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IN DEGREE PROJECT MATERIALS SCIENCE AND ENGINEERING,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2018

Prestressed glue laminated beams reinforced with steel plates

Comparison between prestressed, reinforced and non-reinforced glue laminated beams according to the Eurocode and the Swedish annex

JAFAR AL-DJABER

NAJIB VEGHAR

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ARCHITECTURE AND THE BUILT ENVIRONMENT

Latin lower case letters Parameter Definition Unit 𝑏 Width / Breadth 𝑚𝑚 𝑏𝑒𝑑 Horizontal edge distance to reinforcement 𝑚𝑚 𝑏𝑠 Width of steel reinforcement 𝑚𝑚 𝑏𝑠𝑡𝑠 Carved area where reinforcement is put 𝑚𝑚

2 𝑒 Load eccentricity 𝑚𝑚 𝑓𝑐,0,𝑔,𝑑 Design compression strength parallel to grain 𝑀𝑃𝑎

𝑓𝑐,0,𝑔,𝑘 Characteristic compression strength parallel to grain 𝑀𝑃𝑎

𝑓𝑚,𝑔,𝑑 Design bending strenght 𝑀𝑃𝑎

𝑓𝑚,𝑔,𝑘 Characteristic bending strength 𝑀𝑃𝑎

𝑓𝑡,0,𝑔,𝑑 Design tension strength 𝑀𝑃𝑎

𝑓𝑡,0,𝑔,𝑘 Characteristic tension strength 𝑀𝑃𝑎

𝑓𝑣,𝑔,𝑑 Design shear strength 𝑀𝑃𝑎

𝑓𝑣,𝑔,𝑘 Characteristic shear strength 𝑀𝑃𝑎

𝑓𝑦 Yield strength 𝑀𝑃𝑎

ℎ Depth / height 𝑚𝑚 ℎ𝑙 Depth / height of a lamella 𝑚𝑚 𝑖1 Index for steel for transformed section - 𝑖𝑟 Index for reference material for transformed section - 𝑘 Instability factor - 𝑘𝑐𝑟 Factor accounting for influence of cracks on the shear strength of

member subjected to bending -

𝑘𝑑𝑒𝑓 Deformation factor -

𝑘ℎ Depth factor - 𝑘𝑚𝑜𝑑 Modification factor accounting for load duration and moisture

content -

𝑞𝐸𝑑 Design load with even distribution 𝑘𝑁/𝑚 𝑞𝐸𝑑,𝜔 Evenly distributed load with respect to deformation limit 𝑘𝑁/𝑚 𝑡𝑠 Thickness of reinforcement 𝑚𝑚 𝑧𝑓𝑖𝑐 Fictitious center of gravity, measured from an arbitrarily chosen

point of reference 𝑚𝑚

Latin upper case letters Parameter Definition Unit 𝐴𝑤 Cross sectional area of the web in the section, before

transformation 𝑚𝑚2

𝐺𝑚𝑒𝑎𝑛 Mean value of shear modulus 𝐺𝑃𝑎 𝐶𝐺 Centre of gravity 𝑚𝑚 𝐸𝐵 Stress check at bottom beam for beam type 3 - 𝐸𝑚𝑒𝑎𝑛 Mean value of modulus of elasticity 𝐺𝑃𝑎 ER Stress check at reinforcement level for beam type 3 - 𝐸𝑇 Stress check at top beam for beam type 3 - 𝐼𝑦 Second moment of area about the strong axis 𝑚𝑚

4

𝐿 Beam span 𝑚𝑚 𝑃𝐸𝑑 Design point load 𝑘𝑁 𝑃𝐸𝑑,𝜔 Design load with respect to deformation limit 𝑘𝑁 𝑃𝑝𝑟𝑒 Prestressing axial force 𝑘𝑁

Red Reduction to account for relaxation losses of prestressed steel % 𝑆𝑦 First moment of area about the strong axis 𝑚𝑚

3

𝑊𝑦 Section modulus about the strong axis 𝑚𝑚3

𝑋𝑓𝑖𝑛 Index for concept of final condition, to handle creep problem -

𝑋𝑖 About i-axis, (X-,Y- or Z- axis) - 𝑋𝑖𝑛𝑠𝑡 Index for concept of instantaneous condition, to handle creep

problem -

𝑋𝑆𝐿𝑆 Index for serviceability limit state - 𝑋𝑈𝐿𝑆 Index for ultimate limit state -

Greek lower case letters

Parameter Definition Unit 𝜎𝑐,0,𝑔,𝑑 Design compressive stress 𝑀𝑃𝑎

𝜎𝑀 Design bending stress 𝑀𝑃𝑎 𝜏𝑑 Design shear stress 𝑀𝑃𝑎 𝜔 Deformation 𝑚𝑚 𝜓2 Factor for quasi-permanent value of a variable action - 𝜔𝑚𝑎𝑥 Maximum deformation allowed according to recommendations 𝑚𝑚 𝜇 Parameter for concepts of instantaneous and final conditions - 𝛾𝑀 Partial factors for material properties and resistances - 𝛾𝑀0 Steel resistance of cross-section -

Table of Contents

Acknowledgements .................................................................................................................... 1

Abstract ...................................................................................................................................... 2

1 Introduction ........................................................................................................................ 3

1.1 Background .................................................................................................................. 3

1.2 Aim and goal ................................................................................................................ 3

1.3 Limitation ..................................................................................................................... 3

1.4 Relevant literature review ........................................................................................... 4

2 Methodology ....................................................................................................................... 7

2.1 Theoretical model ........................................................................................................ 8

2.2 Numerical model ....................................................................................................... 15

3 Results ............................................................................................................................... 16

4 Analysis ............................................................................................................................. 17

5 Conclusion ......................................................................................................................... 19

6 Future Research ................................................................................................................ 21

7 References ........................................................................................................................ 22

8 Swedish Summary – Sammanfattning .............................................................................. 23

9 Appendix ........................................................................................................................... 24

Appendix A, Result diagrams ...................................................................... A-1-to A-16

Appendix B, Result data ............................................................................... B-1 to B-30

Appendix C, Raw data .................................................................................. C-1 to C-68

Appendix D, PTC MathCad script ................................................................ D-1 to D-21

1

Acknowledgements We would like to express our gratitude to supervisor Bert Gunnar Norlin for his useful comments and reviews and thank Melissa Rude and Thaddeus Rude from the United States for proofreading the paper. Furthermore, we would like to thank Mr. Adam Ryan, known as user "Ajryan88" in

excelforum.com, for scripting an Excel macro to sort and filter the results of the calculations.

In recognition of his kindness, we have donated to the RiseForAlex Fund in addition to a gift.

Mr. Ryan invested many working hours into scripting a macro that automatically filters the

result database without the need to manually work with excel filtering. His script saved us

valuable time and eased much of our work.

2

Abstract The paper presents details of a numerical analysis and simplified construction of

strengthened glue laminated beams. Glue laminated beams are strengthened through the

use of steel reinforcements embedded between the lamellas of the beams. The study

compares the numerical results from reinforced and prestressed beams, simply reinforced

beams and non-modified beams. Parametric studies were undertaken to evaluate the effects

on reinforcement thickness, beam span, prestressing force and prestressing loss.

Modified and prestressed beams with wide spans and large dimensions had a significantly

higher design load compared to non-modified beams with similar geometry and span. In the

most beneficial cases, a load increase of 438% was observed for point load at midspan and

346% for uniformly distributed load.

3

1 Introduction

1.1 Background Many attempts have been made to improve the load capacity of glulam beams by adding

reinforcement. The reinforcements used have included metallic, organic and polymer

materials. Some reinforcement techniques are applied to existing glue laminated beams to

enhance load capacity by attaching the reinforcement onto the exterior of the wood

material, exposing the reinforcement to weather and fire risk. An alternative method is to

use embedded reinforcements inside the glue laminated beam making it possible to

prestress the reinforcement material and also provide protection from exposure to weather

and fire. Fiber-reinforced polymers (FRP) are ideal to use as a prestressing method due to

the high strength and stiffness of the FRP materials, but have lower prestressing capabilities

compared to steel reinforcement. The analysis of an embedded steel reinforcement is

evaluated in the paper due to most comparable and modern scientific reports covering

materials other than steel (mostly FRP materials).

1.2 Aim and goal The purpose of the paper is to present a large set of data to analyze the benefits of

prestressing and strengthening homogenous glue laminated beams and to provide

manufacturers and structural engineers with a set of data to analyze whether it is viable to

strengthen the beams with the presented method. The goal is to provide results for several

steel reinforcement thicknesses, beam spans and qualities to find the greatest load increase

for the modified beam compared to the non-modified beam.

1.3 Limitation The paper is focused on the structural analytical aspect of the steel reinforcement method.

1.3.1 Limitation in analysis Self-weight of the beam is excluded in the analysis. It is a small fraction of the total load and

the load increase is measured in percent of the original load before applying modifications to

the beam. The load duration was set to permanent without any variable loads to set 𝜓2, the

factor for quasi-permanent value, to “one” giving the same ultimate limit state and

serviceability limit state creep related factors. This is to reduce the results and make it easier

to interpret and draw clear conclusions from the results. No slip is assumed between

reinforcement material and glulam beam, the layers are assumed to have equal deflections.

The materials remain linear elastic and the calculation model is assumed as full composite

action giving overestimated capacity and underestimated deformation. Welding design and

the transfer of shear forces to welds have not been taken into account.

Prestressing a beam is favorable for fatigue resistance according to the authors’ previous

knowledge in concrete design but the effect and magnitude have not been studied in the

paper.

4

Calculations do not account for fire safety according to Eurocode 1995-1-2:2004 (Structural

fire design of timber structures). Practical aspects, manufacturing and cost analysis for the

reinforcement method are not part of the paper.

The script uses a numerical solution to a problem without taking into account whether the

values provided are of a practical use, therefore some design loads may be too small. The

script may obtain design loads below the self-weight of the beam.

1.3.2 Limitation in MathCad model Since there are infinite solutions for the reinforced and prestressed case, due to the

possibility of altering prestressing force, the program code was developed to calculate all

solutions for every 500 Newtons of change in the prestressing force, from 500 N to highest

possible. Some of the limitations in the model are only to decrease the amount of data

obtained and might be adjusted in the Mathcad worksheet to achieve more extended

results, such as scripts for iterations and increments for beam span and steel thickness. Pre-

camber was excluded throughout the calculations and the deformation limit was set to the

length divided by 500 for the instantaneous and divided by 300 for the final condition. The

deformation limit could be changed in the programming script. The result matrix is unitless

and the only way to obtain the corresponding units is to either study the MathCad script or

to export the result to the provided Excel sheet where the units are specified.

