Nonlinear Finite-Element Modeling of HF2V Lead Extrusion Damping Devices: Generic Design Tool

Vishnupriya, Vishnupriya; Rodgers, G. W.; Chase, J. G.

doi:10.1061/(ASCE)ST.1943-541X.0003170

Open access

Technical Papers

Oct 19, 2021

Nonlinear Finite-Element Modeling of HF2V Lead Extrusion Damping Devices: Generic Design Tool

Authors: Vishnupriya Vishnupriya [email protected], G. W. Rodgers https://orcid.org/0000-0003-3907-0308, and J. G. Chase https://orcid.org/0000-0001-9989-4849Author Affiliations

Publication: Journal of Structural Engineering

Volume 148, Issue 1

https://doi.org/10.1061/(ASCE)ST.1943-541X.0003170

PDF

Abstract

Supplemental energy-dissipation devices are increasingly used to protect structures, limiting loads transferred to structures and absorbing significant response energy without sacrificial structural damage. The displacement of the bulged shaft plastically deforms lead in the high force to volume (HF2V) device, dissipating significant energy. HF2V devices are currently designed using limited precision models, so there is variability in force prediction. Further, although the outcome force is predicted, the knowledge of the exact internal mechanisms resulting in these device forces is lacking, limiting insight and predictive accuracy in device design. This study develops a generic finite-element (FEM) model using commercially available software to better understand force generation and aid in precision device design, thus speeding up the overall design and implementation process for uptake and use. The model is applied to 17 experimental HF2V devices of various sizes. The highly nonlinear analysis is run using the software with automatic increments to balance higher accuracy and computational time. The total force output is sum of the friction forces between lead and steel and the contact pressure forces acting between moving shaft and displaced lead. FEM forces and plots of the 17 devices are compared with experimental device forces and test plots. The errors from force comparison for all 17 devices range from

- 8 %

(overprediction) to

+ 39 %

(underprediction) with a mean absolute error of 7.6% and a signed average error of 4.7%, indicating most errors were small. In particular, the standard error (SE) in manufacturing is

SE = \pm 14 %

. Overall, 13 of 17 devices (76%) are within

\pm 1 SE

of 14%; 3 of 17 devices (18%) are within

\pm 2 SE

(

\pm 28 %

), and the last has

- 39 %

error, which is within

\pm 3 SE = \pm 42 %

. These results show low errors and a distribution of errors compared with experimental results that are within experimental device construction variability. The overall modeling methodology is objective and repeatable, and thus generalizable. The results validate the overall approach with relatively very low error, providing a general modeling methodology for accurate design of HF2V devices.

Introduction

It is essentially impossible to exactly predict when an earthquake will occur (Geller 1997). Consequently, to reduce seismic vulnerability and the economic impact of seismic structural damage, it is important to protect structures using structural protective systems such as base-isolation systems, active and semiactive devices, and passive supplemental energy-dissipation devices. Base-isolation systems like lead rubber bearings (LRBs), friction pendulum systems (FPS), superelastic-friction base isolators (S-FBIs), and resilient-friction base isolators (R-FBIs) protect structures by decoupling the structure from horizontal motions of the ground (Ozbulut and Hurlebaus 2012, 2010). Despite the many benefits, base-isolation systems may not fully protect structures from very strong earthquakes generated at near-fault locations. Challenges such as obtaining a sufficient moat-size/rattle-space around a building, managing uplift/overturning demand into isolators, and the ability to generate a flat isolation plane on inclined hill sites can all make implementation of base isolation difficult.

Active and semiactive systems provide variable damping to structures by measuring structural displacements and providing the appropriate force to improve the uncontrolled response of a structure to seismic excitations (Bitaraf et al. 2010b). The performance of base-isolator systems can be improved by the addition of active control methods such as active tuned mass dampers (ATMD), active tendon systems (Fisco and Adeli 2011a), distributed actuators (Fisco and Adeli 2011a), and active multiple tuned mass dampers (AMTMD) (Li et al. 2003).

