Structural behavior of notched glulam beams reinforced by means of plywood and FRP.

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Diploma thesis, structural engineering

Structural behavior of notched glulam beams reinforced by means of

plywood and FRP

Bärförmåga hos limträbalkar med urtag förstärkta med plywood och fiberarmerad plast

Författare: Maha Fawwaz, Adnan Hanna Handledare, LNU: Erik Serrano

Examinator, LNU: Anders Olsson Termin: VT12 15 hp

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Summary

Nowadays, timber is widely used in construction industry thanks to its availability and good properties. The use of solid (sawn) timber is not always proper since it is only available up to certain dimensions. Therefore, the so-called Engineered Wood Products (EWPs) have been introduced to cope with the different design needs of structures.

The Glued laminated Timber (glulam) is a type of EWPs that consists of small sections of timber laminates glued together to form beams and columns. Glulam can be manufactured in almost any size and shape; it can also be tapered or notched.

However, notching a beam at its end leads to a stress concentration at the re-entrant corner of the notch due to the sudden change in the notched beam’s cross section.

The concentration of shear and tensile stresses perpendicular to the grain can lead to a catastrophic brittle failure caused by the crack propagation from the notch corner.

Crack opening due to tensile stresses perpendicular to grain is the most common failure at the notch corner and it is always taken into design consideration. However, shear component is usually exists and must be also considered in design to guarantee the safety of the structure. Currently, only the normal forces perpendicular to the beam’s axis are considered in the design of the reinforcement in design handbooks.

The aim of this thesis was to study the structural behavior of notched glulam beams reinforced by adhered plywood panels and FRP. The carrying capacity of the notched glulam beams at their ends is the main subject of this thesis. In addition, a review of the notched beams design, reinforcements, and analysis theories are included.

Experimental series of three point bending tests with notched glulam beams with different configurations of reinforcement was carried out in lab.

Deformations and forces were measured both with conventional techniques and with

contact-free measurement systems - ARAMIS. On the other hand, a simple model of

two dimensional plane stress element has been created of the unreinforced notched

beam in ABAQUS. The normal and shear stresses were calculated for a horizontal

path of 100 mm in length starting from the notch tip. Afterwards, the mean stresses

were determined for the same path and have been used in calculations. The Mean

Stress Approach has been adopted in the hand calculations to calculate the crack

length and the failure load according to the ABAQUS model. Accordingly, the

failure load was about 40 kN for the unreinforced beams. Also, Eurocode 5 has been

used to calculate the failure load which gave a value of 20.2 kN for the unreinforced

beams. The average maximum applied load in tests was 30 kN for the unreinforced

beams while it reached about two and a half times this value for the CF-reinforced

and the plywood-reinforced beams.

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Sammanfattning

Tack vare sina goda egenskaper används trä i byggnadskonstruktioner i allt store omfattning. Konstruktionsvirke (sågade trävaror) kan dock inte alltid användas på grund av de begränsade dimensioner som finns tillgängliga. På grund av bl a detta har ett flertal så kallade engineer wood products (EWP) utvecklats. Limträ är en typ av EWP som består av sammanlimmade lameller som bygger upp tvärsnitt i balkar eller pelare. Limträ kan tillverkas i nästan godtycklig storlek och form och kan enkelt förses med t ex urtag. Vid urtag i balkändar nära upplag uppstår höga spänningskoncentrationer vid urtagets horn på grund av geometrin. Koncentrationen av normalspänningar och skjuvspänningar kan leda till plötsligt brott på grund av sprickpropagering från urtagets hörn, något som måste tas hänsyn till vid dimensionering. Dagens dimensioneringsmetoder är baserade på att man tar hänsyn till enbart normalspänningarna vinkelrät fiberriktningen.

Målet med detta arbete har varit att studera beteendet hos limträbalkar med urtag vid upplag som förstärkts med fiberarmering eller plywood. Huvudmålet har varit att bestämma balkarnas bärförmåga, vilket skett genom att genomföra försök med olika konfigurationer vad gäller förstärkningsmaterial och dess utformning. Vidare har olika dimensioneringsmetoder från litteraturen studerats.

Kraft och förskjutning under provningarna uppmättes dels med traditionella mätmetoder, men deformationerna mättes även med beröringsfri metod, ARAMIS.

En enkel tvådimensionell finit elementmodell skapades och analyserades i ABAQUS för analys av oförstärkt balk. Normalspänningar och skjuvspänningar beräknades och medelspänningarna längs en på förhand definierad sträcka beräknades.

Medelspänningskriteriet användes sedan för att uppskatta balkens bärförmåga.

Enligt FE-beräkningarna uppskattades bärförmågan för de oförstärkta balkarna till ca 40 kN. Provningarna gav ett medelvärde på balkarnas bärförmåga på ca 30 kN, medan de förstärkta balkarna hade en 2,5 gånger högre bärförmåga. Skillnaden mellan FE-beräkningarna och provningarna kan förklaras med den osäkerhet som finns vad gäller det aktuella trämaterialets egenskaper.

Beräkningar enligt Eurokod 5 gav en karakteristisk bärförmåga på 20,2 kN.

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Abstract

The aim of this thesis was to study the structural behavior of notched glulam beams reinforced by adhered plywood panels and FRP. The carrying capacity of the notched glulam beams at their ends is the main subject of this thesis. In addition, a review of the notched beams design, reinforcements, and analysis theories are included.

Experimental series of three point bending tests with notched glulam beams with different configurations of reinforcement was carried out in lab.

Deformations and forces were measured both with conventional techniques and with contact-free measurement systems - ARAMIS. On the other hand, a simple model of two dimensional plane stress element has been created of the unreinforced notched beam in ABAQUS. The normal and shear stresses were calculated for a horizontal path of 100 mm in length starting from the notch tip. Afterwards, the mean stresses were determined for the same path and have been used in calculations. The Mean Stress Approach has been adopted in the hand calculations to calculate the crack length and the failure load according to the ABAQUS model. Accordingly, the failure load was about 40 kN for the unreinforced beams. Also, Eurocode 5 has been used to calculate the failure load which gave a value of 20.2 kN for the unreinforced beams. The average maximum applied load in tests was 30 kN for the unreinforced beams while it reached about two and a half times this value for the CF-reinforced and the plywood-reinforced beams.

