Finite Element Modeling of Nailed Connections for Low-rise Residential Home Structures Jeffrey Weston 1, Wei Zhang 2

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1 Weston, J., Zhang, W. (2017) Finite element modeling of nailed connections in lowrise residential home structures. Structural Congress 2017, April 6-8, Denver, CO. Finite Element Modeling of Nailed Connections for Low-rise Residential Home Structures Jeffrey Weston 1, Wei Zhang 2 Abstract: Increased frequency and intensity of coastal storms as well as the restructuring of flood insurance policies has renewed interest in the structural performance of coastal residential homes under wind, flooding and storm surge hazards. Varying construction techniques and environmental deterioration often results in nailed connections that are weaker than the structural members they join. Therefore, global performance of light framed residential structures is found to be greatly dependent on the performance of individual connections. Nonlinear behavior of these connections could further complicate accurate characterization of the connection response in the structural system. Traditional finite element approaches such as using multiple nonlinear springs to define directional stiffness often neglect the coupling effects of the multidirectional connection response. Furthermore, the implementation of nonlinear spring connector elements along the corners and edges of a structure could be difficult and time consuming for the user since additional nodes or elements and coupling functions have to be built or used in the modeling process. In the present study, an innovative connection modeling scheme is proposed to model nailed connections using an equivalent element. Experimental data and a nonlinear material model is used to define a single parameterized beam element to represent a connection between structural members. The parameterized beam element retains an intrinsic coordinate system for ease of meshing and couples the axial and transverse connection response for a more conservative displacement solution. 1. INTRODUCTION Low-rise, single family residential homes (SFRHs) are usually non-engineered, lightframed, wooden structures. Typically nails are used to connect structural components such as framing and sheathing. Nailed connections are capable of resisting lateral and withdrawal forces as well as their combined effect (Soltis and Ritter 1996). Connections are of critical importance to structural performance because they are often more vulnerable than the structural members they join. Connection performance depends on the material condition, environmental conditions, and installation techniques. Due to the number of factors influencing their performance as well as their role in the structure s ability to resist loading, connections are considered one of the most important, but least understood components in the SFRH structural system (Soltis and Ritter 1996). Major connections in the SFRHs combine the structural members to form a load path. Within the load path, many connections, such as those joining sheathing to roof 1 Graduate Student, Dept. of Civil and Environmental Engineering, University of Connecticut, Storrs, Connecticut 06269, Jeffrey.weston@uconn.edu. 2 Assistant Professor, Dept. of Civil and Environmental Engineering, University of Connecticut, Storrs, Connecticut 06269, wzhang@engr.uconn.edu.

2 framing and roof framing to wall plates, have been widely studied. Historically, the limits of roof components in SFRHs have been recognized through post hazard observations of failure rather than through analytical or experimental methods (Surry et al. 2005). Although typical SFRH construction planning does not involve an extensive structural analysis, analytical techniques such as the finite element method are well suited to investigate the structural performance of new or existing homes subjected to various loading conditions. To analyze the structural response of a SFRH, the connections should be modeled to effectively predict the global stiffness and response of the entire SFRH. The method of simply sharing of nodes between adjacent structural elements such as sheathing and framing would give the connection an infinite capacity to transfer moments and forces to the members it joins. This simplification, obviously, is not realistic even for the new, well-constructed connections, let alone those connections with damage or decay. To more accurately characterize the connection behavior while not over predicting the global stiffness, spring elements are often used to model the connections in SFRHs. Initially, linear springs models were common and shown to exhibit different performance than that predicted by a classic tributary area approach (He and Hong 2012; Mizzell 1994). Although they performed well, the linear models could only account for very small displacements. The desire to predict the onset of structural failure led to the inclusion of a nonlinear spring model, whose parameters could be obtained experimentally by observing the response of a nailed connection tested in orthogonal directions. Nonlinear springs have been used by many researchers to model connections due to the simplicity of the element formulation and adaptability to nonlinear test datasets (Dao and van de Lindt 2008; Kumar et al. 2012; Pan et al. 2014; Thampi et al. 2011). However, the available nonlinear spring models in finite element software are usually one-dimensional. Three coincident nonlinear spring elements are needed to fully model the translational behavior for a typical connection. Often, for simplicity, nodal rotational effects are not considered. Neglecting nodal rotations may be justifiable when internal sheathing moments are symmetric about the connection, such as in the center of a uniformly loaded panel. Along the corners and edges of a structure, or when loading is not uniform, nodal rotations may take place and it is desirable to capture this effect. Accordingly, Dao and van de Lindt (2008) proposed an updated nonlinear spring element approach to include both rotational and displacement behavior of nailed connections. This approach could require up to six coincident spring elements to represent all nodal degrees of freedom. To characterize softening-type connection behavior, a new equivalent parameterized beam connector (EPBC) model is proposed. The paper is organized in the following way. After a brief introduction of the ENSC and EPBC modeling schemes, a sample procedure to implement the EPBC method is provided. Single element comparisons between EPBC and ENSC formulations are presented. Performance of the two connection modeling schemes is compared in the linear and nonlinear response regimes for one dimensional and multi-dimensional loading.

