Modeling of shear walls using finite shear connector elements based on continuum plasticity

Abstract

Light-frame timber buildings are often stabilized against lateral loads by using diaphragm action of roofs, floors and walls. The mechanical behavior of the sheathing-to-framing joints has a significant impact on the structural performance of shear walls. Most sheathing-to-framing joints show nonlinear load-displacement characteristics with plastic behavior. This paper is focused on the finite element modeling of shear walls. The purpose is to present a new shear connector element based on the theory of continuum plasticity. The incremental load-displacement relationship is derived based on the elastic-plastic stiffness tensor including the elastic stiffness tensor, the plastic modulus, a function representing the yield criterion and a hardening rule, and function representing the plastic potential. The plastic properties are determined from experimental results obtained from testing actual connections. Load-displacement curves for shear walls are calculated using the shear connector model and they are compared with experimental and other computational results. Also, the ultimate horizontal load-carrying capacity is compared to results obtained by an analytical plastic design method. Good agreements are found.

Keywords

shear walls / wall diaphragms / finite element modelling / plastic shear connector / analytical modelling / experimental comparison

Introduction

Background

In many European countries the fire regulations have been changed from prescriptive to functionally based ones. The result of this change is that timber can be used as structural material also for buildings more than two stories high. The increased height of the buildings means that the horizontal stability of the buildings becomes an important issue to be solved.

The traditional way of designing timber frame buildings in the Nordic countries is to regard the building as composed of 2-dimensional structural elements fastened to each other. The shear design of the walls against wind loads is normally carried out using analytical linear elastic methods where each wall is assumed to be fully anchored against vertical uplift. The consequence of this design is often the need to use big tie-downs resulting in complicated and expensive detailing.

To find more economic solutions, more advanced design methods should be applied, e.g., analytical plastic methods and finite element methods that include nonlinear and plastic types of characteristics of, especially, the sheathing-to-framing joints, which are decisive for the shear wall behavior. Since the analytical elastic methods are only capable of analyzing shear walls that are fully anchored (i.e., the leading stud needs to be anchored to the substrate using some kind of hold-downs), it is essential to develop and use methods that are capable of analyzing shear walls that are only partially anchored, which often is the case in practice.

Analytical plastic design methods have been developed by Källsner and Girhammar (see e.g., [-]) that are capable of analyzing both fully and partially anchored sheathed shear walls subjected to arbitrarily distributed vertical point loads. The models are based on the assumption of plastic load-slip relations for the sheathing-to-framing joints. The fasteners can either be modeled as discretely located or uniformly distributed. Due to the “extra” fastener in the corners, the discrete modeling of the fasteners will give higher capacity than that of uniform or smeared modeling. These plastic analysis methods have several advantages and enable more efficient material usage and increased productivity. They are thus capable of taking into account the real tying down conditions for shear walls in practical structures and make it possible to avoid expensive and complicated anchoring to the foundation. In addition, the 3-dimensional behavior of buildings can be utilized through connecting the shear walls to the transverse walls and reduce or eliminate the need for separate hold down devices. With the plastic method it is possible to combine different types of anchoring, e.g., hold-downs and transverse walls, and take the remaining uplifting force through the sheathing-to-framing joints. The method also makes it possible to include the load-bearing capacity of wall segments including openings. The plastic analytical models are general in nature and can thus be used in design of shear walls with different sheet materials, sheathing-to-framing joints, geometric layout, anchoring conditions and load configurations. For further details, see Section 5.3.

Finite element methods for analyzing shear walls have been used by many (see e.g., the early works [,] and the later works [-]. Those models can naturally be applied to the same and even more general condition as mentioned above for the plastic analysis methods. With respect to the sheathing-to-framing joints, they generally use nonlinear elastic springs for discretely located fasteners in their modeling. Although the models are nonlinear, they are employed in terms of total force versus total displacement. Some models are path independent and some path dependent. Only some minor attempts and illustrations have been made to use non-elastic characteristics for the analysis []. For further details, see Sections 2.3.1 and 2.3.2.

In this paper, the authors present a finite element model based on continuum plasticity as an extension to the hitherto nonlinear elastic models. As known to the authors, this is the first time a rigorous model using the solid mechanics theory of plasticity is presented for the analysis of sheathed shear walls. The concept is that the shear forces in the discretely located fasteners of the sheathing-to-framing joints are modeled as shear stresses in a smeared layer between the sheathing and the framing members. This smeared or uniformly distributed layer representing the fasteners corresponds to the model with uniformly distributed fasters in the plastic analysis model mentioned above. This shear layer is then given plastic characteristics with a yield criterion, a hardening rule and a flow rule, and the calculation is performed in terms of increment stress versus increment strain. This shear connector model is path dependent. For further details, see Section 3.1.

Geometry and loading conditions

A typical part of a shear wall is shown in Fig. 1(a); a wall segment consisting of a sheet fastened to a timber frame with mechanical fasteners. The width of the segment is, b, and the height is, h. The spacing of the fasteners (s_i) along the top and bottom rail is denoted s_r, along the perimeter studs s_ps, and along the intermediate stud s_is. In this paper, one-sided sheathing is considered.