Key words: glued laminated timber, glulam, notch, finite element analysis, FRP,

reinforcement

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List of Figures

1 Examples of timber structures [Pictures were taken at Linnaeus University-

V¨axj¨o- building M, N, and Videum]. . . 2

2 Distribution of tensile stresses near the tip of a notched beam. . . 4

3 Different types of notches at a beam⁰s end. . . 6

4 Notched beam at top and bottom. . . 6

5 Compression and tension side notches. . . 7

6 Notched beam reinforced at its end. . . 8

7 Reinforcement by full-threaded lag screw/s. . . 8

8 Reinforcement with glued-in screws. . . 9

9 Formation of fiber reinforced polymer composite. . . 10

10 a. The Plywood used in this project, b. An example of beams reinforcement by plywood [the photo was taken at -Linnaeus University- Building M]. . . 13

11 Notched beam reinforced by Plywood. . . 14

12 Maximum shear stress for rectangular section beam. . . 15

13 The theoretical and the estimated stress perpendicular to the grain. . . . 16

14 Mode I (Tension, Opening), Mode II (In-Plane Shear, Sliding), Mode III (Out-Of-Plane Shear, Tearing). . . 16

15 a. Initial crack in the beam and b. The stresses in an infinitesimal element. 17 16 a. Notched beam reinforced by CF. b. Notched beam reinforced by plywood. 21 17 Three points bending test set up dimensions and the test set up as done in this project (picture taken for one test in lab). . . 23

18 The LVDTs 1 and 2 setup and dimensions. . . 24

19 ARAMIS system as set in the lab during the test (a computer and two cameras). . . 24

20 An example of the crack propagation of glulam beam reinforced with plywood in diagonal configuration (2b, see Table 13). . . 25

21 An overview (early stage) of the notched beam (8a) surface captured by ARAMIS showing the calculation method. . . 26

22 An overview (last stage) of the notched beam (8a) surface captured by ARAMIS showing the calculation method. . . 26

23 The notched beam model mesh in ABAQUS. . . 28

24 Normal stresses distribution (ABAQUS) . . . 28

25 Shear Stresses distribution (ABAQUS) . . . 29

26 Normal stress and mean normal stress for 0 ≤ x_o≤ 100mm. . . 31

27 Shear stress and mean shear stress for 0 ≤ x_o≤ 100mm. . . 31

28 Unreinforced beams vertical displacements measured by LVDT 1. . . 33

29 Unreinforced beams total deflection (lab test). . . 34

30 Unreinforced beams vertical displacements measured by ARAMIS. . . 34

31 Unreinforced beams horizontal displacements measured by LVDT 2. . . . 35

32 Unreinforced beams horizontal displacements measured by ARAMIS. . . 35

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33 Before and after test view of the LVDTs fixed on one beam-1b. . . 36

34 The crack initiation in beam 5a at 8.45-8.74 kN (ARAMIS). . . 36

35 The crack opening in beam 5a at failure (ARAMIS). . . 37

36 The CF-reinforced beams vertical displacements- LVDT 1. . . 38

37 The CF-reinforced beams horizontal displacements LVDT 2. . . 38

38 The CF-reinforced beams total deflections. . . 39

39 The CF-reinforced beams vertical displacements- ARAMIS. . . 39

40 The CF-reinforced beams horizontal displacements- ARAMIS. . . 40

41 Strain Overlay of beam 15a at 70% of the total applied load (ARAMIS). . 40

42 Strain Overlay of beam 15a at failure (ARAMIS) . . . 41

43 The plywood-reinforced beams vertical displacements- LVDT 1. . . 42

44 The plywood-reinforced beams horizontal displacements LVDT 2. . . 42

45 The plywood-reinforced beams total deflections. . . 43

46 The plywood-reinforced beams vertical displacements- ARAMIS. . . 43

47 The plywood-reinforced beams horizontal displacements- ARAMIS. . . . 44

48 Strain Overlay of beam 13a at 70% of the total applied load (ARAMIS). . 44

49 Strain Overlay of beam 13a at failure (ARAMIS) . . . 45

50 Rectangular beam with an end notch. . . 46

51 The Force at section a-a. . . 47

52 Another way of fixing the LVDTs on the beams. . . 48

I.1 Beams used in this project 1,2,4,5. . . ii

I.2 Beams used in this project 6,7,8,9. . . iii

I.3 Beams used in this project 10,12,13,15. . . iv

III.1 The crack initiation in beam 5b (ARAMIS). . . xi

III.2 The crack opening in beam 5b at failure (ARAMIS). . . xi

III.3 The crack initiation in beam 8a (ARAMIS). . . xii

III.4 The crack opening in beam 8a at failure (ARAMIS). . . xii

III.5 The crack initiation in beam 8b (ARAMIS). . . xii

III.6 The crack opening in beam 8b at failure (ARAMIS). . . xiii

IV.1 The strain overlay at 70% of the maximum load in beam 10a (ARAMIS). xiii IV.2 The strain overlay at failure in beam 10a (ARAMIS). . . xiv

IV.3 The strain overlay at 70% of the maximum load in beam 10b (ARAMIS). xiv IV.4 The strain overlay at failure in beam 10b (ARAMIS). . . xiv

IV.5 The strain overlay at 70% of the maximum load in beam 1b (ARAMIS). . xv

IV.6 The strain overlay at failure in beam 1b (ARAMIS). . . xv

IV.7 The strain overlay at 70% of the maximum load in beam 7a (ARAMIS). . xv

IV.8 The strain overlay at failure in beam 7a (ARAMIS). . . xvi

V.1 The strain overlay at 70% of the maximum load in beam 6a (ARAMIS). . xvi

V.2 The strain overlay at failure in beam 6a (ARAMIS). . . xvii

V.3 The strain overlay at 70% of the maximum load in beam 12b (ARAMIS). xvii V.4 The strain overlay at failure in beam 12b (ARAMIS). . . xvii

V.5 The strain overlay at 70% of the maximum load in beam 2b (ARAMIS). . xviii

V.6 The strain overlay at failure in beam 2b (ARAMIS). . . xviii

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List of Tables

1 Mechanical properties of some of the composites [18]. . . 10

2 E-glass fiber properties [3]. . . 11

3 HS and HM carbon fibers properties [3]. . . 11

4 Aramid fibers properties [3]. . . 12

5 Properties of matrix [3]. . . 13

6 Properties of the Glulam Beams used in this project [6] . . . 21

7 Properties of SikaWrap-230 Carbon Fiber [11]. . . 22

8 Mechanical / Physical Properties of Sikadur -300 [11]. . . 22

9 Design Values of Plywood grade P30 [1]. . . 23

10 Material parameters used in ABAQUS [7]. . . 27

11 Using different distances x along the path to estimate x₀. . . 30

12 Crack length x0 iterated process calculations under the combined stresses mode. . . 30

13 Summary of the tests. . . 32

14 The design shear stress according to Eurocode 5 requirements compared to the failure load from lab tests. . . 46

15 The tension forces according to Nordic glulam handbook requirements and the failure load from lab tests for the plywood-reinforced beams. . . 47

II.1 Normal and mean normal stress -ABAQUS. . . v

II.2 Shear stress and mean shear stress -ABAQUS. . . viii

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Acknowledgements

First and foremost, our utmost gratitude to Professor Erik Serrano for his guid- ance that helped us hurdle all the obstacles in the completion of this thesis. We are so happy for getting the chance to work with him in this project and we will never forget his sincere concern and patience.