3 2. MODELING SCHEMES 2.1 Equivalent nonlinear spring connection model (ENSC) A single nonlinear spring element closely replicates experimental force deflection behavior in one dimension through a piecewise linear response. In order to define the unique translational behavior in three dimensions, three elements are typically used for each connection to create a connector set. If the rotational stiffness is to be modeled as well, an additional 3 rotational spring elements are required and up to 6 elements could be necessary to completely define the connection behavior with the ENSC formulation. Since ENSCs lack an intrinsic coordinate system, the nodes connected by ENSCs should have the same orientation, which should agree with the directional stiffness in the element formulation. Judd and Fonseca have addressed this issue with their oriented vs. non-oriented spring pair models, however often times the non-oriented spring pair model is still favored for ease of implementation (Judd and Fonseca 2005; Thampi et al. 2011) potentially compromising results when displacements are large. At the corners and edges of structures where non-collinear elements share common nodes, the nodal misalignment causes a conflicting directional stiffness definition. If more than one set of connector elements are not collinear, the shared node between sets cannot adapt to the axial and transverse directions of both connector sets as illustrated in Fig 1. If axial properties were defined in the global Z direction, node 1 cannot adapt a coordinate system that satisfies the axial direction of the connector from 1 to 3 and 1 to 2 at the same time. Work-around remedies for this situation exist and may include redundant sets of force deflection data such that nonlinear springs with the same global degrees of freedom are assigned different behavior depending on their orientation. Other options include defining duplicate nodes with coupling constraints. In most cases these efforts increase model size, which could reduce calculation efficiency and may even cause loss of modeling fidelity at arguably the most critical locations in the structure. Figure 1: Nodal coordinate misalignment scenarios The stiffness of ENSCs may be decomposed incorrectly in a large deflection analysis due to the reliance on nodal coordinates. After a series of load increments, if the line between nodes which represents the axial direction is no longer parallel to the direction for which the axial properties are defined, or if nodal rotations have occurred, the subsequent axial displacements may include transverse stiffness

