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A NOVEL LVL-BASED INTERNAL REINFORCEMENT FOR HOLES IN

GLULAM BEAMS

Cristobal Tapia Camú1, Simon Aicher2

ABSTRACT: A newly developed reinforcement system for glulam, actually representing a new generic wood com-

pund, is presented. The composite consists on a hybrid cross-section, composed of intercalated layers of GLT and LVL,

glued together along the width-direction of the beam. The specific build-up improves in first instance the mechanical

properties of the glulam in the direction perpendicular to the grain significantly. Hence, the composite is especially well

suited for the reinforcement of arrays of large holes in wide cross-sections. Secondly, the layers were tailored in such

a manner, that the bending load capacity equalls that of the gross-cross-section. A parametric study was performed by

means of the finite element method, to study the redistribution of stresses perpendicular to the main axis of the beam in the

region of stress concentrations at one of the hole corners. Specifically, the load sharing of the vertical tensile force Ft,90

described in the German National Annex to EC5 was analyzed, and an analytical relationship depending on the thickness,

elastic modulus and moment-to-shear-force ratio was developed.

KEYWORDS: holes, LVL, GLT, reinforcements, hybrid build-up, parametric study

1 INTRODUCTION

The drilling of holes in beams made of glued laminated

timber (GLT) represents a frequent necessity in timber

structures, as they are required for the passing-through of

plumbing, electrical and other service-relevant infrastruc-

ture systems. These apertures represent size-dependant

significant weak sections in the beam. The holes can lead

to significant reductions of the cross-section, e.g. up to

40%, and, most importantly, lead to stress concentrations

at the hole corners, resulting in a noticeable decrease of the

maximum load capacities. The failure mechanism of un-

reinforced and reinforced holes is well known and is man-

ifested by the propagation of cracks in the direction of the

grain and beam-length axis, starting from two zones with

high tensile stresses perpendicular to the grain, which are

located at diagonally opposite corner areas of the hole.

Typically, two possible types of reinforcements are con-

sidered to prevent an early propagation of cracks at the

high-stressed corners: internal screw-type reinforcements

and external wood panels, both of which can be designed

according to the German National Annex to EC5 [1].

The two reinforcement alternatives show different advan-

tages and drawbacks when it comes to their mechani-

cal behavior, ease of application and aesthetics, which

play an important role in deciding in favor of one or the

other. Regarding the mechanical aspects, externally ap-

plied plates outperform in general the screw-type rein-

forcements, since they are not only able to redirect the

1Cristóbal Tapia, MPA, University of Stuttgart, Germany,

cristobal.tapia-camu@mpa.uni-stuttgart.de

2Simon Aicher, MPA, University of Stuttgart, Germany,

simon.aicher@mpa.uni-stuttgart.de

stresses perpendicular to the grain, but are also capa-

ble to transfer the shear stresses present in the crack-

endangered region [2]. Although new investigations show

that inclined arrangements of screw-type reinforcements

can help in the transmission of shear stresses too [3, 4],

their application, as well as for non-inclined reinforce-

ments, is in practice limited to small-to-medium sized

beams. Nevertheless, external reinforcements also reveal

a mechanical disadvantage, which becomes evident when

cross-sections of large thicknesses need to be reinforced.

In such a scenario, the panels, fixed on the wide sides of

the beam, have little to no influence on the stresses near the

mid-width zone, which reduces their effectiveness consid-

erably. Even an increase of the plate’s thickness makes no

big difference regarding this issue [2].

To overcome this problem, and based on the necessi-

ties of a specific building project [5], a novel kind of rein-

forcement for holes in glulam beams was developed. The

approach consists in the application of one or several inter-

nal, vertically placed LVL layers between adjacent layers

of GLT, creating a special composite element. This hybrid

configuration enables the reduction of the peak stresses

at the crack-prone corners more effectively, as compared

to simple external plates, and has proven to be especially

well suited for configurations with multiple large holes

placed close to each other [6].