In comparison, semiactive dampers provide stable variable damping using simpler hardware and less power in comparison with active damping systems. Widely used semiactive damping methods include magnetorheological (MR) dampers (Bitaraf et al. 2010a), variable-friction dampers (Ozbulut et al. 2011), semiactive friction dampers (SAFDs) (He et al. 2003), electrorheological (ER) dampers (Gavin et al. 1996), fluid viscous dampers (Datta 2003), semiactive pendulum tuned mass dampers (SAPTMDs) (Setareh et al. 2007), semiactive resettable stiffness dampers (Mulligan et al. 2005, 2010, 2009; Kh Hazaveh et al. 2016), and semiactive viscous fluid dampers (Kh Hazaveh et al. 2016). Some of these active and semiactive damping mechanisms have been widely adapted and applied for vibrational control of buildings in Japan and China (Ikeda 2009; Casciati et al. 2012).

Although active and semiactive systems can adapt to a range of operating conditions and provide structural stability, they generally add significant cost to structure due to the complex hardware and power requirements. Hybrid systems provide improved damping, adaptability, and reliability of buildings by combining base-isolation systems, active or semiactive systems, and passive supplemental energy dampers (Datta 2003; Fisco and Adeli 2011b).

Supplemental passive energy-dissipation devices prevent large base-isolator displacements and do not rely on external power sources for operation (Ozbulut and Hurlebaus 2010). These systems can absorb earthquake response energy, reducing the displacement response. They act as a fuse to limit forces transmitted to structural elements, which can support capacity design procedures. Fluid viscous dampers (VDs), viscoelastic dampers (VEDs), energy-dissipating struts (EDS), ring spring (RS) dampers, and lead extrusion dampers (LEDs) are examples of supplemental damping systems.

Viscous dampers provide rate-dependent dissipative behavior to structures and can be installed in parallel with base isolators, horizontally and connected to conventional brace, or in a conventional diagonal brace. The force range of a VD lies between 30 kN and 2 MN, depending on the size and its design. Force in a VD is produced when incompressible viscous fluid is forced into orifices during earthquake loading. Whereas, shear deformation produced in the viscoelastic material during longitudinal relative displacements results in energy dissipation by the VEDs. A typical VED consists of layers of viscoelastic material bonded between steel plates, which can be connected to some non-load-bearing structural elements. The viscoelastic material properties are complex, and the damping forces produced are dependent on frequency and temperature (Bernal 1994; Zhang and Soong 1992) and provide dynamic and seismic protection to high-rise buildings.

The frictional response force produced by energy-dissipating struts is proportional to displacement. The force and hysteretic behavior of this self-centering device depends on its configuration. Ring spring dampers are low-maintenance fully passive friction dampers with high self-centering capability. RS dampers have been fitted at column–base connections in multistory steel-frame buildings to diminish permanent displacements due to structural rocking (Gledhill et al. 2008) and have force capacities between 5 and 1,800 kN.

In recent years, LEDs have been modified to a smaller volumetric size with high force capacities, called high force to volume (HF2V) devices (Rodgers 2009; Latham et al. 2013; Mander et al. 2009; Bacht et al. 2011; Desombre et al. 2011; Golzar et al. 2017; Rodgers et al. 2010). HF2V devices are generally manufactured for a force capacity of 50–1,000 kN. However, dampers of higher force capacity can be manufactured by varying volume and design. The relative ease of manufacture, retrofit, repeatable performance, and lack of need for replacement makes HF2V devices a viable option for seismic mitigation. The weakly-rate-dependent hysteretic behavior of lead in the LEDs/HF2Vs are desirable and have been applied in large-scale experiments and full-scale field structures (Latham et al. 2013; Rodgers 2009; Rodgers et al. 2012a, b; Vishnupriya et al. 2018). Several methodologies have been developed to incorporate HF2V lead extrusion dampers into structural applications (Rodgers 2009; Latham et al. 2013; Rodgers et al. 2008a, b). However, the HF2V devices do not have recentering capabilities and can used in combinations with the RS dampers to provide significant energy dissipation and structural recentering (Ghahramanian Golzar 2018).

Inside a HF2V device, the bulged shaft displaces a volume of solid lead as the shaft moves within the containing cylinder, creating high contact stresses that distribute the lead out toward the thick fixed-wall condition created by the cylinder. The lead is plastically extruded through the annular restriction created by the bulged central shaft. Providing an accurate prediction of lead extrusion damper forces, stress, and force distribution is essential for device design for any given application (e.g., Latham et al. 2013). To aid design and implementation, empirical models have been proposed to estimate lead extrusion damper force capacities (Vishnupriya et al. 2018), as well as design equations (Rodgers et al. 2007). These models seek to delineate contributions from extrusion and friction, but cannot provide any detailed insight into the flow mechanics as the lead extrudes through the annular orifice.