We would like to thank Bertil Enquist for his time, support, advices. He have taught us a lot regarding the experimental work and lab work would have not been interesting without his valuable thoughts.

Last but not the least, many thanks to our families and friends for their support, inspiration and love.

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1 Introduction

Wood is, thanks to its versatility and ease of use, the oldest material used in construction. Timber structures vary from single family houses to large span structures and buildings. Every building has specific design needs resulting in cross sections of e.g.

beams and columns of different sizes. However, solid (sawn) timber can only be found up to certain dimensions, which makes it necessary to use so-called Engineered Wood Products (EWPs). EWPs such as LVL, Parallam, Glulam, I-joists, and box beams are made of combinations of various forms of wood-based materials and/or sawn timber.

Nowadays, EWPs are used in a wide range of structural applications; from small family houses to large structures, for indoor and outdoor uses, see Figure 1.

Glued laminated timber, or Glulam, consists of small sections of timber laminates that are glued together to form beams and columns, in many cases their position in the final product being dependent on their performance characteristics. These laminates are laid up such that their grain direction is parallel to the longitudinal axis of the beam.

The use of several laminations to form a cross-section, results in decreased variability in material characteristics, thanks to the fact that natural growth characteristics like knots are more dispersed within the glulam volume in comparison to sawn timber [14].

Glulam has been used in Europe since the end of the 19th century. The use of glulam in construction has significantly increased due to its strength, innovative, and versatile properties. The durability of timber combined with the modern industrial techniques provides the glulam with unique qualities; it has an excellent stiffness to weight ratio, its strength is comparable to steel in relation to its self weight. It can be manufactured in almost any size and shape. It can e.g. be manufactured in curved shapes, which makes it easy to achieve high standards in terms of architectural solutions. In roofs, glulam provides large, open spaces without the need for intermediate supports. Glulam is resistant to most corrosive environment agents. It has excellent fire performance with sustained load bearing capacity during fire, and it has good heat insulating properties [14],[12], [3].

Glulam beams can be tapered (single or double), or curved. It can be also notched at the supports. Due to the large sizes and long spans of glulam beams, it is exposed to the risk of connection failure. Therefore, proper detailing of connections is required to transfer design loads to glulam members without causing localized stress concentrations.

The case of notched glulam beam makes a stress concentration at the end of the beam at the tension side. One important aim of timber engineering research is to study the behavior of stresses in such areas, and trying to increase the load-carrying capacity where needed [6].

1.1 Background

The sudden change in a notched beams cross section leads to a stress concentration at the re-entrant corner of the notch. The concentration of shear and tensile stresses perpendicular to the grain can lead to a catastrophic brittle failure of the beam due to

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Figure 1: Examples of timber structures [Pictures were taken at Linnaeus University- V¨axj¨o- building M, N, and Videum].

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crack propagation from the notch corner [8]. Therefore, special design considerations are needed in order to guarantee structural safety.

Generally, notches in beams should be avoided due to their effect of reducing the load carrying capacity. If this is not possible, notched ends of a beam should be reinforced in a way to prevent brittle failure. The reinforcement can be applied on the surface of the member or inside it. Inner reinforcement is often made by applying fully threaded (self-tapping) screws or glued-in steel rods. Outer reinforcement can be realized by adhering panels made of wood based materials like fiber boards, plywood or by using fiber reinforced composites (FRP). Both inner and outer reinforcements have their respective advantages and disadvantages. On the one hand, inner reinforcement does not change the appearance of the beam. Outer reinforcement, on the other hand, has the advantage of providing a higher initial stiffness.

1.2 Problem description

Crack opening due to tension perpendicular to grain stress is the most apparent failure mechanism at the notch corner. However, there exists also a shear component, which usually is neglected in design. Especially for longer cracks the reinforcement is loaded by high shear forces leading to a more pronounced influence of shear stress. The design of the reinforcement in design codes and handbooks [6] up to now has been carried out by taking into account only the normal forces perpendicular to the beams axis. Thus, one way to achieve a more efficient share of loading between the timber element itself and the reinforcement and furthermore to enhance the overall structural behavior of the strengthened element, is to consider also the shear component in design. To do so, improved and more detailed knowledge about the structural behavior and load-carrying capacity of different types of reinforcement is required.

In timber structures in general, it is very important to avoid tensile stresses perpendicular to the grain since the strength in that direction is the lowest [8]. It is highly affected by growth features such as knots and fissures, particularly in early wood, which is the softer part of annual rings [6]. Notching a timber member creates a combination of stress concentration around the notch, and interaction of tension perpendicular to the grain and shear stresses [15]. This interaction causes splitting along the grain, typically, starting at the re-entrant corner of the notch. In addition, notching will highly reduce the area that resist the bending and shear forces. The combination of high stresses will be concentrated in a very small region that makes it hard to determine the load bearing capacity of the beam. The failure in this case will be brittle which is something that for safety reasons needs to be prevented. The distribution of tensile stresses near the tip of a notched beam is shown in Figure 2; it is obvious that already at low levels of external loading the stresses can reach the tensile strength of the beam material [6].

As described above, notching timber flexural members leads to a complex situation that is hard to analyze based on traditional linear elastic theory for beams. Instead, knowledge of fracture mechanics and stress concentrations is required.

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Figure 2: Distribution of tensile stresses near the tip of a notched beam.

1.3 Aim and scope of work

The aim of this thesis is to study the structural behavior of notched glulam beams reinforced by adhered plywood panels and fiber reinforced polymer (FRP). This study will also include a literature survey on reinforcing methods and design approaches both for reinforced notches and reinforcing material. A series of experimental tests with notched glulam beams with different configurations of reinforcement is to be planned and carried out. In order to prevent the beams from failing due to bending, the test set-up will be a three point bending test of short-span notched beams of structural sizes. The dimensions of the beam, the geometry of the notch and the configuration of the reinforcement are chosen such that a combination of stresses at the notch and in the reinforcement occurs being representative for the real structural behavior of the beams. Deformations and forces are to be measured both with conventional techniques and with contact-free measurement systems. The observation made along the tests are to be analyzed by existing design approaches for hand calculation (including the load-carrying capacity of the plywood and the FRP itself and for the reinforcements interaction with the timber) and by using the commercial FE-software ABAQUS.

The work is mainly focused on the analysis of shear and tensile stresses at notch corners and at the crack tip and on the resistance of the reinforcement against these effects. It is the aim of the work to get a deeper understanding on how brittle failure of notched beams can be prevented and how the load carrying capacity and the structural behavior can be optimized by means of outer reinforcement made of plywood or FRP.

2 Literature review

2.1 Design of notched beams

The tension, compression, and shear forces experienced by a uniform cross section beam are primarily oriented along the grain with basically no cross grain tension. This fact is the base of deriving the strength of wooden beam in a spanning application assuming

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that the beam is supported underneath its ends. However, notching is needed in various places to run utilities or to fit with other structural members, see Figure 3 [16].