2 TENSILE FORCE PERPENDICULAR

TO THE GRAIN Ft,90

The present design of holes and hole reinforcements ac-

cording to DIN EN 1995-1-1/NA [1] is based on a fictive

resultant tensile force, Ft,90, representing the integral of

+

+–

–2

1V

M+∆M

V

M

I

I

II

II

x

σ90

Ft,90 =Ft,V+Ft,M

δi

`t,90

σ90 = 0

45°

σ90,max,A

Figure 1: Definition of the vertical tensile force Ft,90 shown for

the case of a rectangular hole. Regions 1 and 2 are subjected to

tensile stresses perpendicular to the grain in the vicinity of the

corners of the hole. The colored field depicts the stress-stresses

perpendicular to the grain σ90.

the stresses perpendicular to the grain in the hole periph-

ery in the crack relevant sections (see Figure 1). This force

is composed of two additive parts: one, Ft,V, accounting

for the shear force, which cannot be transferred in the hole

area, and a second part, Ft,M, related to the bending mo-

ment present in the cross-section:

Ft,90 =Ft,V+Ft,M.(1)

The exact form taken by each of the two terms is, for

this paper, not relevant—although discussion about this

topic can be found in Refs. [2, 7, 8]. The vertical force

acc. to Eq. (1) is used for the design of unreinforced and

reinforced holes. In the first case, the force is compared

against a fictive resulting resistance force, based on the

size-dependent tensile strength of the glulam perpendic-

ular to the grain. In the second case, the force is the ba-

sis of the design of reinforcement elements (screws, rods,

plates). Since the analyzed reinforcement system is used

in combination with holes in GLT beams, the tensile force

Ft,90 becomes a relevant parameter, and is used through-

out this paper for the different analysis. For this purpose,

finite element results are used to numerically compute the

tensile force, following the method depicted graphically in

Figure 1 (the exact procedure is explained in Section 4).

3 REINFORCEMENT DESCRIPTION

AND PARAMETRIC ANALYSIS

The analyzed reinforced system consists on a special

hybrid cross-section, composed of intercalated layers of

GLT

GLT

LVL

t1

t2

t1

(a)

(b)

Figure 2: General description of the analyzed internal rein-

forcement system in two different possible configurations: (a) a

single LVL layer between two GLT elements and (b) two layers

of LVL intercalated in-between three GLT elements.

GLT and LVL, glued together along the width-direction of

the beam (see Figure 2a). The cross-bonded layers with

its fiber direction aligned perpendicular to main axis of

the beam, are chosen in such quantity that they improve

the mechanical properties of the structure in this direction

significantly. On the other hand, the cross-bonded ratio is

set as low as possible, in order not to reduce the length-

wise bending stiffness and strength too much, or even not

at all The latter can obviously only be achieved by use

of material/veneers of the LVL, which are significantly

stronger than spruce GLT (such as beech). This arrange-

ment confers a global strengthening character to the re-

inforcement, making it not only useful near the hole (lo-

cally), but throughout the whole length of the beam. This

continuous, uninterrupted characteristic of the reinforce-

ment makes it a good solution e.g. for beams with multiple

(large) holes, since a single element serves simultaneously

as reinforcement for all the holes.

Depending on the thickness of the needed cross-section,

multiple layers of LVL and GLT can be used, helping to

achieve a more efficient use of both materials (see Fig-

ure 2b). Nevertheless, the optimal configuration of thick-

nesses for a given number of layers and width of the beam

is not a trivial task to solve—not always at least. For the

case in which the system is mainly used as a reinforce-

ment for holes, the percentage of the total tensile force

Ft,90 that goes into each layer needs to be known, in or-

der to perform the required design check at each layer.

However, the composite build-up of the reinforced sec-

tion, added to the complex stress redistribution in the hole-

influenced region of the beam, complicates the determina-

tion of the load sharing between GLT and LVL. Different

variables, like modulus of elasticity (MOE) perpendicular

to the grain (E90), layer thicknesses and section forces ra-

V

M

a= 2hd

hhd

h

2

3hrcorn

X

Y

w

tGLT tGLT

tLVL

Z

Y

Figure 3: Geometry and dimensions of the analyzed configu-

rations; load application and boundary conditions used for the

finite element model.

tios, influence the redistribution of stresses in a complex

manner. An analysis of the influence of these parameters

is, hence, most desirable, since a practical use of the rein-

forcement requires an understanding of the effects of the

different possible combinations. To this end, a parametric

analysis was performed, which is described in the follow-

ing section.

