*Retracted: *Performance of Grouted Coupler Embedded in Reinforced Concrete ABC Bridge Pier at Vehicle Impact

Department of Civil and Environmental Engineering, Utah State University, Logan, Utah, USA

**Academic Editor:** Luciano Ombres

**Special Issue:** Fiber Composite Materials and Civil Engineering Applications

**Received:** September 27, 2022 | **Accepted:** December 23, 2022 | **Published: **January 03, 2023

Recent Progress in Materials **2023**, Volume 5, Issue 1, doi:10.21926/rpm.2301001

**Recommended citation: **Roy S. Performance of Grouted Coupler Embedded in Reinforced Concrete ABC Bridge Pier at Vehicle Impact. *Recent Progress in Materials* **2023**; 5(1): 001; doi:10.21926/rpm.2301001.

© 2023 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

**Abstract**

Increased occurrence of dynamic loading on bridge piers caused by seismic, blast and vehicular impact incidences have been increasingly reported. Statistical data shows that while seismic and blast have received significant attention in the literature despite the frequency of occurrence of vehicular impact is remarkably higher. Unfortunately, pier damage caused by high velocity vehicle impact contemplates a little consideration. With the newer construction methods like accelerated bridge construction (ABC) and its response against vehicle impact performance needs an additional study before widespread use. Static and dynamic performances of the coupler are thoroughly investigated in this study. A representative RC bridge with circular cross-section is considered with hollow cylindrical splice-sleeves filled up with concrete grout is embedded in the foundation. Finite element model (FEM) of single grouted coupler is developed based on the manufacturer data. Apportioned load through area ratio incurred by vehicle impact is considered and applied to the individual coupler. Numerical simulations are also performed using commercial finite element software package, ANSYS. This research is led by mesh independent studies to precisely accomplish sensitivity analyses. The results from numerical simulations are compared with the corresponding analytical solution. Stress-strain resulted from conservative dynamic simulation are plotted followed by integrity analyses to capture material performance at high deformation. Quasi-static strain rates are utilized for steel re-bars with connected coupler and validated with the finite element (FE) results to determine dynamic impact factor (DIF) of steel. This is further validated through utilizing experimental data from published journal. This study instills a good agreement by providing valuable information to aid in the selection of appropriate coupler connections, addressing material properties at various performance level, and contributes useful guidelines for the researchers, practitioners, and structural engineers.

**Keywords **

Accelerated bridge construction (ABC); vehicle impact analysis; finite element modeling (FEM) of grouted coupler; model validation

**1.**** Introduction**

Bridge piers experience highly dynamic impact due to seismic events, blast, and vehicular collisions. This may cause health deterioration of the bridge pier from less to severe, and possibly collapse. Structural evaluation for seismic performance has received significant attention in high earthquake prone states in the region comprising primarily of the western United States of America (USA) [**1**]. The seismic behavior and response of reinforced concrete (RC) piers has been the subject of extensive research efforts, undertaken to improve the capacity of the piers to resist damage during seismic events. Unfortunately, the other mechanisms of dynamic impact loads, vehicular collisions, and blasts have received little attention. This leads an additional insight to investigate the RC bridge pier performing on high strain rate load exerted by dynamic impact [**2**]. Different studies show that vehicular collisions with bridge elements are the most common dynamic impact scenario, especially with the increasing volumes of vehicular traffic [**3**]. The performance of the coupler has investigated for both dynamic and static combined stresses for various vehicle impacts. Quasi-static to dynamic strain rates of steel reinforcement connected to the couplers is also evaluated and published in [**4**]. Characterizing damage incurred by impact is subjected to the concrete strength parameters quantifying a good tradeoff between the shear capacity to primarily resist the impact load, and the flexural capacity controlling principal serviceability of the pier [**5**]. However, this present study intends to quantify the coupler post impact behavior at high strain rate load incurred by semi-trailer on the representative prototyped ABC pier utilizing grouted coupler.

Apart from the dynamic load exhibited by the earthquake, other dynamic loading scenarios, such as blast and impact also warrant investigation and thorough inspection. Precisely, vehicular impact calls for significant attention due to a high frequency of occurrence [**6**]. Recently, this has been observed that crashworthiness by vehicle impact jeopardizes direly the health, and hence reduces serviceability of the pier. With regards to vehicle impact loading encountered by reinforced concrete (RC) bridge piers, majority of the existing literature focuses on identifying damage levels or increasing survivability [**3**,**7**]. While this research is valuable, the performance of RC bridge piers under vehicle impact has yet to be fully investigated for recent developments in terms of materials and construction methods such as accelerated bridge construction (ABC). In ABC, connectors such as splice sleeves and grouted couplers are commonly used to connect different bridge components such as foundations to piers. The introduction of these new materials, splice sleeves and grouted couplers, differ the dynamic response of bridge structures since they typically behave higher stiffness’s than steel rebar, result in discontinuities of the reinforcing steel and change the energy dissipation path [**8**,**9**]. In addition, grouted coupler and splice sleeve connectors facilitate the construction of concrete bridge but their use in plastic hinge location is restricted as specified in seismic bridge design codes [**10**], whereas the performance to withstand high strain rate impact is still relatively unidentified. This study is executed for addressing the material properties and its contribution to the post impact performance of coupler on vehicle impact. In addition to new materials and construction methods, the performance levels of the bridge-piers assessment under multi-hazard effects have yet to be thoroughly researched. Sequential or simultaneous hazardous loading experienced by circular RC bridge pier due to blast and vehicular impact have already been investigated in terms of the performance and resistance reduction method [**11**].

