Development of diaphragm connector elements for three-dimensional nonlinear dynamic analysis of precast concrete structures

Abstract

Diaphragm connector elements were developed for three-dimensional finite element models of precast concrete structures used in nonlinear time history analyses. The use of discrete elements for the diaphragm connectors permits the direct evaluation of local force and deformation demands, information needed in calibrating design factors for a new diaphragm seismic design methodology. This article describes the element formulation. The connector elements consist of assemblages of standard elements readily available in most finite element software libraries. The connector element calibration is based on full-scale testing of common precast diaphragm connectors. In these tests, the connector exhibited hysteretic pinching, stiffness/strength degradation, and slip mechanisms. The diaphragm connector elements were constructed to capture these behaviors to an accuracy sufficient for establishing viable design factors, while still appropriate for insertion into large degree-of-freedom models. The models are validated against the results of simulation-driven tests for critical precast diaphragm joints and a half-scale shake table test.

Keywords

finite elements floor diaphragms nonlinear time history analysis precast concrete

Introduction

A new seismic design methodology is under development for precast concrete floor diaphragms (Building Seismic Safety Council (BSSC) Committee IT6, 2014) based on extensive research (Fleischman et al., 2013). The design methodology employs design force factors indexed to diaphragm performance targets. These design factors are calibrated using nonlinear time history analyses (NLTHAs) of three-dimensional finite element (3D-FE) models of precast structures under suites of earthquakes representing different levels of seismic hazard (Zhang and Fleischman, 2015). Unlike the distributed force transfer in monolithic floor slabs that serve as diaphragms in reinforced concrete (RC) structures for instance (Moehle et al., 2010), forces in precast diaphragms are transferred at the discrete connector locations. The floor joints serve as critical sections in the precast diaphragm, and thus, precast diaphragm response is highly dependent on the characteristics of the diaphragm reinforcement, typically connectors joining the precast floor units (Biondini et al., 2008; Wood et al., 2000). Likewise, the performance targets in the design methodology are directly related to the anticipated force and deformation demands acting on these precast diaphragm connectors. For this reason, a key feature of the NLTHA used to develop the design methodology is the inclusion of discrete representations of the precast diaphragm connectors in the 3D-FE models, termed connector elements. This article describes the construction, calibration, and validation of these elements.

The connector elements are developed using the results of full-scale physical testing of isolated precast diaphragm connectors (Naito et al., 2007). Cyclic loading protocols were imposed that are intended to resemble the demands incurred during seismic action, including combinations of reversing compression and tension in the presence of cyclic shear force. In these tests, the connectors exhibited several important cyclic responses including hysteretic pinching, stiffness degradation, strength degradation, and slip mechanisms.

The connector elements must not only provide accurate hysteretic models of the precast diaphragm connectors, capable of representing precast diaphragm behavior with a level of accuracy sufficient to produce meaningful design factors for the design methodology, but also be appropriate for insertion into large degree-of-freedom (DOF) 3D-FE models with hundreds of these connector elements. Thus, the connector elements are composed of assemblages of standard elements readily available in most commercial FE software package libraries. This approach, successfully used for two-dimensional (2D)-FE models of precast diaphragms in nonlinear pushover analyses (Wan et al., 2015), is extended here for 3D-FE NLTHA.

The connector model developed in this article is experimentally verified by large-scale tests of key precast diaphragm joints (Zhang, 2010; Zhang et al., 2011). The 3D-FE models successfully predicted the response of a shake table test of a three-story half-scale diaphragm-sensitive precast structure (Schoettler et al., 2009) and are being used in simulations to produce the design factors (Fleischman and Wan, 2007; Zhang and Fleischman, 2015) for the precast diaphragm seismic design methodology (BSSC Committee IT6, 2014).

Precast diaphragm modeling approach

Precast diaphragms

Floor diaphragms serve as the horizontal elements of the lateral force–resisting system. In precast building structures, the (in-plane) diaphragm seismic forces must be transferred across joints between precast units (see Figure 1(a)) through in-plane moment and shear, regardless if the precast units are pretopped (dry diaphragm construction) or if a small cast-in-place topping is added (topped diaphragm). This transfer occurs through flange-to-flange connectors that comprise the diaphragm chord and shear reinforcement. The joints serve as planes of weakness, and thus, the response of the diaphragm, in terms of strength and deformation capacity, is highly dependent on the connectors that join the units.

Figure 1.

Precast diaphragm: (a) typical floor plan, (b) 3D-FE model, and (c) discrete connector element.

Discrete connector diaphragm models

Floor diaphragm is typically treated as rigid or elastic element in the design code and design software. However, since the collapse of several precast concrete structures due to failures of the floor systems in the 1994 Northridge earthquake (Iverson and Hawkins, 1994), it has been recognized that the modeling of inelastic diaphragm behavior is required to understand the behavior of precast floor diaphragms for the purposes of improving their seismic design. Diaphragm analytical models used in research for evaluating its seismic response can be categorized into two types: monolithic model and discrete model.