3.1 PARAMETRIC ANALYSIS

The analyzed configuration comprises the basic case,

where only one LVL layer is inserted between two GLT

beams, as shown in Figure 3. Three variables are investi-

gated: (1) the elastic modulus perpendicular to the beam’s

axis of the reinforcement (LVL), (2) the thickness of the

reinforcement, and (3) the ratio of moment-to-shear-force

(M/V) in the region of the hole.

The beam has a depth of h= 450 mm and a total length

of 2700 mm (6h), whereas the total width is set to be

w= 250 mm. The thickness of the LVL layer (tLVL) is var-

ied between 15 mm and 50 mm in steps of 5 mm, while the

thickness of the GLT elements (tGLT) is accordingly ad-

justed to maintain the mentioned constant total width w. A

rectangular hole with a fixed size hd×a= 180 ×360 mm2

is placed at mid-depth with its center at a distance of

3h= 1350 mm from the next (here left) support. The used

corner radius rcorn equals 20 mm.

The MOE perpendicular to the axis of the beam

for the LVL, E90,LVL, is studied for values between

300 N/mm2and 5000 N/mm2in steps of 600 N/mm2,

which, given E90,GLT = 300 N/mm2, gives MOE ratios

E90,LVL/E90,GLT from 1 to 15 in steps of 2. (Note: Phys-

ically, the different E90,LVL values stem from different

cross-bonded ratios of the cross-bonded LVL) The shear-

force Vapplied to the model remains constant for all the

configurations, whilst the moment is varied by means of

an externally introduced moment Mat the position of the

left support. Starting from zero, the external moment is

increase in steps of 3×107Nmm up to 9×107Nmm,

which translates into a total ratio M/Vat the analyzed

(right) edge of the hole of 1.53, 4.53, 7.53 and 10.53 [m].

4 DESCRIPTION OF THE FINITE ELE-

MENT MODEL

The finite element (FE) model was built using the soft-

ware Abaqus v2017 [9] with its standard solver, using 3D

continuum linear elements with reduced integration (ele-

ment type: C3D8R). For the LVL, the size of the elements

along the thickness (Z-direction) was chosen to be 5 mm,

which ensures that for each consecutive thickness ana-

lyzed one extra layer of elements is created. Regarding the

discretization of the GLT, an element thickness of 5 mm

cannot be achieved for all configurations—due to some

values not being multiples of 5—, but a close approxima-

tion was used, defined as (tGLT/dtGLT /5e)mm. This results

in values of exactly 5 mm for LVL-thicknesses multiples

of 10 and approximately 5 mm for the other cases. (Note:

The objective to have the same discretization in both

materials would require thicknesses of 2.5 mm, which

would lead to a considerably increase in the computation

time.) The side lengths of the elements contained in the

XY -planes have a size of 4 mm directly on the periphery

of the hole, increasing up to 10 mm on the region extend-

ing 180 mm to both sides of the hole’s vertical edges (see

Figure 4). After this, the size is progressively increased up

to 30 mm.

Figure 4: Example of the meshing of the finite element model

used for the parametric analysis. The different colors represent

the regions with the two materials used; green: GLT material,

gray: LVL material.

Symmetry conditions are applied at one edge of the

beam, according to Figure 3, where also a shear force

V= 50 kN is applied. A rigid plane (element type: R3D4)

is created on the side of the simple support, placed verti-

cally in the YZ-plane, and is connected to the elements of

the beam by means of tie constraints. The correspond-

ing reference point is placed at the gravity center of the

cross-section ((y,z)=(h/2, w/2)) and is used both to set

the boundary conditions of the simple support and to ap-

ply the external moment Mto the beam.

The material properties used for the LVL and GLT are

presented in Table 1. For the LVL, the shear modulus

(Gxy) was varied with a linear function, based on the

value of E90; it was set to have a value of 650 N/mm2

at E90,LVL = 300 N/mm2and 850 N/mm2for E90,LVL =

3900 N/mm2. For the solution of the model, no geometri-

cal non-linearities were considered.