There are also a few models proposed based on extrusion principles (Tsai et al. 2002; Parulekar et al. 2004) or Mohr-Coulomb criteria (Rodgers et al. 2007, 2006a, b), where, in addition, device forces may include a velocity-dependence component (Pekcan et al. 1999; Rodgers et al. 2006a). Design based models that can precisely predict HF2V force capacity are thus very limited.

As a result, devices are currently designed using models with limited precision. They are then manufactured and tested to determine if the exact force capacities match the design before application, potentially necessitating redesign if initial prototype devices do not provide a good match to the design needs (Rodgers et al. 2019). Thus, there is a major need for a modeling methodology to better estimate device force capacity in the design phase itself, without reliance on experimental testing to correct design. This method would ideally also provide detailed insight into the flow mechanics of the lead during extrusion. This additional insight will also facilitate optimization of design parameters and specific device geometry.

Finite-Element Modeling

Finite-element (FE) analysis is an effective method for simulating complex nonlinear mechanics of device operations and computing resulting force capacities (Lesar 1982; Jin and Altintas 2012). There are numerous FE software packages available. However, ABAQUS is popular for simulating complex contact problems with large deformation, as well as nonlinear problems like cutting and extrusion (Li et al. 2004; Nasr and Ammar 2017; Lei et al. 1999).

FE modeling using ABAQUS has been widely used to simulate the behavior of damping devices and predict damper forces (Titirla et al. 2018; Qiu et al. 2020; Mehrabi et al. 2017). The hysteretic responses achieved from the three-dimensional (3D) FE analysis of metallic dampers under quasi-static cyclic loading were seen to be in agreement with the experimental results for steel based dampers such as comb teeth dampers (Montazeri et al. 2021), two-stage energy-dissipation device (Fan et al. 2016), oval-shaped dampers (Najari Varzaneh and Hosseini 2019), steel memory alloy (SMA) steel dampers (Qiu et al. 2020), annular yield metal damper (Chen et al. 2019), and crawler steel dampers (Deng et al. 2013).

In lead-based dampers, severe lead deformation during loading is analyzed, and the dissipated energy is computed using device specific material properties in ABAQUS for lead extrusion and friction composite damper (Yan et al. 2018), infilled-pipe damper (Maleki and Mahjoubi 2014), and lead-core base isolators (Habieb et al. 2019) via 3D modeling. A two-dimensional (2D) axisymmetric approach is considered for capturing the hysteretic behavior of lead in telescopic lead yield dampers (TLYDs) (Eskandari and Najafabadiab 2018), where adaptive Lagrangian-Eulerian (ALE) meshing formulations are applied to mitigate severe mesh distortions and realistic simulations. Hysteretic behavior of dampers is evaluated and compared with experimental cyclic behavior for quantitative evaluation. However, no studies have previously focused on the finite-element modeling approach for the lead extrusion dampers. Additionally, these studies rely on data from limited numbers of damping devices’ design and strength for modeling and validation, limiting scope of uptake and application. A study encompassing wide variety of devices based on size, design, and force capacity is essential for a modeling and validating a reliable and generic FE tool for the HF2V devices.

Using device-specific material properties and design dimensions of a device, it can be expected to realistically simulate the HF2V lead deformation mechanics within a device, and thus estimate device capacity in the device and building design phases.

The basic behaviors of HF2V devices have been included in larger structural simulations to assess performance (Golzar et al. 2017; Bacht et al. 2011; Desombre et al. 2011; Rodgers et al. 2010, 2012a, b, 2016). In particular, simple device models have been used in larger structural analyses, but lack precision for specific device design (Desombre et al. 2011). This goal thus adds precision to the device design process itself where a detailed device design can be produced to match desired overall response behavior.

Thus, computer simulations could be used to predict device forces to optimize HF2V device design and performance, which has not been done before. Furthermore, such parametric design studies can be undertaken on broader range of devices without the time and cost involved with experimentally testing every configuration. This research uses ABAQUS to present and validate a general modeling nonlinear FE approach to predict HF2V device forces, where a validated general approach allows any user to apply it with confidence and without specialized operator inputs to the modeling.