1.4 Relevant literature review An excessive literature study was carried out to analyze different prestressing methods and

their benefits and restrictions. Since the paper is according to the Eurocode, the literature

was only a tool to identify possible pitfalls that could alter the results in a way that would

make the paper of less scientific value. Many reports, especially when analyzing the

structures in finite element methods, made assumptions that impact the results with great

magnitude without further discussion, leaving the reader less confident in the outcome of

the paper. The reports regarding steel reinforcement were either made according to old

standards or had restrictions in the analysis, making the paper a complement to previous

knowledge, focusing on presenting engineering tools and a large set of numerical data of

interest to glulam manufacturers and structural engineers.

Some authors and researchers have studied the behavior and benefits of reinforced glue

laminated beams through the use of metal reinforcement. The flexural behavior of

unreinforced, reinforced and prestressed-reinforced glue laminated beams were studied

with the use of steel bars at a steel to timber ratio of approximately 1%. For the reinforced

and prestressed beams, the load increased about 40% (De Luca and Marano, 2011).

5

The disadvantage of using the presented reinforcement method is the need for adhesive

between the glulam and the metal rod (or rebar), making the manufacturing process more

time-consuming than friction-based connections due to adhesive dry time before load

application. The adhesive was allowed to set for one month before load application.

A study to analyze different types of metal reinforcement by wire mesh was conducted

where observations were made about the positioning of the reinforcement and its effect on

the properties of the laminated veneer lumber (Hu et al. 2010). The authors of this paper

had already taken into effect the height positioning and the impact of the reinforcement in

the planning phase.

A laboratory experimental study was conducted at the Lebanese American University where

reinforced glue laminated beams were subjected to destructive tests and identified possible

failure modes. One of the reinforcement techniques studied was reinforcement by a flat

surface steel plate with adhesive connected to the bottom of the glue laminated beam. A

bonding failure between the steel plate reinforcement and wood occurred, shown in Figure

1 (A.Issa and Kmeid, 2005).

Figure 1 ”Failure of the steel reinforced glulam beam”, (A.Issa and Kmeid, 2005, 103)

Strengthening solid timber beams through the use of high strength steel cord reinforcement

was executed in a laboratory experiment with different types of cords and adhesives. The

paper discussed some theories adapted in other reports to determine the ultimate bending

moment. One method is Bernoulli’s theorem where plane sections remain plane after being

strained (Borri and Corradi, 2011). Reinforcement methods can be divided into two

categories: those that yield benefits in local effects and those having global effects. Changes

affecting the global behavior of the beam are considered more beneficial due to affecting

the stiffness and overall behavior of the beam.

6

Beams prestressed by inserting compressed wooden blocks with low moisture content into

fitting holes in the glulam beam have been studied. The local prestressing affect is achieved

by swelling along the beam length when humidity equilibrium is achieved. Local effects are

obtained when implementing the modification of a beam with a load capacity increase of

around 11%. Delamination of the beam along with cracking occurred during a destructive

test (Anshari et al. 2011). The method requires a careful selection of wood without natural

defection and heating the reinforcement blocks to 180 °C. Dried wood requires complicated

logistic solutions or risks being exposed to humidity.

7

2 Methodology The concept of the reinforced beam is to use a main steel plate with welded thin toothed

plates, also known as punched metal plates, on each side for a friction-based connection

through hydraulic installment between the bottom lamellas of a glue laminated beam. A full

composite action can be ensured by using a large amount of small evenly distributed teeth,

thereby minimizing the risk of delamination or slip.

Figure 2 Principal construction sketch of the prestressed and modified glulam beam

To avoid any relaxation of the steel, all necessary welding should be performed before the

prestressing process. The reinforcement is added between the lamellas and pressed

hydraulically before connecting the bottom lamellas of a glulam beam.

The steel reinforcement placement is in the tension zone of the beam to manipulate the

fictitious cross section with a significant magnitude and to benefit from the deflection

caused by the prestressing. The fictitious cross section methodology scales the steel on a

geometric basis at a ratio between the elastic modulus for the materials compared. The

steel geometric values are scaled by the tens, giving the difference between the elastic

modulus for steel and glulam.

Two major benefits are protecting the reinforcement from fire exposure and reducing the

risk of bonding failure between the reinforcement and timber as shown in Figure 1.

The construction time is greatly reduced by using a hydraulic solution for the construction of

the reinforced beam instead of a metal to wood adhesive solution, due to lack of adhesive

set time before load application.

8

2.1 Theoretical model A theoretical model of the beam according to the Eurocode was established to evaluate the

increased strength of the glulam beam when modified. The model includes ultimate limit

state (ULS) and final limit state (SLS) with respect to stress and deformation. Three types of

beams were investigated for each homogenous glulam quality, span and dimension available

at a larger European glulam manufacturer according to standard BS EN 1194:1999.

The design shear strength is taken from the Swedish annex related to SS-EN 1194.

Beam type 1: non-modified homogenous glue laminated beam.

Beam type 2: strengthened and unstressed homogenous glue laminated beam, by steel

reinforcement.

Beam type 3: strengthened and pre-tensioned homogenous glue laminated beam, by steel

reinforcement.

2.1.1 Material properties When dealing with timber structures according to the Eurocode, the design value for

resistance (𝑅𝑑) is expressed as

𝑅𝑑 =𝑘𝑚𝑜𝑑𝑅𝑘𝛾𝑀

EN-1995-1-1:2004 (2.14)

Where 𝑘𝑚𝑜𝑑 is a modification factor that takes into account the effect of load duration and

moisture content, 𝑅𝑘 is the characteristic value of the load capacity, and 𝛾𝑀is the partial

factor for material property at the ULS. The values for glulam strength class are given in BS

EN 1194:1999.

At failure condition, ULS values of modulus of elasticity and shear modulus are the fifth

percentage values while at SLS the design values are the mean values.

Equation for mean final value at the SLS condition (𝑋𝑚𝑒𝑎𝑛,𝑆𝐿𝑆,𝑓𝑖𝑛) for shear modulus and

modulus of elasticity are defined as

𝑋𝑚𝑒𝑎𝑛,𝑆𝐿𝑆,𝑓𝑖𝑛 =𝑋𝑚𝑒𝑎𝑛1 + 𝑘𝑑𝑒𝑓

EN-1995-1-1:2004 (2.7)

9

Equation for fifth percentile final value at the ULS condition (𝑋0.05,𝑈𝐿𝑆,𝑓𝑖𝑛) for shear modulus

and modulus of elasticity are defined as

𝑋0.05,𝑈𝐿𝑆,𝑓𝑖𝑛 =𝑋0.05

1 + 𝜓2𝑘𝑑𝑒𝑓 EN-1995-1-1:2004 (2.10)

Where 𝜓2 is the factor for the quasi-permanent value of the variable action causing the

largest stress in relation to the strength, given in Eurocode 0 (EC0). The value is set to 1.0 for

permanent load. Deformation factor 𝑘𝑑𝑒𝑓 is dependent upon the choice of service class and

is given in Table 3.2, Eurocode 5 (EC5). The value is set to 0.6 and corresponds to the

deformation factor for glued-laminated timber in service class 1.

For strength related calculation, the fifth percentile for the shear modulus is defined as

𝐺0.05 =𝐸0.0516

2.1.2 Geometric properties For beam type 1, the second moment of area (𝐼𝑦) and the section modulus (𝑊𝑦) are defined

as

𝐼𝑦 =𝑏ℎ3

12, 𝑊𝑦 =

𝑏ℎ2

6

Beam type 2 and 3

For reinforced beams containing materials with different elastic modulus, such as steel and

glulam beam, the analysis is simplified by transforming the real cross section to a fictitious

cross section where the width of the steel is changed such that the same modulus of

elasticity is achieved through the entire section. The stresses for the fictitious section are

assumed to be uniformly distributed along the fictitious width and must be distributed over

the real width. When dealing with fictitious cross sections, it is recommended to use the

mean values of the modulus for each material instead of the lower fifth percentile.

10

EC5 uses the concepts of instantaneous and final conditions to handle creep phenomenon,

referred to in the paper and MathCad script as “inst” and “fin” (Crocetti et al. 2011, 5.4).

Parameter 𝜇1 for concepts of inst and fin conditions is defined as

𝜇1,𝑈𝐿𝑆,𝑖𝑛𝑠𝑡 = 𝜇1,𝑆𝐿𝑆,𝑖𝑛𝑠𝑡 =𝐸𝑠,𝑚𝑒𝑎𝑛𝐸𝑚𝑒𝑎𝑛

𝜇1,𝑈𝐿𝑆,𝑓𝑖𝑛 =𝐸𝑠,𝑚𝑒𝑎𝑛(1 + 𝜓2𝑘𝑑𝑒𝑓)

𝐸𝑚𝑒𝑎𝑛(1 + 𝜓2𝑘𝑠,𝑑𝑒𝑓)

𝜇1,𝑆𝐿𝑆,𝑓𝑖𝑛 =𝐸𝑠,𝑚𝑒𝑎𝑛(1 + 𝑘𝑑𝑒𝑓)

𝐸𝑚𝑒𝑎𝑛(1 + 𝑘𝑠,𝑑𝑒𝑓)

When permanent load is applied, the factor for quasi-permanent value (𝜓2) is 1.0.

Therefore, 𝜇1 is the same for ULS and SLS under fin condition.

Since the value for 𝜇1 is dependent upon inst and fin state, the center of gravity, second

moment of inertia and bending stiffness must be calculated for both states with Steiner’s

theorem.

