Capacity models for timber under compression perpendicular to grain with screw reinforcement

This paper compares the performance of probabilistic and deterministic capacity models for reinforced timber members under compression perpendicular to the grain. A database collecting approximately 60 test results has been compiled by reviewing research papers and master’s and doctoral theses from the past twenty years. The capacity model proposed for the next generation of Eurocodes assesses the capacity as the minimum between the values associated with two failure modes, one at the contact plate and one at the screw tips. The main drawbacks of the model are the excessive elaborateness, given its limitation in accuracy and the fallacy in predicting the observed failure modes. In detail, the failure by the screw tips seldom occurs, although it was expected in more than half of the selected specimens. The authors attempted to simplify the capacity equation by proposing a generalized expression corresponding to the failure mode at the contact plate, corrected by a factor including the effects of load and screw arrangement and geometric details of the specimen. A deterministic mechanical model obtained by multiplying the timber strength by the contact area with a given coefficient performs better than the Eurocode model, which attempts to include the effect of load diffusion (R2≈0.27\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2\approx 0.27$$\end{document}). A constant factor equal to 2 yields a suitable fitting (R2≈0.76\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2\approx 0.76$$\end{document}). The best performance is achieved with a four-term polynomial, with adimensional addends, leading to an optimum fitting (R2≈0.82\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^2\approx 0.82$$\end{document}).


List of symbols and notations
• A 1 and A 2 : CPG capacities associated with two failure mechanisms: the first by the contact area of the load and the second by the screw tips. • A 11 and A *

11
: Timber contribution to the capacity under CPG. A 11 is calculated with Eq. 5 considering an effec-tive spreading length ( l ef ,1 ). A *

11
, calculated using Eq. 20, neglects any sort of load diffusion. • A 12 : Screw contribution to the capacity under CPG, calculated with Eq. 5. • A ef : Effective contact area. Due to the contribution of adjacent fibres, an increase of 30 mm of the contact length on each side of the contact area is recommended. • a 1 : Spacing between screw or rod reinforcement in the direction parallel to the grain. • a 3,c : Distance between the screw closest to the member edge and the member end in the direction parallel to the grain. • : Angle between screw axis and grain direction of the wood member. • B 1 : CPG capacity of the un-reinforced timber specimen.  Gardoni et al. (2002), it is the capacity, where x are the measurable capacity variables, and = { , } are unknown model parameters with modeling error . • d: Outer thread diameter of the screw; • d 1 : Inner thread diameter of the screw obtained as 0.7d; • E s : Young's modulus of steel equals to 210GPa. The definition of f w,k , reported in Eq. 8, follows Eq.6 in Tab.11.2 of the Eurocode 5 draft (2022). • F c,k : Characteristic axial capacity of the screw. • F exp : Experimental capacity under CPG. • f y,k : Characteristic yielding strength of the steel. • 1 and 2 : Correction factors (Eqs. 25, 26) used in the two probabilistic capacity models. • 3 : Constant coefficient, found equal to 1.2, used in the proposed deterministic capacity model. • h ef : Effective height. For members on continuous support loaded by concentrated forces perpendicular to the grain, the load arrangement factor should be calculated with an effective spreading length of the compressive stresses.
The following equations should be used for continuous and discrete supports respectively: h ef = min{h;280mm} , h ef = min{0.4 ⋅ h;140mm}. • I s = ⋅ d 4 1 64 : Moment of inertia of the screws cross-sections. • k c,90 : load distribution factor and equal to √ l ef ∕l c for unreinforced members. • k = 0.5 1 + 0.49 ̄k − 0.2 +̄2 k : Buckling coefficient for buckling. • k p for un-reinforced compression perpendicular to grain: according to prEN1995 (2021), it accounts for the material behaviour and the degree of deformation perpendicular to the grain. The factor accounts for the increased stiffness when the deformation increases (Leijten et al. 2012), see Table 1.
• k pr parameter adopted for screws reinforced compression perpendicular to the grain (this parameter was originally indicated as k c,90 in EN 1995-1-1): according to prEN1995-1-1, it takes into account the material behaviour and the degree of deformation perpendicular to the grain. The value of k pr according to Blass et al. (2004) and the existing version of (EN1995 2010) can be assumed as 1.75 for glulam members on discrete supports loaded by distributed loads and/or by concentrated loads at a clear distance from the support l s larger or equal to 2h, or 1.5 in case of glulam member on continuous support (sill configuration). For the other cases, the value of k pr can be assumed equal to 1. The tested specimens correspond to the left scenario in Fig. 1.
• k mat = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1.0, n p = 1 1.06, n p ≥ 2 1.10, n p ≥ 3 1.13, n p ≥ 5 1.15, n p ≥ 7 : Material parameter for the number of lamination, where n p is the number of lamination. Parameter for screws and rods with wood-screw thread, where is the angle between the fastener axis and the direction of the grain.