Methods

HF2V Device Mechanics

HF2V device is a simple bulged shaft lead extrusion damper, as shown in Fig. 1. The HF2V device dissipates energy when the central bulged shaft of the device moves under external excitation through the solid-lead working material. As the bulged shaft is displaced through the device, the lead deforms and plastically extrudes through the narrower annular orifice between the bulge and the cylinder wall, producing forces from friction and extrusion (Vishnupriya et al 2018, 2021; Rodgers et al. 2006a, 2007). The displaced lead flows around the bulge and remains in contact with the shaft behind the bulge, where the lead recrystallizes and regains its original properties (Paul 1940; Robinson et al. 1976) at ambient temperatures.

Fig. 1. HF2V lead extrusion damper parts.

Previously, trailing voids were observed behind the bulge after shaft displacement in small devices, which adversely affect force capacity (Rodgers et al. 2006a, 2007). This issue is mitigated by prestressing the lead in the devices [(Rodgers et al. 2006b, March 10–12), 2007, 2019] to remove any air voids created during the manufacturing process. The lead is thus compressed and prestressed in manufacture to provide a more reliable device performance.

FE Model Description

A 2D axisymmetric model is created using ABAQUS/CAE version 6.13. The model is comprised of three parts: (1) an analytical rigid shaft; (2) an analytical rigid wall; and (3) a deformable volume of lead. A 2D model leveraging axial symmetry ensures faster computation. The parts are modeled to experimental device dimensions, based upon the computer-aided design (CAD) models used to manufacture the physical prototypes.

Material properties are assigned only to the lead because the endcaps and thick cylinder walls are effectively rigid to ensure all energy is dissipated by moving lead. The material properties of lead used are recorded in Table 1. The mechanical properties of lead are very sensitive to temperature, chemical composition, strain rate, and loading conditions, among other factors (Rack and Knorovsky 1978; Lindholm 1964; Gondusky and Duffy 1967; Evans 1970). From previous research, the plastic data for pure lead during compression at quasi-static velocities are those given in Table 2 (Loizou and Sims 1953; Lindholm 1964), where the experimental tests were also performed at very slow, quasi-static velocities similar to the experimental data used in this work. The model is not precompressed because the model assumes a perfectly filled lead device, which is the goal of prestressing these devices in actual manufacture (Rodgers et al. 2019).

Table 1. Elastic properties of lead used for modeling (Engineering Toolbox)

Property	Value
Density ( $kg / m^{3}$ )	11,340
Young’s modulus (GPa)	16
Poisson’s ratio	0.44

Table 2. Plastic data of lead used for model

Plastic strain	Yield stress (Pa)
0	689,476
0.01	5,810,000
0.02	89,63,184
0.04	1,24,00,000
0.08	15,100,000
0.12	17,000,000
0.16	18,000,000
0.2	19,000,000
0.24	21,000,000
0.28	22,000,000
0.32	22,750,000

Source: Data from Lindholm (1964).

A velocity of

0.5 mm / s

is applied to a reference node attached to the rigid shaft, which has degrees of freedom only in the longitudinal (

Y

) direction as shown in Fig. 2. The outer ends of the lead and the wall are fixed for rotation (UR3) displacement along the

X

- (U1) and

Y

- (U2) directions. The assembly of parts and the boundary conditions applied are as shown in Fig. 2. A CAX4R/CAX3 mesh is applied on the working material without any element deletion.

Fig. 2. Two-dimensional axisymmetric model and boundary conditions, where RP is the reference point.

The arbitrary Lagrangian-Eulerian (ALE) method, combining the Lagrangian and Eulerian formulations traditionally used for finite-element simulations, is applied on the lead region (Hughes et al. 1981). An unstructured mesh is applied to the lead surface to allow better remeshing of elements under deformation (Tu et al. 2018). The ALE finite-element method is used to simulate large deformation problems, allowing a moving mesh along with the moving part (Gadala and Wang 2000; Zhao et al. 2012; Zhuang et al. 2008; Yildirim et al. 2011). The motion of the mesh is only constrained at the boundaries and allowed to move under high strain within these fixed boundaries. The mesh is smoothed constantly to reduce element distortion without changing the number of elements and their connectivity (Donea et al. 1982). This remeshing allows the simulation of lead flow within the cylinder and around the shaft, providing a visual guide to the evolution of stress distributions with changing strain/strain rates in the devices as the shaft moves and dissipation forces are generated. A finer mesh is applied on lead along the shaft, where large deformations are expected, for higher accuracy in force outputs.

