An Analytical Model for Predicting the Axial Stress Distribution of Self-Tapping Screws Due to Axial Load and Moisture Swelling of Mass Timber Products

Abstract

Self-tapping screws are becoming increasingly popular for use in modern timber structures. The axial stress distribution of self-tapping screws due to a mechanical load has been previously studied. However, the stress distribution of self-tapping screws due to moisture swelling-induced load from wood has not been explored so far. This research presents an analytical model to predict the axial stress distribution in self-tapping screws embedded in mass timber products under the combined effects of axial mechanical loading and wood moisture-induced swelling. The analytical model has been validated with numerical simulation. The input properties of the analytical model can be determined from withdrawal tests of self-tapping screws and the manufacturer’s guide of screw and mass timber products. A simple program has been developed to predict the stress distribution and maximum axial stress in self-tapping screws for a range of effective penetration lengths under a pre-load and moisture content change. Correctly predicting the maximum axial stress in self-tapping screws under the simultaneous action of a pre-load and wood moisture swelling-induced load can help design safer timber structures. This research provides a practical method for practicing engineers to predict the maximum axial stress in self-tapping screws due to pre-load and wood moisture swelling.

1. Introduction

Mass timber construction has gained significant traction in North America as a structural component in constructing tall, complex structures [1]. Self-tapping screws (STSs) are commonly used in mass timber construction for their ease of installation, high connection stiffness, and availability in a wide range of lengths and diameters [2]. Mass timber products such as glulam and cross-laminated timber (CLT) undergo hygroscopic deformation as the wood absorbs or desorbs moisture due to changes in the surrounding moisture conditions. For example, numerical simulations of moisture movement in post-tensioned CLT panels suggest that increasing the relative humidity from 50% to 70% results in an axial strain of around 2% [3]. Moisture swelling of the wood might adversely affect the axial performance of STSs by inducing additional tensile stress before the structural loads come into action. On the other hand, moisture shrinkage of wood does not cause additional tensile stress, as the moisture shrinkage has an overall effect of tensile stress relaxation on STSs.

During construction, exceeding the manufacturer-recommended installation torque for STSs can induce additional axial stress in the screws, similar to post-tensioning in prestressed concrete. This stress, combined with moisture-induced swelling of the surrounding wood, may lead to premature tensile failure if it exceeds the screw’s tensile strength, even before full structural loads are applied. Such failure can occur when over-torquing during installation is followed by fluctuations in the wood’s moisture content over its service life.

The axial stress distribution in dowels [4] and threaded rods installed in predrilled holes [5,6] under only mechanical axial load has been previously studied. The mechanical load might stem from any external source, such as structural loads or over-torquing of the screw. Understanding and quantifying the axial stress distribution in an STS under the simultaneous action of an axial load and moisture swelling of wood is critical to addressing some critical conditions, such as the potential problem of premature tensile failure of STSs installed in an over-torqued condition under increasing moisture conditions. The axial stress distribution of STSs under the simultaneous action of a pre-load and moisture swelling of glulam and CLT was numerically modeled in previous research using the finite element method by the authors of this study [7]. The current study continues the previous study by aiming to develop an analytical model to predict the axial stress distribution of STSs under the same combined loading condition.

The overall stress distribution of the analytical model proposed in this study results from the superposition of the stress distribution from two separate mechanisms: mechanical load (called Mechanism 1 here) and moisture swelling of the wood of mass timber (called Mechanism 2). The analytical model can predict the stress distribution along the longitudinal axis of STSs under the simultaneous action of mechanical load and glulam or CLT moisture swelling. From this stress distribution, the maximum axial stress in a self-tapping screw under a given moisture content changes and the axial load from an external source can be predicted. For simplicity, the proposed analytical model assumes no moisture content gradient in the glulam or CLT.

2. Theoretical Basis of the Analytical Model

The analytical model is inspired by the works presented in [4,6,8], which are based on applying the classical Volkerson theory [9] to axially loaded fasteners. In the analytical model, under the action of an axial load on the screw, it is assumed that pure axial stress arises in the screw and the wood surrounding the screw in the x-direction (Figure 1). Force transfer between the screw and the wood occurs through a shear layer situated at the interface between the wood [4,6,8] and the outer threaded part of the screw [4]. The shear layer is assumed to be in a state of pure shear, while the screw and the wood surrounding the screw are considered to be under pure axial stress. Considering the wood and the screw being under pure axial stress implies that they are more rigid to deformation than reality, which will result in the calculated value of the shear stress of the shear layer being higher than the actual value. Thus, this simplification gives an upper-bound solution to the stresses, and basing the design on these stresses will lead to a conservative design.

The stress–strain relationship of STSs is usually linear elastic up to the yield point, and the post-elastic region is marked by an almost horizontal yield plateau with limited strain hardening [10]. Thus, the material of STSs can be assumed to be elastic-perfectly plastic, and the yield point is taken as the failure point of the STS in the analytical model. As a result, the proposed analytical model falls under the domain of linear elastic stress analysis, which justifies the application of the linear superposition principle of stress presented in the following sections.

2.1. Stress Distribution Due to Mechanical Load

An axially loaded STS embedded in wood under pull–push loading conditions, similar to axially loaded threaded rods [5], is presented on the left of Figure 1. The top of the wood is held down in this loading condition while the screw is pulled in axial tension. The pull–push loading condition is similar to a two-member wood-to-steel connection with an axially loaded self-tapping screw, where the side member is assumed to be sufficiently rigid to support the top surface of the main wood member (Figure 2). Thus, if the stress distribution under the pull–push loading condition is known, the axial stress distribution of the screw in a wood-to-steel connection can be determined under the action of a mechanical load.

A differential section of length dx of the wood–screw interaction zone is considered on the right of Figure 1. The differential section is axisymmetric to the longitudinal axis of the screw. The origin of the coordinate system for the model is taken at the entrant side of the screw in the wood.

When the screw is pulled in axial tension from the wood member, the whole length of the screw is ineffective in imparting withdrawal resistance to the screw due to the tapering geometry of the screw at the tip. The effective penetration length$, L_{eff}$, (Figure 1) of the screw excludes the tip of the screw. Only the effective screw length is effective in imparting withdrawal resistance to the screw. The effective penetration length of the screw can be determined following the screw manufacturer’s guide.

The shear layer displacement due to an axial load (P) is given by

$$\begin{array}{c}{\delta }_1\left(x\right)={\delta }_{s_1}\left(x\right)-{\delta }_{w_1}\left(x\right)\end{array}$$

(1)

Here, ${\delta }_1\left(x\right)$ is the displacement of the shear layer, ${\delta }_{s_1}\left(x\right)$ is the displacement of the screw, and ${\delta }_{w_1}\left(x\right)$ is the displacement of the wood surrounding the screw.

The thickness of the shear layer is assumed to be infinitesimally small, and a linear model describing the constitutive relation of the shear layer is shown in Equation (2):

$$\begin{array}{c}{\tau }_1\left(x\right)={\Gamma }_e{\delta }_1\left(x\right)\end{array}$$

(2)

Here, ${\tau }_1\left(x\right)$ is the shear stress, and ${\Gamma }_e$ is the equivalent shear stiffness parameter of the shear layer. Subscript “e” in the equivalent shear stiffness parameter signifies that the linear elastic domain of the model is considered.

The static equilibrium conditions lead to the following differential equations for the screw and the wood, respectively:

$$\begin{array}{c}{\displaystyle \frac{\mathrm{d}{\sigma }_{s_1}\left(x\right)}{dx}}={\displaystyle \frac{\pi d_{core}{\tau }_1\left(x\right)}{\frac{\pi {d_{core}}^2}{4}}}={\displaystyle \frac{4}{d_{core}}}{\tau }_1\left(x\right)\end{array}$$

(3)

$$\begin{array}{c}{\displaystyle \frac{\mathrm{d}{\sigma }_{w_1}\left(x\right)}{dx}}=-{\displaystyle \frac{\pi d_{core}}{A_{w,eff}}}{\tau }_1\left(x\right)\end{array}$$

(4)

Here, ${\sigma }_{s_1}\left(x\right)$ is the stress in the screw, ${\sigma }_{w_1}\left(x\right)$ is the stress in the wood, $d_{core}$ is the inner core diameter of the screw (excluding screw thread), and $A_{w,eff}$ is the effective area of wood under pure axial stress. The effective area of wood is given by [6], assuming a 3:1 stress dispersion from the support at the top surface of the wood.

$$\begin{array}{c}{A}_{w,eff}=2b\left\{s_s+min\left(e,{\displaystyle \frac{L_{eff}}{6}}\right)+min\left({\displaystyle \frac{s}{2}},{\displaystyle \frac{L_{eff}}{6}}\right)\right\}\end{array}$$

(5)

Here, $b$ is the width of the wood member, $s_s$ is the length of the support at the top surface of the wood member, $e$ is the edge distance from the end of the support, and $s$ is the internal distance between the supports (Figure 3). For a wood-to-steel or wood-to-wood connection, as shown in Figure 2, $A_{w,eff}$ is the area of the whole top surface of the main wood member.