Figure 3 Fictitious cross-section about the y-y axis with the glulam part as the reference material

The fictitious center of gravity about the y-y axis, with the zero point at the bottom of the

homogenous glue laminated beam:

𝑏ℎℎ

2− 𝑡𝑠𝑏𝑠ℎ𝑙 + 𝑡𝑠𝜇1𝑏𝑠ℎ𝑙 = 𝑧𝑓𝑖𝑐(𝑏ℎ + 𝜇1𝑏𝑠𝑡𝑠 − 𝑏𝑠𝑡𝑠)

Center of gravity for each part according to Error! Reference source not found.:

𝐶𝐺𝑧,1 =ℎ𝑙 −

𝑡𝑠2

2 , 𝐶𝐺𝑧,2 = ℎ −

ℎ − ℎ𝑙 −𝑡𝑠2

2 , 𝐶𝐺𝑧,3 = 𝐶𝐺𝑧,4 = ℎ𝑙

11

Fictitious second moment of area about the y-y axis:

𝐼𝑦,𝐶𝐺𝑧,1 =𝑏 (ℎ𝑙 −

𝑡𝑠2)3

12+ 𝑏 (ℎ𝑙 −

𝑡𝑠2) (𝑧𝑓𝑖𝑐 − 𝐶𝐺𝑧,1)

2

𝐼𝑦,𝐶𝐺𝑧,2 =𝑏 (ℎ − ℎ𝑙 −

𝑡𝑠2)3

12+ 𝑏 (ℎ − ℎ𝑙 −

𝑡𝑠2) (𝑧𝑓𝑖𝑐 − 𝐶𝐺𝑧,2)

2

𝐼𝑦,𝐶𝐺𝑧,3 =𝑏𝑒𝑑𝑡𝑠

3

12+ 𝑏𝑒𝑑𝑡𝑠(𝑧𝑓𝑖𝑐 − 𝐶𝐺𝑧,3)

2

𝐼𝑦,𝐶𝐺𝑧,4 =𝜇1𝑏𝑠𝑡𝑠

3

12+ (𝜇1𝑏𝑠)𝑡𝑠(𝑧𝑓𝑖𝑐 − 𝐶𝐺𝑧,4)

2

𝐼𝑦 =∑𝐼𝑦,𝐶𝐺𝑧,𝑖

4

𝑖=1

= 𝐼𝑦,𝐶𝐺𝑧,1 + 𝐼𝑦,𝐶𝐺𝑧,2 + 2(𝐼𝑦,𝐶𝐺𝑧,3) + 𝐼𝑦,𝐶𝐺𝑧,4

The fictitious section modulus about the y-y axis:

𝑊𝑦,𝑚𝑜𝑑,𝑏𝑜𝑡 =𝐼𝑦𝑧𝑓𝑖𝑐

𝑊𝑦,𝑚𝑜𝑑,𝑡𝑜𝑝 =𝐼𝑦

ℎ − 𝑧𝑓𝑖𝑐

Value at steel level defined as;

𝑊𝑦,𝑚𝑜𝑑,𝑟𝑒𝑖 =𝐼𝑦

𝑧𝑓𝑖𝑐 − ℎ𝑙

12

2.1.3 Modification factors Size factor for depth less than 600mm, else the size factor is 1.0:

𝑘ℎ = 𝑚𝑖𝑛 {

1.1

(600𝑚𝑚

ℎ)0.1 EN-1995-1-1:2004 (3.1)

Influence of cracks, according to Swedish Annex:

𝑘𝑐𝑟 = 𝑚𝑖𝑛

{

1.0

(3.0𝑀𝑃𝑎

𝑓𝑣,𝑔,𝑘) BFS 2015: 6-EKS 10 (6.1.7(2))

2.1.4 Actions Design bending stress about y-y axis for all beam types are defined.

For point load at mid span:

𝜎𝑀 =𝑀𝑠𝑑𝑊𝑦

=𝑃𝐸𝑑𝐿

4𝑊𝑦

For uniformly distributed load:

𝜎𝑀 =𝑀𝑠𝑑𝑊𝑦

=𝑞𝐸𝑑𝐿

2

8𝑊𝑦

Design bending stress about the y-y axis caused by prestressing force for beam type 3 is

defined.

𝜎𝑀.𝑝𝑟𝑒 =𝑃𝑝𝑟𝑒𝑒

𝑊𝑦

Where 𝑒 is the fictitious load eccentricity about the y-y axis, defined as

𝑒 = 𝑧𝑓𝑖𝑐 − ℎ𝑙

The smallest fictitious section modulus between bottom value (𝑊𝑦,𝑚𝑜𝑑,𝑏𝑜𝑡) and

top value (𝑊𝑦,𝑚𝑜𝑑,𝑡𝑜𝑝) is chosen for beam type 2. For beam type 3, a stress check is

performed, see 2.1.5 Design criterion.

Design shear stress about y-y axis for all beam types are defined as

𝜏𝑑 =𝑉𝑑𝑆𝑦𝑏𝐼𝑦𝑘𝑐𝑟

13

Where 𝐼𝑦 is the fictitious cross section and 𝑆𝑦 is the first order moment (see Appendix D

page D-6). To simplify the equations, the prestressing force is assumed to have no influence

on the shear stress.

Design compression stress about the y-y axis caused by prestressing force for beam type 3 is

defined.

𝜎𝑐 =𝑃𝑝𝑟𝑒𝑏ℎ

2.1.5 Design criterion Design criterion with respect to moment resistance

𝜎𝑀 ≤ 𝑓𝑚,𝑑𝑘ℎ

Design criterion with respect to shear resistance at support

𝜏𝑑 ≤ 𝑓𝑣,𝑔,𝑑

Design criterion for beams subjected to combined bending moment and compression should

be satisfied.

Stress check at top beam for beam type 3, according to EN 1995-1-1 (6.35):

Equation labeled as 𝐸𝑇 in appendix D page D-8

𝜎𝑀 − 𝜎𝑀.𝑝𝑟𝑒𝑓𝑚,𝑔,𝑑𝑘ℎ

+ (𝜎𝑐

𝑓𝑐,0,𝑔,𝑑)

2

≤ 1

Fictitious section modulus (𝑊𝑦,𝑚𝑜𝑑,𝑡𝑜𝑝) at top beam is used to calculate the design bending

stress.

Stress check at bottom beam for beam type 3:

Equation labeled as 𝐸𝐵 in appendix D page D-8

−𝜎𝑀 + 𝜎𝑀.𝑝𝑟𝑒𝑓𝑚,𝑔,𝑑𝑘ℎ

+ (𝜎𝑐

𝑓𝑐,0,𝑔,𝑑)

2

≤ 1

Fictitious section modulus (𝑊𝑦,𝑚𝑜𝑑,𝑏𝑜𝑡) at bottom beam is used to calculate the design

bending stress.

14

Stress check at reinforcement level for beam type 3:

Equation labeled as 𝐸𝑅 in appendix D page D-8

−𝜎𝑀 + 𝜎𝑀.𝑝𝑟𝑒𝑓𝑦𝛾𝑀0

+ (𝜎𝑐𝑓𝑦𝛾𝑀0

)

2

≤ 1

Fictitious section modulus (𝑊𝑦,𝑚𝑜𝑑,𝑟𝑒𝑖) at reinforcement level is used to calculate the design

bending stress.

The equations 𝐸𝐵 and 𝐸𝑅 may give negative values for allowable bending moment stress

ratio but cannot be considered favorable in equation (6.35). The equations have been

modified in Appendix D page D-8 to obtain absolute values for allowable bending moment

stress ratio.

2.1.6 Deformation Mean values for shear modulus and modulus of elasticity are used in the deformation check,

a SLS problem for both inst and fin conditions.

Bending deflection caused by a point load at midpoint

𝜔 =𝑃𝐸𝑑𝐿

3

48𝐸𝑚𝑒𝑎𝑛𝐼𝑦+

𝑃𝐸𝑑𝐿

4𝐴𝑤𝐺𝑚𝑒𝑎𝑛

Where 𝐴𝑤 is the cross-sectional area of the web in the section before transformation,

assumed as the real cross-section.

Bending deflection caused by a uniformly distributed load

𝜔 =5𝑞𝐸𝑑𝐿

4

384𝐸𝑚𝑒𝑎𝑛𝐼𝑦+

𝑞𝐸𝑑𝐿2

8𝐴𝑤𝐺𝑚𝑒𝑎𝑛

Negative bending deflection, at midpoint caused by prestressing force for beam type 3

𝜔 =𝑃𝑝𝑟𝑒𝑒𝐿

2

8𝐸𝑚𝑒𝑎𝑛𝐼𝑦

15

2.2 Numerical model A numerical model in MathCad was the primary tool used to evaluate the benefit of the

prestressing theory (see appendix D).

2.2.1 MathCad model The functions used in the numerical MathCad script in appendix D are described and

explained on the script sheets. The reader is recommended to view the script to have an

understanding of the functions and the results obtained. The MathCad script is divided into

three different files that are linked with scripting: one for input and output, one for defining

the equations, and the last for executing the full calculations according to the analytical

model with automated scripting functions that loop for every increment change in each of

the varying values.

MathCad sheet “UseMe.xmcd”

The input file is labeled as UseMe.xmcd (see appendix D page D-1 to D-3). It contains the

necessary input values for beam dimension and material properties for both steel and

timber, Eurocode partial factors that may depend on the National Annex and deformation

limits.

The input file executes the calculation command and performs the export of completed data

to Excel. The Output section of the worksheet transforms the full and unitless data matrix

into strings for export to Excel.

Programming script, part 1 “Script_part1.xmcd”

The programming script contains all equations needed to perform the calculation in a

function style scripting method (see appendix D page D-4 to D-11). The script does not

perform any calculations. It only defines functions to be later used. Geometric, modification

factors, action and deformation design equations are defined in the worksheet.

The file contains some programming utilities of significance for creating increments, vectors

and optimizations including matrix related functions to add labels and input data to the

result matrix. The last segment of the file includes output functions to organize the results

obtained.

Programming script, part 2 “Script_part2.xmcd”

With the previously defined equations, a final function is defined for every beam type and

loading condition. The final function performs calculation tasks and returns a set of result

data to the input file. Due to scripting limitations in MathCad, the final result matrix is

unitless and the units have been defined in the Excel sheet (see appendix D page D-12 to

D-21). The resulting matrices are unfiltered with a large set of increments of the prestressing

force.

16

2.2.2 Excel filter macro An Excel macro was made for filtering the results for beam type three and choosing the most

beneficial prestressing force among those calculated. Some manual sorting of data was

performed to compare the different beam types with a non-modified beam with increased

number of lamellas. In the interest of a quick, precise answer, it is recommended to filter

one beam dimension at a time. Appendix B to C are exported from the Excel file.

3 Results Due to the high number of interactions and different beam geometries, beam types and a

variation in steel reinforcement thickness, the results have been included in appendix A

(page A-1 to A-12) as charts and in Appendix B as tables.

17

4 Analysis When applying a filtering function to the results in Excel with respect to highest load

increase, there is a pattern among the beams with the most advantage. The selected cases

studied are beams in the final limit state (SLS).

Comparisons with increased number of lamellas have been made to have a logical

understanding of the improvement for the different reinforcement techniques. Increasing

the number of lamellas may be a less demanding than the reinforcement techniques

presented in the paper. The author and reader can determine if the presented

reinforcement techniques are viable for some geometrical conditions and beam spans.

A limitation occurs when comparing results from increased number of lamellas when

reaching max height of every beam width and span calculated because there is no data to

obtain from the results in Appendix B. In some cases, the maximum beam height in Appendix

B is 495 mm and there is no reference to obtain from a higher beam with the same quality,

width and span. The same logic is applied for the second highest beam. Only one reference

with one extra lamella is possible to obtain from the results in Appendix B.