In the ALE adaptive mesh domain, the values for the frequency of mesh updates and number of iterations per each adaptive mesh increment, called the remeshing sweep per increment, are chosen as 10 and 1, respectively, as a best trade off (Li and Wie 2009; Yildirim et al. 2011). An improved aspect ratio is chosen for the mesh controls where the smoothing algorithm is based on “enhanced algorithm based on evolving geometry” (ABAQUS/CAE 2013). The meshing prediction is obtained based on position from the previous ALE adaptive mesh increment. However, there are trade-offs with this choice, device test velocity, and device capacity or lead strain. Thus, a short moving-average filter is used to eliminate small computational errors and computational noise, which arise due to remeshing each time step. Hence, a filter with exponential moving average, the smooth2 function, is applied to eliminate computational errors and the noise due to remeshing during each time step.

For a surface-to-surface contact between the lead surface and shaft, kinematic contact formulations are defined for the interactions between the rigid shaft and the lead as well as between the rigid wall and lead. The static friction coefficient between the lead and steel surface is given as 0.5 without lubrication (Fuller 1963). The transition from a static friction coefficient to lower values at quasi-static velocities is measured as 0.3 for pure lead (Bowden and Leben 1939). Taking into account the prestressing of lead and the resulting increase in normal forces, the decreased coefficient of friction is taken as 0.25, with no lubrication (Zhou 2014).

The analysis is run using ABAQUS version 6.13 Explicit, in multiple small step times of 1 s with automatic increments, to balance higher accuracy and computational time. The output is obtained from the history output of the contact pressure forces including the normal and friction forces on the lead along the shaft. These values are used to calculate the resistive force on the shaft as it moves through the lead, and thus the device force.

Device Information

Experimental HF2V device dimensions are given in Table 3 for 17 experimental devices of varying sizes and force capacities, where

D_{cyl}

is cylinder diameter;

D_{shaft}

is shaft diameter;

D_{blg}

is bulge diameter; and

L_{cyl}

is cylinder length.

F_{\exp}

indicates the experimental peak force obtained from experiments. Fig. 3 illustrates the dimensions of the HF2V devices as given in Table 3. They are segregated into small, typical, and large devices based on their force capacities and dimensions (Vishnupriya et al. 2018, 2021).

Table 3. Device data used for modeling and analysis

Serial No.	Device	$F_{\exp}$ (kN)	$D_{cyl}$ (mm)	$D_{blg}$ (mm)	$D_{shaft}$ (mm)	$L_{cyl}$ (mm)
Small	3	55	17	13	12	56
Small	4	85	20	17	16	68
Typical	5	160	89	40	30	110
	6	280	89	50	30	110
	7	360	89	58	30	110
	8	200	66	40	30	130
	9	346	66	50	30	130
	10	130	50	32	20	50
	11	150	50	32	20	70
	13	260	60	42	33	160
	14	155	50	35	24	100
	15	250	70	48	30	75
	19	125	40	27	20	100
	20	170	62	45	30	155
Large	16	145	54	35	30	160
	17	200	54	36	30	160
	18	260	54	38	30	160

Analysis

The model is applied the same way to all devices without any changes, a generalized, easily repeatable modeling approach. Results are analyzed for accuracy in prediction of experimental peak device forces, and its capacity to replicate the experimental hysteretic force-displacement behavior of the given HF2V dampers. Specifically, the following are analyzed:

1.

Qualitative validation:

•

The model’s capacity to accurately simulate the lead flow in the device is assessed as a qualitative assessment based on expected behavior. An ideal model should be able to simulate the lead flow around the bulge with the shaft displacement in the HF2V damper in accordance with existing research and provide a visual representation of stress distribution inside the devices.

2.

Quantitative validation:

•

The hysteretic force-displacement plots of the devices obtained from previous experiments are compared with simulated plots for each device.

•

The force capacity of the device is determined by summing the contact pressure forces (extrusion forces) and frictional forces output by the model and compared with the experimental results. This comparison is made relative to

\pm 14 %

standard error seen in testing 96 HF2V devices of 250-kN force capacity, which is thus used as manufacturing variability to assess the results in context (Rodgers et al. 2019). The upper range for the forces can be estimated by using the overstrength factor

φ = 1.2

(Rodgers et al. 2019).