4 contributions as illustrated in Fig. 1. Such a case is likely when ENSCs connect beams to shells. Pressure loading on the shell elements used to represent sheathing remains normal to the deformed shell surface. However, as the shell nodes rotate, the associated beam node connected by an ENSC may not rotate by the same amount causing a directional mismatch between nodal coordinate systems and an uncertain subsequent decomposition of stiffness. Usually, connections are tested purely in the axial or transverse direction to isolate orthogonal load-displacement curves for the ENSC model. In fact, realistic loading always includes a combined axial and transverse response, such as that due to uplift and lateral forces. The decoupled ENSC response may partially explain why some full-scale structural tests show connections failing at only 15% of the observed connection capacity when tested separately (D Costa and Bartlett 2003; Surry et al. 2005). The interaction between the axial and transverse loading should be included in the connection model because it may reduce the capacity of the connection (Judd and Fonseca 2005). Due to the considerable meshing difficulties along corners and edges in SFRHs and the potential for excessive stiffness in nonlinear analyses, it is desirable to consider an alternative to the ENSC. 2.2 Equivalent parameterized beam connection model (EPBC) Due to uncertainty in the wood grain orientation, only axial and transverse directions for the connection deformation are considered in the present study. Experimental work in the literature (Dao, 2008; Hong & Barrett, 2010; Thampi, Dayal, & Sarkar, 2011; etc.), has shown that the axial withdrawal behavior and the transverse deflection both follow typical patterns. A bi-linear elastic-plastic curve reasonably approximates the axial load deflection response of nailed connections. A bar, or axially loaded beam element, with perfect elastic-plastic behavior would exhibit a similar response to axial loading because the entire cross section yields at once. In the transverse direction, the load deflection curves appear to often have an initially linear region followed by steadily decreasing stiffness until the slope becomes negative. In the linear region, and the nonlinear region before the point of zero stiffness, the load deflection curve resembles an isotropic plasticity curve with strain softening behavior. Assuming the former deflection shapes and neglecting the negative stiffness portion of the load deflection curve, it is possible to scale the cross sectional properties of a single beam element such that when coupled with a bilinear elasticplastic material model the resulting element closely resembles observed experimental behavior. The intrinsic element coordinate system attached to the beam will eliminate the shared node and large displacement peculiarities illustrated in Fig. 1, thus reducing the meshing work for the user. The EPBC element will be readily applicable along corners and edges with no constraints on the nodal coordinate system orientation. Additionally, the computation of the element stress will couple the axial and transverse displacement effects. The ability of a single element to handle both axial and transverse deflections could reduce the number of required translational connection elements by a factor of 3. Finally, element solution quantities such as elastic and plastic strain could prove useful in evaluating the potential failure of the connection element due to excessive relative displacement between structural members.

5 To calibrate the EPBC, a beam element with a radially symmetric hollow circular cross section is chosen. Deriving the stiffness matrix for such an element in terms of the cross sectional constants allows the scaling of these constants to force the beam behavior to approximate the experimental data. The radii are sized to guarantee accurate performance in the linear response region. Young s modulus and tangent modulus are scaled to approximate the nonlinear response. The applicability of the EPBC method presented is dependent on the assumption that the experimental data to be represented has an initially linear force-deflection response followed by the typical nonlinear behavior in each respective direction as previously noted. 2.3 Calculation of constants for equivalent parameterized beam connector (EPBC) The finite element form of the stiffness matrix for a single linear Timoshenko beam element is used to extract the force-deflection relationships for one dimensional loading in the axial and transverse directions. Rearranging the force deflection relationship in terms of cross sectional properties and equating the finite element form of longitudinal stress due to axial and transverse loading contributions provides a set of equations. Solution of the equations determines the appropriate radii and Young s Modulus to cause the cross sectional stiffness to support the observed experimental force-deflection relationship. Other important terms to implement this procedure include a user defined shear deflection constant and two criteria defining the range of acceptable input parameters for the procedure to be applicable. Based on the derivation by Weston and Zhang (Weston and Zhang 2016), the appropriate scaling parameters for the EPBC element are given below. r o = Lu(θ+1) 2v (1) r i = L 2v u(θ+1) F x (4F y v F x u(1 + θ)) (2) E = T = 4EI θl 2 σ yd = 2F x 2 v 2 Lπu 2 (θ+1)(f x u 2F y v+θf x u) F x π(r o 2 r i 2 ) = F y L r o 2( π 4 (r o 4 r i 4 )) (3) (4) (5) 4F y > F x (1+θ)u v (6) 2F y (1+θ)u < F x v (7)