Successive differentiation of Equation (1) leads to the following equations:

$$\begin{array}{c}{\displaystyle \frac{d{\delta }_1\left(x\right)}{dx}}={\displaystyle \frac{d{\delta }_{s_1}\left(x\right)}{dx}}-{\displaystyle \frac{d{\delta }_{w_1}\left(x\right)}{dx}}={\epsilon }_{s_1}\left(x\right)-{\epsilon }_{w_1}\left(x\right)\end{array}$$

(6)

$$\begin{array}{c}{\displaystyle \frac{d^2{\delta }_1\left(x\right)}{d{x}^2}}={\displaystyle \frac{d{\epsilon }_{s_1}\left(x\right)}{dx}}-{\displaystyle \frac{d{\epsilon }_{w_1}\left(x\right)}{dx}}\end{array}$$

(7)

Here, ${\epsilon }_{s_1}\left(x\right)$ and ${\epsilon }_{w_1}\left(x\right)$ are the respective strains in the screw and wood, which are related to the stresses by the one-dimension Hooke’s law:

$$\begin{array}{c}{\epsilon }_{s_1}\left(x\right)={\displaystyle \frac{{\sigma }_{s_1}\left(x\right)}{E_s}}\end{array}$$

(8)

$$\begin{array}{c}{\epsilon }_{w_1}\left(x\right)={\displaystyle \frac{{\sigma }_{w_1}\left(x\right)}{E_w}}\end{array}$$

(9)

where $E_s$ and $E_w$ are the Young’s modulii of the screw and the wood in the x-direction, respectively. By proper substitutions and manipulations of Equations (1)–(9), the following governing differential equation can be derived:

$$\begin{array}{c}{\displaystyle \frac{d^2{\delta }_1\left(x\right)}{d{x}^2}}-{\left({\displaystyle \frac{\omega }{L_{eff}}}\right)}^2{\delta }_1\left(x\right)=0\end{array}$$

(10)

where the parameters $\omega$ is defined as [6]

$$\begin{array}{c}\omega =\sqrt{\pi d_{core}{\Gamma }_e\beta {L_{eff}}^2}\end{array}$$

(11)

in which

$$\begin{array}{c}\beta ={\displaystyle \frac{1}{A_s{E}_s}}+{\displaystyle \frac{1}{A_{w,eff}{E}_w}}\end{array}$$

(12)

where $A_s=\left({\displaystyle \frac{\pi }{4}}\times {d_{core}}^2\right)$ is the inner core area of the screw.

Now, the general solution of Equation (10) is given by

$$\begin{array}{c}{\delta }_1\left(x\right)=c_1{e}^{{\displaystyle \frac{\omega x}{L_{eff}}}}+c_2{e}^{-{\displaystyle \frac{\omega x}{L_{eff}}}}\end{array}$$

(13)

Differentiation of Equation (13) and substitution from Equation (6) leads to

$$\begin{array}{c}{\displaystyle \frac{d{\delta }_1\left(x\right)}{dx}}={\displaystyle \frac{\omega }{L_{eff}}}\left(c_1{e}^{{\displaystyle \frac{\omega x}{L_{eff}}}}-c_2{e}^{-{\displaystyle \frac{\omega x}{L_{eff}}}}\right)={\epsilon }_{s_1}\left(x\right)-{\epsilon }_{w_1}\left(x\right)\end{array}$$

(14)

where $c_1$ and $c_2$ are constants in Equation (14), which can be determined by the proper boundary conditions.

According to Figure 4, boundary conditions for the pull–push loading configuration can be written as

$$\begin{array}{c}\mathrm{At} \mathrm{x}=0, {\epsilon }_{s_1}\left(0\right)={\displaystyle \frac{P}{E_s{A}_s}} \left(\mathrm{Tension}\right),{\epsilon }_{w_1}\left(0\right)=-{\displaystyle \frac{P}{E_w{A}_{w,eff}}} \left(\mathrm{Compression}\right)\end{array}$$

(15)

$$\begin{array}{c}\mathrm{At} \mathrm{x}=L_{eff}, {\epsilon }_{s_1}\left(L_{eff}\right)=0,{\epsilon }_{w_1}\left(L_{eff}\right)=0\end{array}$$

(16)

Using boundary conditions (15) and (16) in Equation (14), the values of the constants $c_1$ and $c_2$ are determined as

$$\begin{array}{c}{c}_1={\displaystyle \frac{P\beta L_{eff}{e}^{-\omega }}{\omega \left(e^{-\omega }-e^{\omega }\right)}} ,c_2={\displaystyle \frac{P\beta L_{eff}{e}^{\omega }}{\omega \left(e^{-\omega }-e^{\omega }\right)}}\end{array}$$

(17)

Substituting Equation (17) into Equation (13) gives the displacement function of the shear layer, ${\delta }_1\left(x\right)$:

$$\begin{array}{c}{\delta }_1\left(x\right)={\displaystyle \frac{-P}{\pi d_{core}{L}_{eff}{\Gamma }_e}}{\displaystyle \frac{\omega }{\sinh \omega }}\cosh \left(\omega \left(1-{\displaystyle \frac{x}{L_{eff}}}\right)\right)\end{array}$$

(18)

The shear stress of the shear layer, ${\tau }_1\left(x\right)$ is given by

$$\begin{array}{c}{\tau }_1\left(x\right)={\Gamma }_e{\delta }_1\left(x\right)={\displaystyle \frac{-P}{\pi d_{core}{L}_{eff}}}{\displaystyle \frac{\omega }{\sinh \omega }}\cosh \left(\omega \left(1-{\displaystyle \frac{x}{L_{eff}}}\right)\right)\end{array}$$

(19)

The differential equation for screw stress distribution is given by Equation (3). The stress distribution in the screw at any point x is given by rearrangement and integration of Equation (3):

$$\begin{array}{c}{\sigma }_{s_1}\left(x\right)={{\displaystyle \int }}_0^x{\displaystyle \frac{4}{d_{core}}}{\tau }_1\left(x\right)dx={\displaystyle \frac{4P}{{\pi d_{core}}^2}}{\displaystyle \frac{\sinh \left(\omega \left(1-\frac{x}{L_{eff}}\right)\right)}{\sinh \omega }}\end{array}$$

(20)

Equation (20) gives the stress distribution model inside the screw due to applying a mechanical load P on the screw under the pull–push loading condition. Subscript “1” in the equations signifies the load mechanism under only mechanical load, which is referred to as “Mechanism 1” in this study. Equation (20) is a hyperbolic monotonic function whose maximum value occurs at x = 0. The stress distribution model under a mechanical load presented in this section is based on the work of Stamatopoulos [5], who extended the model initially proposed by Jensen [4] for threaded rods.

2.2. Stress Distribution Due to Wood Swelling

The stress distribution of the STS due to a change in moisture content inside the wood of mass timber members is proposed in this section. The load and stress distribution due to moisture content change is referred to as “Mechanism 2” in this study to maintain consistency with “Mechanism 1” introduced in the previous section. As mentioned before, moisture swelling of the wood in mass timber products might adversely affect the axial performance of STSs by inducing additional axial tensile stress. On the other hand, when wood loses moisture, it shrinks and reduces the confinement around the STS. The loss of confinement leads to a relaxation of any pre-existing tensile stress in the screw, as the surrounding wood contracts away from the screw rather than pressing against it. This loss of confinement has an overall effect of tensile stress relaxation on the STS. Thus, the increased moisture content will lead to wood swelling and contribute to the STS’s total axial tensile stress.

The material of the screw, steel, does not undergo any dimensional changes due to moisture content changes in the wood. Though steel expands/contracts significantly due to temperature changes, the effect of temperature change is not a focus of this research.

Wood can deform freely in unrestrained conditions due to changes in moisture content. However, an STS inserted into a wood member acts as a restraint due to the composite action between the wood material and the thread of the screw, which creates a restrained swelling condition in the wood member. The screw provides the restraint and acts throughout its effective length. In the restrained swelling state, the swelling tendency of wood will lead to additional stress distribution in the screw due to the composite action of the wood and the screw thread.

Chen and Nelson [11] conducted a study to determine the stress distribution in bonded materials induced by thermal expansion, which is analogous to the wood–screw composite system under moisture swelling. Consequently, the shear stress distribution in the shear layer along the length of the screw will be symmetric about the mid-point along the length of the screw, as was found in [11]. The coordinate system of Figure 5 will be considered for the analytical treatment of the screw stress distribution due to wood swelling. The coordinate system is based on the findings of [11], where the shear stress distribution at the junction of two dissimilar materials under thermal expansion was symmetric about the center of the joint. Two independent variables define the coordinate system, $x_1$ and $x_2,$ to leverage the symmetric shear stress distribution and form simplified stress distribution expressions. The effective screw length (L_eff) is the total screw length inside the wood member, excluding the screw tip (L_tip). Each of the variables $x_1$ and $x_2$ span half of the effective screw length.