It is preferable to avoid notches in beams due to their effects of reducing the strength and stiffness by initiating splits or cracks. If a notch is required, it must be at the end support (one or both/ top or bottom) of the beam. Butler has recommended in the Architectural Engineering Design [5] that notches can not be located near the center or in the vicinity of cantileivered supports. And in case of right angled notches, it should be given a radius, e.g. 25 mm. Furthermore and according to Butler [5], the depth of the notch should not exceed 1/10 of the beams depth or 3 inches, it also needs to satisfy one of the two following equations, see Figure 4:

Notch at the bottom of the beam:

dV = 0.67f_vbd²_eb (1)

Where;

V : is the vertical reaction of beam at end support.

f_v: is the safe unit shear stress for species and grade of wood.

b: is the width of beam.

d: is the full depth of the beam.

b_e: is the reduced depth of the beam, b_et when notch at top and b_eb when notched at bottom.

Based on fracture mechanics studies at the inner angle of the notch and as stated in Eurocode 5 [14], the load carrying capacity of a beam with an unreinforced notch can be checked by the following method, see Figure 5:

τ = 1.5V

bh_ef ≤ k_vf_v (2)

where;

k_v = min(1, k_n(1 +1.1i^1.5

√ h ) 1

√ h)(p

α − α²+ o.8e h

r1

α − α²)⁻¹) (3) For beams with a notch at the upper side, kv = 1 and for beams with notch at the bottom side and with h is the total depth of the beam, h_ef is the beam’s depth excluding the notch depth, i = α

h − h_ef, α = h_ef h .

The reason for the design equation being expressed as a shear force design equation is that the notch is located in areas of high shear and effectively zero moment. These equations are limited to single span, simply supported beams [20]. In fact, and as stated before, it is the lateral tension forces that are the critical parameter for the carrying capacity. However, the design equation has been expressed in terms of the shear stresses in the residual cross section for simplicity [6].

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Figure 3: Different types of notches at a beam⁰s end.

Figure 4: Notched beam at top and bottom.

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In the case of notches at the compression side extended into areas of significant moment, the bending capacity should also be checked. This can be performed by using the remaining beam section and the proper allowable stresses for laminations remaining at the notch area [20].

e ≤ d_e, f_v = 3V 2b[d − (d − de

d_e )e]

e > de, fv = 3V 2bd_e

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f_v = 3V 2bd_e(d

d_e)² (5)

Where; fv=shear stress, V =shear force at the notch, b=width of beam, d=depth of beam, de=effective depth, e=length of notch.

Figure 5: Compression and tension side notches.

2.2 Reinforcement of notched beams

Notching a wooden beam leads to two major problems; lower strength across its grain, and dynamics of crack propagation at the apex of an advancing crack where cross grain tensile stresses are concentrated. As a result, the effective strength will dramatically decrease placing the beam in its weakest mode. In this situation, beams need to be reinforced to provide a higher tensile capacity across the grain and preventing any expected crack. To avoid extensive reinforcement of the entire beam, other methods are

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followed to increase the strength of the notched beam, see Figure 6. These reinforcement methods are based on blocking the expected crack or split [16]. The resultant high load-bearing capacity, invisible reinforcement, easy installation, easy design, and the economical factor are all determining the reinforcement method and the choice of reinforcement material. The most common methods and materials used in reinforcing notched wooden beams are; screws, different types of fiber reinforced polymer (FRP), and other materials.

Figure 6: Notched beam reinforced at its end.

2.2.1 Reinforcement by screw/s

Based on one principle, reinforcement by screw/s is performed in two ways; glued-in screw/s, and full-threaded lag screw/s. To resist the tendency of split at the square cornered notches, the full-threaded lag screws can be used, see Figure 7. It is the responsibility of the designer/engineer to decide the methodology, the related fabrication details and sizing of screws [20]. Normally, lag screws of a diameter from 5/8 to 1 inch are used for most reinforcing requirement [2]. There is a specific requirement for the lag screw length and size based on beams width. On the other hand, the use of glued-

Figure 7: Reinforcement by full-threaded lag screw/s.

in screws is limited to indoor construction of service class 1. It must be also avoided in areas of major variations in the moisture content. This method requires that the screw/screws placed as near as possible to the edge of the notch taking into account the

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minimum edge and center distance requirements. As shown in Figure 8, the glued-in length above the notch should be at least half the beams depth at the support. It is not recommended to have more than two rows of screws along the beams length, see Figure 8 [6]. The carrying capacity of the screws is calculated as follows:

The reinforcement shall be designed for a tension force Fs. Fs= R[3(h − h_ef

h )²− 2(h − h_ef

h )³] (6)

With this resistance of the reinforcement kv = 1 will apply.

Figure 8: Reinforcement with glued-in screws.

2.2.2 Reinforcement by fiber reinforced polymer (FRP)

Fiber reinforced plastic or polymer is widely used in construction industries. There are two techniques of strengthening a beam by using FRP. The first one aims to increase the strength and the deflection capacity by pasting FRP plates to the bottom (tension side) of the beam. The other technique is pasting FRP strips in ’U’ shapes around the sides and bottom of a beam providing a higher shear resistance [10]. In this project carbon fiber was glued on the two sides of the beam only and not on the bottom to be comparable to the plywood-reinforcement as will be explained later.

Fiber reinforced polymer is a composite material made of a polymer matrix reinforced with fibers. The fiber and the matrix are bonded at the interface as shown in Figure 9, and each of them have different mechanical properties. By bonding these two materials, the FRP gets its satisfactory performance. The fibers provide FRP with stiffness and strength, while the matrix gives rigidity and environmental protection [18]. The three dominating fibers in civil engineering industry are glass, carbon (graphite), and aramid.

However, there are natural fibers which have been experimented and showed promising results such as; flax, bamboo, hemp, cotton, etc [3].

The Mechanical Properties of the FRP Composite

The mechanical properties of FRP depend on; the ratio of fiber and matrix material, the fiber orientation in the matrix, the manufacture method, and the mechanical properties

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Figure 9: Formation of fiber reinforced polymer composite.

of the constituent materials. Table 1 shows the mechanical properties of some of the composites which are combinations of various reinforcements and matrices [18].

Table 1: Mechanical properties of some of the composites [18].

Material Specific Tensile Tensile Flexural Flexural weight strength modulus strength modulus

(MPa) (GPa) (MPa) (GPa)

E-glass 1.9 760-1030 41 1448 41

S-2 glass 1.8 1690 52

Aramid 58 1.45 1150-1380 70-107

Carbon (PAN) 1.6 1930-2689 130-172 1593 110

Carbon (Pitch) 1.8 1380-1480 331-440

Glass Fiber (GF)

Glass fiber is the most commonly used reinforcement in constructions due to its properties. It has a high strength, low weight, and resistance to the environmental effects, ability to form different shapes, low maintenance, and good durability [10]. The GF is a processed form of glass, which in turn is composed of a number of oxides, such as silica compounds with raw materials, such as limestone, boric acid, fluorspar, and clay.