The racking load is applied either diagonally or horizontally at the upper left corner of the top rail acting in the plane of the frame of the shear wall as illustrated in Figs. 1(b) and (c). The diagonal load corresponds to a horizontal load at the level of the top rail and a vertical load on the leading stud. The vertical component of the diagonal load will hold down the leading stud completely and, thus, make the wall behave as a fully anchored shear wall. For a fully anchored shear wall, the maximum capacity is attained. Horizontal load application will render partially anchored conditions for the wall where there are no rigid tie-downs at the tension studs and the uplifting force is counteracted by a distributed force in the sheathing-to-framing joints along the bottom rail near the leading stud. For further details, see Källsner and Girhammar [,].

Structural behavior

The structural behavior of a shear wall is mainly governed by parameters like boundary conditions (the anchoring of the wall to the surrounding structures), wall geometry and load configuration, and mechanical properties of the different types of joints, especially the sheathing-to-framing joints.

Effect of anchorage

The structural behavior of wall diaphragms is to a large extent determined by the fixing of the wall, especially, to the foundation or floor (Fig. 2, where also the force distribution according to different models are shown). At least two different principles for simplified design are used in Europe: 1) fully anchored shear wall, where it is assumed that the vertical studs are prevented from vertical uplift in relation to the substrate (Fig. 2(a)); and 2) partially anchored shear wall, where the studs are allowed to uplift vertically in relation to the bottom rail, which is anchored to the substrate (Figs. 2(b) and (c)).

According to the first principle the vertical stud on the windward side of the wall diaphragm is fully anchored to the substrate against uplift by a hold down (Fig. 2(a)), i.e., the anchoring force is transferred via the leading stud to the foundation or the floor structure. This principle is usually used in most European countries and means that the overturning moment due to the horizontal forces H will be balanced by a stabilizing moment due to the vertical tension and compression forces at the left and right end of the wall. The tension force at the left end will here be transferred by shear from the left stud to the sheathing material.

The force distribution according to the plastic design method for fully anchored shear walls is shown where the plastic shear flow is denoted f_p [N/m] along the framing members; see Källsner and Girhammar []. To obtain simple expressions for the resistance of the shear walls, the fasteners are assumed to be continuously distributed along the center of the timber members. It is further assumed that the fastener spacing around the perimeter of the sheets is constant. For more details, see section 5.3.

According to the second principle, primarily used in the UK, only the bottom rail is fastened to the substrate (Figs. 2(b) and (c)). In this case the resultant vertical tension force at the leading stud must be transferred as a distributed load via the fasteners, connecting the bottom rail to the sheet material (sheathing-to-framing joints). The force distribution according to two different types of plastic models for partially anchored shear walls is shown in Figs. 2(b) and (c), respectively, where the plastic shear flow is f_p, both parallel along the vertical studs and the top rail, and perpendicular to the grain of the bottom rail, see Källsner and Girhammar [,-,,-].

In model according to Fig. 2(c), the shear forces from the stud-to-rail joints are taken into account. In order for this model to be a true lower bound method, it is required that the shear capacity of the stud-to-rail joints is sufficiently large and can be used for transferring shear forces.

Since the distance between the vertical tension and compression forces becomes smaller in these cases, the stabilizing moment and the load-carrying capacity will be lower. But on the other hand it can save the building from complex and expensive anchoring devices. The plastic method also allows the user to combine partial anchoring of the leading stud, via a simple anchoring device, with anchoring via the sheathing-to-framing joints and connecting the shear wall with the transverse wall.

For more details, see Section 5.3.

Effect of mechanical joints

The mechanical behavior of the sheathing-to-framing joints has a significant impact on the structural performance of a shear wall. Most sheathing-to-framing joints show nonlinear load-displacement curves with more or less plastic behavior. Fairly brittle characteristics may however in some cases occur. An example of such a situation is when the edge distance in the sheathing material is small and the fasteners cause tearing of the sheathing material at the edges. This type of failure is very sensitive to the direction of the fastener force and is most brittle when this force acts perpendicular toward the edge of the sheathing material. However, if only a few of the sheathing-to-framing joints show brittle failure modes, the overall behavior of the shear wall may anyhow be fairly plastic.

Concerning stud-to-rail joints, the characteristics with respect to tension, shear and compression, and combinations thereof are of interest.

Short review of analysis methods for shear walls

Much research work has been conducted in the area of sheathed wood frame shear walls. Most of the studies concern walls or buildings exposed to monotonic and earthquake type of loadings. However, there are relatively few studies dealing with partially anchored shear walls.

As the structural behavior of mechanically jointed shear walls is governed to a large extent by the mechanical properties of the sheathing-to-framing joints, they are in focus in the modeling.

Analytical elastic methods

An analytical method, frequently used in the Nordic countries for fully anchored shear walls, is based on the following assumptions (Källsner and Girhammar []):

• the sheathing-to-framing joints have linear elastic load-displacement characteristics

• the timber members are rigid

• the sheet material is rigid, and

• the timber members are connected to each other by hinges.

The assumption of a linear elastic load-displacement curve for the sheathing-to framing joints is from a practical point of view almost never fulfilled. A typical slip curve is shown in Fig. 4 with a region of relatively high initial stiffness and after the maximum load a region of relatively constant load capacity. The linear elastic model may work satisfactory for small loads in the serviceability limit state but is not always proper for design in the ultimate limit state when the displacements are large.

The last assumption listed above means that the displacements in the joints between the vertical studs and the bottom and top rails are assumed to be equal to zero everywhere in the shear wall. To fulfil this condition the uplift of the vertical stud on the windward side should be prevented by a tie-down fastened to the foundation.

A literature review of the elastic methods is found in Källsner & Girhammar [].