The same assumption of pure axial stress in the screw and the wood surrounding the screw in the x-direction is considered as before. The moisture content change $\left(\Delta u\right)$ is considered uniform throughout the wood member and along the length of the screw. If the swelling coefficient of wood in the x-direction is $\alpha$, under the uniform moisture content change, the strain of the shear layer is given by the following equation:

$$\begin{array}{c}{\displaystyle \frac{d{\delta }_2\left(x\right)}{dx}}={\epsilon }_{s_2}\left(x\right)-{\epsilon }_{w_2}\left(x\right)-\alpha \Delta u\end{array}$$

(21)

All the symbols used in this section have similar meanings to the same symbols used in the previous section, the only difference being the subscript “2” instead of “1” to emphasize Mechanism 2. Differentiation of Equation (21), assuming a constant value of Young’s modulus along x-direction, leads to

$$\begin{array}{c}{\displaystyle \frac{d^2{\delta }_2\left(x\right)}{d{x}^2}}={\displaystyle \frac{d{\epsilon }_{s_2}\left(x\right)}{dx}}-{\displaystyle \frac{d{\epsilon }_{w_2}\left(x\right)}{dx}}={\displaystyle \frac{1}{E_s}}{\displaystyle \frac{d{\sigma }_{s_2}\left(x\right)}{dx}}-{\displaystyle \frac{1}{E_w}}{\displaystyle \frac{d{\sigma }_{w_2}\left(x\right)}{dx}}\end{array}$$

(22)

The stress–strain relationship of the shear layer and the equilibrium conditions given by Equations (2)–(4) for Mechanism 1 apply to Mechanism 2 as well and are given as

$$\begin{array}{c}{\tau }_2\left(x\right)={\Gamma }_e{\delta }_2\left(x\right)\end{array}$$

(23)

$$\begin{array}{c}{\displaystyle \frac{\mathrm{d}{\sigma }_{s_2}\left(x\right)}{dx}}={\displaystyle \frac{\pi d_{core}{\tau }_2\left(x\right)}{\frac{\pi {d_{core}}^2}{4}}}={\displaystyle \frac{4}{d_{core}}}{\tau }_2\left(x\right)\end{array}$$

(24)

$$\begin{array}{c}{\displaystyle \frac{\mathrm{d}{\sigma }_{w_2}\left(x\right)}{dx}}=-{\displaystyle \frac{\pi d_{core}}{A_{w,eff}}}{\tau }_2\left(x\right)\end{array}$$

(25)

Now utilizing Equations (23)–(25), Equation (22) can be rewritten as

$$\begin{array}{c}{\displaystyle \frac{1}{{\Gamma }_e}}{\displaystyle \frac{d^2{\tau }_2\left(x\right)}{d{x}^2}}=\left({\displaystyle \frac{4}{d_{core}{E}_s}}+{\displaystyle \frac{\pi d_{core}}{A_{w,eff2}{E}_w}}\right){\tau }_2\left(x\right)\end{array}$$

(26)

$$\begin{array}{c}{\displaystyle \frac{d^2{\tau }_2\left(x\right)}{d{x}^2}}-\left({\displaystyle \frac{4}{d_{core}{E}_s}}+{\displaystyle \frac{\pi d_{core}}{A_{w,eff2}{E}_w}}\right){\Gamma }_e{\tau }_2\left(x\right)=0\end{array}$$

(27)

Here, $A_{w,eff2}$ is the effective area of the wood under pure axial stress due to wood swelling, which is different from the effective area of wood under pure axial stress under the pull–push loading conditions ($A_{w,eff}$) described in the previous section. The effective area $A_{w,eff2}$ represents the wood area that interacts with the screw thread during wood swelling. The additional stress induced in the screw due to moisture swelling of wood is assumed to be caused by the swelling of the wood in this effective area. This area is given by assuming the same 3:1 stress dispersion from the two ends of the screw (the screw entrant side and the screw tip), as assumed for the pull–push loading conditions in [6]. We do not know the exact stress dispersion angle yet, and in the absence of any better estimates, the 3:1 stress dispersion, similar to the previous case, is assumed. Further research will be conducted to verify the stress dispersion angle in the future. The effective area $A_{w,eff2}$ is given by Equation (28), where $d$ is the outer nominal diameter of the STS, as shown in Figure 6.

$$\begin{array}{c}{A}_{w,eff2}={\displaystyle \frac{\pi }{2}}\left({\left({\displaystyle \frac{L_{eff}}{6}}+{\displaystyle \frac{d}{2}}\right)}^2-{\left({\displaystyle \frac{d}{2}}\right)}^2\right)\end{array}$$

(28)

Equation (27) is a second-order linear differential equation that can be simplified to

$$\begin{array}{c}{\displaystyle \frac{d^2{\tau }_2\left(x\right)}{d{x}^2}}-{K_s}^2{\tau }_2\left(x\right)=0; \left[{K_s}^2=\left({\displaystyle \frac{4}{d_{core}{E}_s}}+{\displaystyle \frac{\pi d_{core}}{A_{w,eff2}{E}_w}}\right){\Gamma }_e\right] \end{array}$$

(29)

The highest shear stress will occur at the screw entrant side and near the end of the screw and will decrease exponentially to zero at the center of the screw inside the wood member. The general solution to Equation (29), which describes this behavior, is given by [8]

$$\begin{array}{c}{\tau }_{1,swelling}\left(x_1\right)=c_3{e}^{-K_s{x}_1}; 0\le x_1\le {\displaystyle \frac{L_{eff}}{2}}\end{array}$$

(30)

$$\begin{array}{c}{\tau }_{2,swelling}\left(x_2\right)=c_4{e}^{K_s\left(x_2-L_{eff}\right)}; {\displaystyle \frac{L_{eff}}{2}}<x_2\le L_{eff}\end{array}$$

(31)

It was mentioned in the previous section that subscript “1” in the equations signifies the load mechanism under only mechanical load. In Equations (30) and (31) and onwards, the subscripts “1” and “2” signify the top half and bottom half of the screw, respectively. The term “swelling” is added to the subscripts to differentiate them from the mechanical load mechanism. Hence, the shear stress in the shear layer from Equations (30) and (31) is compactly given as

$${\tau }_2=\left\{\begin{array}{c}{\tau }_{1,swelling}\left(x_1\right) if 0\le x_1\le {\displaystyle \frac{L_{eff}}{2}}\\ {\tau }_{2,swelling}\left(x_2\right) if {\displaystyle \frac{L_{eff}}{2}}<x_2\le L_{eff}\end{array}\right.$$

In Equations (30) and (31), $c_3$ and $c_4$ are constants that can be determined from the boundary conditions. Differentiation of Equations (30) and (31) leads to

$$\begin{array}{c}{\displaystyle \frac{d{\tau }_{1,swelling}\left(x_1\right)}{d{x}_1}}=-K_s{c}_3{e}^{-K_s{x}_1}\end{array}$$

(32)

$$\begin{array}{c}{\displaystyle \frac{d{\tau }_{2,swelling}\left(x_2\right)}{d{x}_2}}=K_s{c}_4{e}^{K_s\left(x_2-L_{eff}\right)}\end{array}$$

(33)

The boundary condition for the restrained wood swelling case can be written as

$$\begin{array}{c}\mathrm{At} x_1=0, {\displaystyle \frac{d{\tau }_{1,swelling}\left(0\right)}{d{x}_1}}={\Gamma }_e{\displaystyle \frac{d{\delta }_{1,swelling}\left(0\right)}{d{x}_1}}\\ {\Gamma }_e{\displaystyle \frac{d{\delta }_{1,swelling}\left(0\right)}{d{x}_1}}={\Gamma }_e\left({\epsilon }_{s_2}-{\epsilon }_{w_2}-\alpha \Delta u\right)={\Gamma }_e\left(0-0-\alpha \Delta u\right)=-\alpha \Delta u{\Gamma }_e\end{array}$$

(34)

$$\begin{array}{c}\mathrm{At} x_2=L_{eff}, {\displaystyle \frac{d{\tau }_{2,swelling}\left(L_{eff}\right)}{d{x}_2}}={\Gamma }_e{\displaystyle \frac{d{\delta }_{2,swelling}\left(L_{eff}\right)}{d{x}_2}}\\ {\Gamma }_e{\displaystyle \frac{d{\delta }_{2,swelling}\left(L_{eff}\right)}{d{x}_2}}={\Gamma }_e\left({\epsilon }_{s_2}-{\epsilon }_{w_2}-\alpha \Delta u\right)={\Gamma }_e\left(0-0-\alpha \Delta u\right)=-\alpha \Delta u{\Gamma }_e\end{array}$$

(35)

where ${\epsilon }_{s_2}$ and ${\epsilon }_{w_2}$ are the strains in the screw and the wood, respectively, which are zero at the extremities.