The monolithic model treats the diaphragm as inelastic continuous elements including Bernoulli beam formulations (Nakaki, 2000), fiber element and smeared crack models (Fleischman et al., 1998), and inelastic plane stress FEs (Ju and Lin, 1999; Lee and Kuchma, 2008; Tena-Colunga and Abrams, 1996). This monolithic approach for modeling precast diaphragm has two shortcomings: (1) significant computational run times are needed since the entire diaphragm with millions of DOFs is modeled as nonlinear elements and (2) it is difficult for the monolithic model to accurately capture the discrete nonlinear response concentrated at the joint of the precast diaphragm.

Because of the shortcomings of monolithic model, precast diaphragm FE models with discrete representations of the connectors have been adopted in this article. This discrete approach can explicitly model the inelastic response at the diaphragm joint while keeping the precast units as elastic plane stress elements (see Figure 1(b)) to achieve a reasonable computational run time. The nonlinear action of the discrete diaphragm connector can be achieved using a custom element (e.g. user programmable features (ANSYS Version 11, 2007) or a multi-DOF spring elements with user-defined material properties (OpenSees Version 2.4.3, 2013)) or at the macro level using the standard element library available in commercial FE packages. The macro modeling approach was used in this article to model the discrete connector as assemblages of springs, links, and contact elements (see Figure 1(c)). The major reasons for choosing the macro modeling approach are as follows:

Macro modeling approaches have been shown to be effective in the past for modeling other structural members under the earthquake loading, including diagonal truss elements to model the flexural–shear coupling response of RC shear walls (Panagiotou et al., 2012) and 3D diagonal trusses to model the in-plane and out-of-plane coupling responses of masonry infill walls (Kadysiewski and Mosalam, 2009).

The macro model is built on the top of the existing commercial FE program which has robust computational engines and reasonable run times, and effective post-processing program. Thus, this modeling approach can facilitate timely results for developing the diaphragm seismic design methodology which requires versatile parameter modification, agile handling of spatial geometry, and complex boundary conditions in the creation of large DOF models. The macro model uses standard element library available in the existing FE program without significant computer coding or compiling effort. This approach also provides versatility, since the model is easily incorporated into the FE programs using their standard material and element library, and thus should be operable on different programs (e.g. ANSYS, ABAQUS, ADINA¹) and platforms.

Early versions of these “discrete” diaphragm models using the macro modeling approach included those capturing diaphragm flexure response only (Farrow and Fleischman, 2003) and flexure and shear response as uncoupled DOF (Farrow and Fleischman, 2003; Zhang and Oliva, 2007). As a major improvement, the discrete diaphragm model developed by the research program can capture the flexural–shear coupled response. The first versions of the discrete diaphragm models were used for 2D nonlinear static monotonic “pushover” analyses of isolated diaphragms in order to determine precast diaphragm capacity, limit state sequence (Fleischman and Wan, 2007), and load path characteristics (Wan et al., 2012). The connector elements for the 2D discrete diaphragm models were calibrated using monotonic backbones from cyclic tests of isolated connectors as discussed in Wan et al. (2015).

These 2D discrete connector elements included the following features required for accurate pushover response: (1) inclined link elements that provided the interaction of individual components of loading (tension–shear coupling); (2) a contact sub-assemblage composed of a contact element in series with a nonlinear spring element to provide the high stiffness bearing between precast panels and eventual compression limit states (softening/crushing) of the surrounding concrete; and (3) contact element friction capabilities, assigned with a coefficient of friction µ, in order to mobilize the effect of applied compression on shear resistance. Full details of the development of the 2D discrete connector elements are found in Wan et al. (2015).

The 3D-FE models contain the discrete diaphragm models within the structure (see Figure 1(b)). The diaphragm connector element used in the 3D-FE models includes the aforementioned features of the 2D discrete connector element. However, for use in earthquake simulations of structures, the diaphragm connector element is extended to include the hysteretic effects exhibited under cyclic loading (Naito et al., 2007). This article details the construction and calibration of the discrete diaphragm connector elements used for 3D-FE NLTHA, termed herein 3D-FE connector element.

Precast connector characteristics under cyclic loading

Precast diaphragm connector test program

As part of the overall research program, a large number of common diaphragm connectors were tested under cyclic loading to determine their response characteristics (Naito et al., 2006, 2007). A multi-directional test fixture was developed to allow for the simultaneous control of in-plane shear, axial, and bending deformations at the panel joint (shown later in Figure 11(a)). The fixture employs three actuators, two that apply axial displacement to the connector or joint and one that applies a shear displacement. Full details of the specimen are found in Naito et al. (2007).