Different proportions of each element gives different type of glass fibers (E, C, R, S, and T). Each type has different properties and uses. E-glass is the most commonly used due to its good mechanical properties and relatively low cost, see Table 2 [3]. GF is manufactured by drawing very fine filaments from the melt oxides, the filaments ranging from 3 to 24 µm in diameter [18]. To reinforce the matrix, there are several forms of glass fiber strand; chopped fibers, chopped strand mats, surface tissue, and woven fabrics. In civil engineering applications, woven fabrics and glass fiber strands are the most used [18].

The glass fiber reinforcement has more or less same influence on the reinforced members as the plywood reinforcement. However, the glass fiber is preferred due to its

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transparent property and the reminiscent of thick coat of lacquer [6]. Nevertheless, glass fiber reinforcement has drawback that makes the plywood is more proper to use. Some of the glass fiber reinforcement disadvantages are: the limited experience of the glass reinforcement methods and its affect on workers health [6]. The properties of E-glass fiber are shown in Table 2.

Table 2: E-glass fiber properties [3].

E.axial/ Maximum Maximum Poisson’s Density Price

E.radial Stress Strain ratio

(GPa) (GPa) (%) (mg/m3) (Euro/kg)

E-Glass 76/76 2 2.6 0.22 2.6 1.5-3

Fibers

Carbon Fiber (CF)

Compared to glass, carbon has higher strength and much better mechanical properties.

However, it is much more expensive. CF is made by oxidation, carbonisation and graphitization at high temperature of high content carbon precursor materials [3]. There are three precursor choices used in manufacturing CF; pitch, cellulose or polyacrylonitrile (PAN). Among the three choices, PAN precursors are the most used as these provide the CF with the highest mechanical properties and the lowest cost [11,12]. Temperature variation at the graphitization process from 2600 to 3000^◦C, gives high strength (HS) or high modulus (HM) fibers respectively, see Table 3 [3].

Table 3: HS and HM carbon fibers properties [3].

HM Carbon 380/12 2.4 0.6 0.2 1.95 20-60

Fibers

HS Carbon 230/20 3.4 1.1 0.2 1.75 20-60

Fibers

Aramid Fiber

Aramid or aromatic polyamide fiber is manufactured by blending and reaction of aromatic diamines and aromatic diacid chlorides [3]. A solution of aromatic polyamide

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at a temperature between -50 to -80^◦C is extruded into a 200^◦C hot cylinder. Then, fibers left from evaporation are stretched and drawn to get higher strength and stiffness.

This process makes the aramid molecules highly oriented in the longitudinal direction.

Aramid fibers have higher strength and toughness than glass and carbon. They are also very fire, heat, and chemical resistance. However, they are difficult for cutting and machining [18]. Kevlar is a common trade name for aramid fibers. The properties of the aramid fibers are shown in Table 4 [3].

Table 4: Aramid fibers properties [3].

Aramid 130/10 3 2.3 0.35 1.45 20-35

Fibers

Matrix /Resin

The matrix material is a polymer which consists of molecules made from monomer which is simpler and smaller units. Fibers themselves are of little use without the matrix materials. Fibers have a higher modulus and lower elongation than the matrix in order to carry maximum load [18]. The matrices must satisfy the requirements of three essentials features [3]:

1. The mechanical properties: high strength and stiffness, high strain at failure to avoid FRP brittle failure.

2. The adhesive properties: good bonding of fibers and matrix to distribute the loads equally and efficiently.

3. The resistance to environmental degradation: matrix should have a good resistance to protect the fiber from the environmental agents and aggressive substances.

The failure mode in constructions is highly influenced by the type of matrix and its compatibility with the fibers. Several types of matrices can be used in civil engineering applications, however the two major types are thermoplastic and thermosetting polymers. Thermoplastic polymers have lower stiffness and strength than the thermosetting but they are ductile in nature and tougher [18]. The properties of some of the most commonly used matrices are shown in Table 5 [3].

2.2.3 Reinforcement by plywood

Plywood consists of thin sheets of wood veneers glued together with adjacent piles hav- ing their grain at right angles to each other to increase the strength. Plywood resists

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Table 5: Properties of matrix [3].

(GPa) (MPa) (%) (mg/m3) (Euro/kg)

Thermoplastic

Polypropylene 1-1.4 20-40 300 0.3 0.9

(PP)

Polyethere- 3.6 170 50 0.3 1.3

therketone

(PEEK)

Ployamide 1.4-2.8 60-70 40-80 0.3 1.14 5

(PA)

Thermosets

Epoxy^(EP) 2-5 35-100 1-6 0.35-0.4 1.1-1.4 6.5

Polyester 2-4.5 40-90 1-4 0.37-0.39 1.2-1.5 1.5

(UP)

vinylester 3 70 5 0.35 1.2 2.5

cracking, breaking, shrinkage, twisting, and it has a high degree of strength. Figure 10 shows the plywood that has been used in this project and an example of beam reinforcement by plywood. The veneers used to form the plywood layers are normally bonded with grains running against one another and with an odd number of composite parts.

Plywood can be soft or hard and each type has its various applications. Plywood is a widely used wood product and an essential material in construction industry; it is used in floors, roofing and reinforcing of wooden beams [10].

According to Williamson [20], when using the plywood in reinforcing glulam beams, a

Figure 10: a. The Plywood used in this project, b. An example of beams reinforcement by plywood [the photo was taken at -Linnaeus University- Building M].

minimum thickness of 10 mm is glued on each side of the beam as shown in Figure 11.

And the orientation of the outer layer shall be perpendicular to the longitudinal direction of the beams fibers. During the gluing process, a pressure is applied with aid of nails or screws. The penetration of nails or screws in the glulam must be at least twice the

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thickness of the plywood. In addition, Williamson [20] calculated the carrying capacity of the plywood and glue joint FR using equation 7. The strength is reduced by 25 % according to the regulations as the stress distribution in the glue joint is non-uniform [20].

FR= min.







2cd_ef.0, 25f_t 2chef.0, 25fv

2c(h − h_ef).0, 25f_v

(7) Where; 2def is the combined thickness of plywood on both sides of the beam with the direction of the grain perpendicular to the longitudinal direction of the beam, Figure 11 shows the definition of equation 7 parameters.

Design Conditions:

FS≤ F_R (8)

Where FS is the tension forces in the beam, and it is calculated according to equation 6 for both plywood and glued-in screw reinforcement [6].

Figure 11: Notched beam reinforced by Plywood.