Introduction
Timber is a highly anisotropic material: the direction of the grain influences its mechanical properties (Porteous and Kermani 2013). In the direction parallel to the grain, timber is strong and stiff. On the contrary, in the direction perpendicular to the grain, the strength and stiffness are relatively lower (Porteous and Kermani 2013;Augustin et al. 2006). The fibres in timber are tubular and elongated. Therefore, the compression perpendicular to grain (CPG) failure leads to a collapse of the cellular tubes, manifesting as failure along the tubular layers (Persson 2000; Swedish Wood Stockholm, Sweden 2016), see Fig. 2. Beams, trusses, and supports are typical design situations where CPG can occur (Hassan et al. 2014). Failure due to CPG does not generally lead to collapse. The failure in CPG is potentially a concern in the serviceability limit state (SLS). Yet, in Eurocode 5, CPG is regarded as an ultimate limit state (ULS) problem to avoid load failure of structural systems (Forening, Norske Limtreprodusenters Norway 2015). The failure mode due to the CPG occurs on the contact area of the load; see Fig. 3: as the compressive load increases, the fibres buckle. This compression will transmit to neighbouring fibres as a chain reaction. Higher loads can be achieved with higher deformation levels and dispersion lengths (Forening, Norske Limtreprodusenters Norway 2015).
The strength under CPG can be improved by using suitable reinforcements. Past and recent research (Bejtka 2005;Dietsch and Brandner 2015;Harte et al. 2015; De Santis and Fragiacomo 2021) prove that self-tapping screws are the most effective reinforcement system for timber under CPG (Kreuzinger 1999(Kreuzinger , 2001. Screws can also be used to strengthen notches, holes in beams (Aicher andHöfflin 2002, 2003;Kolb 2008;Aicher et al. 2007;Aicher and Höfflin 2009;Aicher 2011), or against tension stress perpendicular to the grain in curved or pitched cambered beams (Eurocode 2010; Tomasi et al. 2010).
Other techniques have been identified in improving the CPG of timber members. Nonetheless, reinforcement, gluedin rods or steel screws are not prescribed in Code (2005). For practical applications, screws recommended for the reinforcement of timber perpendicular to the grain are regulated by European Technical Approvals (ETA 2019), which are supplier-specific (O'Ceallaigh and Harte 2019).
Apart from screws, dowels also proved to be effective reinforcement systems of timber under CPG. Madsen et al. (2000) successfully installed steel dowels perpendicular to grain through the full height of glulam beams. Ed and Hasselqvist (2011) performed similar tests using birch dowels to enhance the compressive strength of glulam beams. Still, using self-tapping screws is the most favoured method for timber reinforcement due to several advantages. They are easy to install and may be used in various applications: to prevent splitting perpendicular to the grain, enhance dowel embedment strength, or enhance strength perpendicular-tograin (Bejtka and Blaß 2004). Screw reinforcements can be used for in-situ and off-site manufactured frames/panels (Kildashti et al. 2021;Alinoori et al. 2020). Bejtka (2005) was among the first to study the effect of reinforcement in  CPG. Pampanin (2013) used self-tapping screws to reinforce laminated veneer lumber (LVL) beams, while Dietsch et al. (2019) performed similar tests on glulam beams.
In the last decade, reinforcement under CPG is also becoming relevant for Cross Laminated Timber (CLT) elements (Mestek 2011). Serrano and Enquist (2010) tested 3-ply CLT elements and found the compressive strength dependent on the bearing area, load orientation to the surface grain, and loading positions. Bogensperger et al. (2011) compared data from glulam and CLT specimens and determined strength under CPG. Lately, Brandner (2018) published a state-of-the-art review on CLT loaded in compression perpendicular-to-plane, where a load-spreading model is proposed (van der Put 2012). Recently, Rothoblaas developed the SPIDER system to enhance the CPG strength of point-supported CLT floor systems using a steel element (Maurer and Maderebner 2021).
Steel is not the sole solution for reinforcing timber under CPG. However, there is growing mainstream research on using timber-based elements as reinforcement. In detail, some research papers propose using hardwood or densified wood dowels as reinforcement perpendicular to the grain (Conway et al. 2020(Conway et al. , 2021Moerman et al. 2021). This solution showed significant potential in improving the load-carrying capacity of timber perpendicular to the grain (Crocetti et al. 2012;Ed and Hasselqvist 2011), despite being outperformed by steel screw alternatives (Orlando et al. 2019).
The use of screw reinforcement for CPG is now an acknowledged solution. The draft of the next generation of Eurocodes, including a section on this topic, confirms this fact. Nonetheless, the knowledge of the actual effect of screws on the timber strength under CPG is still embryonal. The investigation of the CPG without reinforcement has a more extended chronology. Still, Leijten (2016) dashingly states that current CPG models are unreliable, and there is only one accurate model based on yield slip-line theory to be a candidate for future building design codes. He reviewed several past studies, partially itemized below, on CPG tests. They were all discarded due to misconceptions, inaccuracy or lack of adequate methodological rigour; see the studies by Kollmann et al. (2012); Graf (1921); Gehri (1997); Hübner (2013); Gaber (1940);Frey-Wyssling and Stüssi (1948);Rothmund (1944). In North America, Basta (2005) carried out CPG tests according to ASTM-D-143 (ASTM 1991), including the wood species, moisture, and annual ring orientation effects. The research by Basta (2005) was considered irrelevant since the definition of the CPG values obtained with this method was based on the proportional limit. In contrast, it is now based on a 1% deformation limit according to the standard EN 408 (EN408 2012). Hoffmeyer et al. (2000) report tests on 74 sawn timber specimens and 120 glued laminated specimens having a mean CPG strength of 2.9 N/mm 2 . Hoffmeyer et al. (2000) found that CPG strength does not change significantly with the specimen dimensions.
The first studies on CPG without reinforcement were carried out by Suenson (1938a). The tests highlighted that the CPG capacity depends on the loading area and the length of the unloaded part (Leijten 2016). Multiple studies confirmed these aspects in the following years (Reichegger 2004). Leijten (2016) compared different design models for CPG without reinforcement and presented nine load cases, shown in Fig. 4. The load cases distinguish different design configurations, except for load case A, defined by EN 408. In this paper, the authors will adopt the designations in Fig. 4 to identify the different load cases.
If, according to Madsen et al. (1982), CPG did not receive sufficient attention from the scientific community, CPG with reinforcement received even less (Thelandersson and Mårtensson 1997;Kevarinmäki 1992). No paper attempts to review and compare existing capacity models for CPG with reinforcement. Additionally, the deficiency of organic research in this area is further fed by exiguous experimental data. This paper attempts to fill the mentioned gap in knowledge by investigating the following aspects: • Collecting a database of tests of timber specimens under CPG, which could be used in future studies for validation and calibration purposes. The database can be downloaded as supplementary electronic material. • Critical review of the current CPG capacity model, included in the draft of the next generation of Eurocodes. • Comparison between eight probabilistic and deterministic capacity models to verify if the assumptions of the Eurocode model (especially those on the load spreading) have a solid foundation. • Improvement and possible simplification of the Eurocode model, given the optimal fitting of the simplistic mechanical model and the probabilistic ones. This paper recognizes two candidate models for possible future code developments.
The paper has the following organization. Sec.3 presents the mechanical model and some literature background. Sec.4 shows the four studies used for the database compilation, further discussed in Sec.5, where a sensitivity analysis is carried out. Finally, Sec.6 compares several mechanical models by highlighting each formulation's possible pros and cons.