6 Based on the inequalities in equations 6 and 7, the region of acceptable input parameters can be determined at the chosen shear deflection constant. Figure 2: Feasibility region for EPBC method In the event that the experimentally observed force deflection data does not fall into the region specified by equations 6 and 7, the imposed shear deflection constant can be adjusted to rotate the acceptable region specified by the inequality. As long as the shear deflection constant remains of the form presented in equation 4 where only a constant is modified, the inequality bounding the upper region of acceptable values will have twice the slope of that bounding the lower region. The modification of the shear deflection constant has no effect on the longitudinal stress computation due to the formulation of standard Timoshenko beam elements. 3. IMPLEMENTING PARAMETERIZED BEAM CONNECTORS 3.1 Sample procedure to calibrate EPBC: Figs. 3 and 4 show some sample experimental curves for axial and transverse force deflection data, taken from Thampi et al. (2011). First, the coordinates at the limit of the experimentally observed linear region for each plot are estimated. Next, the length of the connection element is determined based on the distance between the nodes of the two structural members to be connected. The inputs and outputs for the scaling calculation procedure are summarized in Tables 1 and 2. The contents of Table 2 are used to define the appropriate hollow cylindrical section, the bilinear isotropic elastic-plastic material model and the transverse shear deflection constant necessary to calibrate an EPBC element.

7 Table 1: Inputs for EPBC calculation Inputs: Value: Comment: L 2 in -As req d in analysis model F y 370 lbs -Approximate coordinates at v in limit of linear region of F x 180 lbs transverse and axial u in deflection curves. θ 1 -Chosen constant Table 2: Outputs from EPBC calculation Calculated Parameters: Value: Comment: r o in -Equation 1 r i in -Equation 2 E psi -Equation 3 E t psi -Initial value of E used 10 σ yd 4.85 psi -Equation 5 T lbf -Equation 4 ν 0 -Poisson s Ratio set to zero 3.2 Single element comparison: ENSCs and EPBCs are both used to approximate the experimental load deflection behavior. Using a single ENSC or a single EPBC, the analytical force deflection performance can be compared to the experimental data that the element seeks to represent. The ENSC and EPBC models are cantilevered at the left end and then have either a pure axial or purely transverse load applied at the tip. The resulting load deflection curves are presented in Figs. 3 and 4 respectively. Figure 3: Single element comparison between Experimental data, ENSC and EPBC, axial loading

8 Fig. 3, presenting the axial load deflection, shows very good agreement between the two connection modeling schemes and the experimental results in the linear region. Negative stiffness is noted in the experimental data after the initial linear limit. Use of zero, or negative stiffness in the ENSC definition is possible. EPBCs cannot model negative stiffness, however perfectly plastic behavior is possible. Defining a negative or zero slope force deflection relationship after the initial linear limit is likely to cause convergence issues regardless of the connection modeling scheme especially in analyses of the entire SFRH structural system. In practice, a small tangent modulus is chosen for the EPBC formulation to aid in convergence of the solution. Based on sensitivity analysis carried out for this study, the recommended starting value is E t = E 10. Figure 4: Single element, 2D loading response comparison of connectors and experimental data Fig. 4, presenting the transverse load deflection, also shows very good agreement between the two connection modeling schemes and the experimental results in the initial linear region. There is fair agreement between EPBCs and experimental input data at the beginning of the nonlinear region. As deflections become larger, the agreement appears to improve. In the transverse direction, the choice of tangent modulus influences how quickly the deflection curve bends over, in other words, the onset rate of plastic deformation. The choice of tangent modulus is then a balance between approximating the form of the nonlinear region for both axial and transverse deflections. This is primarily why the current EPBC method is only applicable where bi-linear and softening behavior is observed in the axial and transverse directions respectively. ENSC elements may be more versatile in the sense that there are no restrictions on the shape of the piecewise force deflection curve defined. However, the coarse definition of the piecewise stiffness may lead to excessive solution times. Since the nonlinear stiffness in the EPBC formulation is dictated by the yielding of the cross section propagating inward from the outermost fiber, there is a functional relationship between force and deflection for all values of deflection. In contrast, the