Appendix A page A-13 to A-16 contains charts to analyze the behavior of some beam

geometries and spans for all satisfactory solutions with a prestressing force increment of one

kilo Newton. The script was engineered to export the most beneficial results to minimize the

number of exported values. For some geometries and spams thousands of satisfactory

solutions where calculated.

A higher quality steel reinforcement is preferred due to increased design resistance and

decreased deflection when engineering steel structures, according to the Eurocode standard

for steel structures (i.e. BS EN 1993-1-1:2005 (E) equation 6.55).

Changing the yield strength of steel does not affect the elastic modulus when adapting the

fictitious cross section theory for the steel reinforcement, as the yield strength of the steel

material no longer affects the design strength other than improving the stress check at

reinforcement level, according to EN 1995-1-1 (6.35).

A stress analysis is performed between load and resistance in the wood engineering, and the

fictitious cross section theory takes into effect the elastic modulus ratio between the glulam

and steel. The elastic modulus for a steel material remains the same if another steel quality

is chosen.

When transmuting the steel reinforcement to a fictitious material with increased geometry

values to be considered as a glulam material, the effect of the yield strength is lost. A

laboratory experiment could be performed to take into consideration the effect of improved

steel quality and to redefine the theory for future research. Another method is to use a well-

defined finite element solution to the problem where the focus is to accurately and

reasonably determine the parametric values and coefficients needed for the simulation of

the steel to wood behavior.

18

4.1 Prestressing analysis The prestressing analysis is included in Appendix A page A-13 to A-16 as charts for some

beam geometries and spans.

Figure 4 Prestressing analysis chart for beam GL24h 65x225 with 5m span, point load at midspan, see Appendix A.

Some observations are made in the prestressing analysis that are valid for every chart and

beam type analyzed. A larger steel reinforcement thickness is beneficial for smaller

prestressing forces between 0 kN up to a certain point, here called point A. A smaller steel

reinforcement thickness is most beneficial for prestressing forces above point A. The location

of point A in the prestressing force increment spectrum varies from one beam geometry to

another. As Figure 4 shows, point A is approximately at a prestressing force of 40 kN for

beam GL24h 65x225 mm and with the span of 5 meters with a point load at midspan and

steel reinforcement types as described in the paper.

In some of the analyzed beams, the increase in loading capacity from prestressing force

equal to zero up to point A is linear (𝑦 = 𝑚𝑥 + 𝑐), while in other cases the increase is

parabolic (𝑦 = −𝑚𝑥2 + 𝑛𝑥 + 𝑐). The shape of the increase may be affected by what failure

mode is achieved first in the satisfactory solution.

The most beneficial result (maximum point on the y-axis) and highest loading capacity in the

analyzed beams occurred always beyond point A, here called point B. Point B may

hypothetically overlap with point A if no satisfactory solutions are obtained for further

prestressing increments. Point B may occur before a fictitious location of point A. This

scenario may hypothetically occur if the beam capacity of different steel thickness cases

merge outside the satisfactory solution boundary.

19

In all of the analyzed cases, the most beneficial result was the one with steel reinforcement

thickness of 10 mm for beam type 3. A larger reinforcement thickness brings the fictitious

center of gravity of the beam closer to the reinforcement level causing a smaller lever, and a

smaller design bending stress caused by prestressing force and eccentricity 𝜎𝑀.𝑝𝑟𝑒. If the

design stress check (according to 2.1.5 Design criterion) is at top beam then a smaller

reinforcement thickness will give the most beneficial results (maximum y-value in Figure 4)

and that was the case for all prestressing analysis.

4.2 Most beneficial results Some observations are made by studying the visualized results in the charts provided in

appendix A page A-1 to A-12 or the numerical values provided in appendix B.

A larger steel reinforcement thickness is advantageous to the beam capacity for all studied

dimensions and spans for beam type 2. Increasing the beam span for a specific geometry

resulted in a higher load capacity increase for beam type 2 in percentage to an unmodified

glulam beam with corresponding geometry and span.

A smaller steel reinforcement thickness is advantageous to the beam capacity for all studied

dimensions and spans for beam type 3, with a steel thickness of 10 mm (see 4.1 Prestressing

analysis). Increasing the beam span resulted in a higher beam capacity increase for all beams

in beam type 3 in comparison to beam type 1.

The most beneficial load capacity increase among all beam type 2 data is found at large

dimensions and spans. Beam GL28h 115x405 mm with the span of 9 m had a load increase of

106% for point load at midspan and 105% for uniformly distributed load. The load increase

values are calculated in Appendix B.

Results for beam type 3 were always higher than adding one lamella except for beam quality

GL24h, dimension 65x225 mm and a span of 2 to 2.5 meters.

The most beneficial load capacity increase among all beam type 3 data is found at large

dimensions and spans. Beam GL28h 90x360 mm with the span of 11m had a load increase of

438% for point load at midspan and 346% for uniformly distributed load.

4.3 Least beneficial results The load capacity increase for beam type 2 is smaller than the load capacity increase when

adding a lamella (where comparison to a larger height is applicable) in the case of a small

beam geometry, span and a low beam quality (see results for glulam beams with the quality

GL24h, dimensions 65x225 mm and 65x270 mm around the span of 2 meters in Appendix A

page A-1 and A-7).

20

5 Conclusion With the defined methodology, benefits of up to 438% to load increase is achieved for

beams with wider spans affected by point load at midpoint and up to 346% for similar beams

affected by a uniformly distributed load. Adding an extra lamella proved more beneficial

than reinforcing the beam in some cases with short spans and slim cross sections.

The adapted fictitious cross section theory does not take into account the yield strength of

the chosen steel quality other than in the stress check equations, giving all results in the

lowest possible steel grade from the input data (due to the MathCad functions used). The

main benefit of beams type 2 and 3 is obtained through the fictitious cross section due to

the elastic modulus of the steel being larger than the glulam elastic modulus by

approximately 20 times.

The prestressed strengthening method studied in the presented paper is assumed to not

improve the shear capacity of a glue laminated beam. Design loads provided in appendix A

may exceed the design criterion with respect to shear resistance at the support. There are

several implemented reinforcement methods not discussed in the paper that are available to

strengthen a glue laminated beam with respect to shear resistance at the supports.

A slender beam combined with the edge fire safety distance to reinforcement limits the

fictitious cross section increase. Therefore, a limited steel reinforcement width is possible.

The edge distance to steel reinforcement with respect to fire safety is a larger percentage of

the total width of a slender beam compared to a wider beam. A logical conclusion based on

the safety edge distance aspect, is that wider beams will benefit from steel reinforcement.

A unified conclusion cannot be made for all cases. In some cases where comparison is

possible to increased beam height in appendix A, the reinforcement method for beam type 3

had a significantly greater value than adding extra lamellas to the beam height, making it a

viable reinforcement option from a structural engineering standpoint. In other few cases,

the use of the reinforcement methods in beam type 2 or 3 had a smaller value than adding

one or two extra lamellas but may have a practical use if there are requests in other

disciplines that limit the cross sections of a beam to be engineered.

21

6 Future Research More accurate prestressing loss, expand the model to account for reinforced non-

homogenous glue laminated beams. Expand the model to calculate variable loads and

different load cases. Expand the model to adapt to fire safety according to

Eurocode 1995-1-2:2004 (or newer versions). Expand the model to approximate the real

behavior between steel and timber with possibility for slip. Expand the model to adapt for

the risk of the lower lamella fracturing due to prestressing force if a certain amount of the

lower lamella is removed. Modeling of the prestressing force, with respect to economic

analysis, to determine the value of increasing the beam dimension versus the work load and

cost of the reinforcement technique presented in the paper.

By assuming full composite action, the capacity is overestimated and the deformation is

underestimated for the reinforced cases. The error is in reality too large to be disregarded

but using many small tooth-plated connectors with the glued area would decrease the error.

The main aim of the thesis is to study if the modification method used gives any noticeable

result. A more accurate analysis could be performed later by modifying the MathCad script.

A modification of the numerical methodology presented, mainly the fictitious cross section

equations, is strongly advised to take into account the benefit of a higher-grade steel to

obtain more beneficial results for beam type 3 due to utilization from a higher yield strength.

The MathCad script can be improved by adding the type of the chosen design criterion on

each beam case to help analyze the data.

Studying the effect of fatigue behavior on modified glulam beams compared to a non-

modified glulam beams may provide an alternative use of the beam concept.

22

7 References Anshari, Guan, Kitamori, Jung and Komatsu. 2011. Structural behavior of glued laminated

timber beams pre-stressed by compressed wood. Construction and Building Materials 29

(2012) 24-32. University of Liverpool (UK), Kyoto University (Japan), University of Mataram

(Indonesia) and Shizuoka University (Japan). Elsevier Ltd.

BFS 2015:6 – EKS 10 (2015), Boverkets författningssamling (Swedish Annex to the Eurocode),

Boverket.

Borri and Corradi. 2011. Strengthening of timber beams with high strength steel cords.

Composites: Part B 42 (2011) 1480-1491. University of Perugia, Italy. Elsevier Ltd.

Crocetti, Johansson, Kliger, Mårtensson, Norlin, Pousette and Thelandersson. 2011. Design

of timber structures. Swedish wood.

De Luca and Marano. 2011. Prestressed glulam timbers reinforced with steel bars.

Construction and Building Materials 30 (2012) 206-217. University of Basilicata, Italy. Elsevier

Ltd.

EN 1995 (2004), Eurocode 5 – Design of timber structures, European Committee for

Standardization.

Hu, Li, Cheng and Zhang. 2010. Design and property analysis of the metal mesh reinforced

LVL. Advanced Materials Research Vols 113-114 (2010) 2145-2149. Northeast Forestry

University, Harbin, China. Trans Tech Publications.

Issa and Kmeid. 2004. Advanced wood engineering: glulam beams. Construction and Building

Materials 19 (2005) 99-106. Lebanese American University, Byblos, Lebanon. Elsevier Ltd.

Porteous Jack & Kermani Abdy. 2009. Structural Timber Design to Eurocode 5. Edition 3.

Blackwell Science Ltd.

23

8 Swedish Summary – Sammanfattning Vetenskapsrapporten analyserar numeriska nyttan av armerade och förspännda limträbalkar

med stålband försedda med ståltänder. Jämförelse mellan förspända och armerade samt

endast armerade balkars görs med icke modifierade limträbalkar. Resultatpåverkan av

varierande spännvidder, tvärsnittsgeometrier och armeringstjocklekar utvärderas med hjälp

av en Mathcadskript för två lastfall; fritt upplagd balk med punktkraft i balkmitt och en jämnt

utbredd last över balklängden. En jämförelse mellan den procentuella lastökningen

tillämpas även för ett ökat antal lameller för respektive balk.