Standard proposal and literature background
The design model in the current EN 1995-1-1 is presented as follows (EN1995 2010): where c,90,d is the stress level given by the effective contact area. The factor k c,90 accounts for the material behaviour and the deformation perpendicular to the grain. Generally, this factor is considered equal to 1.0. Still, for specific support conditions (when higher deformation perpendicular to grain can be considered acceptable), the CPG capacity can be increased (the different support conditions are further specified in Eurocode 5 (EN1995 2010)).
In the new standard proposal (prEN1995 2021), a factor k p is introduced, which also accounts for the material behaviour and the deformation perpendicular to the grain. In the new standard proposal, a coefficient k c,90 is introduced, but with a different physical meaning related to the effects of the load arrangement. To avoid confusion when discussing the different standard proposals, the factor k c,90 of the current EN 1995-1-1 version is indicated as k pr (to be distinguished from k p , which in the new standard proposal prEN 1995 is proposed only for the un-reinforced . The effective contact area, A ef , accounts for the contribution of adjacent fibres. A 30mm increase in the contact length on both sides of the contact area is recommended. The effective length considers the load distribution of the CPG stress, where the stress distributes to the unloaded timber parts. The synoptic table in Table 3 shown below resumes the proposed design models for un-reinforced and reinforced timber members under CPG. A detailed definition of the notation is given at the beginning of the paper (Fig. 5). Eq. 3 shows the design model of the current Eurocode 5 proposal for non-reinforced specimens (EN1995 2010).
The new standard draft (prEN1995 2021) enclosed a design equation (Eq. 5) for reinforced timber members under CPG. The bearing capacity can be primarily increased by reinforcing the member with fully threaded self-tapping screws. The screws can vary in diameter, head, and length. Threads along the whole length reduce the risk of pushing in due to the bond between the screws and timber. The design model represents an evolution of the one in Eq. 3 for nonreinforced members. Three failure modes may occur: failure due to the withdrawal of the screw, failure due to buckling of the screw, or failure of timber, as confirmed by the investigations by Bejtka and Blaß at the University of Karlsruhe (Bejtka and Blaß 2004).  Leijten (2016). A-E define symmetric loading configurations. F-J correspond to situations where the loading is eccentric to the reaction force.
The first failure mode (Pushing-in of the screw) corresponds to the pushing-in capacity of the screws (withdrawal capacity F w,k ). Exceeding the bearing strength leads to penetration of the screws into the timber member. Numerous studies at the University of Karlsruhe have shown that the capacity to push in is equal to the withdrawal capacity. This failure mode occurs mainly using short screws (Bejtka 2005). The second failure mode corresponds to the buckling of the screws. This failure mode occurs mainly for long and slender screws. The third failure mode corresponds to timber failure at the screw tips. The feature of this failure mode is lateral expansion perpendicular to the grain, resulting in cracks at the screw tip. Bejtka (2005) observed this failure mode with small loading areas and short screws. The maximum stress in the timber specimens by the screw tip depends on the effective dispersion length and width. The effective length depends on the size of the screw, the screw arrangement, and the design situation; see  (2) Note k pr is originally indicated as k c,90 in EN 1995-1-1 as in the Eq. 1. The new term is here adopted to be coherent with the new proposal.

Characteristic strength of non-reinforced members under CPG
Characteristic strength of reinforced members under CPG (prEN1995 2021) for intermediate support can be assumed equal to one for estimating the mean value of the buckling resistance effective length, l ef ,2 in the third failure mode. The study investigated the effect of self-tapping screws as singleside reinforcement in timber (Dietsch et al. 2019). The study showed that the stresses varied extensively along the screw length, with a stress concentration by the tips of the screws, as shown in Fig. 6. However, the stress concentration is compensated for by stress redistribution due to the elastic-plastic behaviour of timber. Figure 6 depicts the distribution of stresses in a horizontal plane along l ef ,2 with different screw groups (four, six, and ten). Figure 6 reveals that the stress increases by the screw tips. The study concludes with a recommendation for using screws with overlap to transmit the stress concentration efficiently. The first and second terms of Eq.3 correspond to the contributions by screws and timber, respectively. The withdrawal resistance, F w,k , the compression resistance, F c,k and the number of screws, n, provide an estimate of the characteristic strength of the screws. Conversely, the effective contact length, l ef ,1 , is increased by up to 30 mm, assuming the load will disperse at a 45 • angle. The effective length at the screw tip is defined by l ef ,2 , as depicted in Fig. 7.
PrEN 1995 is currently under development. Therefore, the reported equations represent an intermediate step toward a definite standard proposal. The synoptic table also provides the expressions for the withdrawal resistance, F w,k , and compression resistance, F c,k , of the screws. The authors adopted the following abbreviations to indicate the different contributions to the capacity. A 1 indicates the failure of the screw, where A 11 and A 12 separately express the contributions of timber and screws. A 2 represents the capacity related to the failure of the timber at the screw tip.