Applying boundary conditions (34) and (35) in Equations (32) and (33), the values of the constants $c_3$ and $c_4$ are determined as

$$\begin{array}{c}{c}_3={\displaystyle \frac{\alpha \Delta u{\Gamma }_e}{K_s}}\end{array}$$

(36)

$$\begin{array}{c}{c}_4=-{\displaystyle \frac{\alpha \Delta u{\Gamma }_e}{K_s}}\end{array}$$

(37)

Substituting Equations (36) and (37) into Equations (30) and (31) and using Equation (23), the displacement (${\delta }_{1,swelling} \mathrm{and} {\delta }_{2,swelling}$) and shear stress distribution function of the shear layer due to wood swelling are

$$\begin{array}{c}{\delta }_{1,swelling}\left(x_1\right)={\displaystyle \frac{\alpha \Delta u}{K_s}}{e}^{-K_s{x}_1}\end{array}$$

(38)

$$\begin{array}{c}{\tau }_{1,swelling}\left(x_1\right)={\displaystyle \frac{\alpha \Delta u{\Gamma }_e}{K_s}}{e}^{-K_s{x}_1}\end{array}$$

(39)

$$\begin{array}{c}{\delta }_{2,swelling}\left(x_2\right)=-{\displaystyle \frac{\alpha \Delta u}{K_s}}{e}^{K_s\left(x_2-L_{eff}\right)}\end{array}$$

(40)

$$\begin{array}{c}{\tau }_{2,swelling}\left(x_2\right)=-{\displaystyle \frac{\alpha \Delta u{\Gamma }_e}{K_s}}{e}^{K_s\left(x_2-L_{eff}\right)}\end{array}$$

(41)

Now, the differential equation for screw stress distribution is given by

$$\begin{array}{c}{\displaystyle \frac{d{\sigma }_{s_2}\left(x\right)}{dx}}={\displaystyle \frac{4}{d_{core}}}{\tau }_2\left(x\right)\end{array}$$

(42)

The stress distribution in the screw due to wood swelling is given by re-organization and integration of Equation (42):

$$\begin{array}{c}{\sigma }_{s_{1,swelling}}\left(x_1\right)=\int {\displaystyle \frac{4}{d_{core}}}{\tau }_{1,swelling}\left(x_1\right)dx=-{\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}{e}^{-K_s{x}_1}+c_5\end{array}$$

(43)

$$\begin{array}{c}{\sigma }_{s_{2,swelling}}\left(x_2\right)=\int {\displaystyle \frac{4}{d_{core}}}{\tau }_{2,swelling}\left(x_2\right)dx=-{\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}{e}^{K_s\left(x_2-L_{eff}\right)}+c_6\end{array}$$

(44)

Hence, the axial stress in the screw from Equations (43) and (44) is compactly given as

$${\sigma }_{s_2}=\left\{\begin{array}{c}{\sigma }_{s_{1,swelling}}\left(x_1\right) if 0\le x_1\le {\displaystyle \frac{L_{eff}}{2}}\\ {\sigma }_{s_{2,swelling}}\left(x_2\right) if {\displaystyle \frac{L_{eff}}{2}}<x_2\le L_{eff}\end{array}\right.$$

In Equations (43) and (44), $c_5$ and $c_6$are constants of integration, which can be determined from the following boundary conditions:

$$\begin{array}{c}{\sigma }_{s_{1,swelling}}\left(0\right)=0=-{\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}+c_5=0\end{array}$$

(45)

$$\begin{array}{c}{c}_5={\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}\end{array}$$

(46)

$$\begin{array}{c}{\sigma }_{s_{2,swelling}}\left(L_{eff}\right)=-{\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}+c_6=0\end{array}$$

(47)

$$\begin{array}{c}{c}_6={\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}\end{array}$$

(48)

Thus, the stress distribution in the screw due to wood swelling (Mechanism 2) is given by

$$\begin{array}{c}{\sigma }_{s_{1,swelling}}\left(x_1\right)={\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}\left(1-e^{-K_s{x}_1}\right); 0\le x_1\le {\displaystyle \frac{L_{eff}}{2}}\end{array}$$

(49)

$$\begin{array}{c}{\sigma }_{s_{2,swelling}}\left(x_2\right)={\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}\left(1-e^{K_s\left(x_2-L_{eff}\right)}\right); {\displaystyle \frac{L_{eff}}{2}}<x_2\le L_{eff}\end{array}$$

(50)

The stress distribution model under wood swelling presented in this section is derived based on the principles adopted from [8,11], which dealt with the stress distributions in dissimilar bonded materials caused by differential expansion and contraction.

2.3. Shear Stiffness Parameter Determination and Superposition of Stress Distribution

The unknown parameter in the analytical model is the constant ${\Gamma }_e$, which is required to determine the constant $\omega$ given by Equation (11). Setting $x=0$ and ignoring the negative sign in Equations (18) and (38) give the withdrawal displacements of the screw at the top surface of the wood. The withdrawal displacement under mechanical load is given by

$$\begin{array}{c}{\delta }_{{withdrawal}_1}={\displaystyle \frac{P\omega }{\pi d_{core}{L}_{eff}{\Gamma }_e}}{\displaystyle \frac{\cosh \omega }{\sinh \omega }}={\displaystyle \frac{P}{\pi d{L}_{eff}{\Gamma }_e}}{\displaystyle \frac{\omega }{\tanh \omega }}\end{array}$$

(51)

The withdrawal displacement due to wood swelling is given by

$$\begin{array}{c}{\delta }_{{withdrawal}_{1,swelling}}={\displaystyle \frac{\alpha \Delta u}{K_s}}\left(1\right)={\displaystyle \frac{\alpha \Delta u}{K_s}}\end{array}$$

(52)

The withdrawal stiffness ${\mathrm{K}}_w$ is given by

$$\begin{array}{c}{\mathrm{K}}_w={\displaystyle \frac{P}{{\delta }_{{withdrawal}_1}+{\delta }_{{withdrawal}_{1,swelling}}}}\end{array}$$

(53)

Here, $P$ is the axial load acting in the screw at $x=0$.

Now, ${\delta }_{{withdrawal}_1}$ is the withdrawal displacement due to the application of an external axial load and ${\delta }_{{withdrawal}_{1,swelling}}$ is the withdrawal displacement of the screw due to moisture swelling of the wood, which physically represents the gradual emergence of the screw from the wood member with moisture ingress. The withdrawal displacement due to an external “active” load acting directly on the screw should be higher than the withdrawal displacement due to moisture ingress, which is somewhat of a “passive” effect since it stems from the wood material. Thus, ${\delta }_{{withdrawal}_1}\gg {\delta }_{{withdrawal}_{1,swelling}}$, and ignoring ${\delta }_{{withdrawal}_{1,swelling}}$ in Equation (53), the withdrawal stiffness becomes

$$\begin{array}{c}{\mathrm{K}}_w={\displaystyle \frac{P}{{\delta }_{{withdrawal}_1}}}=\pi d_{core}{L}_{eff}{\Gamma }_e{\displaystyle \frac{\tanh \omega }{\omega }}\end{array}$$

(54)

The withdrawal stiffness ${\mathrm{K}}_w$ can be determined from a withdrawal test of the STS inserted into glulam or CLT, and Equation (54) can be solved to determine the value of ${\Gamma }_e$. A program was developed in MATLAB R2023b to determine the value of ${\Gamma }_e$.

Finally, the stress distribution in the screw is given by the superposition of the stress distribution from two mechanisms due to mechanical load and moisture swelling of the wood of mass timber. The superposition of Equations (20) (Mechanism 1), (49), and (50) (Mechanism 2) are given by the following equations:

$$\begin{array}{c}{\sigma }_s\left(x\right)={\sigma }_{s_1}\left(x\right)+{\sigma }_{s_{1,swelling}}\left(x_1\right)={\displaystyle \frac{4P}{{\pi d_{core}}^2}}{\displaystyle \frac{\sinh \left(\omega \left(1-\frac{x}{L_{eff}}\right)\right)}{\sinh \omega }}+{\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}\left(1-e^{-K_s{x}_1}\right)\end{array}$$

(55)

$$\begin{array}{c}{\sigma }_s\left(x\right)={\sigma }_{s_1}\left(x\right)+{\sigma }_{s_{2,swelling}}\left(x_2\right)={\displaystyle \frac{4P}{{\pi d_{core}}^2}}{\displaystyle \frac{\sinh \left(\omega \left(1-\frac{x}{L_{eff}}\right)\right)}{\sinh \omega }}+{\displaystyle \frac{4\alpha \Delta u{\Gamma }_e}{d_{core}{K_s}^2}}\left(1-e^{K_s\left(x_2-L_{eff}\right)}\right)\end{array}$$

(56)

where $0\le x_1\le {\displaystyle \frac{L_{eff}}{2}}$, ${\displaystyle \frac{L_{eff}}{2}}<x_2\le L_{eff}$ and $0\le x\le L_{eff}$.