Results and Discussion

HF2V Simulations and Qualitative Validation

During simulations, the shaft is displaced upward in small steps of 1 s. The flow of lead in a device is shown in Fig. 4. In the first steps, the lead is strained under compression by the bulge, causing a rise of stress on the top surface of the bulge and along the shaft above the bulge. As the shaft moves further through the lead, the mesh is observed to move downward along the shaft. The mesh/nodes move around the shaft and attach to the shaft behind the bulge as the shaft moves further up through the lead. The stresses build behind the shaft as more lead is pushed behind the bulge through the annular orifice between the bulge and the wall.

Fig. 4. Stress distribution in lead with shaft displacement (upward) at the input displacements noted for Device 17: (a) 2 mm; (b) 4 mm; (c) 10 mm; and (d) 15 mm.

The simulations replicate the expected actual behavior of lead in the lead extrusion damper under loading. With no element deletion, the mesh is not distorted to failure, and the number of elements before and after the analysis is the same. Thus, the overall qualitative validation matched expected plastic flow mechanics and similarly was good for this device and all others simulated (not shown). The overall modeling approach thus appears credible with no specific means to measure internal stresses in the HF2V devices from experiments for direct qualitative comparison.

The small separation between lead and the shaft seen in Figs. 4(b–d) is due to the delay in remeshing with each increment and can be avoided by employing fine meshing and higher remeshing frequencies in the model, with a corresponding increase in computation time.

Force-Deflection Plot Comparisons: Quantitative Validation

Force-displacement hysteresis plots are made for each device, with six shown in Fig. 5. The experimental plots are compared with the plots from FEM results. The FEM plots for Devices 6 and 19 show a very good match with the experimental plots, with overall similarity in the plot shapes and average forces achieved.

Fig. 5. Comparisons of plots between experimental results and finite-element models for HF2V devices.

The plots for Devices 7 and 18 show experimental plots similar to the FEM plots, but the peak forces do not perfectly match. In this case, the cause is less certain, but likely due to relatively small differences between real and assumed material properties. Also, the potential friction force coming from shaft to endcap friction is not captured in the FEM model.

The regular very small noise in the FEM plots are due to the stiction or stick-slip mechanism of lead at the cylinder walls or due to “snagging” along the analytical rigid shaft surfaces (ABAQUS/CAE 2013). This computational noise in the plots can be reduced by adding filters to the history output after identifying the correct cut-off frequencies required for the analysis, such that the peak forces are not modified. The comparison of plots with and without the filter are shown in Fig. 6 to illustrate the small computational noise and impact of filtering.

Fig. 6. Plots of the FEM results for Device 6 without and with filter.

In event of an earthquake, the devices undergo cyclic loading due to cyclic structural response. The FEM can be applied to study the fully reversed loading behavior of the dampers under loading. The plots in Fig. 7 illustrate the example of the FEM response of a HF2V device under fully reversed loading compared with the experimental results from variable-amplitude loading of the damper. Due to the high computation costs associated with a variable-loading protocol similar to the experimental testing, FE analysis is performed to match only the smaller amplitudes from the experimental testing.

Fig. 7. Comparison of fully reversed loading response of experimental results and FEM plots for the HF2V devices.

The key analysis challenge is using the FEM approach is to predict the onset of the yield plateau. Given the weakly-rate-dependent elastoplastic nature of the device response, additional loading cycles or larger input displacements add significant computational time, but limited additional scientific insight. Therefore, the FEM analysis presented within Fig. 7 excludes the larger displacement cycles.

It is worth noting the significant stiffness disparity between the FEM and experimental results in Figs. 5 and 7. This disparity is due to the fact that the FEM only considers elastic deformation of the lead working material and that the steel components of the device are modeled as analytically rigid in the computational model. Due to the elastic deformation of all steel components within the experimental test, as well as test machine compliance, the real device stiffness is much lower than the FE model. This analytical-rigid simplifying assumption for the steel components was applied to reduce computational effort, and the comparison in Figs. 5 and 7 is intended to show a comparison of peak device forces, but should not be used to predict initial device stiffness.

The differences in initial stiffness of the FEM plots in comparison with the experimental plots could also be attributed to differences in actual lead data from the lead stress-strain data considered in this research or variation in lead material properties. Reduced-integration elements are almost exclusively used in ABAQUS/Explicit computations for nonlinear problems. Stiffness in the elements may be overestimated or underestimated due to use of fewer integration points for displacement-based FE formulations. Although computational costs are minimized by the use of reduced-integration hourglass control elements, it could have an effect on the accuracy of elements.