I huvuddel består teorin av att transmutera stålets geometri till en kvotrelaterad storhet där

elasticitetmodulen för ingående stål- och trämaterial beaktas. Vidare antas den armerade

balken vara homogen till följd av tillämpningen av den fiktiva tvärsnittsteorin.

Rapporten beaktar endast den analytiska metodiken utan hänsyn till kostnads- eller

arbetsrelaterade aspekter.

Fördelar i bärförmåga noterades där mest fördelaktiva värden för punktlast i balkmitt

uppgick till 438% i bärförmågeökning och 346% i bärförmågeökning vid jämnt fördelad last.

Noteringarna gjordes för långa balkar med grövre dimensioner och tunnare stålförstärkning.

En förspänningsanalys dokumenterade att tunnare stål gav högre bärförmåga än tjockare

armeringsförstärkning för analyserade förspända och modifierade limträbalkar. Vilket

förklaras av att det fiktiva tvärtsnittets tyngdpunkt hade en mindre värdeminskning än

balkar med grövre stålarmering och därmed nyttjades en högre hävarm till den excentriska

stålförspänningen, något som var gynnsamt för det aktuella spänningsbrottmoden.

24

9 Appendix Relevant information about the attached appendixes

Appendix A, Result diagrams

All charts are based on final limit state values. For beam type 3, values are without reduction

for prestressing losses due to relaxation. Red is set to 0%.

Page A-1 to A-12 are final solution charts providing the most beneficial prestressing force for

each beam dimension and span.

Page A-13 to A-16 are charts for some of the beam dimensions and spans showing the effect

of prestressing with the increment of 1 kN. The number of satisfactory solutions calculated

for each chart are presented in Table 1.

Table 1 Number of solutions calculated to create the charts in appendix A page A-13 to A-16.

Number of solutions presented in the charts

Quality Dim. Span (m) Point load at midspan

Uniformly distri. load

GL24h 65x225 2 409 409

GL24h 65x225 5 403 405

GL24h 65x315 2 563 563

GL24h 65x315 5 567 550

GL28h 90x270 5 748 738

GL28h 90x270 8 744 744

GL28h 90x495 5 1203 1220

GL28h 90x495 8 1215 1216

GL28h 115x270 5 952 952

GL28h 115x270 8 948 948

GL28h 115x495 5 1583 1583

GL28h 115x495 8 1573 1579

Appendix B, Result data

Appendix B is a comparison of the raw data in appendix C for beam type 1, 2 and 3

presented in the same row, including values when adding one or two lamellas to the beam

(if data is available at appendix C). Values that are faded out in color are identical to the

value on the row above. New beam dimensions (width and height) are marked with a color

highlight. The load increase is given, and a value of 100% means double the capacity.

Appendix C, Raw data

Raw data where the most beneficial solution for the prestressing increment on each beam

dimension and span is exported to the appendix. Appendix C is used to calculate percentage

gain in appendix B and to make the final solution charts in appendix A page A-1 to A-12.