Equations (55) and (56) give the stress distribution in an axially loaded STS under pull–push loading conditions due to moisture swelling of a wood member. The coordinate system for the analytical model is shown in Figure 5.

In a two-member wood-to-steel connection with an STS (Figure 2), if the screw is tightened with a torque more than that required to make the connection snug, an axial load will be induced in the screw. The side member is assumed to be sufficiently rigid to provide support at the top surface of the main wood member. Then, the axially loaded screw in the main wood member is similar to an axially loaded screw under pull–push loading conditions. In this over-torqued condition of the screw, if there is a change in moisture content in the main wood member, the total stress distribution in the screw is given by Equations (55) and (56).

3. Analytical Model Input Properties

It is essential to choose representative values of the input properties for the analytical model to model the screw stress distribution reliably. The equivalent shear stiffness parameter can be found by solving Equation (54) if the withdrawal stiffness is known from STS withdrawal tests. For this purpose, a withdrawal test program was conducted under varying moisture conditions. The specimens shown in Figure 7 were tested after exposure to the moisture content changes listed in the second and third columns of

Table 1. Withdrawal stiffness and equivalent shear stiffness properties of different test groups.

Mass Timber Product and Self-Tapping Screw	Target Initial and Final EMC	${\mathbf{K}}_{\mathit{w}}$ (kN/mm)	CoV (%)	${\mathbf{\Gamma }}_{\mathit{e}}$ (MPa/mm)	Withdrawal Strength (MPa)
160 × 170 mm CLT, 8 mm Screw	12%	22.86	10.92	23.84	9.39
	16%	18.69	38.35	19.01	7.19
	21%	13.98	25.06	13.71	6.45
260 × 270 mm CLT, 13 mm Screw	12%	22.09	23.18	6.62	6.18
	16%	21.23	19.73	6.34	6.04
	21%	15.55	10.29	4.55	4.72
80 × 160 mm Glulam, 8 mm Screw	12%	19.41	24.67	20.21	9.31
	16%	20.32	12.52	21.64	9.13
	21%	17.41	10.82	18.03	7.71
130 × 260 mm Glulam, 13 mm Screw	12%	26.13	24.09	8.15	8.83
	16%	22.54	22.10	7.01	7.13
	21%	20.46	20.12	6.29	5.64

. As shown in Figure 7, screws with nominal outer diameters of 8 mm and 13 mm were centrally inserted into the broad face of CLT and glulam of various sizes. The glulam used in the tests was Douglas Fir–Larch, with a stress grade of 16c-E, while the CLT was Spruce–Pine–Fir (SPF) with a stress grade of V2 [12]. The initial moisture condition in Table 1 represents the wood moisture content at the time of self-tapping screw installation. In contrast, the final moisture content corresponds to the equilibrium moisture content after specimen conditioning. For simplicity, this study assumes a uniform moisture content throughout the CLT or glulam in both stages, disregarding the effects of moisture gradients [7].

The specimens underwent a two-stage conditioning process to achieve the targeted moisture conditions. The first stage established the initial equilibrium moisture content (EMC) for self-tapping screw installation, while the second stage conditioned the specimens to reach the final target EMC. This approach ensured that the screw installation did not significantly disrupt the uniform moisture content within the glulam or CLT. After achieving the final target EMC, withdrawal tests were conducted on the specimens in displacement control under pull–push loading conditions. From the withdrawal tests, the withdrawal stiffness of the different specimen configurations under the different moisture conditions was determined. Details of the test program can be found in [13].

If the withdrawal stiffness values are expressed in units of force per unit length of the screw, Equation (54) can be solved by using the MATLAB program developed by the authors to determine the equivalent shear stiffness parameter at 12%, 16%, and 21% EMC. The equivalent shear stiffness parameter (${\Gamma }_e$) for the different withdrawal test specimen configurations were determined from the withdrawal stiffness (${\mathrm{K}}_w$) values using Equation (54) and the MATLAB program developed by the authors (shown in Table 1). Higher moisture contents are known to have a softening effect on wood [14] and are marked by lower values of the withdrawal stiffnesses, as seen in Table 1. Though the equivalent shear stiffness parameter changes with the change in the moisture content of the wood, a constant value of the shear stiffness parameter was considered in the analytical model to obtain simple forms of the analytical expressions. It is suggested that the shear stiffness parameter be taken at the initial moisture content of the wood since that gave more conservative predictions on par with the numerical analysis, as exhibited in the following section.

Young’s modulus values of glulam and CLT at 12% and 21% EMC were determined from the product manufacturers’ guides on the glulam and CLT and the published literature [15]. The Young’s modulus values of the glulam and CLT are summarized in

Table 2. Young’s modulus of CLT and glulam.

Mass Timber Product	Layer Type	Wood EMC	${\mathit{E}}_{\mathit{R}}$ (MPa)	${\mathit{E}}_{\mathit{T}}$ (MPa)
160 × 170 mm CLT	Longitudinal	12%	1193.4	631.8
	Transverse		918	486
	Longitudinal	21%	918.9	486.5
	Transverse		706.9	374.2
260 × 270 mm CLT	Same property for all layers	12%	969	513
		21%	746.1	395
80 × 160 mm Glulam	Same property for all layers	12%	843.2	620
		21%	649.3	477.4
130 × 260 mm Glulam	Same property for all layers	12%	843.2	620
		21%	649.3	477.4

. Similar to the shear stiffness parameter, it is recommended to use the value of Young’s modulus at the initial moisture content as the input for the analytical model, as this results in higher axial stress values in the screw. In the absence of experimental methods to validate the analytical model, we have opted for the more conservative approach, which provides higher predicted maximum stress values. The swelling coefficient values along the radial and tangential anatomic direction of the laminates of glulam and CLT were determined experimentally from swelling tests described in [13]. The average swelling coefficient values of all layers of each CLT and glulam product are given in

Table 3. Effective swelling coefficient values.

Type of Wood Product	Layer Type	${\mathit{\alpha }}_{\mathit{L}}\left({\mathit{\%} }^{-1}\right)$	${\mathit{\alpha }}_{\mathit{R}} \left({\mathit{\%} }^{-1}\right)$	${\mathit{\alpha }}_{\mathit{T}} \left({\mathit{\%} }^{-1}\right)$
160 × 170 mm CLT	Longitudinal	0	0.0016	0.0024
	Transverse	0.0001	0.0017	0.0028
260 × 270 mm CLT	Longitudinal	0	0.0016	0.0027
	Transverse	0.0002	0.0019	0.0029
All Glulam	-	0.0001	0.0017	0.0029

. In verifying the analytical model with numerical analysis, Young’s modulus and swelling coefficient values of glulam in the tangential direction were taken as the input values ($E_W=E_T$ and $\alpha ={\alpha }_T$), as the moisture swelling of wood is the highest in the tangential direction.

The effective area of wood under axial stress in the pull–push loading condition ($A_{w,eff}$) was calculated for the withdrawal test specimens using Equation (5), and the results are given in

Table 4. Wood effective areas.

Wood Products and Self-Tapping Screw	${\mathit{A}}_{\mathit{w},\mathit{e}\mathit{f}\mathit{f}}$ (mm2)
160 × 170 mm CLT, 8 mm Screw	32,064
260 × 270 mm CLT, 13 mm Screw	60,424
80 × 160 mm Glulam, 8 mm Screw	16,032
130 × 260 mm Glulam, 13 mm Screw	30,212

.

The core diameter of the screws ($d_{core}$) and the effective penetration length ($L_{eff}$) were determined from the screw manufacturer’s guide, according to the geometry of the withdrawal test specimens (

Table 5. Screw properties.

Screw Type	${\mathit{d}}_{\mathit{c}\mathit{o}\mathit{r}\mathit{e}}$ (mm)	${\mathit{L}}_{\mathit{e}\mathit{f}\mathit{f}}$ (mm)	$\mathbf{Mean} {\mathit{E}}_{\mathit{S}}$ (GPa)
8 mm nominal diameter screw	5	72	208.2
13 mm nominal diameter screw	9.6	120	226.6

). Young’s modulus of the screws were taken from the screw tensile tests conducted by [13] and shown in Table 5. Once all the properties mentioned in Table 1, Table 2, Table 3, Table 4 and Table 5 are known, the flowchart in Figure 8 illustrates how the analytical model can be used to model the stress distribution and the maximum stress in the STS. The two MATLAB programs developed by the authors are identified as “gammasolver.m” and “maxstress.m” in the flowchart and can be obtained from the first author (.m is the file extension for the program script in MATLAB). The first program, “gammasolver.m” is used to solve for the equivalent shear stiffness parameter. Once the equivalent shear stiffness parameter is determined for a particular combination of screw and wood product for the moisture content change range of interest, the value can be used to model the stress distribution of the same screw in a wood-to-wood or wood-to-steel connection in which the main member consists of the same wood product, using the second program, “maxstress.m”. Further implementation of the analytical model is described in the following sections.