The connector characteristics were provided through loading protocols for cyclic tension/compression and shear. The cyclic protocol consisted of three cycles at each increasing level of tension (opening) or shear (sliding) deformation. For the cyclic tension/compression tests, the shear actuator provided compensation displacements to generate a constant shear force. For the cyclic shear tests, the axial actuator provided compensation displacements to generate a constant axial force. Baseline characteristics were determined with the constant secondary force in each case held to zero. Note that loading histories of combined shear, axial, and bending loading, as would be produced in response to earthquake ground motions, are used in the verification testing (see section “Comparison of connector models to test results”).

Precast diaphragm connector test results

Figure 2 shows three typical diaphragm connectors, used in this article to demonstrate the 3D-FE connector element construction: (a) a dry chord connector (2#5 bars), (b) an untopped flange-to-flange shear connector (JVI Vector²), and (c) a topped shear connector (ductile mesh ladder). Figure 3 shows the cyclic test results and backbone curves for these diaphragm connectors: (a) cyclic tension with zero shear force test for the dry chord connector, (b) cyclic shear with zero axial force for the JVI Vector, and (c) cyclic shear with zero axial force for the ductile mesh connector. Full details of these connectors and the testing are found in Naito et al. (2007).

Figure 2.

Typical precast connectors: (a) dry chord, (b) JVI Vector, and (c) ductile mesh.

Figure 3.

Connector cyclic tests results: (a) dry chord, (b) JVI Vector, and (c) ductile mesh.

Observed hysteretic behaviors

As can be observed in Figure 3, four basic hysteretic effects occurred in the response of the typical precast concrete diaphragm connectors, sometimes in combination (Naito et al., 2007):

A pinching effect due to bond slip (see Figure 4(a)). This effect is generated by the loss of bond and subsequent slip of the reinforcement in the concrete during reversal loading. This effect occurs after large joint inelastic opening that causes cracking in the surrounding concrete, but prior to compression transfer associated with joint closing (Naito et al., 2007). This effect is mostly seen in connectors under predominantly cyclic tension loading.

A stiffness degradation effect due to cracking of the surrounding concrete (also in Figure 4(a)). This effect is observed during a reloading cycle following a large inelastic tension deformation cycle that causes the concrete surrounding the steel elements to crack (Naito et al., 2007). Instead of following the initial elastic stiffness, the reloading path will “shoot through” at a lower stiffness, until the maximum opening point of the previous cycle is reached. This effect is most noticeable under tension load.

A strength degradation effect due to loss of surrounding concrete (see Figure 4(b)). This effect is observed during the reloading cycle following a large inelastic shear deformation cycle that causes local spalling of the concrete surrounding the connection region (Naito et al., 2007). This effect is most noticeable in connectors under primarily shear loading.

A slip–catch effect due to slip and re-engagement of the steel element anchorages (see Figure 4(c)). This phenomenon is exhibited during large shear deformation after a major cyclic shear load event sufficient to fully degrade the dowel action through cracking and spalling of the concrete surrounding the steel element(s). A region of near-zero stiffness (slip) through the ± sliding displacement range occurs in subsequent cycles where the steel elements can rotate prior to re-engaging due to catenary (“kinking”) action. This effect occurs under shear loading of connectors employing reinforcing bars.

Figure 4.

Diaphragm connector hysteretic effect: (a) pinching/stiffness degradation, (b) strength degradation, and (c) slip–catch.

Hysteretic models for diaphragm connectors

In this section, construction of the hysteretic models for the 3D-FE connector elements is presented. As noted previously, the 3D-FE connector elements use as a basis the 2D discrete diaphragm connector model (Wan et al., 2015) developed for monotonic pushover analysis and extend the properties of the nonlinear sub-assemblage elements to incorporate hysteretic rules suitable for 3D-FE NLTHA.

The hysteretic models for the 3D-FE connector element are calibrated to match cyclic tests of isolated diaphragm reinforcement (Naito et al., 2007). The four basic hysteretic effects discussed in previous section are captured using modeling techniques readily available in the commercial software package ANSYS. The use of ANSYS facilitated the handling of multi-step iterative solutions of large DOF structural models (pre-processing, solution, and post-processing), but necessitated the use of available elements within the element library to create the desired behavior. The behaviors exhibited by the connections are different from that produced by standard elastic–plastic kinematic or isotropic hardening models directly available in the software (ANSYS Version 11, 2007). As such, the 3D-FE connector elements are created through assemblages of multiple elements in series and/or parallel with different properties or unloading rules, to create the desired cyclic response features.

In the following, each effect is demonstrated individually, by matching a given connector sub-assemblage to an idealized test backbone. In the construction of the actual connector elements, which involves combinations of the hysteretic effects, as well as the tension–shear coupling derived for the 2D connector models (Wan et al., 2015), complex element assemblages are created (see section “Comparison of connector models to test results”).