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3 Theory

Theoretical methods for strength analysis are:

3.1 Constant cross-section area beams

This is the simplest possible glulam beam model and the one used most often in engineering applications. This type of beam can be analyzed by the classical beam theory (Bernoulli-Euler) combined with linear elastic material behavior and single point maximum stress failure criterion. When this type of model is subject to bending, the shear stresses at the top and the bottom faces of the beam will be zero. Furthermore, the maximum shear stress will arise at the neutral axis position [17]. For rectangular sec-

Figure 12: Maximum shear stress for rectangular section beam.

tion of width b and depth h the maximum shear stress will be calculated according to equation 9, see Figure 12.

τ = 3V

2bh (9)

Where V is the shear force.

In structural design of timber structures it is important to avoid tensile stresses perpendicular to the grain since the strength in that direction is very low [8]. This type of tensile stresses in case of notched beams at the end will cause brittle failure. In Figure 13 the distribution of tensile stresses near the tip of the notched is presented and it is obvious that the stresses can reach the strength of the material even for low load levels, assuming a linear elastic material behaviour the theoretical stress will typically be much larger than the tensile strength [14]. Since the very high stress is often concentrated at a very small region it is difficult to determine the load bearing capacity of a notched beam with the conventional stress criteria i.e. the one described above. Instead it is necessary to rely on tests or on concepts based on fracture mechanics.

3.2 Linear elastic fracture mechanics (LEFM)

Linear Elastic Fracture Mechanics (LEFM) first assumes that the material is linear elastic. Consequently, the stress field near the crack tip is calculated using the theory of elasticity. Instead of maximum stress, energy criteria are used to determine the load

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Figure 13: The theoretical and the estimated stress perpendicular to the grain.

bearing capacity of the structure. In Linear Elastic Fracture Mechanics, most formulas are derived for either plane stresses or plane strains, associated with the three basic modes of loadings on a cracked body: opening, sliding, and tearing, see Figure 14 [8].

In fact, it is not possible to use LEFM to determine where a crack will develop in a stressed body but it can be used for analysis of whether an existing crack will propagate or not by considering the energy balance of the system [8].

Figure 14: Mode I (Tension, Opening), Mode II (In-Plane Shear, Sliding), Mode III (Out-Of-Plane Shear, Tearing).

3.2.1 Stress Intensity Factor

Three linearly independent cracking modes are used in fracture mechanics. These load types are categorized as Mode I, II, or III as shown in Figure 14. Mode I is an opening (tensile) mode where the crack surfaces move apart in the direction perpendicular to the crack surface. Mode II is a sliding (in-plane shear) mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Mode III is a tearing (Out-Of-Plane Shear, Tearing) mode where the crack surfaces move relative to one another and parallel to the leading edge of the crack. Mode I and II are the most common load type encountered in engineering design.

The stress intensity factor for mode I is KI and applied to the crack opening mode. The mode II stress intensity factor, K_II, applies to the crack sliding mode and the mode III

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stress intensity factor, KIII, applies to the tearing mode. These factors are formally defined as follow:

K_I = lim_r→0√

2πrσ_yy(r, θ) (θ = 0) K_II = lim_r→0√

2πrτ_yx(r, θ) (θ = 0) K_III = lim_r→0√

2πrτ_yz(r, θ) (θ = 0)

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Figure 15: a. Initial crack in the beam and b. The stresses in an infinitesimal element.

Critical stress intensity factor

A crack will grow when the stress state at the crack tip exceeds a certain critical value.

The stress intensity factor determines the amplitude of the crack tip stress state for certain geometry and loading case. We may thus conclude that a crack will grow when K reaches a critical value. This implies that a crack growth criterion can be formulated, where the stress intensity factor for a certain situation is compared to this critical value.

The value of the actual stress intensity factor has to be calculated, e.g. by the use of finite element analysis [8]. Figure 15 shows an initial crack in a beam and the stresses in an infinitesimal element. One approach is then to calculate the state of stress close to the crack tip and substitute the stress component values obtained into equations 10.

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The critical value has to be known from experimental measurements. It is called the Fracture Toughness and denoted as Kc.

KI = KIC; KII = KIIC; KIII = KIIIC (11) 3.2.2 Energy Release Rate

In 1920s, Grifith proposed the basic concept of the energy release rate approach. He stated that the difference between a cracked and uncracked body is the additional surface associated with a crack. In order for the crack to open i.e. to create new surfaces, energy is required. According to Grifith, a stressed body remains stable if it has the sufficient energy to opens a crack while still maintain equilibrium [19]. Thus, at crack propagation the energy release rate is given by:

G = −∂Π

∂A (12)

Where Π is the potential energy of the system and A is the crack area. The crack is just about to propagate when the energy release rate equals its critical value[19].

G = Gc (13)

As stated before, three possible modes of loading can be defined each of these being associated with its respective critical energy release rate.

3.3 Mean Stress Approach

The conventional method discussed in 3.1 is not valid to use in case of stress singularities at pre-existing crack tip, instead Linear Elastic Fracture Mechanics is used in such cases as mentioned in 3.2. However, LEFM is only valid in the case of so-called square root singularities. The mean stress approach is a linear elastic failure criterion method of more general applicability than the previously mentioned methods in sections 3.1 and 3.2. The mean stress approach idea is to consider the mean stress acting over an area instead of the stress in a point. The size of the failure area considered in turn depends on the material properties [7]. A commonly made assumption for strongly orthotropic material like wood is that the tensile forces perpendicular to grain and/or shear forces along grain develop a fracture along grain. Below the basic expression needed for the mean stress approach calculations are given without any detailed derivation, such a derivation can be found in Gustafsson⁰s paper [7].Two criteria have to be chosen in this approach; a basic stress criterion which is here the stress criterion of Norris [4], and a basic LEFM crack propagation criterion, here chosen as the crack propagation criterion of Wu:

(σ

f_t)²+ (τ

f_v)² = 1 (14)

K_I

K_IC + ( K_II

K_IIC)² = 1 (15)

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Where σ and ft are the perpendicular to grain tensile stress and strength respectively, and τ and fv are the longitudinal shear stress and strength respectively. K_I and K_II are the stress mode I and II intensities, K_IC and K_IIC are the corresponding critical stress intensities.

When replacing the tensile and shear stresses by the mean stresses, then:

(¯σ

f_t)²+ (τ¯

f_v)² = 1 (16)

Where ¯σ and ¯τ are the mean stress acting on an area of length x₀, a length along the assumed crack path. The length x₀ is now chosen such that for a linear elastic structure with an pre-existing sharp crack, the mean stress approach and the traditional LEFM approach should give the same prediction of load bearing capacity. This gives for pure mode I and mode II.