4.1 COMPUTATION OF THE TENSILE FORCE

Ft,90

In order to numerically obtain the vertical tensile force

Ft,90 from the results of the FE model, a horizontal path

is defined at each Z-position where element nodes are

Table 1: Material properties used for the finite element model

Material ExEy/zGxy/xz Gyz νxy/xz νyz

[N/mm2] [N/mm2] [N/mm2] [N/mm2] [–] [–]

GLT 11500 300 650 65 0.02 0.2

LVL 11800 variable f(Ey)Gxy ·0.1 0.02 0.2

present. The paths start at the corner 2 of the hole, sub-

jected to tensile stresses perpendicular to the beam’s axis,

at an angle ϕ= 45°, as depicted in Figure 1, and extend

for 400 mm. A numerical integration of the tensile stresses

(σ90 > 0) is performed and the result is then multiplied by

the thickness of the element (if it is between other ele-

ments) or by half of the thickness (if the path lays on one

of the outer faces of each layer (GLT or LVL)). Finally,

the values obtained in each layer are summed up and the

total vertical tensile force for each layer is obtained.

5 RESULTS

The results from the FE-models were analyzed in order

to gain some insight into the effect produced by the vari-

ation of each parameter on the structure. Following, the

isolated influence of each one of the three analyzed pa-

rameters is presented. Firstly, the influence of the ratios of

MOEs perpendicular to the beam’s axis (E90,LVL/E90,GLT)

is discussed. Secondly, the effect of the relative thickness

of the LVL (tLVL/w), and thirdly the impact produced by

the different moment-to-shear-force ratios is revealed.

All the results were analyzed for corner 2 (see Figure 1),

as in this region the tensile stresses, induced by the interac-

tion of moment and shear-force, have their maximum ef-

fect. Note: on corner 1 the effect of the moment produces

compressive stresses perpendicular to the grain and, thus,

reduces the level of the vertical tensile force Ft,90 com-

puted at this location.

5.1 EFFECT OF THE ELASTIC MODULUS PER-

PENDICULAR TO THE BEAM AXIS OF THE

LVL

The effect of the MOE perpendicular to the main axis is

presented in Figure 5 for tLVL = 35 mm (tLVL/w= 0.14)

and a moment-to-shear-force ratio M/V= 1.53 m. The

horizontal x-axis shows the different MOE ratios investi-

gated

βE=E90,LVL

E90,GLT

,(2)

whilst the left vertical y-axis depicts the percentage of

the force Ft,90 (bars) being taken up by the LVL element

(ηFE). On the right vertical y-axis (orange dots) the val-

ues according to Eq. (3) are shown, which represent the

LVL stiffness ratio vs. the total stiffness perpendicular to

the grain:

η90,LVL =E90,LVL ·tLVL

PiE90,i·ti

(3)

Equation (3), representing a load sharing according to an

ideal parallel spring/stiffness system, was chosen as a ref-

erence, since this is the direction in which the computed

Ft,90 force is acting. In this sense, it should in theory re-

flect the behavior of the redistribution of stresses in the re-

gion of stress concentrations perpendicular to the beam’s

principal axis.

1.0

3.0

5.0

7.0

9.0

11.0

13.0

15.0

βE=E90,LVL/E90,GLT [-]

0

10

20

30

40

ηFE,LVL =Ft,90,LVL/Ft,90,tot [%]

14.5

23.9 28.2 30.9 32.8 34.4 35.7 36.8

tLVL/w = 0.14

M/V = 1.53 [m]

0

20

40

60

η90, acc. to Eq. (2) [%]

14.0

32.8

44.9 53.3 59.4 64.2 67.9 70.9

Figure 5: Tensile force Ft,90 taken by the LVL as a func-

tion of the ratio of MOEs perpendicular to the beam’s main

axis E90,LVL/E90,GLT, for constant LVL/beam thickness ratio and

moment-to-shear-force ratio. The bars (corresponding to the left

y-axis) show the FE-results, while the orange dots (right y-axis)

show the values according to Eq. (3).

It can be observed from Figure 5, that the FE-results

follow in principle a similar evolution as the one obtained

with Eq. (3). However, only the scale differs, which indi-

cates that the stresses do not have enough space to redis-

tribute according to an ideal parallel system assumption.