Beam type 2 Beam type 3, no relaxation

GL2

4h

65

x2

25

GL2

4h

65

x2

70

GL2

4h

65

x3

15

0

25

50

75

100

2 3 4 5

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

2 3 4 5

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

2 3 4 5 6

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

2 3 4 5 6

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

2 3 4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

2 3 4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

Appendix A, Result diagrams Point load at midpoint

A-1


GL2

8h

90

x2

25

GL2

8h

90

x2

70

GL2

8h

90

x3

15

0

25

50

75

100

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m


A-2


GL2

8h

90

x3

60

GL2

8h

90

x4

05

GL2

8h

90

x4

50

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m


A-3


GL2

8h

90

x4

95

GL2

8h

11

5x2

25

GL2

8h

11

5x2

70

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m


A-4


GL2

8h

11

5x3

15

GL2

8h

11

5x3

60

GL2

8h

11

5x4

05

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m


A-5


GL2

8h

11

5x4

50

GL2

8h

11

5x4

95

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m


A-6


GL2

4h

65

x2

25

GL2

4h

65

x2

70

GL2

4h

65

x3

15

0

25

50

75

100

2 3 4 5

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

2 3 4 5

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

2 3 4 5 6

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

2 3 4 5 6

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

2 4 6

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

2 4 6

Loa

d i

ncr

ea

se %

Beam span m

Appendix A, Result diagrams Uniformly distributed load

A-7


GL2

8h

90

x2

25

GL2

8h

90

x2

70

GL2

8h

90

x3

15

0

25

50

75

100

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m


A-8


GL2

8h

90

x3

60

GL2

8h

90

x4

05

GL2

8h

90

x4

50

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

450

500

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

450

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m


A-9


GL2

8h

90

x4

95

GL2

8h

11

5x2

25

GL2

8h

11

5x2

70

0

25

50

75

100

125

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

4 5 6 7 8 9 10 11

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

4 5 6 7 8

Loa

d i

ncr

ea

se %

Beam span m


A-10


GL2

8h

11

5x3

15

GL2

8h

11

5x3

60

GL2

8h

11

5x4

05

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

400

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

50

100

150

200

250

300

350

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m


A-11


GL2

8h

11

5x4

50

GL2

8h

11

5x4

95

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0.0

50.0

100.0

150.0

200.0

250.0

300.0

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0

25

50

75

100

125

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m

0.0

50.0

100.0

150.0

200.0

250.0

300.0

4 5 6 7 8 9

Loa

d i

ncr

ea

se %

Beam span m


A-12

Point load at midspan Uniformly distributed load

GL2

4h 6

5x22

5 L

= 2m

GL2

4h 6

5x22

5 L

= 5m

GL2

4h 6

5x31

5 L

= 2m

0

5

10

15

20

0 50 100 150 200

P.Ed

kN

P.pre kN

0

5

10

15

20

0 50 100 150 200

q.Ed

kN

/m

P.pre kN

0

2

4

6

8

10

0 50 100 150 200

P.Ed

kN

P.pre kN

0

1

2

3

4

5

0 50 100 150 200

q.Ed

kN

/m

P.pre kN

0

10

20

30

40

50

0 50 100 150 200

P.Ed

kN

P.pre kN

0

10

20

30

40

50

0 50 100 150 200

q.Ed

kN

/m

P.pre kN

Appendix A, Result diagrams Prestressing force increment

A-13


GL2

4h 6

5x31

5 L

= 5m

GL2

8h 9

0x27

0 L

= 5m

GL2

8h 9

0x27

0 L

= 8m

0

5

10

15

20

0 50 100 150 200

P.Ed

kN

P.pre kN

0

2

4

6

8

10

0 50 100 150 200

q.Ed

kN

/m

P.pre kN

0

5

10

15

20

0 100 200 300

P.Ed

kN

P.pre kN

0

2

4

6

8

10

0 100 200 300

q.Ed

kN

/m

P.pre kN

0

5

10

15

20

0 100 200 300

P.Ed

kN

P.pre kN

0

1

2

3

4

5

0 100 200 300

q.Ed

kN

/m

P.pre kN


A-14


GL2

8h 9

0x49

5 L

= 5m

GL2

8h 9

0x49

5 L

= 8m

GL2

8h 1

15x2

70 L

= 5

m

0

20

40

60

80

100

0 100 200 300 400 500

P.Ed

kN

P.pre kN

0

10

20

30

40

50

0 100 200 300 400 500

q.Ed

kN

/m

P.pre kN

0

10

20

30

40

50

0 100 200 300 400 500

P.Ed

kN

P.pre kN

0

5

10

15

20

0 100 200 300 400 500

q.Ed

kN

/m

P.pre kN

0

5

10

15

20

25

30

0 100 200 300 400 500

P.Ed

kN

P.pre kN

0

2

4

6

8

10

0 100 200 300 400 500

q.Ed

kN

/m

P.pre kN


A-15


GL2

8h 1

15x2

70 L

= 8

m

GL2

8h 1

15x4

95 L

= 5

m

GL2

8h 1

15x4

95 L

= 8

m

0

5

10

15

20

0 100 200 300 400 500

P.Ed

kN

P.pre kN

0

1

2

3

4

5

0 100 200 300 400 500

q.Ed

kN

/m

P.pre kN

0

20

40

60

80

100

0 200 400 600 800 1000

P.Ed

kN

P.pre kN

0

10

20

30

40

0 200 400 600 800 1000

q.Ed

kN

/m

P.pre kN

0

20

40

60

80

100

0 200 400 600 800 1000

P.Ed

kN

P.pre kN

0

5

10

15

20

0 200 400 600 800 1000

q.Ed

kN

/m

P.pre kN


A-16

Beam grade L b h ts Red P.pre V.ed

P.ed inst

P.ed fin

V.ed inst

V.ed fin

P.ed inst

P.ed fin

P.ed inst

P.ed fin 1x45 2x45

UNIT: m mm mm mm % kN kN kN kN kN kN % % % % % %

GL24h 2 65 225 10 0 63.0 25.3 13.9 13.9 23.8 23.2 13.8 17.5 38.0 34.0 41.8 90.0GL24h 2 65 225 10 20 79.0 25.3 13.9 13.9 23.8 23.2 13.8 17.5 39.0 34.0 41.8 90.0GL24h 2 65 225 20 0 41.0 25.3 13.9 13.9 23.1 22.5 19.5 23.0 31.8 30.3 41.8 90.0GL24h 2 65 225 20 20 51.0 25.3 13.9 13.9 23.1 22.5 19.5 23.0 32.6 30.3 41.8 90.0GL24h 2 65 225 30 0 30.0 25.3 13.9 13.9 22.8 22.5 23.5 27.4 30.9 31.4 41.8 90.0GL24h 2 65 225 30 20 37.0 25.3 13.9 13.9 22.8 22.5 23.5 27.4 31.5 31.4 41.8 90.0GL24h 2.5 65 225 10 0 62.0 25.3 9.72 10.1 23.8 23.2 30.1 29.0 57.6 47.1 55.7 109GL24h 2.5 65 225 10 20 78.0 25.3 9.72 10.1 23.8 23.2 30.1 29.0 59.0 47.1 55.7 109GL24h 2.5 65 225 20 0 40.0 25.3 9.72 10.1 23.1 22.5 36.6 35.0 50.5 43.1 55.7 109GL24h 2.5 65 225 20 20 50.0 25.3 9.72 10.1 23.1 22.5 36.6 35.0 51.6 43.1 55.7 109GL24h 2.5 65 225 30 0 30.0 25.3 9.72 10.1 22.8 22.5 41.3 39.8 49.7 44.3 55.7 109GL24h 2.5 65 225 30 20 37.0 25.3 9.72 10.1 22.8 22.5 41.3 39.8 50.4 44.3 55.7 109GL24h 3 65 225 10 0 63.0 25.3 7.00 7.29 23.8 23.2 33.2 44.1 82.6 70.3 66.7 142GL24h 3 65 225 10 20 79.0 25.3 7.00 7.29 23.8 23.2 33.2 44.1 84.1 70.3 66.7 142GL24h 3 65 225 20 0 41.0 25.3 7.00 7.29 23.1 22.5 49.2 56.3 74.4 65.6 66.7 142GL24h 3 65 225 20 20 51.0 25.3 7.00 7.29 23.1 22.5 49.2 56.3 75.6 65.6 66.7 142GL24h 3 65 225 30 0 30.0 25.3 7.00 7.29 22.8 22.5 59.8 61.8 73.3 67.1 66.7 142GL24h 3 65 225 30 20 38.0 25.3 7.00 7.29 22.8 22.5 59.8 61.8 74.2 67.1 66.7 142GL24h 3.5 65 225 10 0 62.0 25.3 5.26 5.48 23.8 23.2 34.2 45.6 108 94.2 68.2 159GL24h 3.5 65 225 10 20 78.0 25.3 5.26 5.48 23.8 23.2 34.2 45.6 110 94.2 68.2 159GL24h 3.5 65 225 20 0 40.0 25.3 5.26 5.48 23.1 22.5 50.9 62.6 89.8 89.0 68.2 159GL24h 3.5 65 225 20 20 50.0 25.3 5.26 5.48 23.1 22.5 50.9 62.6 99.5 89.0 68.2 159GL24h 3.5 65 225 30 0 33.0 25.3 5.26 5.48 22.8 22.5 62.0 73.5 87.3 90.6 68.2 159GL24h 3.5 65 225 30 20 41.0 25.3 5.26 5.48 22.8 22.5 62.0 73.5 93.4 90.6 68.2 159GL24h 4 65 225 10 0 62.0 25.3 4.09 4.26 23.8 23.2 34.9 46.6 129 119 69.2 162GL24h 4 65 225 10 20 78.0 25.3 4.09 4.26 23.8 23.2 34.9 46.6 136 119 69.2 162GL24h 4 65 225 20 0 58.0 25.3 4.09 4.26 23.1 22.5 52.1 64.1 117 111 69.2 162GL24h 4 65 225 20 20 73.0 25.3 4.09 4.26 23.1 22.5 52.1 64.1 124 111 69.2 162GL24h 4 65 225 30 0 56.0 25.3 4.09 4.26 22.8 22.5 63.5 75.4 113 109 69.2 162GL24h 4 65 225 30 20 70.0 25.3 4.09 4.26 22.8 22.5 63.5 75.4 119 109 69.2 162GL24h 4.5 65 225 10 0 72.0 25.3 3.26 3.40 23.8 23.2 35.4 47.3 159 143 69.9 165GL24h 4.5 65 225 10 20 90.0 25.3 3.26 3.40 23.8 23.2 35.4 47.3 163 143 69.9 165GL24h 4.5 65 225 20 0 71.0 25.3 3.26 3.40 23.1 22.5 52.9 65.2 143 130 69.9 165GL24h 4.5 65 225 20 20 89.0 25.3 3.26 3.40 23.1 22.5 52.9 65.2 143 130 69.9 165GL24h 4.5 65 225 30 0 71.0 25.3 3.26 3.40 22.8 22.5 64.5 76.8 136 125 69.9 165GL24h 4.5 65 225 30 20 88.0 25.3 3.26 3.40 22.8 22.5 64.5 76.8 133 125 69.9 165GL24h 5 65 225 10 0 80.0 25.3 2.66 2.77 23.8 23.2 35.7 47.8 188 166 70.4 166GL24h 5 65 225 10 20 99.0 25.3 2.66 2.77 23.8 23.2 35.7 47.8 187 166 70.4 166GL24h 5 65 225 20 0 81.0 25.3 2.66 2.77 23.1 22.5 53.5 66.0 168 148 70.4 166GL24h 5 65 225 20 20 101 25.3 2.66 2.77 23.1 22.5 53.5 66.0 160 148 70.4 166GL24h 5 65 225 30 0 81.0 25.3 2.66 2.77 22.8 22.5 65.3 77.8 156 140 70.4 166GL24h 5 65 225 30 20 101 25.3 2.66 2.77 22.8 22.5 65.3 77.8 145 140 70.4 166GL24h 5.5 65 225 10 0 86.0 25.3 2.21 2.30 23.8 23.2 36.0 48.2 217 189 70.8 168GL24h 5.5 65 225 10 20 107 25.3 2.21 2.30 23.8 23.2 36.0 48.2 211 189 70.8 168GL24h 5.5 65 225 20 0 88.0 25.3 2.21 2.30 23.1 22.5 53.9 66.6 191 165 70.8 168GL24h 5.5 65 225 20 20 110 25.3 2.21 2.30 23.1 22.5 53.9 66.6 176 165 70.8 168GL24h 5.5 65 225 30 0 89.0 25.3 2.21 2.30 22.8 22.5 65.9 78.6 176 154 70.8 168GL24h 5.5 65 225 30 20 111 25.3 2.21 2.30 22.8 22.5 65.9 78.6 155 154 70.8 168

Type 3Input: Steel grade S235, bs = b - 20mm Type 1 Extra lamellaType 2Modified

Appendix B, Result data Point load at midpoint

B-1


P.ed inst

P.ed fin

V.ed inst

V.ed fin

P.ed inst

P.ed fin

P.ed inst

P.ed fin 1x45 2x45



GL24h 2 65 270 10 0 89.0 30.3 19.7 19.7 29.5 28.9 16.9 21.9 50.8 45.6 34.0GL24h 2 65 270 10 20 111 30.3 19.7 19.7 29.5 28.9 16.9 21.9 52.2 45.6 34.0GL24h 2 65 270 20 0 59.0 30.3 19.7 19.7 28.7 28.0 24.1 28.8 42.3 39.9 34.0GL24h 2 65 270 20 20 74.0 30.3 19.7 19.7 28.7 28.0 24.1 28.8 43.5 39.9 34.0GL24h 2 65 270 30 0 44.0 30.3 19.7 19.7 28.3 27.7 28.6 33.1 39.9 39.5 34.0GL24h 2 65 270 30 20 55.0 30.3 19.7 19.7 28.3 27.7 28.6 33.1 40.8 39.5 34.0GL24h 2.5 65 270 10 0 89.0 30.3 15.8 15.8 29.5 28.9 16.9 21.9 50.8 45.6 34.0GL24h 2.5 65 270 10 20 111 30.3 15.8 15.8 29.5 28.9 16.9 21.9 52.2 45.6 34.0GL24h 2.5 65 270 20 0 59.0 30.3 15.8 15.8 28.7 28.0 24.1 28.8 42.3 39.9 34.0GL24h 2.5 65 270 20 20 73.0 30.3 15.8 15.8 28.7 28.0 24.1 28.8 43.5 39.9 34.0GL24h 2.5 65 270 30 0 44.0 30.3 15.8 15.8 28.3 27.7 28.6 33.1 39.9 39.5 34.0GL24h 2.5 65 270 30 20 55.0 30.3 15.8 15.8 28.3 27.7 28.6 33.1 40.8 39.5 34.0GL24h 3 65 270 10 0 88.0 30.3 11.7 12.2 29.5 28.9 31.7 31.7 69.6 57.4 44.9GL24h 3 65 270 10 20 110 30.3 11.7 12.2 29.5 28.9 31.7 31.7 71.3 57.4 44.9GL24h 3 65 270 20 0 59.0 30.3 11.7 12.2 28.7 28.0 39.8 39.2 60.2 51.3 44.9GL24h 3 65 270 20 20 74.0 30.3 11.7 12.2 28.7 28.0 39.8 39.2 61.6 51.3 44.9GL24h 3 65 270 30 0 44.0 30.3 11.7 12.2 28.3 27.7 44.9 43.9 57.5 50.8 44.9GL24h 3 65 270 30 20 55.0 30.3 11.7 12.2 28.3 27.7 44.9 43.9 58.6 50.8 44.9GL24h 3.5 65 270 10 0 89.0 30.3 8.84 9.21 29.5 28.9 35.9 48.5 92.0 78.0 53.9GL24h 3.5 65 270 10 20 111 30.3 8.84 9.21 29.5 28.9 35.9 48.5 93.8 78.0 53.9GL24h 3.5 65 270 20 0 59.0 30.3 8.84 9.21 28.7 28.0 54.3 57.4 81.1 71.1 53.9GL24h 3.5 65 270 20 20 73.0 30.3 8.84 9.21 28.7 28.0 54.3 57.4 82.7 71.1 53.9GL24h 3.5 65 270 30 0 44.0 30.3 8.84 9.21 28.3 27.7 63.8 62.7 78.2 70.6 53.9GL24h 3.5 65 270 30 20 55.0 30.3 8.84 9.21 28.3 27.7 63.8 62.7 79.3 70.6 53.9GL24h 4 65 270 10 0 88.0 30.3 6.91 7.20 29.5 28.9 36.9 50.1 115 99.2 55.0GL24h 4 65 270 10 20 110 30.3 6.91 7.20 29.5 28.9 36.9 50.1 117 99.2 55.0GL24h 4 65 270 20 0 58.0 30.3 6.91 7.20 28.7 28.0 56.1 70.1 103 91.5 55.0GL24h 4 65 270 20 20 73.0 30.3 6.91 7.20 28.7 28.0 56.1 70.1 104 91.5 55.0GL24h 4 65 270 30 0 44.0 30.3 6.91 7.20 28.3 27.7 68.5 82.0 99.4 91.0 55.0GL24h 4 65 270 30 20 54.0 30.3 6.91 7.20 28.3 27.7 68.5 82.0 101 91.0 55.0GL24h 4.5 65 270 10 0 88.0 30.3 5.54 5.77 29.5 28.9 37.6 51.2 138 121 55.7GL24h 4.5 65 270 10 20 110 30.3 5.54 5.77 29.5 28.9 37.6 51.2 141 121 55.7GL24h 4.5 65 270 20 0 59.0 30.3 5.54 5.77 28.7 28.0 57.4 71.9 120 112 55.7GL24h 4.5 65 270 20 20 73.0 30.3 5.54 5.77 28.7 28.0 57.4 71.9 127 112 55.7GL24h 4.5 65 270 30 0 47.0 30.3 5.54 5.77 28.3 27.7 70.2 84.2 110 112 55.7GL24h 4.5 65 270 30 20 58.0 30.3 5.54 5.77 28.3 27.7 70.2 84.2 119 112 55.7GL24h 5 65 270 10 0 88.0 30.3 4.54 4.73 29.5 28.9 38.2 52.0 162 143 56.3GL24h 5 65 270 10 20 110 30.3 4.54 4.73 29.5 28.9 38.2 52.0 164 143 56.3GL24h 5 65 270 20 0 69.0 30.3 4.54 4.73 28.7 28.0 58.3 73.2 140 133 56.3GL24h 5 65 270 20 20 87.0 30.3 4.54 4.73 28.7 28.0 58.3 73.2 149 133 56.3GL24h 5 65 270 30 0 67.0 30.3 4.54 4.73 28.3 27.7 71.4 85.9 135 130 56.3GL24h 5 65 270 30 20 83.0 30.3 4.54 4.73 28.3 27.7 71.4 85.9 143 130 56.3GL24h 5.5 65 270 10 0 89.0 30.3 3.78 3.94 29.5 28.9 38.6 52.6 186 165 56.7GL24h 5.5 65 270 10 20 111 30.3 3.78 3.94 29.5 28.9 38.6 52.6 189 165 56.7GL24h 5.5 65 270 20 0 82.0 30.3 3.78 3.94 28.7 28.0 59.1 74.2 166 152 56.7GL24h 5.5 65 270 20 20 102 30.3 3.78 3.94 28.7 28.0 59.1 74.2 170 152 56.7GL24h 5.5 65 270 30 0 81.0 30.3 3.78 3.94 28.3 27.7 72.4 87.1 157 146 56.7GL24h 5.5 65 270 30 20 101 30.3 3.78 3.94 28.3 27.7 72.4 87.1 159 146 56.7