4. Validation with Numerical Analysis

4.1. Finite Element Model

The proposed analytical model was validated through finite element analysis (FEA) simulations using ABAQUS/Standard Solver 2021 [16]. The theoretically predicted stress distributions in self-tapping screws were compared with FEA results for a wood-to-steel connection. The connection included a glulam timber specimen with dimensions of 80 mm × 160 mm and an 8 mm self-tapping screw with a total length of 160 mm. The screw was inserted centrally into the glulam product, perpendicular to the grain direction, with a penetration length of 10d (excluding the screw tip), as shown in Figure 9. In the wood-to-steel connection, the steel component was considered rigid. A fixed boundary condition was applied at the top of the glulam instead of explicitly modeling the steel plate to simplify the model.

Although three-dimensional (3D) finite element models can fully capture the orthotropic behavior of wood, they impose significant computational demands. A two-dimensional (2D) axisymmetric approach offers substantially reduced computational cost while maintaining solution accuracy [17]. Previous investigations have demonstrated that for axially loaded fasteners, the differences in the predicted withdrawal capacity and connection stiffness between 3D and 2D axisymmetric models are negligible [18]. Based on these considerations and the axial symmetry of the loading configuration, a 2D axisymmetric modeling approach was implemented in the current study. The geometric configuration of the axisymmetric model is depicted in Figure 9, with the axis of symmetry coinciding with the longitudinal axis of the self-tapping screw.

The self-tapping screw in a connection is typically installed as a tight fit, with the wood material filling the screw pitch and leaving no clearance between the screw thread surfaces and the wood. Under combined loading conditions—comprising external axial loads from torque and forces due to wood swelling—failure may occur in both the wood and the screw. Possible failure mechanisms in wood include the initiation and propagation of cracks at the root of the internal threads formed by the screw. These cracks may lead to the creation of a withdrawal failure surface along the screw thread path, depending on the magnitude of the moisture content (MC) change and the torque-induced load. The finite element model incorporates hard contact with a friction-type model to simulate the interaction between the screw and the wood, accounting for a tight-fit connection. A cohesive zone model was employed to capture the initiation and propagation of damage along the potential withdrawal failure path. Cohesive surfaces were defined along the screw length and around the threaded region, enabling the simulation of crack growth and development, as illustrated in Figure 10.

The finite element model is based on the foundations laid in previously established models [7,13,19]. The geometry of the connection was created using the Part and Assembly module in ABAQUS/CAE [16], with a refined meshing strategy to accurately capture stress concentrations in the screw-threaded region. Four-node bilinear axisymmetric quadrilateral elements (CAX4R) were employed. The finite element analysis comprised approximately 10,700 elements, with the mesh density determined through a convergence study. The mesh size was gradually increased with distance from the screw–wood interface to optimize computational efficiency without compromising accuracy, as illustrated in Figure 11.

4.2. Material Properties

Isotropic material properties, given in

Table 6. Glulam input properties for finite element analysis.

${\mathit{E}}_{\mathit{L}}$ (MPa)	${\mathit{E}}_{\mathit{T}}$ (MPa)	${\mathit{E}}_{\mathit{R}}$ (MPa)	${\mathit{\upsilon }}_{\mathit{R}\mathit{L}}$	${\mathit{\upsilon }}_{\mathit{T}\mathit{L}}$	${\mathit{\upsilon }}_{\mathit{R}\mathit{T}}$	${\mathit{G}}_{\mathit{R}\mathit{T}}$ (MPa)	${\mathit{G}}_{\mathit{L}\mathit{T}}$ (MPa)	${\mathit{G}}_{\mathit{L}\mathit{R}}$ (MPa)	EMC (%)
12,400	620	843.2	0.036	0.029	0.39	86.8	967.2	793.6	12
11,656	558	758.88	0.036	0.029	0.39	78.99	880.15	722.18	15
11,532	551.8	750.45	0.036	0.029	0.39	78.12	870.48	714.24	16
11,160	502.2	682.99	0.036	0.029	0.39	73.78	822.12	674.56	18
10,788	477.4	649.26	0.036	0.029	0.39	69.44	773.76	634.88	21

, were assigned to the STS, and the tensile strength was taken as 1100 MPa [10]. The wood of the glulam member was assigned orthotropic material properties, reflecting its varied characteristics along the local longitudinal, radial, and tangential directions. The input material properties for the glulam were determined according to the stress grade of the glulam, as shown in Table 6. Accurate stress analysis requires incorporating the moisture-dependent variations in elastic properties. Accordingly, these material properties at different moisture contents were determined from previous studies [13,15,20]. Since all layers of the glulam were of the same grade, it was modeled as a single unit without separately representing individual layers and glue lines. The results indicate this approach is appropriate for standard-sized glulams [21].

The interaction between the screw threads and the surrounding wood was defined using “Hard contact” for normal behavior and a “Penalty” approach for tangential behavior. A friction coefficient of 0.2 was used for the tangential behavior [7]. Potential cracks in the wood due to withdrawal were modeled using cohesive surfaces along the screw thread, as introduced in the previous section. The fundamentals of the constitutive behavior and the traction–separation law of the cohesive surface, called the Cohesive Zone Model [19], are provided in the following section.

4.3. Cohesive Surface

The Cohesive Zone Model (CZM) simulates fracture behavior in materials and their interfaces through a traction–separation law. This law characterizes the response between cohesive traction and separation across the fracture surface, capturing the progressive degradation of material properties during the fracture process [19,22].

In fracture mechanics theory, crack development in materials can follow three principal modes: Mode I (opening), Mode II (sliding), and Mode III (tearing). The CZM can handle interface failure under both pure and mixed-mode loading conditions. To define the cohesive contact interaction in ABAQUS/CAE, three essential parameters must be specified: elastic stiffness, which characterizes the initial elastic response of the interface through normal and shear stiffness components; damage initiation criteria, which determine the onset of interface degradation based on traction or separation thresholds; and damage evolution law, which governs the progressive deterioration of the interface properties after damage initiation, typically through energy-based or displacement-based approaches.

The initial linear elastic behavior of the traction–separation model, which relates the normal and shear stresses to normal and shear separation across the interface, is given by

$$\left\{t\right\}=\left\{\begin{array}{c}{t}_n\\ t_s\\ t_t\end{array}\right\}=\left[\begin{array}{c}{K}_{nn}{K}_{ns}{K}_{nt}\\ K_{ns}{K}_{ss}{K}_{st}\\ K_{nt}{K}_{st}{K}_{tt}\end{array}\right]\left\{\begin{array}{c}{\delta }_n\\ {\delta }_s\\ {\delta }_t\end{array}\right\}=\left[K\right]\left\{\delta \right\}$$

(57)

The nominal traction stress vector $t$ consists of $t_n$ (normal traction) and $t_s$, $t_t$ (shear tractions), with corresponding separations δ_n, δ_s, and δ_t. $K$ is the interface stiffness matrix.

Figure 12 illustrates the cohesive constitutive law for pure mode loading conditions. The constitutive response can be expressed as

$$\mathrm{t}=K\delta \delta <{\delta }_m^0$$

(58)

$$\mathrm{t}=(1-D)K\delta {\delta }_m^0\le \delta < {\delta }_m^f$$

(59)

$$\mathrm{t}=0 \delta \ge { \delta }_m^f$$

(60)

where ${\delta }_{\mathrm{m}}^0$ is separation at the initiation of damage, and ${ \delta }_{\mathrm{m}}^{\mathrm{f}}$ is the effective separation at complete failure.

A scalar damage variable, D, represents the damage at the contact point and is defined as

$$\begin{array}{c}D={\displaystyle \frac{{\delta }_m^f ( {\delta }_m^{\mathit{max}}{-\delta }_{m }^0)}{{\delta }_m^{\mathit{max}}{ (\delta }_m^f{-\delta }_{m }^0)}}; D\in \left[0,1\right]\end{array}$$

(61)

In Equation (61), ${\delta }_{\mathrm{m}}^{\max }$ refers to the maximum value of the effective separation.