FEM Force Prediction: Quantitative Validation

The force prediction results from the FE model and experimental data are given in Table 4. The total device forces are calculated by summing the contact pressure forces and friction forces formed during interaction between the rigid shaft and deformable shaft. For context, the standard error (SE) of manufacturing for HF2V devices is based on other unrelated experimental data for 64 identical devices (Rodgers et al. 2019).

Table 4. Comparison of forces from experiments and finite-element modeling

Serial No.	Device	$F_{\exp}$ (kN)	$F_{model}$ (kN)	$F_{contact pr}$ (kN)	$F_{friction}$ (kN)	Error (%)
Small	3	55	46	12	34	16
Small	4	85	52	8	44	39
Typical	5	160	167	54	106	$- 4$
	6	285	310	130	170	$- 8$
	7	390	400	135	265	$- 3$
	8	200	185	60	125	8
	9	346	345	145	190	0
	10	130	125	67	58	4
	11	150	152	78	74	$- 1$
	13	260	245	68	177	6
	14	155	155	45	110	0
	15	250	200	115	85	20
	19	125	130	59	71	$- 4$
	20	190	213	70	150	$- 12$
	16	170	175	27	148	3
	17	200	220	65	155	10
	18	260	245	45	140	6

The forces obtained from the FEM have forces within 10% for 13 of 17 devices (76%), which is within

\pm 1 Standard d e v i a t i o n (SD)

of manufacturing variability of 14%. Between 14% and 28% or

\pm 2 SEs

, there are 3 of 17 devices (18%). The last 1 of 17 devices has

- 39 %

error, which is within

\pm 3 SE = \pm 42 %

. This outcome and each

\pm SD

variation is visually represented in Fig. 7 where experimental forces are plotted against the forces obtained from the FEM analyses. The solid line in the center signifies a perfect-case 1:1 line where experimental forces and FEM results exactly match. Each line gray line away from the 1:1 line represents

\pm 1 SE

. The dots represent the experimental device forces plotted against the FEM model forces of the HF2V devices. Thus, all 17 devices are within

\pm 3 SE

(99.4%) of possible experimental variation in manufacture. These outcomes match reasonable statistical expectations of being within the expected variability due to manufacture for this smaller sample number, providing a further validation. The spread accounts for variability in device force due to manufacturing variability by

\pm 1

, 2, and 3 SEs.

In particular, this SE can be attributed to assembly and manufacturing variance. The model does not have manufacturing or assembly variability. The errors in experiments should be within the same distribution. Hence, a perfect 1:1 match would not be expected, but the spread shown in Fig. 8 meets statistical variation given this variability. It thus validates the model’s accuracy because each

\pm SE

variability captures expected numbers of devices. Specifically, if the model had a notable bias error, this result would not have occurred.

Fig. 8. Experimental device forces against FEM analysis results: SD representation of the devices. $R^{2} = 0.95$ for the perfect case line.

Some of the errors in the results can be attributed to insufficient device design data and testing data available for finite-element modeling, such as the specific device testing velocity, specific exact bulge profile data, bulge curvature radius, bulge angles, and bulge length. Previous studies suggested these parameters can potentially influence the extrusion forces outcomes (Lontos et al. 2008; Fereshteh-Saniee et al. 2013; Golondrino et al. 2012a, b). The understrength of the experimental results in comparison with the FEM results can be explained by irregularity in prestress and can be achieved by further prestressing.

Empirical and analytical modeling approaches for the HF2V lead extrusion dampers have provided an estimation of specific device forces and limit load capacities (Vishnupriya et al. 2018), but not the full trajectory as in this work. The peak forces achieved from the FE analysis are in good agreement with the force predicted using empirical model for the HF2V devices in previous research. Analytical models of the HF2V devices using first principles predicts the upper and limit loads of the HF2V devices (Vishnupriya et al. 2021). The computed FEM forces lie between the analytically predicted HF2V upper- and lower-bound limit force ranges. Unlike these models, the FEM approach provides insight into the internal mechanisms of the LEDs and computes the device forces automatically by generating a hysteresis plot for the device under loading, providing novel added information for design.