9 ENSC formulation relies on a discontinuous piecewise definition of the force displacement curve potentially causing solution difficulties if a load step is executed near one of the defined discontinuities. It should also be noted that a better calibration of the EPBC may be achieved by choosing a different point as the linear limit in Table 1. EPBC performance is directly dependent on the point used for the linear limit which is not always well defined. Figure 5: Loading and boundary conditions used to illustrate EPBC load coupling To illustrate the coupling effects of EPBCs, consider Fig. 5. Based on Figs 3 and 4 the applied loads in Fig. 5 are within the linear response of the connection if the axial and transverse effects are independent. If the true response of a nailed connection is coupled, then loads in the axial and transverse directions that are each only slightly below the linear response limit would likely lead to nonlinear deformation. Fig. 6 shows the magnitude of the deflection for a single ENSC and EPBC subject to the boundary conditions and loading of Fig. 5. Figure 6: 2D loading comparison, 150lb and 350lb in axial and transverse directions respectively The results for the simple combined loading of a single element show a decoupled ENSC response. For the ENSC, the stiffness in the resultant direction of the loading is the vector sum of the stiffness in each orthogonal direction. In contrast, the EPBC element couples the axial and transverse loads through the computation of element stress and begins exhibiting nonlinear behavior at roughly 55% of the applied loading. These results support the claim made in Fig. 1 showing erroneous decomposition of the connector stiffness under large deflections. ENSCs generally

10 have very few solution outputs available in the post processor. As such, evaluating simple criteria such as the relative displacement between structural members can be difficult. When EPBCs are utilized, solution quantities such as element strain are available. Although meaningless in the traditional sense of judging material failure, the axial EPBC strain provides a means of quickly determining relative displacements across the connection. Experimental tests that apply combined transverse and axial loading are needed to better evaluate validity of the performance predicted by the EPBC formulation. The present results can confirm only the ability to couple directional effects, thus suggesting a more conservative solution compared to the ENSC. The ability to couple directional stiffness effects may lead to more realistic displacement solutions in critical regions of the structure such as along corners and edges where failure often initiates. 4. CONCLUDING REMARKS Connection performance greatly influences the structural response of SFRHs. Accordingly, a reliable and robust method for modeling connections in global structural models is desirable. Currently ENSCs offer simple element formulation, scalability and accurate representation of one-dimensional loading, however there may be inaccuracies in the way they handle multi-dimensional loading especially at larger deflections. Furthermore the reliance of ENSCs on a nodal coordinate system creates implementation challenges along corners and edges for the user when the finite element mesh is constructed. The current connection modeling schemes may cause a loss of modeling resolution at these critical locations. In the present study, an alternative method of modeling nailed connections with a single equivalent parameterized beam element was proposed. When modeling sheathing to framing connections for the interior sections of a panel, use of a single EPBC reduces the number of elements required to represent the translational behavior by a factor of 3 compared to ENSCs. Along corners and edges where the need for non-collinear connection elements arise, the EPBC method is readily applicable due to the use of an intrinsic coordinate system. There is no need to consider other methods such as those involving duplicate nodes, redundant sets of directional stiffness or coupling constraints thus further reducing the number of elements in a global structural model. In a single element comparison, EPBC performance closely matches ENSC behavior for one dimensional axial or transverse loading. Under combined loading, EPBCs couple axial and transverse effects through the computation of element stress leading to a more conservative solution compared with ENSCs. Additionally, elemental solution quantities available in the EPBC output, such as axial strain, may be useful in evaluating structural failure. Future work will be conducted to compare the combined loading response of EPBCs and ENSCs to experimental results. 5. ACKNOWLEDGEMENT Funding for this work was partially provided by Connecticut Sea Grant, University of Connecticut through Award No. NA14OAR , Project Number R/CH-1 and the Graduate Assistance in Areas of National Need (GAANN) fellowship, Grant number P200A The support is greatly appreciated. Any opinions, findings,

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