Analytical plastic methods

The principles for a new plastic design method, that is capable of including the effects of the influences mentioned above, have been developed by the authors, see for example Källsner and Girhammar [,,,], Vessby et al. [], and Girhammar and Källsner [,]. The method is based on the static theorem (Neal []) and renders lower bound values for the plastic capacity of the sheathed wood frame shear walls and may be used for different geometrical configurations (e.g., openings), boundary conditions (e.g., partial anchorage, inter-component connections) and loading configurations (e.g., vertical loads). The method assumes plastic properties of the sheathing-to-framing connections and is a simple engineered method, which efficiently makes use of the constituent materials and takes practical supporting conditions for the wall into account.

Finite element methods

Earlier developments

Foschi [] was one of the first to develop a finite element model for wood shear walls. In his analysis he considered four basic structural elements: the sheathing, the framing members, the stud-to-rail joints, and the sheathing-to-framing connections. With respect to the connections between the frame members, he modeled their axial, shear, and moment load-deformation behavior with three-parameter nonlinear relationships. Concerning the connections between sheathing and frame members, he modeled them as path independent single spring elements with the same type of three-parameter nonlinear load-deformation relationship as for the stud-to-rail joints. The joint force was based on the total displacement of the joint and not on the local incremental change in deformation. The load to grain direction dependency was included in the three parameters by using a Hankinson type of formula for an arbitrary angle between force and grain direction knowing the parallel and perpendicular joint characteristics. It is worth noting that the nonlinear load-deformation relationship used for the sheet connector is of a monotonically increasing kind with no ultimate capacity.

Dolan & Foschi [] extended this model to include the bearing effects between adjacent sheathing panels, the out-of-plane behavior of sheathing panels, and the prediction of the ultimate load capacity of the shear wall by modeling the sheathing-to-framing connections with an ultimate capacity using a softening straight line after a chosen maximum capacity of the joint. The sheathing-to-framing connector was modeled using three independent nonlinear springs (relative movement, slip of the connection, in the directions parallel and perpendicular to the framing axis, and the relative out-of-plane movement).

Later, Foschi [] modified this softening branch of the joint characteristics by using an exponential function instead of a straight line.

Modeling of sheathing-to-framing joints by spring elements

Judd and Fonseca [] discussed various models for sheathing-to-framing connections: a so-called Single Spring (SS) Model, a Non-oriented Spring Pair (NSP) Model and an Oriented Spring Pair (OSP) Model. The action of a connector is in these models represented by the forces in x- and y-directions transferred between a point in the sheathing and the adjoining point in the frame. The models are made for 2D analysis, with the two structural elements to be connected located in the x-y-plane and overlapping. The two adjoining points have before deformation of the connector the same x and y coordinates. The notations of connector forces and deformations are shown in Fig. 3.

Although nonlinear, the models under consideration are discussed in terms of total force versus total displacement,

(1)P=Ku,

where

(2)K=\left[\begin{array}{llll}{K}_{\mathrm{11}}& K_{\mathrm{12}}& -K_{\mathrm{11}}& -K_{\mathrm{12}}\\ K_{\mathrm{21}}& K_{\mathrm{22}}& -K_{\mathrm{21}}& -K_{\mathrm{22}}\\ -K_{\mathrm{11}}& -K_{\mathrm{12}}& K_{\mathrm{11}}& K_{\mathrm{12}}\\ -K_{\mathrm{21}}& -K_{\mathrm{22}}& K_{\mathrm{21}}& K_{\mathrm{22}}\end{array}\right];\hspace{1em}P=\left[\begin{array}{l}{P}_{\mathrm{1}x}\\ P_{\mathrm{1}y}\\ P_{\mathrm{2}x}\\ P_{\mathrm{2}y}\end{array}\right];\hspace{1em}\mathrm{and}\hspace{1em}u=\left[\begin{array}{l}{u}_{\mathrm{1}x}\\ u_{\mathrm{1}y}\\ u_{\mathrm{2}x}\\ u_{\mathrm{2}y}\end{array}\right].

The SS-model is characterized by

(3)K_{\mathrm{11}}=K_{\mathrm{22}}=K_r=K_r({\Delta }_r),

(4)K_{\mathrm{12}}=K_{\mathrm{21}}=\mathrm{0},

where

(5){\Delta }_r=\sqrt{{\Delta }_x^{\mathrm{2}}+{\Delta }_y^{\mathrm{2}}}.

This corresponds to the action of a single nonlinear spring between the two nodal points. The orientation, \varphi, of this single spring changes if the ratio between {\Delta }_x and {\Delta }_y changes during the course of loading. The function K_r({\Delta }_r) is determined experimentally by a uni-axial monotonic test of load P versus deformation {\Delta }_r. The SS-model is relevant for connectors with isotropic characteristics, loaded monotonically in a constant direction.

The NSP-model is characterized by

(6)\begin{array}{l}{K}_{\mathrm{11}}=K_x({\Delta }_x),\\ K_{\mathrm{22}}=K_y({\Delta }_y),\\ K_{\mathrm{12}}=K_{\mathrm{21}}=\mathrm{0.}\end{array}

This means an independent action of two independent nonlinear springs, one for the force in x-direction and one for the force in y-direction. This gives the possibility to take possible orthotropic properties of the connector into account. The NSP- and SS-models coincide for monotonic loading in the x- and y-directions if K_x = K_y. For connections displaced monotonically in other directions, for instance along a trajectory of 45° with respect to the x-direction, it is found that the NSP-model overestimates the stiffness if the connector performance is nonlinear with decreasing tangential stiffness.