Modeling of pinching effect

The pinching effect is captured in the connector element through two nonlinear axial springs in parallel (see insert in Figure 5(a)), each with different unloading protocols available in the software (ANSYS Version 11, 2007): an elastic–plastic spring that unloads at its initial stiffness and a nonlinear elastic spring that unloads along its loading path (see Figure 5(a)). As seen in Figure 5(a), the input characteristic of these springs is weighted by a factor α from the target response backbone curve, here a generic cyclic tension test backbone, similar to the one shown in Figure 3(a).

Figure 5.

Pinching effect: (a) input, (b) symmetric, and (c) asymmetric characteristic.

The degree of pinching is controlled by the factor α. As seen in Figure 5(b), when α = 1.0, no pinching occurs in the model under unloading. When α = 0, a “fully” pinched condition (no energy dissipation) is obtained. For values in between, the pinching effect is produced. Note that when α < 0.5, pinching will start in the unloading regime (see Figure 5(b)), and when α > 0.5, pinching starts in the reloading regime. Therefore, as α decreases, more pinching, and thus less energy dissipation, is obtained.

For a connector model with complex assemblages, an asymmetric target response, that is, T_y ≠ C_y (see Figure 5(c)), can be used for the pinching model to produce the desired connector response. Such an approach is used to create the chord connector, as will be described in section “Comparison of connector models to test results.”

Modeling of stiffness degradation

The stiffness degradation effect is achieved through the use of a sub-assemblage of nonlinear links in parallel possessing different strain hardening rules (see Figure 6(a)). A set of links with isotropic hardening produce the stiffness degradation. The isotropic hardening rule produces expansion or shrinking of the yield surface (ANSYS Version 11, 2007), and hence, stiffness degradation can be created by the loss of stiffness associated with portions of the assemblage losing strength. A link with kinematic strain hardening, which possesses a constant translating yield surface (ANSYS Version 11, 2007), is used to maintain the connector element overall strength as the stiffness degrades.

Figure 6.

Stiffness degradation: (a) model, (b) input characteristic, and (c) hysteretic response.

As shown in Figure 6(b), the N isotropic hardening links in the sub-assemblage all possess descending branch behavior and are in parallel with one non-degrading kinematic hardening link. The sum of the input characteristic curves of the N + 1 links at any point is set to equal to the target backbone response, here a generic cyclic tension test backbone, similar to the one shown in Figure 3(a). For complex assemblages, the target response can be a portion of the test backbone (see section “Comparison of connector models to test results”).

Each isotropic link is assigned its own portion of the strength (F_ti) and an elastic limit deformation (δ_ti), where t refers to the tension and i refers to the ith link (see Figure 6(b)). After reaching F_ti at its elastic limit deformation, the strength of a given link will degrade to a pre-assigned residual strength R. This descending branch response produces the stiffness degradation (see Figure 6(c)). Note that the full degradation for link i occurs at the limit deformation of the next link, δ_t_(i + 1), in order to achieve a smooth piecewise curve in the total response. Each successive link is provided with a greater elastic limit deformation but a lower percentage of the overall strength. The single perfectly plastic kinematic strain hardening link prevents strength degradation when the stiffness degrades, consistent with test observations (e.g. Figure 3(a)).

It should also be noted in Figure 6(c) that the connector model unloading stiffness also reduces as the loading stiffness degrades. The reduction in unloading stiffness observed is higher than the unloading stiffness degradation observed in the test (e.g. Figure 3(a)). This deviation in unloading stiffness can be compensated by contributions from other sub-assemblages during construction of the full connector model, as discussed in section “Comparison of connector models to test results.”

The characteristic inputs for the link elements are determined based on N + 1 control points identified on the target response backbone. The individual link yield deformations δ_ti are aligned to each control point displacement and correspond to a target connector force T_i. The number of control points selected should be sufficient to match the stiffness degradation pattern observed in the test. At a minimum, the control points should encompass the following four states: (1) within the elastic range (e.g. δ_t₁, T₁), (2) the yield point (δ_ty, T_y), (3) the ultimate strength (δ_tu, T_u), and (4) a post-yield state intermediate to yield and ultimate (e.g. δ_t₄, T₄). Additional control points between yield and ultimate can be used to increase the degree of stiffness degradation.

The strength (F_ti) of each link element is calculated by solving the following N + 1 linear equations, based on equating the cumulative strength of the links to the target strength at each control point

T_{i} = δ_{ti} \sum_{j = i}^{N + 1} (\frac{F_{tj}}{δ_{tj}}) + (i - 1) R

(1)

where i is incremented from 1 to N + 1 representing each link, and j is a dummy index for summation. The kinematic strain hardening link is provided with a strength of $F_{t (N + 1)}$ =T_u − ΣR at a yield deformation $δ_{t (N + 1)}$ equal to the deformation δ_tu at ultimate strength of the overall target strength, T_u.