Coming from a rather bending-dominated cross-section,

the internal forces are internally distributed based on the

stiffnesses parallel to the axis of the beam, and only in

a very close proximity of the corner they start to redis-

tribute according to the stiffnesses perpendicular to the

main axis. In this sense, the real load sharing is composed

of a weighting of η90 on the one hand, and the theoretical

load sharing parallel to the grain, η0on the other hand, (as

proposed in [6])

η0,LVL =E0,LVL ·tLVL

PiE0,i·ti

,(4)

which means

η90,LVL =f(η90,η0).(5)

The discussed behavior is observed in an exact manner

for all the thicknesses and M/Vratios investigated. For

this reason, the presentation of a single case is sufficient

for the purpose of a general understanding. It is of impor-

tance to notice, that for the base case (E90,LVL/E90,GLT =

1), the FE-results conform very closely to the theoretical

values, which should be equal to the thickness ratio. This

serves as a verification of the FE-model and the applied

post-processing methodology.

5.2 EFFECT OF REINFORCEMENT THICKNESS

The influence produced by the variation of the thick-

ness of the LVL layer is depicted in Figure 6 for a con-

stant MOE ratio E90,LVL/E90,GLT = 11 and a constant

M/Vratio of 1.53 m, i.e. equal to M/Vin Figure 5. In

the figure, the x-axis represents the different thickness ra-

tios βt=tLVL/w, whilst in the left and right y-axis the

same variables as in Figure 5 are presented (percentage

of Ft,90 in the LVL and η90 acc. to Eq. (3), respectively).

The load sharing behavior obtained with the FE-model fol-

lows an almost perfectly linear relationship with increas-

ing thickness ratios (R2= 0.999 for the observed case;

compare also with Figure 7). This does not match with the

predictions according to Eq. (3), which predicts a signifi-

cantly higher nonlinear force relationship as for the previ-

ous case.

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

βt=tLVL/w [-]

0

10

20

30

40

ηFE,LVL =Ft,90,LVL/Ft,90,tot [%]

22.0 25.5 28.7 31.6 34.4 37.0 39.6 42.1

E90,LVL/E90,GLT = 11.0

M/V = 1.53 [m]

0

20

40

60

η90, acc. to Eq. (2) [%]

Figure 6: Tensile force Ft,90 taken by the LVL as a function of the

relative thickness of the LVL plate. The bars (corresponding to

the left y-axis) show the FE-results, while the orange dots (right

y-axis) show the theoretical values according to Eq. (3).

An analysis of the configurations with different MOE

ratios shows that this linear relationship is present in all

the studied cases. Figure 7 shows the different percent-

ages of total Ft,90 in the LVL for the stiffness ratios (ηFE)

used in the parametric study, and makes the mentioned lin-

earity evident by showing the R2values. This effect is ex-

pressed for all the ratios M/Vas well, reason for which the

presentation of a single case is deemed sufficient.

The phenomenon observed here can be explained by the

fact that, added to the previous mentioned effects, the LVL

layer has a limited area of influence to its sides regard-

ing the redirection of load/stresses on the width direction

(z-direction), which does not evolve at the same rate as its

thickness. This can be observed in Figure 8, where the

stresses perpendicular to the beam’s axis, σ90, are plotted

vs. the width, i.e. z-direction, at an angle of 45°. The dis-

tance of influence of the LVL layer is a nonlinear function,

and adds up to the deviation from the analytical Eq. (4).

Additionally, it can be seen in Figure 8 that the stresses σ90

exhibit a large variation within the thickness of the LVL,

having a peak at the surface in contact with the GLT, then

decreasing parabolically to the mid-thick of the plate. A

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

tLVL/w [-]

0

10

20

30

40

Ft,90,LVL/Ft,90,tot [%]

M/V = 1.53 [m] E90,LVL/E90,GLT

1.0; R2=1.000

3.0; R2=0.999

5.0; R2=0.999

7.0; R2=0.998

9.0; R2=0.998

11.0; R2=0.998

13.0; R2=0.999

15.0; R2=0.999

Figure 7: Tensile force Ft,90 taken by the LVL as a function of

the thickness ratio of the LVL, for different MOE ratios. The lin-

ear regression for the MOE ratios are represented by the dashed

lines and their R2values are displayed on the right legend.