Also in the OSP-model is the connector represented by two springs, but the springs are in general not oriented in the x- and y-directions but instead oriented parallel and perpendicular to the initial displacement trajectory, {\varphi }_{\mathrm{0}}. The stiffness of the two springs may be different and they are

(7)\begin{array}{l}{K}_{//}=K_{//}({\Delta }_{r,\mathrm{0}}),\\ K_{\perp }=K_{\perp }({\Delta }_{s,\mathrm{0}}),\end{array}

where {\Delta }_{r,\mathrm{0}} and {\Delta }_{s,\mathrm{0}}is the deformation parallel and perpendicular to the {\varphi }_{\mathrm{0}} direction, respectively. By conventional coordinate transformation it is for the OSP-Model found that

(8)\begin{array}{l}{K}_{\mathrm{11}}=K_{//}{\cos }^{\mathrm{2}}{\varphi }_{\mathrm{0}}+K_{\perp }{\sin }^{\mathrm{2}}{\varphi }_{\mathrm{0}},\\ K_{\mathrm{22}}=K_{//}{\sin }^{\mathrm{2}}{\varphi }_{\mathrm{0}}+K_{\perp }{\cos }^{\mathrm{2}}{\varphi }_{\mathrm{0}},\\ K_{\mathrm{12}}=K_{\mathrm{21}}=K_{//}\cos {\varphi }_{\mathrm{0}}\sin {\varphi }_{\mathrm{0}}-K_{\perp }\cos {\varphi }_{\mathrm{0}}\sin {\varphi }_{\mathrm{0}}.\end{array}

The OSP-model does not, unlike the NSP-model, allow for initially orthotropic connector properties. But it can for approximately isotropic connectors in general, be expected to be more accurate than the NSP-model, at least in the cases of constant deformation orientations, \varphi ={\varphi }_{\mathrm{0}}\ne \mathrm{0}.

The SS-, NSP- and OSP-models are mathematically all defined as for nonlinear elastic performance. Judd and Fonseca [] did, however, illustrate and investigate also non-elastic performance during loading reversals. Consideration to non-elastic performance requires experimental tests of the connector force-displacement performance during unloading. In the analysis it is then required that the sign and the history of the deformations are considered when spring stiffness is evaluated from the experimental result.

New finite shear connector element based on continuum plasticity

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2D connector spring elements of the type discussed in the preceding section can also be thought of as shear elements. A shear element represents the compliance of a thin layer between the two structural elements that are connected. The connector force P_x corresponds to a shear stress {\tau }_{xz} and the connector deformation {\Delta }_x to a shear strain {\gamma }_{xz}, etc. To make the analogy complete a bond layer area associated with the connector and a bond layer thickness must be defined. The choice of this area and thickness is, however, arbitrary. The SS-model is in terms of shear elements an isotropic element, the NSP- and the OSP-models are orthotropic with the principal material axis oriented in the x and {\varphi }_{\mathrm{0}} directions, respectively.

The 2D spring elements discussed can be described as tailored engineering models that can give accurate results for loading and deformation in an approximately constant direction. By means of the analogy with shear elements it is, however, possible to define the performance of connectors also within the solid mechanics theory of plasticity for homogeneous continuous materials. For calculation of increment in stress versus increment in strain within the theory of plasticity a yield criterion, a hardening rule and a flow rule are needed. Having these quantities defined, the elasto-plastic stiffness matrix that relates the stress and strain increments can be determined as described in textbooks on plasticity, e.g., Ottosen and Ristinmaa []. This shear connector model is path dependent. The incremental stress versus strain relation in conventional general theory of plasticity is

(9)\mathrm{d}{\sigma }_{ij}=D_{ijkl}^{\mathrm{ep}}\hspace{0.17em}\mathrm{d}{\epsilon }_{kl},

where the tangential elastic-plastic stiffness tensor is given by

(10)D_{ijkl}^{\mathrm{ep}}=D_{ijkl}^e-{\displaystyle \frac{D_{ijkl}^e{\displaystyle \frac{\partial g}{\partial {\sigma }_{st}}}{\displaystyle \frac{\partial f}{\partial {\sigma }_{mn}}}{D}_{mnkl}^e}{H+D_{pqst}^e{\displaystyle \frac{\partial f}{\partial {\sigma }_{pq}}}{\displaystyle \frac{\partial g}{\partial {\sigma }_{st}}}}}.

D_{ }^e is the elastic stiffness tensor, f represents the yield criterion, g the plastic potential and H the plastic modulus. Commonly it is assumed that f=g, giving an associated flow rule for a given yield criterion.

In analysis of the shear element under consideration there are only two stress components, {\tau }_{xz}, and, {\tau }_{yz}, or the two corresponding connector forces, P_x and P_y, to consider. A scalar effective connector force can, as an example, be defined as

(11)P=\sqrt{P_x^{\mathrm{2}}+P_y^{\mathrm{2}}},

and a yield function f can then be defined as

(12)f=P-P^y,

indicating yielding if f>\mathrm{0}. The yield force P^yis defined according to some hardening rule, for instance

(13)P^y=P^{yi}+k{({\Delta }^{\mathrm{p}})}^n,

where P^{yi}, kandnare material parameters and {\Delta }^{\mathrm{p}} is the effective plastic strain defined by

(14){\Delta }^p={\displaystyle \int d{\Delta }^p},

where

(15)d{\Delta }^p=\sqrt{{\mathrm{(d}{\Delta }_x^{\mathrm{p}})}^{\mathrm{2}}+{\mathrm{(d}{\Delta }_y^{\mathrm{p}})}^{\mathrm{2}}}.