Database compilation
The database used to validate and calibrate capacity models for reinforced specimens under CPG with steel plates is based on the following experimental tests resumed in the appendix (Electronic Supplementary Material): (2004); • Tomasi et al. (2023). Bejtka (2005) wrote Volume 2 of the Karlsruhe reports on CPG with reinforcement using fully threaded screws. He investigated the failure mechanisms and the influence of the screws based on multiple laboratory tests. The table in the appendix reports the geometry of specimens, screws, experimental capacity and the relative capacity increment compared to the non-reinforced case. The test result revealed that the capacity under CPG can increase up to 329 % using  six screws. Furthermore, it is possible to improve the capacity further by using more screws. FE models in Ansys and experimental tests supported the development of a design model for reinforced members. The design model, known as the Model of Karlsruhe, is described in Eq. 10 (Bejtka 2005). Where: The definition of the parameters is detailed in the initial list of symbols and notation. The model is analogous to the one in prEN 1995, except for minor discrepancies. Additionally, Bejtka (2005) validated the model using both the characteristic and mean strength perpendicular to the grain, f c,90,k = 3.0 N/mm 2 and f c,90 = 5.0 N/mm 2 obtained from experimental tests. As expected, the predicted capacity is minor compared to the test results when using the characteristic value f c,90,k = 3.0 N/mm 2 . A good correlation is observed with f c,90 = 5.0 N/mm 2 .
Nilsson (2002) wrote a master thesis on the influence of reinforcement on CPG based on experimental tests. Multiple glulam specimens with the dimensions 500 × 90 × 315 mm (length x width x height) were tested using the screws SFS WT-T 8.2 with different lengths. She considered two load cases: Torx and load case B with either one, four, six, or  Fig. 8, affecting the magnitude and location of the lateral expansion. Nevertheless, the test specimen is not split to verify the failure mode of the screw. In this paper, the authors will use the sole experimental data corresponding to samples loaded by steel plates. The specimens loaded with timber plates are insufficient to develop a probabilistic capacity model. Reichegger (2004) investigated the effect of self-tapping screws for members under CPG. The analysis is based on experimental tests and comparisons with the design model of Karlsruhe. He used spruce Glulam with strength class GL 24h according to EN 1194, with compressive strength f c,90,k = 2.7 N/mm 2 . The test specimens had dimensions 400 × 120 × 200 mm and 600 × 120 × 400 mm (length × width × height). The tests resemble load case B, where the load is applied by steel and timber plates with dimensions 80 × 120 and 120 × 120 mm, respectively. He used two types of screws: SFS WT-T and SPAX-S, where the mutual spacing is based on the ETA (European Technical Assessment) recommendation. The compressive capacity of the non-reinforced and reinforced test specimens is obtained following EN 1193, similar to the procedure described in EN 408. Furthermore, the study compares the test result to the Karlsruhe model. With a steel plate, the capacity increases up to 33 % and 167 %. Conversely, using a timber plate leads to an increment equal to 3 % and 124 % using WT-T 6.5 × 130 mm and SPAX-S 8 × 200 mm screws, respectively. Tomasi et al. (2023) tested the accuracy of the design model for reinforced glulam members subjected to CPG, according to the new proposal of Eurocode 5 based on the Model of Karlsruhe in Eq. 10 (Bejtka 2005). Tomasi et al. (2023) present the experimental tests of multiple timber specimens with screw reinforcement under CPG by considering different geometry, load cases, and screw arrangement. Furthermore, the predicted capacity and failure modes are compared with the test results. The experimental results show that using threaded screws as reinforcement effectively increases the capacity of timber subjected to CPG. The experimental tests do not confirm the predictions regarding failure modes and capacities for all configurations.

Comparison with prEN 1995 model predictions
This section compares the design model of (prEN1995 2021) and the test results from past research. Nilsson (2002) and Reichegger (2004) transmitted the load by either a steel or timber plate, while Bejtka transferred the load with just a steel plate. The steel plate evenly distributes the force on the contact area and the screw heads. With a timber plate, the screws penetrate the timber plate, and the increase of capacity due to reinforcement may not occur (Bejtka 2005). As remarked above, since most of the tests are executed with steel plates, the investigation will neglect the CPG with timber plates for the load transfer.
The compiled database, to be downloaded as supplementary electronic material, compares the predicted capacity according to the two failure mechanisms ( A 1 and A 2 ) and the estimated force at 1% deformation. The authors could note the experimental failure mode in most specimens, missing in the Karlsruhe report (Bejtka 2005) and Nilsson (2002). The k pr assumed in the analyses is 1 for load case H, 1.5 for A, and 1 for B and C, following the definition in the initial list of notations. The additional parameters needed to calculate A 1 and A 2 , especially the geometrical parameters of the screw and plate arrangements, are reported in the appendix (Electronic Supplementary Material).
There is a significant deviation between the test results obtained by Bejtka and the predicted load-carrying capacity. He investigated two load configurations, indirect (H) and direct (C) load arrangements. The spacing between the load and the support is less than two times the height for load case H. Thus, k pr is considered equal to 1.0 for both cases. The observed failure mode is only reported in the preliminary tests. The failure mode for D_8a_6 is buckling, while the design model predicts the timber failure. Elsewhere, the failure modes agree with the predictions. In the tests by Bejtka, the highest capacity is achieved with six screws or four screws with a 10mm diameter. The significant variation in capacity depends on the different lengths and diameters of the screws. For most configurations, the A 1 is lower than the test results. The forces and predictions in load case H are generally lower than in case C. In H, the load dispersion is reduced considerably since the member is loaded near the edge.
In the tests by Nilsson (2002), the predicted failure modes follow A 2 for all configurations. Regrettably, Nilsson (2002) did not split the specimens to observe the failure mode. Additionally, the predicted capacity is consistently lower than the experimental values. Interestingly, the resistance for the test configuration with eight screws is lower than that with six. However, the recommended minimum spacing is not satisfied. This fact might increase the risk of splitting and result in a capacity reduction. The load situation is B, and A 2 is the estimated failure mode for all specimens.
There is a significant gap between A 1 and A 2 due to the k pr factor, assumed equal to 1.5 for A 1 , whilst for A 2 k pr is not considered.
In the study by Reichegger (2004), the specimens attained failure in the screws due to either buckling or withdrawal. The predicted and estimated failure modes are inconsistent since A 1 was consistently observed while A 2 was predicted. According to the design model, A 2 is the decisive capacity and is consistently lower than A 1 . The A 1 capacity agrees with the experimental observations. The capacity of the screws is the minimum between buckling and withdrawal resistance. The predicted failure mode of WT-T 6.5 x 130 and WT-T 8.2 x 160 is withdrawal, while buckling occurred. The remaining screws fail due to buckling due to their significant slenderness.
The test data by Tomasi et al. (2023) is the more comprehensive, including three load cases, A, B and C. The experimental results show that using threaded screws as reinforcement effectively increases the capacity of timber subjected to CPG. However, the tests did not confirm the predictions on failure modes and capacities for all configurations. The A 2 failure mode never occurs despite the model predicting its occurrence in more than half of the tested specimens. Conversely, A 1 estimates conform with the test results for load case B. The predicted capacity is consistently lower in load case C. This might depend on the k pr coefficient, assumed 1.5 and 1 for B and C, respectively. The overall predicted failure mode of A 1 is mainly confirmed, except for the shorter screws.