¯ σ =

s 2K_I²

πx0

; τ =¯ s

2K_II² πx0

(17) And by substituting the mean stresses in equation 18 into equation 17 , the length x₀ is derived:

x0 = 2K_I² πf_t²(1 +

(K_II KI

)² (f_v

ft

)²

) (18)

Where; KI is the magnitude of KI at the instant of fracture at the actual mixed loading ratio k. KI can be calculated from equation 19 as follow:

K_I = −K_IIC² 2KICk² +

s K_IIC⁴

4K_IC² k⁴ +K_IIC²

k² (19)

Where the actual mixed loading ratio is:

k = KII

K_I = τ¯

¯

σ (20)

Then the crack length x0 in equation 18 expressed by the stress intensity factor and the critical energy release rate will be as follow:

x0 = 2K_IIC² πf_t² (K_IIC

KIC

)² 1 4k⁴(

r

1 + 4k²( K_IC KIIC

)²− 1)²(1 + k² (f_v

f_t)²

) (21)

x0= 2EIGIC

πf_t² Ex

E_y(GIIC

G_IC )² 1 4k⁴(

s

1 + 4k²r Ey

E_x GIC

G_IIC − 1)²(1 + k² (fv

ft

)²

) (22)

Where;

K_I =√

E_IG_I; K_II =√

E_IIG_II (23)

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1 E_I = 1

E_x s E_x

2E_y v u u t

sE_x

E_y + E_x

2G_xy − v_yxE_x

E_y; 1 E_II = 1

E_x r 1

2 v u u t

sE_x

E_y + E_x

2G_xy − v_yxE_x E_y (24) The computation of the crack length in equation 21 and 22 is iterative because the two equations contain the mixed mode ratio k which requires x₀ to be computed. The mean stress approach is the adopted method in the calculations of this project.

3.4 Finite Element Method

The finite element method is a numerical approach to solve general differential equations in an approximate manner [13]. The considered physical problem is described by a differential equation or equations that are assumed to hold over a certain one-, two-, or three-dimensional region. By using the finite element method, the entire region is divided into smaller parts (finite elements) and the approximation is carried out over each element instead of the approximation for the entire region. The finite element software ABAQUS has been used in this project.

4 Methodology

4.1 Experimental

4.1.1 Material and specimens

Twelve Glulam beams of type GL40C and size (90*315*4000mm) were used in this project. Specimens of length 50 mm were cut out from the ends of each beam and used to calculate the wet and the dry densities and subsequently the moisture content (MC).

The beams were notched at each end and the notch dimensions are (400*110mm). The reinforcement has been done using carbon fiber and plywood in two different configurations; vertical and diagonal with an angle of 45 degrees, see Figure 16. The two sides of the beams were reinforced with dimensions of 100*315 mm as shown in Figure 16. The carbon fiber that has been used is SikaW rap^r− 230CN W with Sikadur^r− 330 glue.

SikaW rap^r− 230CN W is a unidirectional, carbon fiber fabric (sheet) with mid-range strengths. Sikadur^r − 330 is two part epoxy, thixotropic, mid-viscous, filled epoxy based primer / impregnation resin / adhesive with standard pot life and curing speed [11]. It is designed to be used for SikaW rap^r fabric installation. The plywood is P30 K-PLY of a 10 mm thickness which is glued with an adhesive that is a mixture of Casco Adhesive 1711 and 2622 hardener of mixing ratio 100:25 respectively. In addition to the glue, pressure was applied by using nails to attach the plywood to the beams. For the CF reinforcement the beams were left in the climate room of (65 % humidity and 20^oC) for seven days to cure, while it took two days to cure in the plywood reinforcement due to the different glue requirements. The properties of Glulam beams, carbon fiber, Sikadur^r− 330, plywood are all shown in Tables 6, 7, 8, 9 respectively.

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Figure 16: a. Notched beam reinforced by CF. b. Notched beam reinforced by plywood.

Table 6: Properties of the Glulam Beams used in this project [6]

Strength values in Mpa Glulam CE L40c

bending strength f_m,k 30.8

Tension strength parallel to grain ft,0,k 17.6 Tension strength perpendicular to grain f_t,90,k 0.4 Compression strength parallel to grain f_c,0,k 25.4 Compression strength perpendicular to grain fc,90,k 2.7

Shear strength f_v,k 2.7

Modulus of elasticity ^Mpa

Parallel to grain E0,mean 13 000

Characteristic E_0,05 10 500

Perpendicular to grain E_90,mean 410

Shear modulus Gmean 760

Density in kg

m³ ρ_k 400

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Table 7: Properties of SikaWrap-230 Carbon Fiber [11].

Dry fiber properties

Values in longitudinal direction of the fiber

tensile modulus minimum value 234000 N/mm tensile strength minimum value 4300 N/mm² Laminate properties

Values in longitudinal of the fibers single layer direction related to effective

,10 samples per test series

laminate thickness impregnation resin Sikadur-300

laminate thickness 1 mm

(nominal)

design cross section 1000 mm²

per 100 mm width

tensile modulus Average 29 KN/mm²

Characteristic 26.3 N/mm²

tensile strength Average 440 N/mm²

Characteristic 363 N/mm² Laminate properties

Values in longitudinal of the fibers single layer direction

related to fiber

,10 samples per test series

thickness impregnation resin Sikadur-300

laminate thickness (nominal) 0.131 mm

design cross section per 100 mm width 131 mm²

tensile modulus Average 222 KN/mm²

Characteristic 201 N/mm²

tensile strength Average 3367 N/mm²

Characteristic 2778 N/mm² Design values Tensile resistance

Average 415 KN/m

Characteristic 365 KN/m Tensile force at 0.4 % elongation

Average 113 KN/m

Characteristic 104 KN/m Tensile force at 0.6% elongation

Average 169 KN/m

Characteristic 156 KN/m

Table 8: Mechanical / Physical Properties of Sikadur -300 [11].

Tensile Strength 30 N/mm2 (7 days at 23^oC) E-Modulus

Flexural 3800 N/mm2 (7 days at 23^oC) Tensile 4500 N/mm2 (7 days at 23^oC)

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Table 9: Design Values of Plywood grade P30 [1].

Strength values in (Mpa) PLY P30

Tension strength parallel to grain f_t,0,k 16.7 Tension strength perpendicular to grain f_t,90,k 8.3

Planer shear strength fv,k 1

Panel shear strength f_p,k 3

Modulus of elasticity in Mpa

Parallel to grain E0,mean 8000

Characteristic E_0,05 10 500

Perpendicular to grain E_90,mean 4000

Figure 17: Three points bending test set up dimensions and the test set up as done in this project (picture taken for one test in lab).

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Figure 18: The LVDTs 1 and 2 setup and dimensions.

4.1.2 Test set up

Three point bending tests have been carried out for short-span notched beams in order to avoid bending failure. The test setup and the distances between the supports and the applied load are shown in Figure 17. Two LVDTs were fixed on the notch tip to measure the vertical and the horizontal displacements, see Figures 17 and 18. In addition, one sensor was placed under the beam where the load is applied to measure the total deflection of the beam, see Figure 17. ARAMIS, a non-contact three dimension full field displacement measuring system was also used for each test. ARAMIS measures the displacements on a surface and calculates the surface strains using two cameras simultaneously; it tracks the 3D coordinates of an object surface under loading for every stage of the load and then calculates the deformations at a large number of points on the surface, see Figure 19. The resolution used in ARAMIS was set to be 15 by 15 pixels, meanng that each point reported in ARAMIS represents a surface of approximately 4 by 4 mm².