Modeling of strength degradation

The strength degradation effect is modeled using a nonlinear degrading link element (see Figure 7(a)) possessing an isotropic strain hardening rule. This rule produces a yield surface that changes (increases or reduces) based on cumulative plastic strain (ANSYS Version 11, 2007). The strength degradation effect is achieved by providing the link a descending branch characteristic, thereby producing reduction in the yield surface with total cumulative plastic strain. The link element input characteristics, in terms of force F_vj and deformation δ_vi, are determined based on control points i = 1,…, N on a target response backbone. Figure 7(b) shows this approach for the full cyclic shear test backbone curve of Figure 3(b). Each control point on the cyclic backbone curve corresponds to a shear deformation (Δ_vi) and a shear force (V_i). Sufficient control points (N) should be provided to accurately reproduce the ascending and descending branches of response.

Figure 7.

Modeling of strength degradation effect: (a) model, (b) input characteristic, and (c) plastic deformation.

Control points prior to the peak load V_peak can be directly assigned to the characteristic input for the link element since no strength degradation has yet occurred

F_{vi} = V_{i}

(2)

δ_{vi} = Δ_{vi} for Δ_{vi} \leq Δ_{peak}

(3)

However, the characteristic input for the link element after the peak load must be modified relative to the post-peak control points on the cyclic backbone because the strength degradation in the model is controlled by cumulative rather than absolute plastic deformation. For this reason, the link characteristic at each post-yield control point is provided with the same strength (F_vj = V_i) but a larger deformation than represented on the target response backbone, that is, the target response backbone is shifted to the right as indicated in Figure 7(b). The increase in deformation at each control point is determined by mapping the absolute deformation to an accumulated deformation using the following rule

\begin{matrix} δ_{vi} = & 3 n_{1} (Δ_{vi} - Δ_{peak}) \\ + \sum_{j = 2}^{i} [4 n_{j} (Δ_{vi} - Δ_{peak})] + Δ_{peak} for Δ_{vi} > Δ_{peak} \end{matrix}

(4)

where n_i is the number of repeated cycles at the ith control point and j is a dummy index for summation. In this simple rule, cumulative plastic deformation is idealized as four times the absolute plastic deformation for each loading cycle, except in the first inelastic cycle where it is three times (see Figure 7(c)).

This approach produces hysteretic strength degradation aligned to the target response backbone for a single symmetric test loading protocol, and thus will only approximate other loading histories. Comparison of this model to large-scale testing of precast diaphragm joints under estimated (irregular) seismic load histories produced only slight differences (see section “Comparison of connector models to test results”), thereby suggesting the adequacy of this approach. For numeric stability in a large DOF model, it is necessary to introduce a non-degrading spring with a small strength and stiffness in parallel with the degrading link element in the connector model assemblage (see Figure 7(a)).

Modeling slip–catch response

A “hook” mechanism (see Figure 8(a)) is used to capture the slip–catch hysteretic effect. This mechanism is produced by connecting the precast panels with a pair of identical element groups consisting of a nonlinear spring and contact element in series with a rigid link element. Each nonlinear shear spring, whose unloading path is parallel to its initial loading path, provides a stiffness contribution only when its associated contact element is closed. Therefore, only one group of elements will be activated when shear loading is applied in a single direction at the joint between the panels.

Figure 8.

Modeling of slip–catch effect: (a) model, (b) spring response, and (c) overall response.

Figure 8(b) shows individual responses of the nonlinear springs from each group, corresponding to the overall element response through stages 1–8, as shown in Figure 8(c). In a given shear loading cycle, as the precast panels slide relative to each other in one direction, one contact element will close and the other will open. Therefore, only one of the nonlinear springs, the one connected to the closed contact element (e.g. spring1), will provide stiffness contribution, while the one connected to the open contact element (spring2) remains inactive. After spring1 yields, its unloading path will not pass through the origin and a residual deformation occurs at the load reversal point (see Figure 8(b)). When spring1 passes the load reversal point, its associated contact element serves as a “hook,” possessing memory of the location of this point (see Figure 8(b)) and begins to open. Thus, spring1 no longer provides stiffness contribution after passing the load reversal point. Meanwhile, spring2 begins to provide a stiffness contribution. In the subsequent shear loading cycle, spring1 will develop stiffness again only after the reloading displacement reaches the load reversal point from the previous cycle, at which time the contact element closes (dotted line in Figure 8(b)). Thus, there is a zero stiffness “slip” in the overall response of the connector element after spring2 passes through its reversal loading point, but prior to spring1 regaining its stiffness again (see Figure 8(c)).

Comparison of connector models to test results

In this section, the response of connector elements constructed using the models described in the previous section is compared to the experimental results. The cyclic response of precast diaphragm reinforcement typically exhibits more than one of the hysteretic effects. Thus, the connector elements are constructed as complex assemblages of the different model sub-assemblages.