Having a yield criterion and a hardening rule defined and assuming plastic flow according to associated plasticity, the following incremental load versus deformation relation is obtained,

(16)\left[\begin{array}{l}d{P}_x\\ d{P}_y\end{array}\right]=\left[\left[\begin{array}{ll}G& \mathrm{0}\\ \mathrm{0}& G\end{array}\right]-{\displaystyle \frac{G^{\mathrm{2}}}{(H+G)(P_x^{\mathrm{2}}+P_y^{\mathrm{2}})}}\left[\begin{array}{ll}{P}_x^{\mathrm{2}}& P_x{P}_y\\ P_x{P}_y& P_y^{\mathrm{2}}\end{array}\right]\right]\hspace{0.17em}\left[\begin{array}{l}d{\Delta }_x\\ d{\Delta }_y\end{array}\right],

where G is a material property, the elastic shear stiffness of the connector, meaning that for ,

(17)\left[\begin{array}{l}d{P}_x\\ d{P}_y\end{array}\right]=\left[\begin{array}{ll}G& \mathrm{0}\\ \mathrm{0}& G\end{array}\right]\hspace{0.17em}\left[\begin{array}{l}d{\Delta }_x\\ d{\Delta }_y\end{array}\right].

The plastic modulus H is

(18)H={\displaystyle \frac{d{P}^y}{d{\Delta }^{\mathrm{p}}}}=k\hspace{0.17em}n\hspace{0.17em}{({\Delta }^{\mathrm{p}})}^{n-\mathrm{1}}.

The above derivation of incremental force versus incremental deformation is just a simple example to illustrate a possible application of plasticity theory in connector modeling. Test results or detailed failure mechanism analysis may of course suggest other connector performances, for instance some other hardening rule as in the application discussed in the following.

Modeling of shear walls with plastic shear connector elements

Material properties and shear wall geometry

In this paper, shear walls with a single segment and one-sided sheathing according to Fig. 1 are studied. They are designed as follows:

• Frame members: Pine (Pinus Silvestris), C24, 45 mm × 120 mm. Rails are 1245 mm and studs are 2310 mm long. Stud spacing is 600 mm.

• Sheathing: Hardboard, C40, 8 mm (wet process fiber board, HB.HLA2, Masonite AB), 1200 × 2400 mm.

• Sheathing-to-framing joints: Annular ringed shank nails, 50 mm × 2.1 mm, (Duofast, Nordisk Kartro AB). The joints were hand-nailed and the holes were pre-drilled, 1.7 mm. Nail spacing was 100 mm along the perimeter and 200 mm along the vertical center lines of the sheets. Edge distance was 11.25 mm along the vertical studs (the edge of the sheet was located in the center of the stud) and 22.5 mm along the bottom and top rails (the edge of the sheet was located at the outer edge of the rail).

• Stud-to-rail joints: Two annular ringed shank nails of dimension 90 mm × 3.1 mm were applied in the grain direction of the vertical studs.

The sheathing was fastened to the timber frame only on one side and the bottom rail was anchored to the foundation. No tie-downs were used.

In the experiments, the diagonal load was always directed to the lower right corner of the shear walls during the test. In the computer analyses, the direction of the diagonal load was kept constant at the initial angle during the simulations. This difference results in a somewhat lower horizontal component in the testing situation than in the analysis situation and, therefore, results in a somewhat higher horizontal load capacity in the experimental than in the calculation situation. The exact location of the applied point load is at the intersection between the center lines of the two framing members (the nodal point).

In the numerical and experimental results presented below, only the horizontal component of the diagonal load is shown. The global displacement measured in the tests and calculated here was the horizontal displacement of the upper rail of the shear wall (in the tests, the horizontal displacement of the upper right corner was measured, but in the FE simulations, the horizontal displacement of the upper left corner was calculated; the difference is believed to be negligible).

The framing members were in the calculations assigned linear elastic (orthotropic) properties: E = 12000 MPa and G = 800 MPa (in the grain direction) and the sheathing (isotropic) properties: E = 6000 MPa and G = 2300 MPa.

The properties of the sheathing-to-framing connections and the stud-to-rail joints are given in the following sections.

Relationships for sheathing-to-framing joints

The load-displacement curves presented in Fig. 4 are based on tests of sheathing-to-framing joints with single fasteners loaded parallel and perpendicular to the grain of the timber members (Girhammar et al. []). In the figure, also the adapted average curves, on which the analyses discussed below are based, are shown as bold curves for the different series (cf. also Fig. 5). The average curves are based on the average load for a given displacement and in order to achieve a smooth curve each value on the average curve is obtained by weighting the value in question with respect to the closest two data points with higher and lower displacements, respectively. The testing situation in Fig. 4(a) corresponds to the conditions for the joints along the bottom rail in case of partially anchored shear walls. In the studs, the main force direction in the joints is along the length of the stud according to Fig. 4(b).

Two experimental and four different piecewise linear shear load-displacement curves for the sheathing-to-framing joints according to Fig. 5 are applied in the different FE calculations. The six curves studied represent different kinds of post-peak connector performance: ductile, brittle, gradual and rapid plastic softening or fracture, and test characteristics. The curve for gradual plastic fracture softening is taken as the mean from the tests results at loading parallel and perpendicular to grain.