B-2


P.ed inst

P.ed fin

V.ed inst

V.ed fin

P.ed inst

P.ed fin

P.ed inst

P.ed fin 1x45 2x45



GL24h 6 65 270 10 0 90.0 30.3 3.19 3.33 29.5 28.9 38.9 53.1 208 187 57.0GL24h 6 65 270 10 20 112 30.3 3.19 3.33 29.5 28.9 38.9 53.1 213 187 57.0GL24h 6 65 270 20 0 91.0 30.3 3.19 3.33 28.7 28.0 59.6 75.0 189 170 57.0GL24h 6 65 270 20 20 114 30.3 3.19 3.33 28.7 28.0 59.6 75.0 188 170 57.0GL24h 6 65 270 30 0 91.0 30.3 3.19 3.33 28.3 27.7 73.1 88.1 178 162 57.0GL24h 6 65 270 30 20 114 30.3 3.19 3.33 28.3 27.7 73.1 88.1 173 162 57.0GL24h 2 65 315 10 0 117 35.4 26.4 26.4 35.1 34.6 18.7 24.5 61.9 55.6GL24h 2 65 315 10 20 147 35.4 26.4 26.4 35.1 34.6 18.7 24.5 63.6 55.6GL24h 2 65 315 20 0 80.0 35.4 26.4 26.4 34.4 33.7 27.0 32.6 51.3 48.0GL24h 2 65 315 20 20 100 35.4 26.4 26.4 34.4 33.7 27.0 32.6 52.8 48.0GL24h 2 65 315 30 0 60.0 35.4 26.4 26.4 33.9 33.2 32.1 37.3 47.6 46.4GL24h 2 65 315 30 20 75.0 35.4 26.4 26.4 33.9 33.2 32.1 37.3 48.8 46.4GL24h 2.5 65 315 10 0 117 35.4 21.1 21.1 35.1 34.6 18.7 24.5 61.9 55.6GL24h 2.5 65 315 10 20 147 35.4 21.1 21.1 35.1 34.6 18.7 24.5 63.6 55.6GL24h 2.5 65 315 20 0 80.0 35.4 21.1 21.1 34.4 33.7 27.0 32.6 51.3 48.0GL24h 2.5 65 315 20 20 99.0 35.4 21.1 21.1 34.4 33.7 27.0 32.6 52.8 48.0GL24h 2.5 65 315 30 0 61.0 35.4 21.1 21.1 33.9 33.2 32.1 37.3 47.7 46.4GL24h 2.5 65 315 30 20 76.0 35.4 21.1 21.1 33.9 33.2 32.1 37.3 48.8 46.4GL24h 3 65 315 10 0 118 35.4 17.6 17.6 35.1 34.6 18.7 24.5 62.0 55.6GL24h 3 65 315 10 20 146 35.4 17.6 17.6 35.1 34.6 18.7 24.5 63.6 55.6GL24h 3 65 315 20 0 80.0 35.4 17.6 17.6 34.4 33.7 27.0 32.6 51.3 48.0GL24h 3 65 315 20 20 99.0 35.4 17.6 17.6 34.4 33.7 27.0 32.6 52.8 48.0GL24h 3 65 315 30 0 60.0 35.4 17.6 17.6 33.9 33.2 32.1 37.3 47.6 46.4GL24h 3 65 315 30 20 75.0 35.4 17.6 17.6 33.9 33.2 32.1 37.3 48.8 46.4GL24h 3.5 65 315 10 0 118 35.4 13.6 14.2 35.1 34.6 31.6 32.5 79.7 65.6GL24h 3.5 65 315 10 20 147 35.4 13.6 14.2 35.1 34.6 31.6 32.5 81.4 65.6GL24h 3.5 65 315 20 0 80.0 35.4 13.6 14.2 34.4 33.7 40.9 41.2 67.8 57.5GL24h 3.5 65 315 20 20 100 35.4 13.6 14.2 34.4 33.7 40.9 41.2 69.5 57.5GL24h 3.5 65 315 30 0 60.0 35.4 13.6 14.2 33.9 33.2 46.5 46.2 63.6 55.8GL24h 3.5 65 315 30 20 75.0 35.4 13.6 14.2 33.9 33.2 46.5 46.2 65.0 55.8GL24h 4 65 315 10 0 117 35.4 10.7 11.2 35.1 34.6 36.7 47.3 99.6 84.1GL24h 4 65 315 10 20 146 35.4 10.7 11.2 35.1 34.6 36.7 47.3 102 84.1GL24h 4 65 315 20 0 79.0 35.4 10.7 11.2 34.4 33.7 56.6 57.0 86.4 75.1GL24h 4 65 315 20 20 99.0 35.4 10.7 11.2 34.4 33.7 56.6 57.0 88.4 75.1GL24h 4 65 315 30 0 60.0 35.4 10.7 11.2 33.9 33.2 62.9 62.5 82.0 73.3GL24h 4 65 315 30 20 75.0 35.4 10.7 11.2 33.9 33.2 62.9 62.5 83.4 73.3GL24h 4.5 65 315 10 0 117 35.4 8.63 8.99 35.1 34.6 37.6 51.8 120 103GL24h 4.5 65 315 10 20 146 35.4 8.63 8.99 35.1 34.6 37.6 51.8 123 103GL24h 4.5 65 315 20 0 80.0 35.4 8.63 8.99 34.4 33.7 58.3 73.3 106 93.2GL24h 4.5 65 315 20 20 99.0 35.4 8.63 8.99 34.4 33.7 58.3 73.3 108 93.2GL24h 4.5 65 315 30 0 59.0 35.4 8.63 8.99 33.9 33.2 71.8 79.3 101 91.2GL24h 4.5 65 315 30 20 74.0 35.4 8.63 8.99 33.9 33.2 71.8 79.3 102 91.2GL24h 5 65 315 10 0 117 35.4 7.09 7.38 35.1 34.6 38.4 53.0 141 123GL24h 5 65 315 10 20 147 35.4 7.09 7.38 35.1 34.6 38.4 53.0 144 123GL24h 5 65 315 20 0 80.0 35.4 7.09 7.38 34.4 33.7 59.7 75.9 126 112GL24h 5 65 315 20 20 99.0 35.4 7.09 7.38 34.4 33.7 59.7 75.9 128 112GL24h 5 65 315 30 0 61.0 35.4 7.09 7.38 33.9 33.2 73.6 89.4 120 110GL24h 5 65 315 30 20 76.0 35.4 7.09 7.38 33.9 33.2 73.6 89.4 122 110