Sensitivity analysis
Before developing a probabilistic capacity model for reinforced members under CPG, a sensitivity analysis of the design model highlights the effect of variable dimensions on specimen capacity. This study and the previous experimental data will support a discussion on the limits of the current capacity model and the need for a more advanced proposal. In 2019, the CEN committee compared two models proposed in Eurocode 5 for non-reinforced members. The sensitivity study compared two approaches by Leijten and Blaß by varying the height and length of the specimen (Niebuhr and Sieder 2019). The current design model for CPG in prEN 1995 is a modified approach after Leijten. The author replicated the study in prEN 1995 for non-reinforced and reinforced members under load cases B and C, see Fig.4. Figure 9 displays load cases B and C used as reference configurations for the sensitivity analysis. The contact length is varied between 20 mm and 300 mm, while the height is between 20 mm and 1200 mm. The width and total length are set constant and equal to 140 mm and (2000 + l c ) mm, respectively. The characteristic properties of GL 30c are used in the calculations. Figure 10 plots the bearing capacity corresponding to 1.0 %, 2.5 %, and 10 % deformation. The design model considers a non-linear increase in capacity with increasing deformation. The increase is significantly higher between 2.5 % and 10 % deformation, compared to 1.0 % and 2.5 % deformation. For 1.0 % deformation, the material behaviour factor, k p , is equal to 1.0. For 2.5 % and 10 %, k p is equal to 1.4 and 2.1, respectively (prEN1995 2021). Figure 10 highlights the importance of the assumed limit deformation in the predicted resistance. The resistance is a highly conventional value, generally estimated at 1% deformation. The difference between the two cases is the load condition, which affects the load arrangement factor, k c,90 . In non-reinforced members, the load arrangement factor depends on the load dispersion due to the effective height and contact length. In discrete supports, the effective height is limited to 140 mm, while in continuous supports to 280 mm. Consequently, discrete supports have a lower k c,90 factor. In discrete supports, the beam will exhibit additional deformation due to bending. Figure 10 shows a significant increase in capacity for load case B until the 280mm height is reached. After this point, the resistance is constant, despite the increase in height. After that, the capacity increases linearly with the contact length, although the increment rate is higher for case B than C. For case C, the member's height is reduced by using 0.4 ⋅ H until the limit is reached. Fig. 9 Reference configuration of the non-reinforced members, with details on the geometry in load cases B and C. According to EN1995 (2010), k pr is assumed to be equal to 1.5 for case B and 1 for case C.

Fig. 10
Sensitivity analysis of the bearing capacity of the reference configuration in load cases B (a) and C (b). According to EN1995 (2010), k pr is assumed to be equal to 1.5 for case B and 1 for case C. Figure 11 shows the geometric details of the reference configurations in load cases B and C. In this subsection, the authors assess the sensitivity of the reinforced members to the screw length and diameter. The two variables have a decisive impact on capacity. In particular, the diameter and length affect A 1 , while A 2 is affected by the screw length. The screw length varies between 80mm to 500mm. Following the standards' recommendations, the diameter varies between 6mm and 12mm. Other parameters, such as the contact length, member height, screw distance, and the number of screws, are considered constant. The characteristic values of timber correspond to GL 30c. Figure 12 plots the resistance as the minimum between A 1 and A 2 . Load case B leads to higher resistance than C.

Reinforced specimens under CPG
The capacity increases linearly with the screw length up to a specific value in both cases. Within this region, the screw diameter does not affect the capacity. The capacity increases with larger diameters when the screw length exceeds a specific length. Figure 13 separates the contributions A 1 and A 2 .
A 2 does not change between the two load cases, increasing linearly with the length of the screw, while the screw diameter does not influence the capacity. When the screw length is small, the load-bearing capacity is driven by A 2 . Conversely, when the screw length increases, the spreading length will also do. Therefore, beyond a specific screw length, A 1 becomes the predicted failure mode. The curves corresponding to A 1 are similar, but the capacities are different for the two load cases. Estimating A 1 is more convoluted since the capacity depends on both timber and screw contributions. Fig. 11 Description of the reference case studies for reinforced members in load cases B and C. According to EN1995 (2010), k pr is assumed to be equal to 1.5 for case B and 1 for case C.

Fig. 12
Sensitivity analysis of the bearing capacity of the reinforced members for load cases B and C. According to EN1995 (2010), k pr is assumed to be equal to 1.5 for case B and 1 for case C. Figure 14 illustrates the capacity of one screw. The resistance of the screw is the minimum resistance between buckling and withdrawal. In Fig. 14a, F k shows the same trend as A 1 . As clarified above, the capacity stabilizes if the screw length reaches a specific level based on the screw diameter. Before this level is reached, the withdrawal capacity provides the minimum value. Subsequently, the buckling resistance gives the minimum. The buckling depends on the slenderness ratio of the screw, determined by the characteristic yield capacity of screws and the buckling load. The only parameter that varies in these formulas is the diameter. The buckling capacity is constant since it is not dependent on the screw length. In contrast, the withdrawal capacity depends on the screw length and diameter.

