

Connection group 1 Connection group 2 Fdowel [kN]

Angle []

FRd [kN]

Fdowel/FRd Fdowel [kN]

Angle []

FRd [kN]

Fdowel/FRd

1 49.82 -170.27 115.30 0.432 43.45 -9.19 115.56 0.3762 53.95 172.25 116.21 0.464 9.54 -120.41 74.92 0.1273 82.12 153.85 102.51 0.801 50.81 -166.97 113.39 0.4484 13.58 -95.43 66.83 0.203 42.06 5.17 117.10 0.3595 0.32 68.35 70.76 0.004 4.62 -177.83 17.70 0.2616 21.34 74.80 68.62 0.311 48.13 179.39 117.82 0.4087 60.77 -26.34 102.33 0.594 58.22 21.77 106.55 0.5468 62.24 -5.80 116.91 0.532 15.32 120.73 75.10 0.2049 95.89 9.17 115.58 0.830 67.42 169.82 115.07 0.586

Figure 7.25: The dowel forces acting on the wood material

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8 Comparing hand and FEM-calculations

An important part of this analysis is the difference between the resulting dowel forces found by the conventional hand calculation method described in section 6 and the FEM-analysis done in Abaqus, and in order to check that the same amount of force is transferred through the left frame corner in the two calculation method the reaction forces at the frame supports are compared. In the table below the reaction forces in the x-, y- and z-direction are listed along with the differences given as absolute values and in percentage. It is seen that the differences between the reactions in the x- and y-direction are less than 1% which must be in sufficient consistent.

Table 8.1: Comparison of reaction force found by Abaqus and hand calculation.

Support A (left) Support B (right) i Abaqus

[kN] Hand [kN]

Diff. [kN]

Diff./RAbaqus [%]

Abaqus[kN]

Hand[kN]

Diff. [kN]

Diff./RAbaqus [%]

Rx,i 32.59 32.59 0.007 0.02% -15.99 -16.11 0.116 -0.73%Ry,i 45.78 46.04 0.264 0.58% 45.97 46.07 0.100 0.22%Rz,i 2.93E-3 0 2.93E-3 100% -7.34E-6 0 7.34E-6 100%

The illustrations of the dowel force shown in Figure 6.7 and Figure 7.25 are collated in Figure 8.1. As expected from the hypothesis put forward in section 2 the dowel forces found by the FEM-analysis have a clear tendency to lean more towards the fibre directions than the corresponding forces found by the hand calculation method. With exception of those for the middle dowels (dowel 5 in both CG1 and CG2), some of the largest angle differences are marked on the figure. The reason for ignoring the middle dowels are that they are only loaded to something between 0.4% and 6.3% of the load carrying capacity depending on the calculation method used.

When looking at connection group 1 the magnitude of the dowel forces from the hand calculation and FEM-analysis in dowel 3 are of almost the same magnitude (79.4 kN and 82.1 kN, respectively) but they are attacking with an angle difference of 17.6 which actually increases the load carrying capacity from 86.1 kN to 102.5 kN of this particular dowel when using the FEM. Generally the dowel force to load carrying capacity ratio is lower for the FEM-results but not in dowel 9 in connection group 1. The Abaqus dowel force is here leaning more in the fibre direction than the hand calculation dowel force but it is almost twice the magnitude (95.9 kN as opposed to 53.5 kN). This means that even though the load carrying capacity from the FEM-analysis is increased the dowel force is increased more and here the dowel force is much closer to the load carrying capacity (83% compared to 50%).

The difference in magnitude of the dowels loaded perpendicular to the fibre direction is clearly seen in dowel 4 and 6 in connection group 1 and dowel 2 and 8 in connection

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group 2. The hand calculation dowel forces are here much larger (about 2 to 4 times) than those found by Abaqus. This is also seen from the ratio between the largest and smallest of the dowel forces within the connection groups; here the ratio about 2 for the hand calculations and 7 for the FEM-analysis. I.e. according the the FEM-simulation the dowels loaded in the fibre direction are taking up much more load than those loaded perpendicular to the fibre direction.

By assessing the radial stresses shown in Figure 7.22 it is, furthermore, unrealistic that the four dowels mentioned should be able to take up more load than they already are in the finite element analysis. This flaw in the hand calculation method would be even more distinctive if the connection group were more widespread in the fibre direction. I.e. if the distances from the do