Figure 19: ARAMIS system as set in the lab during the test (a computer and two cameras).

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4.1.3 Work procedure

The plan was to test the twelve notched beams on both sides which would make a total of 24 tests. However, only seventeen tests were carried out and only fifteen of these tests were successful, see Table 13. Two beams were unreinforced, and ten were reinforced; half with carbon fiber and the other half with plywood. Four successful tests were performed for the unreinforced beams and the crack each time stopped before it reached the center of the beam. Six successful tests were done for the five CF-reinforced beams. In the case of the diagonal configuration, one side failed due tension and shear forces (7a) while the opposite side has failed due to bending (7b). Therefore, only one side was tested for the rest of the beams of the same configuration. For the vertical configuration, the crack propagation stopped before it reached the center of the beam in one test (10) while it went the whole way to the other support in the other beams (15,4), see Figure 20. Finally, six tests were carried for the five plywood-reinforced beams, see Table 13. The crack crossed the loading area in all beams except one (9); consequently, five successful tests were done for the five PLY-beams, see Table 13. The vertical and

Figure 20: An example of the crack propagation of glulam beam reinforced with plywood in diagonal configuration (2b, see Table 13).

horizontal displacements were measured during the tests with the two LVDTs; 1 and 2 respectively, see Figure 18. The same measurements have been calculated by ARAMIS.

Calculations in ARAMIS were done by placing four points on a certain distance from the notch⁰s vertical and horizontal edges as shown in Figure 21 where point 3 and 4 are placed vertically and point 5 and 6 horizontally. In addition, another two points were placed on the two reference triangular plates, one point on each; see point 1 and 2 in Figure 22. Later on, a line through the two vertical points and another line through the two horizontal points were drawn to calculate the angle of the notch during testing which is assumed to be a right angle at the notch tip before testing, see Figure 22. In order to measure the vertical and horizontal displacements, two planes were created using the two points on the reference triangular plates and the drawn vertical and horizontal lines so that each plane contains one reference point and normal to the adjacent-line, see plane 1 and 2 in Figure 22. Afterward, the vertical and horizontal displacements were

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obtained by measuring the distance between each plane and the fronting parallel line.

The reentrant angle at the notch tip was also calculated at the early and late stages of the test and the calculations show noticeable changes as presented in Figures 21 and 22.

Figure 21: An overview (early stage) of the notched beam (8a) surface captured by ARAMIS showing the calculation method.

Figure 22: An overview (last stage) of the notched beam (8a) surface captured by ARAMIS showing the calculation method.

4.2 Computational 4.2.1 ABAQUS Model

Based on the finite element method approach, a simple model of the notched beam was performed using the software ABAQUS 6.11. The model is a two dimensional plane

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stress modal that consists of two parts: the first part represents the unreinforced beam as shown in Figure 18, and the second part represents the two supports (steel plates) as analytical rigid surfaces. An interaction has been created between the two parts of type surface to surface contact were the steel plate part is the master surface and the beam is the slave surface to simulate the support conditions that were used in the lab tests.

Two properties were defined for the interaction: the normal behavior-(hard contact) and the tangential property -(penalty with friction coefficient equal to 0.6). Then, the three instances (part 1 and two instances of part 2) were assembled. The beams material elastic properties were defined according to Table 10 and the elastic type was chosen to be Lamina which is a two dimensional orthotropic material type. The material parameters that have been used in this model are taken from Gustafsson⁰s paper [7], and not the glulam L40C properties used in this project since the latter are characteristic values to be used in design (include high safety factors). In Gustafsson⁰s paper [7] these parameters have been used in fracture mechanics modelling for strength analysis of timber beams with a hole or a notch. Besides the initial step that is already created in ABAQUS,

Table 10: Material parameters used in ABAQUS [7].

Modulus of elasticity parallel to grain Ex 12000 MPa Modulus of elasticity perpendicular to grain Ey 400 MPa

Shear Modulus Gxy 750 MPa

Tensile strength perpendicular to grain ft,90 3 MPa

Shear strength fv 9 MPa

Fracture energy Mode 1 GIC 300 J/m²

Fracture energy Mode 2 GIIC 1050 J/m²

Poissons ratio vxy 0.5 -

stress intensity factor for mode IC KIC 0.507476 M P a.m^0.5 stress intensity factor for mode IIC KIIC 2.22192 M P a.m^0.5

an analytical step was added of type (Static, General) and the increment lengths were defined to be from 5% to 10% of the total loading range and with a maximum number of increments of 100. To calculate the stresses in the beam, two equal concentrated loads were applied of a total magnitude of 30 kN. The boundary conditions were set at the two supports to constrain the motion in vertical and horizontal direction. To mesh the beam in a suitable way, the beam was partitioned and a structured mesh was used in major parts of the beam. The mesh element length for the whole beam was set to 10 mm while an element length of 1 mm was used for the area around the notch tip, see Figure 23. In that area an unstructured mesh was used to simplify the mesh transition.

The two element lengths provide an accurate approximation to calculate the stresses at the notch tip and reduce at the same time the calculation time in ABAQUS. The normal stress and shear stress distributions in the notched beam are visualized in Figures 24 and 25.

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Figure 23: The notched beam model mesh in ABAQUS.

Figure 24: Normal stresses distribution (ABAQUS)

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Figure 25: Shear Stresses distribution (ABAQUS)

4.2.2 Method

To investigate where the crack is initiating, the normal (σ or S22) and shear stresses (τ or S12) were calculated from the notch tip along a path length of 100 mm. The stresses were calculated at the nodal points of the elements of size 1 mm. Based on the Mean Stress Method, the mean normal and shear stresses are considered instead of the maximum single point normal and shear stress. Each respective mean stress assumed acting on an area of b times x₀ [7]. The mean stress for a certain length x₀ is calculated in ABAQUS by integrating the stress along x0 and then dividing the result by x0, see Figures 26 and 27. As mentioned above, the calculation of x₀ is in reality an iterative procedure, since its calculation involves the mixed mode ration k, which in turn is a function of x0. To illustrate that the determination of x0 in practice is fast converging and rather insensitive for the initial guess arbitrary distances were chosen along the crack path to calculate x₀by use of equations 22 and/or 23. The results are given in Table 11, showing that the ratio k is rather insensitive for the length x0 chosen. The parameters listed in Table 10 were used in calculations [7]. Consequently, the stresses at x₀ equal to 18 mm were chosen as a start values for calculations. For simplification if pure mode I is assumed, then the crack growth occurs when the mean stress ¯σ exceeds the tensile strength ftfor the choice of x0 = 18 mm and the failure criterion reads:

¯

σ18= ft (25)

Structural behavior of notched glulam beams reinforced by means of plywood and FRP.

Diploma thesis, structural engineering