Up to the peak force all assumed curves are equal and very close to the experimentally determined curves for loading parallel and perpendicular to grain. For total shear displacement less than 1.0 mm, the performance is assumed to be elastic with a shear stiffness of 500 N/mm. Thereafter plastic deformation develops with the plastic strain hardening or softening modulus Hdetermined from the effective plastic deformation calculated according to the preceding section and from a curve showing the effective force versus the effective plastic deformation, determined from Fig. 5 with the elastic part of the deformation subtracted from the total deformation.

The curves in Fig. 5 show the performance for uniaxial shear loading in the x or y directions. For other orientations of a load increment, effective shear force and effective plastic deformation according to Eqs. (11), and 14 and 15, respectively, were used. In case of unloading of a joint, a linear elastic unloading path is assumed (parallel to the initial stiffness).

Relationships for stud-to-rail joints

The stud-to-rail joints are also important in analyzing the performance of the shear wall with respect to shearing, compression and tension. In many models, the stud-to-rail joints are modeled as frictionless hinges, not allowing for any horizontal translation, uplift or plastic compression; see for example the analytical model of Källsner and Girhammar [] and the finite element model of Collins et al. []. The different properties of the stud-to-rail joints assumed in this study are shown in Fig. 6 as piecewise linear relations. These estimates are based on experimental results performed by Girhammar and Palm []. No rotational stiffness of the joint is assumed. In the finite element model, these characteristics in shear, compression and tension are assumed to be uncoupled.

FE model

The one-sided sheathed shear wall was analyzed as a two dimensional structure. Thus, only the in-plane behavior of the shear wall was considered. The finite element modeling of the sheathed light-frame timber shear wall was divided into the following parts:

• Framing members

- the 2 rails and the 3 studs are modeled by 2*20+ 3*40 Timoshenko beam elements

• Sheathing

- the sheet is modeled by 20*40 4-node plain stress plate elements

• Sheathing-to-framing connections

- the joints are modeled by 2*20+ 3*40 shear layer elements

• Stud-to-rail joints

- the joints are modeled by six joint elements.

The shear layer elements used to model the sheathing-to-framing connections were simple special purpose elements having 4 nodes and 8 degrees of freedom, and representing the performance of the connectors by an equivalent nonlinear elastic-plastic shear layer as described in the preceding sections. The stud-to-rail joints were each made up of two nonlinear elastic un-coupled springs representing the normal and shear properties of the connection.

The nonlinear structural performance was analyzed by incremental increase in prescribed displacement at the point of load application. This increase was made only in the direction of the load, i.e., horizontally or diagonally, while displacement in the perpendicular direction was free. Each increment, 0.25 mm, comprised of calculation of tangential stiffness at the starting point of the increment and calculation of the element unbalance forces at the end of the increment. The boundary conditions were represented by zero prescribed displacement for all beam elements modeling the bottom rail. The choice of a sufficiently small displacement increment size was based on convergence calculations, using different increment sizes. For the present FE-mesh and structure, increment size 0.25 mm was found to be sufficiently small to affect the computational results only marginally.

The finite element simulations were made by a code developed by means of Matlab and the finite element toolbox Calfem []. The modeling of shear walls using spring connector elements with stiffness properties described in analogy with elastic-plasticity theory for continuum media is validated by comparison to test results and to results obtained by the use of the assumption of ideal plastic performance. The numerical method used for the nonlinear calculations (incremental application of load with calculation of the unbalance force after each increment) is well established, and particularly straight forward in the present case with load application in terms of prescribed displacement of a single degree of freedom. To find a sufficiently small increment size, convergence calculations were carried out, as mentioned above.

Computational results – simulations and comparisons

In this section results from finite element simulations using the shear connector elements are presented together with corresponding results using the Single Spring Model (SS-model) obtained by Vessby et al. [14; concerning the SS-model, see also Ref. [11]] and from experimental tests conducted by Girhammar et al. [,]. Load-displacement curves for shear walls with one segment are studied.

Shear connector model

The two kinds of loading illustrated in Fig. 1b were considered: horizontal and diagonal. The calculated structural response to these loadings is for the gradual plastic softening curve shown in Fig. 5(c) indicated in Fig. 7 together with the corresponding experimental recordings [] and the analytical plastic load capacities according to Eqs. (23) and (24), respectively, in section 5.3 (dashed lines).

It can be noted that the peak horizontal load component is about 4 times higher in the case of diagonal loading than for horizontal loading. This is because failure in the case of horizontal loading essentially is due to uplift of the leading stud and the left part of the sheathing. For diagonal loading the failure is essentially due to shear failure along the rails. It is clear from Fig. 7 that there is a very good agreement between the maximum computational and analytical load capacities. However, it is also evident that the computational results underestimate the experimental load capacity.

The differences between theoretical results and experimental ones have earlier been discussed by Vessby et al. []. First, all values are as they should on the conservative side. There are some reasons for the discrepancy. One is that the fastener is modeled as smeared shear layer and not as a discrete point. The “extra” discrete fastener in the corners will render a shear capacity that, for an ordinary shear wall segment, is about 4%–8% higher than the corresponding uniformly distributed shear flow in the smeared shear connector element (compare also the differences between the curves in Fig. 9, where the single spring model of Vessby et al. [] uses discretely located fasteners). The other reason is likely associated with the differences in the manufacturing of the sheathing-to-framing joints. During the manufacturing of the test specimens for single joints, only the hardboard was predrilled. During the manufacturing of the walls, the hardboard was placed on the timber frame when the predrilling took place, resulting in that both the hardboard and the timber members were predrilled. To study this effect some additional sheathing-to-framing joint tests were carried out using predrilled and non-predrilled timber members. A preliminary evaluation indicates that the moisture content has a significant influence on the capacity. For example, for a moisture content of 10%, the test results indicate that predrilling of the timber members may increase the capacity of the joints by up to about 20%. For high moisture contents, the corresponding increase in capacity is only a few per cent.