This study adapted material parameters for the cohesive surface model from [7]. Due to slight differences between the current model and that of [7], particularly in the modeling of the screw thread geometry, the suitability of these parameters for the current model was validated with the experimental results. Withdrawal tests were simulated with the finite element model at three distinct equilibrium moisture contents (EMC) of 12%, 16%, and 21%. The finite element results were compared with the experimental data reported by [7].

The stiffness values in the normal and the two transverse directions define the elastic regime of the bilinear traction–separation law of the cohesive layer. In this study, the focus was on axial loading conditions, making the normal stiffness perpendicular to the longitudinal axis of the screw (${\mathrm{K}}_{nn}$) the least significant. The two transverse stiffness values, shear stiffness (${\mathrm{K}}_{ss}$) and tangential stiffness (${\mathrm{K}}_{tt})$, are crucial since they are oriented parallel to the longitudinal axis and tangent to the cylindrical surface of the screw, respectively. The two transverse stiffness values are assumed to be equal since, for the 2D axisymmetric modeling adopted here, differentiating the stiffness values does not lead to any significant difference [18]. The shear stiffness values were then compared with experimental data from withdrawal tests, as shown in

Table 7. Cohesive surface properties for finite element analysis.

EMC (%)	Knn, Kss/Ktt (N/mm3)	${\mathit{K}}_{\mathit{s}\mathit{e}\mathit{r}}$ from Test (N/mm3)	${\mathit{K}}_{\mathit{s}\mathit{e}\mathit{r}}$ from Finite Element Analysis (N/mm3)	$\mathbf{Difference} \mathbf{in} {\mathit{K}}_{\mathit{s}\mathit{e}\mathit{r}}$ (%)	Fmax from Test (kN)	Fmax from Finite Element Analysis (kN)	Difference in Fmax (%)
12	200, 118	19.41	19.7	1.48	16.85	17.1	1.48
16	200, 146	20.32	20.04	1.38	16.52	16.4	0.73
21	200, 57.73	17.41	17.12	1.67	13.96	14.2	1.72

.

Damage begins when the cohesive interaction between two interfaces starts degrading [16]. Several damage initiation criteria are available; in this simulation, it is assumed that damage initiates when a quadratic traction function involving the nominal stress (traction) ratios reaches a value of one (Equation (62)).

$$\begin{array}{c}{\left\{{\displaystyle \frac{\langle t_n\rangle }{t_n^0}}\right\}}^2+{\left\{{\displaystyle \frac{t_s}{t_s^0}}\right\}}^2+{\left\{{\displaystyle \frac{t_t}{t_t^0}}\right\}}^2=1 \end{array}$$

(62)

The quadratic traction damage initiation criterion requires three parameters. $t_n^0$ represents the maximum traction when separation occurs normally at the screw interface and $t_s^0$ and $t_t^0$ represent the maximum tractions under pure shear in the two shear directions. As the screw’s axial loading makes the maximum normal traction the least critical parameter in this finite element model, it was arbitrarily set to 100 N/mm² [7]. The maximum traction in the shear directions was assigned the mean withdrawal strength value from the specimen’s withdrawal test results at 12%, 16%, and 21% EMC (Table 1).

A linear energy-based softening model, excluding mode-mixing, was employed as the damage evolution model. The area under the curve in Figure 12 represents the fracture energy, also called fracture toughness. This quantity reflects the energy dissipated during the complete separation of the two initially bonded surfaces. A constant fracture energy value characterized damage evolution and ultimate failure in all three directions. The fracture energy was calculated from the area under the force–displacement curves from the withdrawal tests. The reader is referred to [7]. The details of the withdrawal tests and the determination of numerical model parameters are described here.

The suitability of the properties adopted from [7] for the threaded screw model used in this study was verified by conducting displacement control simulations and comparing the numerical results with the experimental withdrawal test results at 12%, 16%, and 21% EMC (Table 7). Two metrics were used to judge the equivalence of the numerical and experimental results: the slope of the linear fit between 10% and 40% of the maximum load in the load–displacement response (K_ser) and the maximum force reached (F_max). It can be seen from Table 7 that there was a negligible difference in $K_{ser}$ and F_max, confirming the suitability of the properties adopted from [7].

4.4. Numerical Simulation Strategy and Comparison with Analytical Model

The EMC change in the glulam member was modeled using a hydro-thermal analogy in the ABAQUS Standard solver, simulating moisture content increases from 12% to 18% and 12% to 21%, as thermal stress formulations are analogous to the moisture swelling process in wood. The hydro-thermal analogy uses the mathematical similarity between Fourier’s law of heat conduction and Fick’s law of diffusion to simulate moisture transport in materials. In this analogy, heat flux is analogous to moisture flux, thermal conductivity is analogous to the diffusion coefficient, and temperature is analogous to moisture concentration [23,24].

Fourier’s law is expressed as

$$\begin{array}{c}q=-k𝛻T \end{array}$$

(63)

which is analogous to Fick’s law:

$$\begin{array}{c}J=-D𝛻C \end{array}$$

(64)

q and J are the heat and moisture fluxes, k and D are the thermal conductivity and diffusion coefficients, and T and C are the temperature and moisture concentrations, respectively.

The analogy applies to the governing equations for both transient and steady-state conditions. The equations describe how the driving variables change over time in transient conditions. For heat conduction, the transient equation is

$$\begin{array}{c}{\displaystyle \frac{\partial T}{\partial t}}=𝛻\left(k𝛻T\right)\end{array}$$

(65)

For moisture diffusion, it takes the following form:

$$\begin{array}{c}{\displaystyle \frac{\partial C}{\partial t}}=𝛻\left(D𝛻C\right)\end{array}$$

(66)

For constant density, this simplifies to

$$\begin{array}{c}{\displaystyle \frac{\partial u}{\partial t}}=𝛻\left(D𝛻u\right)\end{array}$$

(67)

Here, $u$ is the moisture content (MC). Under steady-state conditions, and in the case of uniform moisture content or temperature changes, both Fourier’s law of heat conduction and Fick’s law of diffusion simplify to Laplace’s equation:

$$\begin{array}{c}{𝛻}^{2}T=0 \end{array}$$

(68)

$$\begin{array}{c}{𝛻}^{2}u=0\end{array}$$

(69)

In these conditions, the solution depends entirely on the boundary conditions applied to the system, such as fixed temperatures or moisture content.

The total strain in wood subjected to moisture content changes can be expressed as the sum of four primary components [23]:

$$\begin{array}{c}\epsilon ={\epsilon }_e+{\epsilon }_s+{\epsilon }_{ms}+{\epsilon }_c \end{array}$$

(70)

The elastic strain (${\epsilon }_e$) represents the material’s mechanical response and is determined through the compliance matrix, incorporating the moduli of elasticity, shear moduli, and Poisson’s ratios. The shrinkage/swelling strain (${\epsilon }_s$) accounts for dimensional changes caused by variations in moisture content and is defined as

$$\begin{array}{c}{\epsilon }_s=\mathrm{\alpha }\Delta u \end{array}$$

(71)

Here, α is the hygro-expansion coefficient (moisture swelling coefficient), and Δu represents the change in moisture content. Mechano-sorptive creep (${\epsilon }_{ms}$) describes the deformation resulting from the interaction of mechanical loads and moisture changes, while creep (${\epsilon }_c$) accounts for deformation under sustained loads over time. Given that this study considers loading within a few days of installation of the screw, the effects of mechano-sorptive creep and normal creep were not included in the analysis due to their minimal impact.

An analogy with thermal expansion is employed to model moisture-induced strain. In thermal analysis, the strain caused by temperature changes is expressed as

$$\begin{array}{c}{\epsilon }_s=\mathrm{\beta }\Delta \mathrm{T} \end{array}$$

(72)

where β is the thermal expansion coefficient, and ΔT is the temperature change [16].

A comparison between Equations (69) and (70) highlights the analogy between the two phenomena. Both types of strain are governed by a material-dependent coefficient (β and α). They are directly proportional to temperature change (ΔT) in the case of thermal strain and moisture content change (Δu) for shrinkage/swelling strain. Therefore, the thermal-moisture analogy was used to simulate the MC change in ABAQUS. In this analogy, temperature corresponds to the wood’s moisture content, while the coefficient of thermal expansion represents the swelling coefficient. The wood moisture swelling coefficient along the three orthotropic directions (Table 3) was defined in the finite element model with material orientation per the specimen’s orthotropic directions.

To simulate the effect of moisture and torque on screws in steel-to-wood connections, the numerical analysis was conducted in two steps. A constant predefined temperature field was applied throughout the glulam in the initial step. In the first step, following the initial step, an axial load was introduced to simulate the torque effect. The second step involved modifying the predefined temperature field to represent moisture content variations and simulate wood swelling. Two moisture changes were analyzed, from 12% to 18% and 12% to 21%. The moisture content was assumed to be uniform inside the glulam member.