0 50 100 150 200 250

z-direction [mm]

0.25

0.50

0.75

1.00

1.25

1.50

Stresses perp. to main

axis σ90 [N/mm2]

tLVL/w =0.06

tLVL/w =0.12

tLVL/w =0.18

E90,LVL/E90,GLT =11.0

Figure 8: Stresses perpendicular to the main axis of the beam,

obtained under 45°at corner 2 for three different ratios LVL to

beam widths. The presented path crosses the three different lay-

ers (GLT-LVL-GLT). The effect of the thickness of the LVL on the

GLT stresses is depicted.

detailed investigation of these two effects are outside of

the scope of this paper, but the respective influences will

be taken into account by means of a coefficient, when try-

ing to obtain an analytical equation for the load sharing of

the system.

5.3 EFFECT OF THE MOMENT-TO-SHEAR-

FORCE RATIO M/V

A study of the parameters “moment-to-shear-force ra-

tio” (M/V) is relevant, since the internal forces in the re-

gion containing the hole are expected to have an effect

in the mentioned weighting of the load shares η90 and

η0. Figure 9 presents the FE-results for a constant thick-

ness ratio tLVL/w= 0.14. The x-axis shows the different

MOE ratios, whilst the y-axis presents the percentage of

the force Ft,90 in the LVL. An interesting behavior is ob-

served when comparing the effect of the ratio M/Vfor dif-

ferent MOE ratios: while for βE< 5 a higher M/Vratio

produces a decrease of the tensile force taken by the LVL

(ηFE), the opposite happens for βE> 5, i.e. ηFE grows

with higher moment-to-shear-force ratios.

This phenomenon can be further analyzed by perform-

ing a linear regression on each set of results with different

MOE ratios. This is, for every group of ratios M/Vcorre-

sponding to the same thickness and MOE ratio (e.g. first

four columns of the Figure 9) the slope is computed. Since

the percentage of the tensile force Ft,90 that is taken by the

LVL was defined as

ηFE,LVL =Ft,90,LVL

Ft,90,tot

,(6)

the slope needed is the partial derivative of ηFE,LVL with

respect to (M/V), i.e.

θ=∂ηFE,LVL

∂(M/V).(7)

Figure 10 presents the values of the slope θ, normalized

by the MOE ratio βE, for different thickness ratios βt, as a

function of the MOE ratio. This figure confirms the initial

observation regarding the effect of the moment-to-shear-

force ratio, since a negative slope is observed for all the

thickness ratios when βE< 5 (shaded area). The second,

and more relevant information, consists in the fact that,

the curve, i.e. the section force ratio effect on ηFE,LVL , ob-

tained for all the configurations with βE> 5 behave in a

similar manner.

5.4 ANALYTICAL ASSESSMENT OF THE LOAD

SHARING RATIO

With the gathered information from the previous sec-

tions, an analytical model for the computation of the load

share between LVL and GLT of the tensile force Ft,90 will

be presented. In essence, the proposed model considers

the determination of an effective thickness, teff (Eq. (8)),

and an effective MOE perpendicular to the grain for the

LVL, E90,eff (Eq. (9)), which are then used to compute the

values η90 and η0according to Eqs. (3) and (4), respec-

tively. (Then: E90,LVL =E90,eff,LVL and tLVL =teff,LVL.)

These two values are weighted (parameter p3(see Eq. 10))

and then multiplied by a factor dependent on the M/Vratio

(Eq. (10)). The model contains four parameters p1to p4

to be derived from an optimization procedure (see below),

and takes the following form:

1.0 3.0 5.0 7.0 9.0 11.0 13.0 15.0

βE=E90,LVL/E90,GLT [-]

0

10

20

30

ηFE,LVL =Ft,90,LVL/Ft,90,tot [%]

tLVL/w = 0.14

M/V [m]

1.53

4.53

7.53

10.53

Figure 9: Tensile force Ft,90 taken by the LVL as function

of the ratio of MOEs perpendicular to the axis of the beam

E90,LVL/E90,GLT. The force ratios are specified for different

moment-to-shear-force ratios M/V.

1 3 5 7 9 11 13 15

βE=E90,LVL/E90,GLT [-]

−0.100

−0.075

−0.050

−0.025

0.000

∂ηFE,LVL

∂(M/V )·E90,GLT

E90,LVL

tLVL/w

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Figure 10: Dependency of Ft,90 share on different ratios M/V

(θ), normalized by the MOE ratio E90,LVL /E90,GLT, plotted

against the stiffness ratio. All the studied thickness ratios are

presented as well.

teff =(tLVL)p1(8)

E90,eff =E90,LVL ·(βE)p2(9)

ηanalyt =[η90 ·p3+η0·(1 – p3)]·γM(10)

γM=M

V

1

βEp4

.(11)

It is important to notice that in the computation of η90

and η0only the variables corresponding to the LVL have

to be changed, i.e. thickness and E90,GLT remain the same.