The experimental comparisons presented here include (1) cyclic increasing amplitude tests of full-scale isolated connectors (Naito et al., 2007); (2) simulation-driven large-scale testing of critical precast diaphragm joints under expected seismic loading histories (Zhang, 2010; Zhang et al., 2011); and (3) a shake table test (Schoettler et al., 2009). These tests involved the common precast diaphragm reinforcement details as shown in Figure 4.

Diaphragm dry chord connector

The cyclic tension/compression test response of the dry chord connector (2#5) exhibits the following effects (refer to Figure 3(a)): (1) pinching, (2) stiffness degradation, and (3) high stiffness in compression provided by the surrounding concrete. Accordingly, the dry chord connector element (see Figure 9(a)) is constructed by (1) combining the sub-assemblages for pinching and stiffness degradation described in section “Hysteretic models for diaphragm connectors”) and (2) adding the contact sub-assemblage introduced in section “Precast diaphragm modeling approach” and is described in detail in Wan et al. (2015). It should be noted that the assemblage considered here is attempting to match test results under cyclic axial force alone (Figure 3(a)) as is appropriate for a chord in a high flexure zone (e.g. midspan of a simple diaphragm layout). Modeling of chord connectors in other regions requires coupled shear–tension behavior, which involves orienting a portion of the elements at an angle based on the level of coupling evidenced in tests under combined axial and shear actions, as described in Wan et al. (2015).

Figure 9.

Dry chord connector: (a) model and (b) model–test comparison.

The pinching sub-assemblage requires the selection of an α factor to control the initiation of pinching. The cyclic test response curve (see Figure 9(b)) indicates that this occurs approximately at the abscissa crossing (zero load) point. Thus, a value α = 0.5 is selected (refer to Figure 5(b)). The pinching sub-assemblage also requires the following adjustments within a complex element assemblage:

The pinching sub-assemblage strength characteristic is biased toward compression such that the tension strength is near zero (e.g. Figure 5(c)). This approach permits the stiffness degradation sub-assemblage (described next) to act on the full tension response, rather than only a portion, to better match behavior. The compression-only springs in ANSYS do not possess the nonlinear feature. Therefore the nonlinear spring elements are used. However they do not permit assigning a zero-tension strength input (ANSYS Version 11, 2007). Thus, the nonlinear spring elements are assigned a small nominal tension strength of 4.45 kN (1 kips). It is noted that while this adjustment shifts the pinching initiation into the compression region for the pinching sub-assemblage itself (refer to Figure 5(c)), the desired pinching initiation point near the zero crossing is maintained for the full assemblage because the stiffness degradation sub-assemblage provides a portion of unloading stiffness, as shown in Figure 9(b).

The compression strength assigned to the springs is only a portion of the connector full compression yield strength (C_y) in recognition that the link elements used for stiffness degradation also contribute to compression strength (and stiffness). This proportion is assigned as 0.75 to match the unloading pattern observed in the cyclic test response (see Figure 9(b)). This assignment results in compression strength input of 0.75αC_y and 0.75(1 − α)C_y for the elastic–plastic and inelastic springs (refer to Figure 5(a)), respectively (note C_y = T_y). The assignment will decrease the degree of the unloading stiffness degradation originated from the stiffness degradation sub-assemblage as discussed in section “Hysteretic models for diaphragm connectors.”

For the stiffness degradation sub-assemblage, seven control points (N = 6) are assigned on the cyclic test backbone curve (see gray dots in Figure 3(a)). These control points, used to calculate the input characteristic of the link elements (refer to Figure 6(b)), are listed in the first two rows of Table 1. The strength of each link element is determined using equation (1) as shown in the bottom row of Table 1. In equation (1), the residual strength R for each degrading link is set to the small nominal value of 4.45 kN (1 kips).

Table 1.

Link element characteristic input: 2#5 dry chord connector.

Control force	T ₁	T ₂ = T_y	T ₃	T ₄	T ₅	T ₆	T ₇ = T_u
(kN)	85	183	208	231	223	198	167
Control opening	δ_t ₁	δ_t ₂ = δ_ty	δ_t ₃	δ_t ₄	δ_t ₅	δ_t ₆	δ_t ₇ = δ_tu
(cm)	0.060	0.203	0.475	0.632	0.922	1.217	1.875
Calculated strength: equation (1)	F_t ₁	F_t ₂	F_t ₃	F_t ₄	F_t ₅	F_t ₆	F_t ₇
(kN)	32.1	93.5	35.8	76.9	72.2	84.8	140.5

The input characteristics of the contact sub-assemblage (contact element and nonlinear spring in series) match the compression response of an equivalent strut of the surrounding concrete (Wan et al., 2015). As shown in Figure 9(b), the dry chord connector model results exhibit good agreement with the experimental results. The stiffness degradation, pinching, and high stiffness contact effects observed in the cyclic test response are successfully captured by the dry chord connector model. One discrepancy between the test and model is the contact path in some loading/unloading cycles: the test results show the contact engages before the panel closes while the model only has contact activated after the panel completely closes. However, this behavior only occurs for a few intermediate cycles in the test and only effects the local high-frequency accelerations in the dynamic analysis, which are not the results of importance. Thus, the connector response shown is considered acceptable.