B-3


P.ed inst

P.ed fin

V.ed inst

V.ed fin

P.ed inst

P.ed fin

P.ed inst

P.ed fin 1x45 2x45



GL24h 5.5 65 315 10 0 116 35.4 5.92 6.17 35.1 34.6 38.9 53.8 163 142GL24h 5.5 65 315 10 20 145 35.4 5.92 6.17 35.1 34.6 38.9 53.8 165 142GL24h 5.5 65 315 20 0 80.0 35.4 5.92 6.17 34.4 33.7 60.7 77.3 146 130GL24h 5.5 65 315 20 20 100 35.4 5.92 6.17 34.4 33.7 60.7 77.3 148 130GL24h 5.5 65 315 30 0 59.0 35.4 5.92 6.17 33.9 33.2 74.9 91.2 127 128GL24h 5.5 65 315 30 20 74.0 35.4 5.92 6.17 33.9 33.2 74.9 91.2 141 128GL24h 6 65 315 10 0 117 35.4 5.01 5.22 35.1 34.6 39.4 54.5 184 162GL24h 6 65 315 10 20 146 35.4 5.01 5.22 35.1 34.6 39.4 54.5 187 162GL24h 6 65 315 20 0 80.0 35.4 5.01 5.22 34.4 33.7 61.5 78.5 157 149GL24h 6 65 315 20 20 100 35.4 5.01 5.22 34.4 33.7 61.5 78.5 168 149GL24h 6 65 315 30 0 77.0 35.4 5.01 5.22 33.9 33.2 76.0 92.6 151 146GL24h 6 65 315 30 20 96.0 35.4 5.01 5.22 33.9 33.2 76.0 92.6 161 146GL24h 6.5 65 315 10 0 116 35.4 4.30 4.48 35.1 34.6 39.7 55.0 206 182GL24h 6.5 65 315 10 20 145 35.4 4.30 4.48 35.1 34.6 39.7 55.0 209 182GL24h 6.5 65 315 20 0 92.0 35.4 4.30 4.48 34.4 33.7 62.1 79.3 181 168GL24h 6.5 65 315 20 20 114 35.4 4.30 4.48 34.4 33.7 62.1 79.3 189 168GL24h 6.5 65 315 30 0 90.0 35.4 4.30 4.48 33.9 33.2 76.9 93.8 172 162GL24h 6.5 65 315 30 20 113 35.4 4.30 4.48 33.9 33.2 76.9 93.8 178 162GL24h 7 65 315 10 0 117 35.4 3.73 3.88 35.1 34.6 40.0 55.5 228 202GL24h 7 65 315 10 20 146 35.4 3.73 3.88 35.1 34.6 40.0 55.5 231 202GL24h 7 65 315 20 0 101 35.4 3.73 3.88 34.4 33.7 62.6 80.1 204 186GL24h 7 65 315 20 20 126 35.4 3.73 3.88 34.4 33.7 62.6 80.1 208 186GL24h 7 65 315 30 0 101 35.4 3.73 3.88 33.9 33.2 77.5 94.7 193 177GL24h 7 65 315 30 20 126 35.4 3.73 3.88 33.9 33.2 77.5 94.7 193 177GL28h 4 90 225 10 0 107 38.9 6.14 6.40 36.6 35.7 35.5 47.3 142 131 69.2 162GL28h 4 90 225 10 20 134 38.9 6.14 6.40 36.6 35.7 35.5 47.3 147 131 69.2 162GL28h 4 90 225 20 0 105 38.9 6.14 6.40 35.5 34.6 52.8 64.8 130 120 69.2 162GL28h 4 90 225 20 20 132 38.9 6.14 6.40 35.5 34.6 52.8 64.8 131 120 69.2 162GL28h 4 90 225 30 0 104 38.9 6.14 6.40 35.1 34.5 64.2 76.1 124 117 69.2 162GL28h 4 90 225 30 20 130 38.9 6.14 6.40 35.1 34.5 64.2 76.1 123 117 69.2 162GL28h 5 90 225 10 0 132 38.9 4.00 4.17 36.6 35.7 36.4 48.6 202 178 70.4 166GL28h 5 90 225 10 20 165 38.9 4.00 4.17 36.6 35.7 36.4 48.6 195 178 70.4 166GL28h 5 90 225 20 0 134 38.9 4.00 4.17 35.5 34.6 54.2 66.8 178 156 70.4 166GL28h 5 90 225 20 20 168 38.9 4.00 4.17 35.5 34.6 54.2 66.8 163 156 70.4 166GL28h 5 90 225 30 0 136 38.9 4.00 4.17 35.1 34.5 66.1 78.5 165 146 70.4 166GL28h 5 90 225 30 20 170 38.9 4.00 4.17 35.1 34.5 66.1 78.5 144 146 70.4 166GL28h 6 90 225 10 0 147 38.9 2.81 2.92 36.6 35.7 36.9 49.3 258 223 71.1 169GL28h 6 90 225 10 20 184 38.9 2.81 2.92 36.6 35.7 36.9 49.3 240 223 71.1 169GL28h 6 90 225 20 0 152 38.9 2.81 2.92 35.5 34.6 55.0 67.8 225 190 71.1 169GL28h 6 90 225 20 20 190 38.9 2.81 2.92 35.5 34.6 55.0 67.8 192 190 71.1 169GL28h 6 90 225 30 0 155 38.9 2.81 2.92 35.1 34.5 67.2 79.9 204 173 71.1 169GL28h 6 90 225 30 20 193 38.9 2.81 2.92 35.1 34.5 67.2 79.9 163 173 71.1 169GL28h 7 90 225 10 0 158 38.9 2.07 2.16 36.6 35.7 37.2 49.7 311 267 71.6 170GL28h 7 90 225 10 20 197 38.9 2.07 2.16 36.6 35.7 37.2 49.7 284 268 71.6 170GL28h 7 90 225 20 0 164 38.9 2.07 2.16 35.5 34.6 55.6 68.5 266 222 71.6 170GL28h 7 90 225 20 20 205 38.9 2.07 2.16 35.5 34.6 55.6 68.5 219 222 71.6 170GL28h 7 90 225 30 0 168 38.9 2.07 2.16 35.1 34.5 67.8 80.8 239 200 71.6 170GL28h 7 90 225 30 20 206 38.9 2.07 2.16 35.1 34.5 67.8 80.8 184 198 71.6 170


B-4


P.ed inst

P.ed fin

V.ed inst

V.ed fin

P.ed inst

P.ed fin

P.ed inst

P.ed fin 1x45 2x45



GL28h 4 90 270 10 0 126 46.7 10.4 10.8 45.3 44.4 37.6 50.9 125 110 55.0 125GL28h 4 90 270 10 20 158 46.7 10.4 10.8 45.3 44.4 37.6 50.9 127 110 55.0 125GL28h 4 90 270 20 0 84.0 46.7 10.4 10.8 44.1 43.1 56.9 70.9 108 104 55.0 125GL28h 4 90 270 20 20 105 46.7 10.4 10.8 44.1 43.1 56.9 70.9 116 104 55.0 125GL28h 4 90 270 30 0 64.0 46.7 10.4 10.8 43.5 42.6 69.3 82.8 100 104 55.0 125GL28h 4 90 270 30 20 79.0 46.7 10.4 10.8 43.5 42.6 69.3 82.8 108 104 55.0 125GL28h 5 90 270 10 0 127 46.7 6.82 7.11 45.3 44.4 38.9 52.9 169 156 56.3 129GL28h 5 90 270 10 20 159 46.7 6.82 7.11 45.3 44.4 38.9 52.9 177 156 56.3 129GL28h 5 90 270 20 0 125 46.7 6.82 7.11 44.1 43.1 59.2 74.1 156 145 56.3 129GL28h 5 90 270 20 20 156 46.7 6.82 7.11 44.1 43.1 59.2 74.1 160 145 56.3 129GL28h 5 90 270 30 0 123 46.7 6.82 7.11 43.5 42.6 72.3 86.7 149 140 56.3 129GL28h 5 90 270 30 20 154 46.7 6.82 7.11 43.5 42.6 72.3 86.7 150 140 56.3 129GL28h 6 90 270 10 0 150 46.7 4.80 5.00 45.3 44.4 39.7 54.0 225 202 57.0 131GL28h 6 90 270 10 20 188 46.7 4.80 5.00 45.3 44.4 39.7 54.0 225 201 57.0 131GL28h 6 90 270 20 0 153 46.7 4.80 5.00 44.1 43.1 60.5 75.9 204 181 57.0 131GL28h 6 90 270 20 20 191 46.7 4.80 5.00 44.1 43.1 60.5 75.9 195 181 57.0 131GL28h 6 90 270 30 0 154 46.7 4.80 5.00 43.5 42.6 74.0 89.0 190 170 57.0 131GL28h 6 90 270 30 20 193 46.7 4.80 5.00 43.5 42.6 74.0 89.0 176 170 57.0 131GL28h 7 90 270 10 0 166 46.7 3.56 3.71 45.3 44.4 40.2 54.7 278 246 57.5 133GL28h 7 90 270 10 20 207 46.7 3.56 3.71 45.3 44.4 40.2 54.7 271 246 57.5 133GL28h 7 90 270 20 0 172 46.7 3.56 3.71 44.1 43.1 61.3 77.0 250 215 57.5 133GL28h 7 90 270 20 20 215 46.7 3.56 3.71 44.1 43.1 61.3 77.0 227 215 57.5 133GL28h 7 90 270 30 0 174 46.7 3.56 3.71 43.5 42.6 75.1 90.4 229 198 57.5 133GL28h 7 90 270 30 20 218 46.7 3.56 3.71 43.5 42.6 75.1 90.4 199 198 57.5 133GL28h 8 90 270 10 0 177 46.7 2.74 2.85 45.3 44.4 40.5 55.2 329 289 57.8 134GL28h 8 90 270 10 20 222 46.7 2.74 2.85 45.3 44.4 40.5 55.2 316 289 57.8 134GL28h 8 90 270 20 0 185 46.7 2.74 2.85 44.1 43.1 61.9 77.8 292 249 57.8 134GL28h 8 90 270 20 20 231 46.7 2.74 2.85 44.1 43.1 61.9 77.8 259 248 57.8 134GL28h 8 90 270 30 0 189 46.7 2.74 2.85 43.5 42.6 75.9 91.4 266 225 57.8 134GL28h 8 90 270 30 20 236 46.7 2.74 2.85 43.5 42.6 75.9 91.4 221 225 57.8 134GL28h 4 90 315 10 0 167 54.4 16.1 16.8 54.0 53.2 37.4 51.3 108 93.2 45.2 101GL28h 4 90 315 10 20 209 54.4 16.1 16.8 54.0 53.2 37.4 51.3 110 93.2 45.2 101GL28h 4 90 315 20 0 113 54.4 16.1 16.8 52.9 51.7 57.5 69.1 96.6 85.9 45.2 101GL28h 4 90 315 20 20 141 54.4 16.1 16.8 52.9 51.7 57.5 69.1 98.4 85.9 45.2 101GL28h 4 90 315 30 0 86.0 54.4 16.1 16.8 52.1 51.0 70.4 75.0 93.2 84.9 45.2 101GL28h 4 90 315 30 20 107 54.4 16.1 16.8 52.1 51.0 70.4 75.0 94.6 84.9 45.2 101GL28h 5 90 315 10 0 166 54.4 10.7 11.1 54.0 53.2 39.1 53.9 151 134 46.6 104GL28h 5 90 315 10 20 208 54.4 10.7 11.1 54.0 53.2 39.1 53.9 154 134 46.6 104GL28h 5 90 315 20 0 113 54.4 10.7 11.1 52.9 51.7 60.6 76.9 133 125 46.6 104GL28h 5 90 315 20 20 142 54.4 10.7 11.1 52.9 51.7 60.6 76.9 140 125 46.6 104GL28h 5 90 315 30 0 90.0 54.4 10.7 11.1 52.1 51.0 74.5 90.3 122 124 46.6 104GL28h 5 90 315 30 20 112 54.4 10.7 11.1 52.1 51.0 74.5 90.3 133 124 46.6 104GL28h 6 90 315 10 0 166 54.4 7.54 7.85 54.0 53.2 40.2 55.5 196 175 47.3 107GL28h 6 90 315 10 20 208 54.4 7.54 7.85 54.0 53.2 40.2 55.5 199 175 47.3 107GL28h 6 90 315 20 0 144 54.4 7.54 7.85 52.9 51.7 62.4 79.5 175 163 47.3 107GL28h 6 90 315 20 20 179 54.4 7.54 7.85 52.9 51.7 62.4 79.5 181 163 47.3 107GL28h 6 90 315 30 0 141 54.4 7.54 7.85 52.1 51.0 77.0 93.6 167 157 47.3 107GL28h 6 90 315 30 20 176 54.4 7.54 7.85 52.1 51.

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