Justification for probabilistic models
The sensitivity analysis and the experimental results highlight the following possible limits of the current capacity model.
• The current capacity model estimates the capacity as the minimum between two failure modes. However, despite the model predicting the failure by the screw tips, the observed failure generally occurs by the contact area of the applied load. Additionally, the sensitivity analyses proved that the discrepancy between A 1 and A 2 reduces significantly if the minimum capacity is A 2 . The inaccuracy of the A 2 failure mode might depend on the k pr factor, affecting A 1 , which increases the gap Fig. 13 Sensitivity analysis of the bearing capacity of the reinforced members for load cases B and C, where both A 1 and A 2 are plotted. According to EN1995 (2010), k pr is assumed to be equal to 1.5 for case B and 1 for case C. 1 3 between A 1 and A 2 . The authors believe that the model can be simplified with a single capacity equation based solely on the first failure mode ( A 1 ), properly corrected to include the effect of the screw length on the timber contribution. • The model is deterministic. However, the availability of more experimental data might support the development of a probabilistic capacity model with a correction factor dependent on the geometric arrangement of the specimens. • The model might be too complex, given its level of accuracy. The experimental data showed that the failure mode of the screws generally agrees with its prediction. Therefore, it might be helpful to correct the timber contribution with a single correction factor, dependent on adimensional geometrical parameters, given the significant uncertainty of this term. • The differences between load cases can be expressed by synthetic geometric parameters rather than relying on the classification in Fig. 4. Avoiding such a classification and introducing a correction term, including the geometric feature characterizing the load configurations, might enhance the generality of the capacity model. Besides, the sensitivity analysis highlights a significant gap in the resistances for load cases B and C. However, the experimental predictions do not entirely confirm this gap. Therefore, a data-driven probabilistic capacity model might highlight the most significant parameters in the capacity model.
The following section will discuss a novel probabilistic capacity model for reinforced timber specimens under CPG obtained by calibrating a factor correcting the timber contribution characterized by higher uncertainty. The model will be simplified by replacing the effecting length with the contact area of the steel plate. Given the geometric limits by the standard, the correction factor will account for the uncertainty in the effective length and k pr by calibrating a suite of explanatory variables dependent on the geometric arrangement of the specimen.

Comparison between capacity models
This section compares the performance of eight capacity models, resumed in the following synoptic Table 4.
To properly compare experimental data with model predictions, the mean values of the mechanical parameters replaced the characteristic ones. The first two models labelled F 1 c,90,m and F 2 c,90,m , correspond to the first and the second failure mechanisms, known as A 1 and A 2 . The authors separately compare each model prediction with the data to understand whether the proposed A 1 expression performs better when used without evaluating the minimum between A 1 and A 2 ( F 3 c,90,m ). The fourth model, F 4 c,90,m , is the simplistic capacity model. The timber contribution is obtained by multiplying the contact area by the strength of the timber perpendicular to the grain without any coefficient. The 5th

Capacity model according to the EC5 proposal
4. Simplistic model for the 1st failure mode 5. Probabilistic model with adimensional explanatory functions and variance stabilizing transformation (P.M.1) 6. Probabilistic model with adimensional explanatory functions and variance stabilizing transformation (P.M.2) The variables with subscript m have replaced those with k to obtain the mean values. and the 6th models, labelled F 5 c,90,m and F 6 c,90,m , identify two probabilistic capacity models. The experimental data and existing capacity models prove that the higher uncertainty in the equation stands in the timber contribution. Several coefficients and arbitrary assumptions about the load diffusion affect this term, while the screw addend is generally considered as it is. Therefore, the authors calibrated a correction factor for the sole timber contribution as a function of a suite of explanatory variables. In the first probabilistic model (P.M.1), the experimental data were pre-processed with a variance stabilizing transformation. No variance stabilizing transformation is applied in the second one (P.M.2). The relevance of these models also depends on their generality. The multiplication factors, named 1 and 2 , model an adimensional expression, the ratio between the expected and estimated timber contribution. Therefore, the authors selected adimensional geometric functions as candidate explanatory functions. A step-wise model reduction is then implemented to reduce the model, thus reaching a trade-off between complexity and accuracy. Lastly, the authors compare two deterministic models where the timber contribution is amplified by factors 2 and 1.4, respectively. The first factor is obtained from an ordinary least squares estimation, while the second is based on the formulation for CPG without reinforcement, where 1.4 is adopted.
This section might help the scholar and the practitioner understand the pros and cons of existing capacity models. This research's principal limitation is the modest number of test data. Approximately 60 samples might not be adequate for statistics. Despite the constraints, the authors propose probabilistic capacity models, which may be subjected to further calibration if more experimental data is added to the collected database. For this reason, the authors did not implement a Bayesian calibration, which should be more appropriate when having more data samples. The model calibration has been carried out with a maximum likelihood estimation. This section will first introduce the probabilistic capacity model and then compare the eight models.