In addition, it has to be noticed that the unit of the moment-

to-shear-force is in [m].

15 20 25 30 35 40 45

Ft,90,LVL,FE/Ft,90,tot [%]

20

30

40

Ft,90,LVL,analyt/Ft,90,tot [%]

R2= 0.9939

M/V

1.53 [m]

4.53 [m]

7.53 [m]

10.53 [m]

Figure 11: Linear regression (dashed line) of all the analyzed

data used for the calibration of the model. The individual data

for the FE-results and analytical model are shown as scattered

dots, differenced by color for their moment-to-shear-force ratio.

The model (ηanalyt) was calibrated with the data ob-

tained from the parametric analysis (ηFE,LVL), by means

of an optimization process based on the least-square-roots

method. To this end, a function was written using Python

and its scientific library SciPy [10], which contains the

needed optimization algorithms. For the calibration pro-

cess, only the results for MOE ratios βE≥5were consid-

ered, as it was shown that for the lower ratios the effect of

the moment-to-shear-force ratio has the contrary effect on

6%

8%

10%

12%

14%

16%

18%

20%

0

10

20

30

40

Ft,90,LVL/Ft,90,tot [%]

M/V =1.53

a)

6%

8%

10%

12%

14%

16%

18%

20%

0

10

20

30

40

M/V =4.53

b)

Analytical

E90,LVL/E90,GLT

5.0

7.0

9.0

11.0

13.0

15.0

6%

8%

10%

12%

14%

16%

18%

20%

tLVL/w

0

10

20

30

40

Ft,90,LVL/Ft,90,tot [%]

M/V =7.53

c)

6%

8%

10%

12%

14%

16%

18%

20%

tLVL/w

0

10

20

30

40

M/V =10.53

d)

FE-results

E90,LVL/E90,GLT

5.0

7.0

9.0

11.0

13.0

15.0

Figure 12: Comparison of the FE-results (ηFE,LVL) and values obtained with the calibrated analytical model (ηanalyt ) regarding the

Ft,90,LVL load ratio. The subplots (a)-(d) relate to different moment-to-shear-force ratio.

the load sharing, and is therefore not trivial to model with

an analytical equation. Nevertheless, the authors are of

the opinion that for the investigated use case, lower ratios

of MOEs perpendicular to the beam axis are of little rel-

evance, since a reinforcement of the cross-section in this

direction requires a relatively high MOE value.

Table 2: Parameters obtained through the optimization process

for the analytical model

p1p2p3p4

1.021 0.135 0.346 0.00647

The results from this process are presented in Figure. 11

and 12, and the obtained parameters p1,p2,p3and p4are

given in Table 2. Figure 12 shows the percentage of the

force Ft,90 taken by the LVL, computed both by means

of the FE-results (bars) and by the calibrated analytical

model (dots). It can be observed that for all the studied

cases a satisfactory agreement between the FE-results and

analytical model is achieved. In other words, the proposed

model is adequate to represent the load sharing of the ten-

sile force Ft,90 between LVL and GLT.

Figure 11 depicts the results of a linear regression be-

tween all the FE-obtained results for the load sharing and

the corresponding analytical ones. From the figure, it can

be seen that a good correlation between both values is ob-

served, which is represented by an R2value of 0.994. Nev-

ertheless, the fit is not perfect, and some values (mostly

for the extremes of the thickness ratios analyzed) express

some differences with the FE-results (up to 11 %, but nor-

mally moving around 1 %).

6 CONCLUSIONS

The parametric study of the GLT-LVL composite shows

that the load share for the vertical tensile force Ft,90 is

a non-trivial value to obtain. The analyzed parameters

showed that

• a load sharing purely based in the stiffness ratios per-

pendicular to the grain, η90, does not fully explain the

numerical results. Due to the relatively short distance

ahead of the hole, in which the stresses are redis-

tributed to produce the tensile stress concentrations

at the corners, the redistribution according to the the-

ory is only partially fulfilled. Moreover, the real load

share is bound to a weighted combination of η90 and

η0.