Typically, glulam members are connected to thick side members using self-tapping screws, where the thick side member can be considered rigid. A fixed boundary condition was applied to the top surface of the glulam member to represent its rigid support. The bottom surface was constrained against vertical movement in the initial analysis step. After the initial step, this boundary condition was deactivated, primarily to ensure convergence of the finite element model during the moisture swelling and shrinkage phase. The finite element model and boundary conditions are illustrated in Figure 13. A concentrated load of 5 kN was applied to simulate the axial force generated by the screw torque. This load was implemented through a reference point (RP) located at the top of the screw core, which was kinematically coupled to the screw’s upper surface to ensure proper load distribution.

In the finite element model, the vertical (y) direction, aligned with the screw’s longitudinal axis, was considered as the tangential direction of the glulam. Young’s modulus and swelling coefficient values of glulam in the tangential direction were taken as the input values ($E_W$ and $\alpha$) in the analytical model to maintain consistency with the numerical model. The reader is referred to [7] for further details on the finite element model.

The contour plots of the stress distribution from the numerical simulations, shown in Figure 14, illustrate the effects of different axial loads and moisture content (EMC) changes. Figure 15 compares the analytical and numerical stress distributions along the length of the screw, starting from the screw entrant side in the glulam. It is important to note that the stress distribution shown in Figure 15 follows the screw path marked in Figure 13, which represents the center of the screw. This path was defined using the ABAQUS/CAE visualization module.

The contour plots in Figure 14 reveal that higher stresses occur at the screw threads, as these areas experience stress concentrations compared to the stresses along the screw center. Both the analytical and numerical stress distributions follow similar trends. The undulations in the numerical stress distribution are attributed to the inclusion of screw thread geometry, which creates a stress concentration that propagates across the screw length. This phenomenon cannot be captured in the analytical model.

The differences between the maximum stresses predicted by the numerical and analytical models were 4.6% for EMC changes from 12% to 18% and 3.1% for EMC changes from 12% to 21%. The critical conditions of the screw are the higher moisture content changes and, to that end, the difference in the predicted maximum stresses from the numerical and analytical model are minimal.

5. Analytical Model Implementation and Discussion

For the withdrawal test specimen comprising 130 × 260 mm glulam and a 13 mm screw (Figure 7), the total average stress distribution in the screw was decomposed into the stress distributions from the two separate mechanisms of mechanical loading and moisture swelling of the wood. The total stress distribution was decomposed using Equations (20), (49) and (50) and illustrated in Figure 16. The mechanical axial load considered for the illustration of stress decomposition was 15 kN. The 15 kN load was arbitrarily chosen for illustration purposes only. The moisture content change considered for the stress decomposition in Figure 16 is 9% (for example, a change in EMC of wood from 12% to 21%). Although the screw stress distribution in Mechanism 1 is non-linear, as given by Equation (20), the non-linearity is not reflected if the effective length of the screw is relatively small. Thus, for the effective length of the screw of 120 mm, as shown in Figure 16, although the stress distribution in Mechanism 1 appears to be linear, it is not. With a larger effective penetration length of the screw, the shape of the stress distribution curve in Mechanism 1 will appear convex.

The axial load alone induces a maximum stress of approximately 210 MPa in the screw, while moisture-induced swelling of the wood increases this value to about 250 MPa. The typical tensile strength of 13 mm self-tapping screws used in this study is about 1200 MPa [13]. Therefore, by comparing the axial stress in the screw from Figure 16 to the tensile strength, it can be inferred that moisture swelling alone is unlikely to generate axial stress values high enough to cause axial tensile failure in screws since the moisture content change considered is already greater than typical moisture content variations in wood structures in Canadian climates [12]. According to Table 12.2.1.6 of CSA O86 [12], a moisture content greater than 19% is considered a green condition, which the interior of wood structures rarely experiences. The initial moisture content in this study is 12% and a moisture content change close to 7% is generally not expected. In summary, a sufficiently high mechanical load combined with moisture-induced wood swelling can cause tensile failure in long self-tapping screws.

In two-member connections, such as the one shown in Figure 17, an axial load can develop if the screw is tightened beyond the necessary torque to secure the connection. If the side member is rigid enough to support the upper surface of the primary member, this situation becomes analogous to screw withdrawal under pull–push loading. If the primary wood member swells due to moisture while in this over-torqued state, the analytical model presented in this study can be used to predict the axial stress distribution in the screw within the primary wood member, provided the induced axial load is known. The axial load resulting from over-torquing can be estimated by correlating it with the screw installation torque, which the second author of this study is currently investigating. Nevertheless, the mechanical axial load in the analytical model might stem from any external source, like the over-torquing of the screw or load transfer between two or more structural member connections.

From the screw stress distribution provided by the analytical model, the maximum axial stress in screws with different effective penetration lengths (L_eff) under the combined effects of axial load and moisture swelling of the wood can be determined. This maximum axial stress corresponds to the peak point in the “Total Stress” curve shown in Figure 16. The maximum axial stresses in screws with 8 mm and 13 mm outer nominal diameters at various penetration lengths inserted into the glulam main member, as shown in Figure 17 (where d is the outer nominal diameter of the screw), were calculated. The induced nominal stresses in an STS at different penetration lengths, subjected to axial loads of 5 kN and 7 kN and three moisture content changes in the main glulam member, are plotted in Figure 18, Figure 19, Figure 20 and Figure 21. These plots were generated using the method outlined in the flowchart of Figure 8. The axial load values of 5 kN and 7 kN were based on reasonable estimates from an ongoing test program conducted by the second author, which investigates the relationship between screw over-torque and the induced axial load.

In Figure 18, Figure 19, Figure 20 and Figure 21, the tensile strengths of the screws [13] are shown as horizontal lines. For the 8 mm screw, a 9% change in EMC results in the maximum axial stress exceeding the tensile strength when the effective penetration length exceeds approximately 225 mm for a 5 kN axial load and 200 mm for a 7 kN load. In contrast, for the 13 mm self-tapping screw, the maximum axial stress exceeds the tensile strength at an effective penetration length of about 425 mm for a 5 kN load and 400 mm for a 7 kN load. A 9% EMC change is unlikely in timber connections unless they are exposed to outdoor conditions, and the same holds for a 6% EMC change. For a 6% EMC change, the critical screw lengths for the 8 mm screw are approximately 325 mm and 300 mm for 5 kN and 7 kN loads, respectively. For the 13 mm screw, the critical lengths are around 600 mm and 575 mm for 5 kN and 7 kN loads, respectively.

The curves shown in Figure 18, Figure 19, Figure 20 and Figure 21 can be interpolated for intermediate values of EMC changes. However, new plots must be made for different axial load values. Curves similar to Figure 18, Figure 19, Figure 20 and Figure 21 can be produced for self-tapping screws of different nominal outer diameters (d), moisture content changes (u), axial load values (P), and effective penetration lengths (L_eff) from the analytical model. As shown in Figure 8, the input parameters of the analytical model are the withdrawal stiffness (K_w) from a screw withdrawal test, the screw geometry (d, d_core and L_eff), the screw’s modulus of elasticity (E_s), the connection geometry (A_w,eff), the main wood member’s modulus of elasticity (E_w), and the shrinkage/swelling coefficient ($\alpha$) parallel to the screw axis. The two MATLAB programs developed by the first author can generate these curves and provide an easy implementation of the analytical model, which is available on the first author’s GitHub page. These curves can form the basis of design guidelines for STSs in terms of the induced axial load stemming from initial screw installation torque or any other sources and the maximum expected moisture content change in the wood member from the time of screw installation.

6. Conclusions

This research presents an analytical model for predicting the axial stress distribution in self-tapping screws embedded in a wood member. The model accounts for externally applied axial mechanical loads and additional loads induced by wood moisture swelling. Validation was performed through numerical analysis using the finite element method. The analytical model could not be validated with experimental data, as we currently do not have a method available to measure the stress distribution of the screw within the wood member.

This study primarily focuses on screws embedded in mass timber products such as glulam and CLT, but the proposed model can be applied to any wood member. For simplicity, the model assumes a uniform moisture content change throughout the wood member. Future research could extend this approach to incorporate non-uniform moisture variations. Additionally, the effective area of the wood in swelling, a key parameter in the analytical model, would benefit from further exploration.

Despite the limitations, this study establishes a foundation for analyzing the critical stress state in wood screws under the combined effects of axial loads and moisture swelling of the wood. The proposed model is mechanics-based and does not rely on empirical coefficients or experimental calibration. The input properties for the model can be determined from simple screw withdrawal tests. A computational tool has been developed to predict the stress distribution and maximum axial stress in self-tapping screws across various penetration lengths, given a specific axial load and a range of moisture content changes. By integrating moisture-induced axial loads into the stress analysis of axially loaded self-tapping screws, this research can contribute to safer design practices and the development of standards for the structural use of self-tapping screws in timber structures.