Probabilistic capacity models
The simplistic mechanics-based model of reinforced timber under CPG can be written as follows: where the contact area replaces the effective length and k pr is assumed equal to 1. C(x, ) is the capacity, x are the measurable capacity variables, and = { , } are unknown model parameters. Following Gardoni et al. (2002), the proposed form of the capacity can be written as follows: where T(⋅) is a variance stabilizing transformation, (x, ) is a correction term based on mechanics rules and evidence derived from the experimental data, F exp is the experimental resistance. The product is the model error, with model standard deviation and normally distributed random variable . The model is based on three assumptions: additivity (i.e., the additivity of ); homoskedasticity (i.e., the independence of from x ); normality (i.e., the normality of ). Through a suitable choice of T(⋅) , such assumptions can be approximately satisfied in the transformed space within the range of the data used to calibrate the model.
The correction term (x, ) is selected as a linear combination of n dimensionless explanatory functions h i (x) and reads The set of explanatory functions h(x) = {h 1 (x), .., h n (x)} is constructed starting from physical variables not included in A * 1 that may be relevant for the described physical phenomenon, but also from those included in A * 1 and the effect of which should be recalibrated in light of the available experimental data. The following set of explanatory functions is used, also reported in Table 5.   H : This term is the only one taking into account the screw geometry. It expresses the ratio between the threaded length of the screw and the specimen height.
The experimental data cannot be compared with the characteristic resistance F c,90,k . Therefore, the mean values of the strength perpendicular to the grain and yielding strength of steel replaced the characteristic values. The mean value f c90,m , in Eq.23, is obtained by assuming a coefficient of variation equal to 0.61, following Bejtka and Blaß (2004), who indirectly identified 5MPa as the mean value for the compression perpendicular to grain since it yields a higher correlation with the experimental parameters. Therefore a Coefficient of Variation (CoV) equal to 0.12 is estimated by assuming f c90,m = 3 MPa and f c90,k = 2.5 MPa following Leijten et al. (2010) and Bogensperger et al. (2011).
The mean yielding strength of steel is obtained by assuming a CoV equal to 0.08, based on the experimental data by Bejtka and Blaß (2005).
At this stage, the estimates of the mean strength are compared with the experimental values, as in Eq. 21. In the first step, all the parameters are used for calibration. To facilitate the use of the model and its possible implementation into technical standards, proposing a model with a limited number of explanatory functions should be advisable without significantly compromising accuracy. Nonetheless, reducing the number of terms usually leads to a higher value of .
Stepwise deletion allows finding a compromise between parsimony and accuracy. There are several procedures to apply stepwise deletion, and they mainly differ in the deletion criteria, such as the stepwise deletion based on p-values (Stone 1996). The stepwise deletion process used in this paper starts with a model that includes all the candidate explanatory functions and, at each step, removes the explanatory function with the highest coefficient of variation (CoV) of the corresponding i , as proposed in Gardoni et al. (2002). Once an explanatory function is removed, the model is re-calibrated, and the deletion process is repeated. The deletion process ends when either grows beyond an undesirable threshold (24) f y,m = f y,k (1 − 1.64 ⋅ 0.08) Fig. 15 Stepwise deletion process for the CPG model with (a) and without (b) variance stabilizing transformation. or the increment of is too large compared to the reduction of the model complexity. Figure 15 displays the CoV of the explanatory variables as a function of the number of steps for the two models with and without variance stabilizing transformation. The logarithm to base ten is used as variance stabilizing transformation in the first model, Fig. 15(b). In the same plot, the right y-axes show the standard deviation of the error as a function of the number of steps of the deletion process. The authors stopped at steps five and seven for the two models.
Figure15 highlights that after steps 5 and 7, there is a significant increment of the modelling error. Eqs.25, 26 report the final truncated expressions for the 1 and 2 correction terms in the two probabilistic models. Additionally, Tabs 6, 7 show the statistics of the coefficients of the explanatory functions of the two models, respectively. (25) The selected explanatory functions of the two models are not entirely the same. The 1 requires more terms, as discussed in the following paragraphs, to achieve a similar level of accuracy to the second model. The second model is more accurate than the first one. The lower values of the standard deviation of the first model, displayed in Fig. 15, depend on the different metrics, with and without the variance stabilizing transformation. In the first model, the authors are modelling the logarithm of the ratio. Table 8, discussed in the following paragraphs, compares the models' performances. It shows that the second model leads to a much better fitting despite 1 having seven terms and 2 just five. Interestingly, the second 2 does not include the terms expressing the distance of the load plate from the specimen edges. The relevant terms are just the density of the screws ( a 1 ∕b c ), the ratio between the plate width and the specimen height ( b c ∕H ), the percentage number of screws oriented parallel to grain ( n 0 ∕n ), and the ratio between the threaded length of the screw and the specimen height ( l r ∕H) . The fact that the stepwise deletion neglects the distances from the edges is coherent with the good performance of the simplistic mechanical model, as discussed later.
It must be remarked that the dependence of the resistance on the explanatory functions is not straightforward to understand from visual inspections; see Fig. 16. This fact entails the development of more complex capacity models, as carried out in this paper. Fig. 17 and Table 8 compare the performances of the considered eight capacity models.