Still another reason is partly due to the higher density of the framing members in the shear wall tests than in the sheathing-to-framing joint tests. See also section 4.1 for additional contributing factors for this effect.

The curves in Fig. 7(a) were terminated when there was a big drop of the load or that the testing was stopped because the load had decreased large enough for capturing the behavior and not destroy the elements unnecessarily (some elements were repaired and used for other purposes). Figure 8(a) gives an illustration of the development of shear deformation between the frame and the sheathing during the course of loading. The shear deformation is for horizontal loading shown for a number of points along the bottom rail. It is evident that the vertical deformation is much greater than the horizontal deformation. It is also clear that the deformation develops approximately along a straight line with about constant ratio between the vertical and horizontal deformation. This means that a corresponding nonlinear elastic model would probably give about the same computational result as the present nonlinear plastic model.

Figure 8(b) shows for the bottom rail the horizontal (solid line) and vertical (dashed line) shear stress components in the shear layer that represents the action of the nails. The stress components are shown at the instance of maximum global load (circles) and at the global horizontal displacement 60 mm (squares), respectively. It is evident that the uplifting force is carried by the nails to the left, while the horizontal force by the nails to the right for both cases of peak global load and a global displacement of 60 mm.

Figure 9 shows the influence of the post peak characteristics of the sheathing-to-framing joints (cf. Fig. 5). The horizontal load-displacement curves for partially anchored shear walls with a single segment and subjected to a horizontal loading are shown for the different connector types. For the shear connector model discussed in this section, the curves for the joint types, (a) ductile, (c) gradual plastic fracture softening, (e) rapid plastic fracture softening, and (f) brittle are shown with thick lines. The single spring model according to Section 5.2, was applied for the joint types, (a) ductile, (b) test parallel (from Fig. 4(b)), (d) test perpendicular (from Fig. 4(a)), and (f) brittle and the results are shown with thin lines []. The dashed line represents the analytical plastic load capacity according to Eq. (22) in Section 5.3.

It is evident from Fig. 9 that the global ductility is much affected by the type of connector characteristics, but not so much the load capacity. The curve representing the rapid plastic fracture softening connector behavior (thick curve (e)) was terminated at a displacement of about 37 mm due to unstable computations. In case of the brittle connector properties, the global post peak load performance is unstable and, therefore, only symbolically illustrated in Fig. 9 with a vertical line (thick line (f)).

According to Fig. 9 and as mentioned above, all curves obtained from the single spring (SS) model show a higher load-carrying capacity than those from the shear connector (SC) model. In the SS-model, the connectors are modeled as discrete points, while in the SC-model they are smeared out as a continuous shear layer. Basing the ultimate capacity of the sheathing-to-framing joint on the same peak value from the experimental results (Fig. 4) leads to a higher effective capacity in case of discrete connector representation due to the effect of the connectors in the corners (cf. Källsner and Girhammar []). Both the capacity (one more fastener contribute) and the moment effect (longer lever arm) of the fasteners in the corners is increased.

It is evident from Fig. 9 that for the joint type (a) the results of the single spring and the shear connector models approach one another for large displacements. The same goes for the joint type (e). Also, as expected, the thick curve (c) in Fig. 9 obtained by the shear connector model lies between the thin curves (b) and (d) given by the single spring model (Vessby et al. []) for large displacements. The (c)-curve is based on an average curve between the parallel (b)-curve and the perpendicular (d)-curve for the sheathing-to-framing joints according to Fig. 5.

Single spring model

The single spring model has been presented in Section 2.4.2, for further details see e.g., Judd and Fonseca []. The use of this model implies that only one load-displacement curve applies to each of the fasteners. This model has been applied by Vessby et al. [11,14] and they performed the finite element simulations using the commercial software Abaqus. All properties and parameters they used were approximately the same as those employed in the shear connector model.

According to Vessby et al. [,], a linear elastic unloading path is assumed (parallel to the initial stiffness) in case of unloading of a joint. In the plastic model, a von Mises plasticity model with isotropic hardening behavior is used. This means that the yield surface increases from the initial yield load until maximum load is attained and, thereafter, decreases as the load decreases.

The results are inserted in Fig. 9.

Analytical plastic method

The plastic analytical models are general in nature and can be used in design of shear walls with different sheet materials, sheathing-to-framing joints, geometric layout, anchoring conditions and load configurations.

The plastic design model will be applied to fully and partially anchored shear walls (in this paper with a single segment). The model is based on the following assumptions: 1) framing members and sheets are rigid; 2) stud-to-rail joints act as hinges; 3) sheathing-to-framing joints have ideally plastic load-slip characteristics, have the same stiffness in all joints, and the stiffness is independent of the force direction and the mutual orientation of the sheets and framing members; 4) displacements of the wall are small compared to the width and height of the sheets; 5) edge distances of sheathing-to-framing joints are small compared to the width and height of the sheets, i.e., the fasteners are approximately located along the edges of the sheets; 6) spacing between the fasteners in the sheathing-to-framing joints are relatively small and a continuous shear flow from the fasteners are assumed.