Untopped diaphragm shear connector

The untopped flange-to-flange shear connector (JVI Vector) exhibits strength degradation under cyclic shear loading (refer to Figure 3(b)), as well as a tension–shear coupling effect described in Wan et al. (2015). These two effects should be isolated in the model because the strength degradation is due to the damage of surrounding concrete (see section “Precast connector characteristics under cyclic loading”), while the tension–shear coupling manifests itself in yielding of connector steel elements. To isolate these effects, the connector model contains two elements sub-assemblages in series (see Figure 10(a)): one for the strength degradation and the other for tension–shear coupling. Both sub-assemblages are assigned the full connector strength since they are connected in series. However, the stiffness of each sub-assemblage has to be doubled to achieve the correct overall stiffness. This is accomplished by scaling the displacement inputs by a factor of a 1/2. In addition, a contact/friction sub-assemblage is placed in parallel to the other two elements sub-assemblages for modeling high stiffness in compression and the associated friction.

Figure 10.

JVI Vector: (a) model and (b) model–test comparison.

The tension–shear coupling sub-assemblage requires two primary parameters to obtain the desired coupling observed in connector tests under proportional tension–shear loading (Wan et al., 2015): the angle of inclination θ for the coupled link elements (see Figure 10(a)), and the ratio (ω_v) of the shear strength in the uncoupled spring to the full shear strength of the connector. The calibrated values of these coupling parameters, as determined from tests of isolated JVI Vector connectors under cyclic shear, cyclic tension, and tension–shear proportional loading, are θ = 33° and ω_v = 0.6, respectively. The full description of model construction and parameters’ calibration for the coupled behavior appears in Wan et al. (2015).

The link element input characteristics for the strength degradation sub-assemblage (refer to Figure 7) are calculated from equations (2) to (4) based on the control points as shown in the first two columns of Table 2. These control points are selected as the points on the cyclic shear test backbone which correspond to the peak displacement loading points of each loading cycle (refer to gray dots in Figure 3(b)). The strength of each control point from the test backbone is subtracted from the strength contribution of the weak and flexible non-degrading shear spring as shown in the third and fourth columns of Table 2. The calculated link element input characteristics using equations (2) to (4) are shown in the last two columns of Table 2. As discussed above, the input displacements for the degrading link element (column 7) and for the non-degrading spring (column 1) are scaled by 1/2 as shown in Table 2.

Table 2.

Strength degradation sub-assemblage input: JVI Vector connector.

Control points				Plastic deformation		Link
Δ_vi (cm)	Test (kN)	Non-deg. spring (kN)	V_i (kN)	Δ_vi − Δ_vy (cm)	n	Δ_vi (cm)	F_vi (kN)
0.005	11.1	0.1	10.9	–	–	0.005	10.9
0.055	48.2	1.2	47.1	–	–	0.055	47.1
0.105	82.3	2.2	80.1	–	–	0.105	80.1
0.115	57.8	2.5	55.3	0.03	1	0.14	55.3
0.205	44.5	4.4	40.1	0.20	2	0.935	40.1
0.305	40.0	6.5	33.5	0.40	2	2.54	33.5
0.42	26.7	9.0	17.7	0.63	1.5	4.43	17.7
0.62	13.3	13.3	0.0	1.04	1.5	7.54	0.0

In the contact/friction sub-assemblage, the level of friction is obtained by assigning a coefficient of friction µ to the contact element. The µ is selected as 0.55 for the connector model on the basis of calibration from cyclic shear tests of the JVI Vector connector with and without axial compression force, as discussed in Wan et al. (2015). Figure 11 compares this model with the cyclic shear response from test of the JVI Vector when also subjected to constant 44.48 kN compressive force. A good agreement between the model and the test is obtained.

Figure 11.

Calibration of friction coefficient for JVI model: (a) cyclic shear test under compression and (b) cyclic shear response.