Comparison between capacity models
The performance of each model is discussed in a bullet point following a logic flow. Then, the comparison between models backs a conclusive discussion, highlighting the most suitable mechanical models and future research perspectives.
• Adopting A 2 as mechanical models leads to an extremely poor fitting. This fact is confirmed by the significantly high error and negative correlation coefficients occurring in rare circumstances in case of poor-quality regressions. This fact agrees with the experimental evidence, which showed that A 2 seldom occurs despite being predicted in multiple circumstances. Conversely, A 1 is in discrete agreement with the experimental data. The third capacity model, named F 3 c,90,m , is in the current EC5 proposal. Although the experimental data showed that it does not predict the failure modes, its performance is discrete. Therefore, using A 2 in some cases, rather than A 1 , improves the model performance. The reason behind this might be the following: A 1 takes the load diffusion into account, which leads to a strength amplification not occurring in practice. Conversely, A 2 neglects this phenomenon, since l ef ,2 is generally lower than l ef ,1 . Therefore, the discrete performance of the EC5 model does not depend on its mechanical background, contradicted by the experiments, but on error compensation. In some cases, A 2 better predicts failure mechanisms associated with A 1 because it neglects the model's assumed load diffusion. • The fourth capacity model, F 4 c,90,m , is the simplistic model, where the timber contribution is obtained by multiplying the strength perpendicular to grain by the contact area of the steel plate. The performance is worse than the EC5 model, proving that the load diffusion occurs, despite not being entirely grasped by the EC5 model. The  performance of this model demonstrates that the load diffusion is not vertical, although lower than 45 • , as proved by the first model. Nonetheless, the plate distances from the edges have a limited effect, and the stress bulb might have a steeper diffusion angle. • The two probabilistic models have both satisfactory performance. However, the first one ( F 5 c,90,m ) with a variance stabilizing transformation has a modest performance compared to the number of terms involved. The relative error also increases compared to the simplistic mechanical model. Therefore, the first probabilistic model exhibits a good correlation, but its higher complexity does not yield a satisfactory accuracy gain. Conversely, the sixth model ( F 6 c,90,m ) has an outstanding performance. It is the best model among the eight for all error metrics with an R 2 = 0.82 . An R 2 higher than 0.8 can be a good fitting for a mechanical model. Besides, the correction term for F 6 c,90,m has just four addends, excluding the intercept. The four addends have a clear mechanical meaning and do not comprise the distances from the edges. This fact further confirms that the assumed diffusion model could be improved. • The second model with the best performance is the F 7 c,90,m , where the correction term is assumed constant and equal to 2, as estimated from a least-squares optimization. This model can be recommended in standard applications due to its simplicity and high accuracy. Besides, this model is consistent with the one for CPG without reinforcement, which assumed an amplification factor equal to 1.4. F 7 c,90,m and F 8 c,90,m assume as coefficients for the timber contribution 1.2 and 1.4, respectively. Although the second coincides with the one for CPG without reinforcement, a 2 factor provides better performance in error metrics with R 2 = 0.76 . The comparison between F 7 c,90,m and F 8 c,90,m proves that, if there are screws, the timber contribution amplifies and should be equal to 2.
In conclusion, the two models exhibiting the best performances are F 6 c,90,m and F 7 c,90,m , in Eq.17 and Eq.18, respectively. The performance of the second probabilistic model is the best and highlights the most relevant variables to be included in a capacity model, neglecting those related to load diffusion. The deterministic model in F 7 c,90,m , obtained by assuming the correction factor constant and equal to 2, represents the optimal compromise between complexity and accuracy and might be the most suitable for standard implementation. Figure 17 plots the predicted vs the estimated resistance values for the eight capacity models. The figure establishes the above comments, proving the excellent fitting obtained with the 6th and 7th models.

Discussion
The basis of the Eurocodes includes both mechanical principles and axiomatic truths. Therefore, in some instances, the formulations are not rigorous but strongly conventional. For example, that is the case of the assumptions behind the A 1 failure mechanism, where the load diffusion hypothesis results from conventional assumptions rather than mechanics. Still, a rigorous mechanical approach to this problem would be challenging to tackle with an analytical approach. Therefore, the main answer to this aspect should be found in the experimental data rather than in reasonable assumptions on the most likely structural response, i.e. a load diffusion at 45 • . This assumption would be fair for isotropic materials but apodictic when applied to orthotropic materials with structural discontinuities (the screws, e.g.).
The probabilistic capacity model in Eq.17 demonstrates that a good regression can be obtained using a few adimensional parameters without considering those expressing the distances from the edges. However, the main limitations of the model are twofold: it is calibrated on a limited dataset, which might be inadequate for a good probabilistic formulation. Additionally, its validity can only be ensured within the limits of the adimensional parameters used for calibration. Therefore, Table 9 lists the selected explanatory functions with the maximum and minimum values from the chosen database.
However, given the uncertainty of this model, which could be further tested against more experimental data, hopefully coming from now on, the deterministic one in Eq. 18 could be a good compromise between complexity and accuracy. The model in Eq. 18 is analogous to the one for compression perpendicular to grain without reinforcement except for 3 , equal to 2 rather than 1.4. Simplistic deterministic models might be preferable if they exhibit excellent accuracy. Besides, more experimental data would be needed for a more reliable probabilistic formulation.

Conclusion
This paper reviews existing capacity models for timber elements under compression perpendicular to the grain (CPG) with screw reinforcement. Eight mechanics-based probabilistic and deterministic models have been compared against experimental values collected by the authors on test data from the past twenty years. The database consists of approximately 60 test data, which, despite being insufficient for an ideal probabilistic formulation, proved some weaknesses of existing formulations currently included in the draft of the next generation of Eurocodes. The main aspects that arise from the model's comparison and the analysis of the experimental data are: • The formulations for predicting the assumed two failure mechanisms, one by the contact load ( A 1 ) and the other by the screw tips ( A 2 ), can be improved. The predicted failure mechanism ( A 2 ) does not occur in more than half of the experimental data. Consequently, the A 1 predictive model performs better than min(A 1 ,A 2 ). Still, the model defined as min(A 1 ,A 2 ) possesses a discrete accuracy compared to A 1 . • The above observation might prove that the main shortcoming of the A 1 model stands in the assumed diffusion mechanism of the load. Contrary to the EC5 assumptions, the load diffusion is not at 45 • , as confirmed by the good prediction of the mechanical model No 7 obtained by summing the screw and timber contribution amplified by a constant coefficient. The timber contribution is obtained by multiplying the timber strength by the contact load without any assumption about the stress spreading and the relative position of the loading plate to the specimen edges. The simplistic model yields an R 2 =0.76 vs an R 2 =0.59 for the model equal to A 1 . • The best fitting with the experimental data, with an R 2 =0.82, is obtained with a probabilistic model, where a factor corrects the timber contribution to resistance. The adimensional explanatory functions, selected by a stepwise deletion process, do not depend on the distances of the plate from the edges, proving that other parameters affect the resistance. Namely, the selected parameters are the density of the screws ( a 1 ∕b c ), the ratio between the plate width and the specimen height ( b c ∕H ), the percentage number of screws oriented parallel to grain ( n 0 ∕n ), and the ratio between the threaded length of the screw and the specimen height ( l r ∕H). • The paper proposes a deterministic version of the probabilistic model, where the correction factor is assumed constant and equal to 2 after a least-squares optimization. This model represents a compromise between complexity and accuracy for future code developments. • The experimental results do not confirm a significant difference between the analysed load cases classified in Fig. 4. Therefore, a unified coefficient k pr is recommended for future standardisation.
This paper highlights the need for more experimental data for more reliable calibration of the probabilistic capacity model to extend its validity to a broader range of parameters.