Overall Assessment

The material property values and boundary conditions were the same for all devices modeled, with only the specific device geometry changing. No calibrations or user inputs were made for simulating any devices for fitting or excluded due to large errors. The results from the FEM model are comparable to the forces from the design based force prediction model for the same HF2V devices modeled in this study (Vishnupriya et al. 2018), and the statistical distribution of results around the 1:1 line in Fig. 8 indicates no specific error or bias. Hence, the approach was entirely general and replicable.

Another possible limitation is high nonlinearity of the model. Remeshing computational noise and the choice of meshing frequency and sweeps per increment of 10:1 are not necessarily optimal. However, this choice is good enough given the good results presented here. An optimal value or methodology to estimate the best value for these simulation parameters would possibly improve results.

ABAQUS requires the input of stress-strain data for lead to be input based upon uniaxial test data. However, during the plastic deformation of extrusion, triaxial states of stress will exist within the lead working material. Therefore, additional factors such as the transverse confinement of the lead during plastic deformation may influence results. This limitation requires a significant change to the stress-strain data input into the model and would also require availability of experimental data for lead that were obtained from a triaxial stress tests.

One further possible limitation is temperature dependence and high-velocity loading of the devices, which is not captured in the model. However, there is no record of the testing temperature during the experimental HF2V testing or the heat produced during these device experiments because none were measured. HF2V damper behavior can possibly be more accurately modeled by considering temperature dependent Young’s modulus (Wen et al. 2002), stress-strain rate-temperature-load-dependent yield stresses (Wong et al. 1999), an experimentally derived coefficient of friction value (Childs 2006; Hora et al. 2013) or friction factor (Flitta and Sheppard 2003), and strain-softening and strain-hardening models (Kong et al. 1999; Ertürk 2009; Kong and Hodgson 2000). However, the built-in models in many FE packages are incapable of capturing many of these phenomena. Given such values, user-defined materials can be implemented through user subroutines inside VUMAT and FE coding (ABAQUS/CAE 2013) to improve model performance. VUMAT is a user subroutine that allows implementation of constitutive material behavior law for large-deformation processes that are complex to model (Ming and Pantalé 2018). However, this requires considerable expertise for effective implementation (Ming and Pantalé 2018; ABAQUS/CAE 2013).

A last limitation is computational time. Each simulation required 8–10 h using a 3.60 GHz Intel Core i7-470 computer with 32 GB of RAM. Highly nonlinear models are computationally expensive. However, more powerful computers or faster algorithms would improve this issue significantly.

Thus, the resulting finite-element model is a generic model that can predict well within the range for all types of HF2V lead extrusion dampers. Therefore, it can be used as a design tool along with the design-based model (Vishnupriya et al. 2018, 2021) to obtain the precise force capacity range of the desired device, limiting the need for extensive prototype validation and possible device redesign. However, due to the safety-critical nature of the implementation of these devices, final experimental testing to prove the design is likely to remain a necessary step before installation in the field.

Conclusion

Finite-element modeling and analysis of the internal reaction mechanisms of HF2V lead extrusion damper is done for the first time. The paper presents a finite-element modeling methodology for HF2V lead extrusion dampers that can predict device force capacities within the 14% standard error derived from experimental testing. The paper gives in detail all the finite element modeling parameters and methodology used in modeling the HF2V lead extrusion dampers.

The model is generic, and modeling parameters are kept consistent. The same modeling approach is applied to all devices, and only device geometry is changed. The model predicts device forces most precisely for typical and large devices. The model can be used as a design tool and can be used as a reference of expected HF2V device behavior and force capacities before manufacturing. The device parameters can be modified to observe corresponding force changes, and a design can be generated that gives optimum force outputs as required for the particular application.

Data Availability Statement

Some or all data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Acknowledgments

This project was partially supported by QuakeCoRE, which is a center funded by New Zealand Tertiary Education Commission. This is QuakeCoRE Publication No. 0690.

References

ABAQUS/CAE. 2013. ABAQUS users manual 2013 version 6.13-2: Documentation collection. Providence, RI: Dassault Systèmes Simulia.

Abstract

Introduction

Finite-Element Modeling

Methods

HF2V Device Mechanics

FE Model Description

Device Information

Analysis

Results and Discussion

HF2V Simulations and Qualitative Validation

Force-Deflection Plot Comparisons: Quantitative Validation

FEM Force Prediction: Quantitative Validation

Overall Assessment

Conclusion

Data Availability Statement

Acknowledgments

References

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