Topped diaphragm shear connector

The topped shear connector (ductile mesh) exhibits the following effects under cyclic shear loading: (1) slip–catch effect and (2) strength degradation effect (refer to Figure 3(c)). The techniques introduced in section “Hysteretic models for diaphragm connectors” to model the slip–catch (refer to Figure 8) and strength degradation effect (refer to Figure 7) are used to create the connector model for the ductile mesh. For this, strength degradation is built directly into the slip–catch “hook” mechanism of Figure 8(a) by replacing the nonlinear springs in Figure 8(a) with the strength degradation sub-assemblage as used in JVI Vector (see Figure 10(a)). The resulting ductile mesh connector element is shown in Figure 12(a). The input characteristic for the degrading link element in the strength degradation sub-assemblage is determined using equations (2) to (4) applied to the cyclic test backbone curve of the ductile mesh connector (see Figure 3(c)), following the same procedure detailed in the previous section for the JVI Vector connector. Figure 12(b) compares the model to the test results of the ductile mesh connector under cyclic shear. As seen in Figure 12(b), the connector model captures the slip–catch and strength degradation effects observed in the test.

Figure 12.

Ductile mesh connector: (a) model and (b) model–test comparison.

Diaphragm critical flexure and shear joints

The connector model within the 3D-FE model has been verified using the results of simulation-driven physical tests on half-scale precast diaphragm joints representing the critical flexure joint (Zhang et al., 2011) and the critical shear joint (Zhang, 2010) in the precast diaphragm. In both cases, the diaphragm joint tested contained the dry chord connector as the flexural reinforcement and the JVI Vector connector as the shear reinforcement. The former applied a predetermined displacement history (PDH) to the specimen based on a 3D-FE model; the latter was a full-fledged hybrid (adaptive) test (Mercan and Ricles, 2005). The results shown are for response to a maximum considered earthquake to assure that the connector model is tested under its full range of possible demand.

Figure 13 compares the response prediction of fully simulated 3D-FE model built using the connector models to the test results. These include the following: (a) the moment–rotation (M–Θ) response at the critical flexure joint for the PDH test and (b) shear–sliding (V–δ_v) response at the critical shear joint for the hybrid test. As seen, the model prediction shows a good agreement with the test results.

Figure 13.

Model verification: (a) critical flexural joint and (b) critical shear joint, following Zhang and Fleischman (2015).

Diaphragm global response

A three-story half-scale precast structure was also tested in this research program (Schoettler et al., 2009). Although direct measurement of the local response can only be approximated, the 3D-FE model using the connector models presented in this article has also successfully predicted the global response of shake table tests (Fleischman et al., 2013). As an example, Figure 14 shows comparisons between the FE model and test results for (a) diaphragm inertial force and (b) diaphragm joint opening profile for a design basis earthquake motion. Good agreement is seen between the test and the analytical model.

Figure 14.

Diaphragm response comparison: (a) inertial force and (b) joint opening, following Fleischman et al. (2013).

Diaphragm connector response under seismic loadings

The importance of the tension–shear coupling effect for the diaphragm connector can be observed in the 3D NLTHA of a precast structural model (refer to Figure. 1(b)) calibrated from the test results. The diaphragms in the structural model contain the dry chord and JVI Vector connector elements. The hysteretic response (both shear and tension) of one JVI Vector connector elements obtained from the NLTHA is shown in Figure 15. The cyclic tension or shear test backbones are marked as dash lines. The JVI Vector connector element response in tension shows a strength reduction (see Figure 15(a)) due to the tension–shear coupling effect. The connector element also shows a reduction in shear strength (relative to the peak strength indicated on the backbone) due to tension–shear coupling effect and exhibits cyclic shear strength degradation after peak connector force (see Figure 15(b)).

Figure 15.

Hysteresis response of JVI Vector model: (a) tension and (b) shear.

Conclusion

Hysteretic models for precast diaphragm connectors have been developed. These models have been incorporated into connector elements used in the large DOF 3D-FE models of precast structures for earthquake simulations using NLTHA. The connector elements are composed of sub-assemblages of elements from libraries of commonly available FE software packages. These hysteretic connector models have been calibrated to match cyclic tests of isolated precast diaphragm connectors. The following characteristics of precast diaphragm connectors observed in tests have been captured by the models:

Pinching effect under reverse loading.

Stiffness degradation with increase in inelastic opening.

Strength degradation with increase in inelastic shear deformation.

Slip–catch effect under reverse loading.

Tension–shear coupling effect under combined loading components.

High compression stiffness due to panel contact and the effect of friction effect under compression.

The precast diaphragm connector models were verified by large-scale physical tests of precast diaphragm joint containing multiple diaphragm connectors. 3D-FE models using these connector elements have successfully predicted the response of a shake table test for a three-story half-scale precast structure. The diaphragm connector models developed in this article are being used in 3D NLTHA of large DOF FE models of diaphragm-sensitive precast structures in order to determine diaphragm design factor for the new seismic design methodology for precast concrete diaphragms.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Precast/Prestressed Concrete Institute (PCI), the Charles Pankow Foundation, and the National Science Foundation (NSF) under Grant CMS-0324522 and SGER Supplement CMMI-0623952. The authors are grateful for this support. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Notes

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