• A change in the ratios of elastic modulus perpendic-

ular to the main axis of the beam produces a similar-

shaped curve as the theoretical η90 does, however, a

difference in scale is observed. However, a change in

the thickness ratio is characterized by a rather linear

behavior, which is not explained by the theoretical

η90. The most relevant reason is the decaying influ-

ence of the LVL on the GLT with increasing distance

in the width direction; also, there is a large variation

of the stresses within the LVL, which seems to get

stronger with thicker cross-sections.

• The moment-to-shear-force ratio has a very complex

influence in the load sharing of the tensile force Ft,90,

producing a decrease in the load sharing for MOE

ratios smaller than 5, and increasing the same value

for MOE ratios larger than 5. Nevertheless, since

the expected use as a reinforcement requires relative

high MOE ratios, this effect can be ignored and as-

sume that a higher M/Vratio will produce higher load

shares.

• Based on the observations made during the analysis

of the results, an analytical model for the computa-

tion of the loading share was developed. The model

considers modifications for the thickness and MOE

values of the LVL, which are used to compute the the-

oretical load shares η90 and η0; after this, both values

are weighted and multiplied by a factor dependent on

the moment-to-shear-force.

• The calibrated model is able to reproduce the finite

element results in a good manner, exhibiting relative

small errors of about 1%.

• The model parameters p1–p4were calibrated to FE

results for a specific rectangular hole configuration,

with a hole-to-depth ratio of 0.4 and a hole aspect

ratio (length/depth) of 2. Further hole configurations

shall be studied to assure the generality of the chosen

approach.

REFERENCES

[1] DIN EN 1995-1-1/NA, National Annex – Nation-

ally determined parameters – Eurocode 5: Design

of timber structures – Part 1-1: General – Common

rules and rules for buildings, German Institute for

Standardization, Berlin, Germany, 2013.

[2] S. Aicher and C. Tapia, “Glulam with laterally re-

inforced rectangular holes,” in World Conference

on Timber Engineering, Auckland, New Zealand,

2012.

[3] M. Danzer, P. Dietsch, and S. Winter, “Reinforce-

ment of round holes in glulam beams arranged ec-

centrically or in groups,” in CD-ROM Proceed-

ings of the World Conference on Timber Engineer-

ing (WCTE 2016), J. Eberhardsteiner, W. Winter,

A. Fadai, and M. Pöll, Eds., Vienna, Austria: Vi-

enna University of Technology,Austria, 2016, :

978-3-903039-00-1.

[4] C. Tapia and S. Aicher, “Holes in glulam – Orienta-

tion and design of internal reinforcements,” in CD-

ROM Proceedings of the World Conference on Tim-

ber Engineering (WCTE 2016), J. Eberhardsteiner,

W. Winter, A. Fadai, and M. Pöll, Eds., Vienna,

Austria: Vienna University of Technology, Austria,

2016, : 978-3-903039-00-1.

[5] T. Butler, “International house sydney,” in Pro-

ceedings (Part II) 22nd. Wood Construction Forum

(IHF 2016), Garmisch, Germany, 2016, pp. 35–45.

[6] S. Aicher and C. Tapia, “Novel internally LVL-

reinforced glued laminated timber beams with

large holes,” Construction and Building Materials,

vol. 169, pp. 662–677, 2018, : 0950-0618. :

10.1016/j.conbuildmat.2018.02.178.

[7] S. Aicher and L. Höfflin, “Runde Durchbrüche in

Biegeträgern aus Brettschichtholz. Teil 1: Berech-

nung,” Bautechnik, vol. 78, no. 10, pp. 706–715,

2001.

[8] C. Tapia and S. Aicher, “Improved design equations

for the resultant tensile forces in glulam beams with

holes,” in International Network on Timber En-

gineering Research — Meeting 50, Kyoto, Japan,

2017.

[9] Abaqus v2017, Dassault Systèmes Simulia Corp,

Johnston, RI, USA, 2017.

[10] E. Jones, T. Oliphant, P. Peterson, et al.,SciPy:

Open source scientific tools for Python, [Online;

accessed 2018-04-25], 2001–. [Online]. Available:

http://www.scipy.org/.