The plastic analysis model for fully anchored sheathed shear walls has been presented before by Källsner and Girhammar []. The plastic method is a so called lower bound method, i.e., based on the static theorem requiring that the static equilibrium is fulfilled for all parts of the structure and no brittle failures occur. The method gives a capacity that is lower than or equal to the true plastic load-carrying capacity (in the ultimate limit state).

The plastic distribution of the forces in the fasteners is illustrated in Fig. 2(a). The plastic force in the fastener is distributed as a constant shear flow per unit length f_p. The horizontal load-carrying capacity H for a fully anchored shear wall is given by Källsner and Girhammar [] as

(19)H=f_{p}b,

where b is the width of the wall segment.

For partially anchored timber shear walls, two different plastic models will briefly be presented here. The difference between the models is that in the second one, the shear forces from the stud-to-rail joints are taken into account. In order for the latter model to be a true lower bound method, it is required that the shear capacity of the stud-to-rail joints is sufficiently large and can be used for transferring shear forces. But even if this condition concerning the horizontal equilibrium is not fully met, this method gives values that are close to test results.

The basic assumptions of the plastic analytical models for a shear wall, where the studs are partially anchored, but the bottom rail anchored to the substrate, are as follows: 1) the sheathing-to-framing joints, referring to the vertical studs and the top rail, are assumed to transfer shear forces only parallel to the timber members; 2) the sheathing-to-framing joints, referring to the bottom rail, are assumed to transfer forces either parallel or perpendicular to the bottom rail; 3) the stud-to-rail joints are not assumed to transfer any tensile or shear forces in the first model, but do so in the second one, where some shear forces can be transferred to the bottom rail; and 4) compressive forces can be transferred in the stud-to-rail joints. To obtain simple expressions for the resistance of the shear walls, the fasteners are assumed to be continuously distributed along the center of the timber members. It is further assumed that the fastener spacing around the perimeter of the sheets is constant.

The plastic distribution of the forces in the fasteners is illustrated in Figs. 2(b) and (c), respectively. According to Källsner and Girhammar [,] and Källsner et al. [], the horizontal plastic load-carrying capacity H for a partially anchored shear wall is for the first plastic model (Fig. 2(b)) given by

(20)H=f_{p}b\left[\sqrt{\mathrm{1}+{\displaystyle \frac{h^{\mathrm{2}}}{b^{\mathrm{2}}}}}-{\displaystyle \frac{h}{b}}\right],

and for the second plastic model (Fig. 2(c)) by

(21)H=f_{p}b{\displaystyle \frac{b}{\mathrm{2}h}}.

The notations are shown in Figs. 1 and 2.

For the analytical model the distributed plastic capacity f_p of the sheathing-to-framing joints can be taken from the curves representing the perpendicular and the parallel directions in Fig. 4. Using the peak values and a center distance between the fasteners of 100 mm leads to f_p = 9.64 and 9.94 N/mm, respectively.

For the horizontal load case in Fig. 7(a), partially anchored, the capacity of the wall is determined by the capacity of the fasteners in the bottom rail, which are mainly loaded in the perpendicular direction, and is from Eqs. (20) and (21) obtained respectively as

(22)H=\mathrm{0.236}{f}_{\mathrm{p}}b=\mathrm{0.236}\hspace{0.17em}\times \hspace{0.17em}\mathrm{9.64}\hspace{0.17em}\times \hspace{0.17em}\mathrm{1200}=\mathrm{2730}\mathrm{ N,}

(23)H=\mathrm{0.250}{f}_{\mathrm{p}}b=\mathrm{0.250}\hspace{0.17em}\times \hspace{0.17em}\mathrm{9.64}\hspace{0.17em}\times \hspace{0.17em}\mathrm{1200}=\mathrm{2890}\mathrm{ N}.

For the diagonal load case in Fig. 7(b), fully anchored, the wall is assumed to be loaded in pure shear and it is reasonable to assume that the strength of the joints in the parallel direction governs the capacity resulting in

(24)H=\mathrm{1.0}{f}_{\mathrm{p}}b=\mathrm{9.94}\hspace{0.17em}\times \hspace{0.17em}\mathrm{1200}=\mathrm{11900}\mathrm{ N}

.

The values according to Eqs. (22) and (24) are inserted in Figs. 7 and 9, respectively.

Conclusions

In this paper a new finite shear connector element based on the theory of plasticity is presented. The derivation of the relationship between incremental force and incremental deformation is given. The actual connector behavior and the pertaining different plastic properties are determined from the test results.

The starting point is the incremental stress versus strain relation as given by the general theory of plasticity, where the tangential elastic-plastic stiffness tensor includes the elastic stiffness tensor, the plastic modulus, a function representing the yield criterion, and function representing the plastic potential. The two functions mentioned are usually assumed to be equal giving an associated flow rule for a given yield criterion.

The new plastic shear connector model is evaluated with respect to the horizontal load-displacement behavior of a single segment shear wall and compared with corresponding test results and with other computer results presented in the literature (using nonlinear spring models). Also, the ultimate horizontal load-carrying capacity of the shear wall is compared to the new analytical plastic design method developed by the authors.

The calculated curves are on the safe side with respect to the test results and there is a very good agreement between the load-carrying capacity between that obtained by the shear connector model and that by the analytical plastic design method.

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