However, precise assessment of the performance of splice-sleeve along with grouted coupler used in ABC needs an additional attention to be carried out on the ‘Grouted Coupler and Splice Sleeve Connections’ proposed by Utah Department of Transportation (UDOT) [**12**] before its extensive use. To analyze the impact characteristics of grouted coupler, it is placed in the pier- foundation connection to evaluate the performance standard of the splice sleeve and grouted coupler mechanism as a composite material [**13**]. The connection of splice sleeve is further studied in the report proposed by ‘Idaho Transportation Department’ using grouted couplers [**14**]. A study of the structural reliability of RC piers subjected to sequential loading exerted by blast and vehicular impact were also studied. The study showed that the structural performance of the piers is particularly sensitive to its dimensions [**11**]. This implies that an increase in the stiffness of the pier could possibly help it withstand the external forces caused by vehicular impact. Coupler sections have been studied as a means of increasing the stiffness of the RC piers to improve their seismic performance and overcome the development of plastic hinges [**11**,**15**]. However, the effect of these couplers on the response of RC bridge piers subjected to vehicle impact loading is a complex mechanism and still relatively unknown. This study is an attempt to evaluate the performance of individual grouted coupler section against short duration impact to predict the coupler behavior, and material properties as well. In the present study, splice-sleeves along with high grade concrete grouted coupler are embedded into the pier-foundation, placing the foundation top and the coupler cross-section in the same level, as shown in Figure 1. However, plastic hinges are expected to form in the weakest part of the pier, predicting dissipation of maximum energy [**8**], particularly at the pier-foundation junction as predicted and specified [**16**]. Performance of single pier due to dynamic impact exerted by vehicle collision on it, undergoing high frontal overpressure [**2**] as well. As such, failure mechanism of each coupler along with the material properties need rigorous prior investigation before recommending its widespread use in foundation-pier connection and other places for ABC non-traditional RC pier under axially compressive stress, and the combined stresses resulting from residual flexure caused by impact load, transferring it at the pier base [**17**].

**Figure 1** Splice-sleeve in Figure 1 [**18**].

The importance of this holistic present study is to examine static and dynamic characteristics of the single coupler material used in RC bridge pier to withstand vehicle impact. For assessing post high velocity vehicle impact behavior and enhancing the performance level, study of grouted coupler on dynamic load needs further scrutiny. The objective of this study is to determine post impact performance reliability of the ABC pier under vehicular impact by utilizing dynamic impact factor (DIF), carried out from finite element models (FEM) using manufacturer supplied material properties, and experimental testing results of the coupler. The FEM simulations are used in this study to determine the DIF of the single coupler under axial compression and shear load typically experienced from vehicle impact The FEM is used extensively in this study to determine DIF of the coupler. This present study is an accomplishment to investigate the performance of the connectors used in the non-traditional RC pier at vehicle impact to direct necessary bridge calibration followed by post impact performance before recommending it for widespread use in ABC.

**2. Methodology**

In this study, a prototypical splice-sleeve, grouted coupler, embedded in a foundation cap that connects to reinforcing steel from a bridge pier is evaluated for its performance under vehicular impact loading. Quasi-static to dynamic strain loading has been incorporated to investigate the grouted coupler performance at high vehicle impact load experienced from semi-trailer. Prototyped ABC pier with lower concrete strength has been utilized [**7**] to estimate damage of the coupler-rebar region incurred by impact. The impact of the pier specimen is subjected to the concrete strength parameters due to its high vulnerability to vehicle impact due to its exposed surface area quantifying a good composition between the shear capacity to primarily resist the impact load, and the flexural capacity predominately control the serviceability of the pier [**5**]. However, this present study intends to quantify the coupler high velocity post vehicle impact behavior at high strain rate load incurred by semi-trailer on the representative prototyped half sized ABC pier.

Performance level comprising of material properties and post impact behavior is determined by fractioning the load per coupler. The investigation is carried out through numerical simulations via FEM utilizing the material properties from manufacturer’s data. To examine the material behavior and failure pattern against vehicle impact, short duration post impact performances are evaluated through static and dynamic numerical simulations of the single coupler and validated with the manufacturer provided experimental data to compute the dynamic impact factor (DIF). The numerical models and respective simulations are exclusively carried out using the commercially available software package, ANSYS. DIFs computed using numerical simulations and analytical methods are compared, and validated with the experimental results published in the journals.

*2.1 **Material and Geometric Properties of Splice-sleeve*

*2.1*

*Material and Geometric Properties of Splice-sleeve*

Various connection types have been studied for precast concrete bridge piers in seismic areas and designated in two major categories of emulative and rocking connections [**19**]. The emulative connection for precast components is specified as a connection that includes special detailing. This type of splice-sleeve predicts better performance to withstand dynamic impact for reproducing a monolithic cast-in-place component.

For this study, geometrical details of splice sleeve recommended, and typically used in ABC connector is 8U-X and is as shown in the Figure 1 [**20**] and Table 1. Splice-sleeve used for grouted coupler in the column rebar embedded and placed (Figure 2) within the pier foundation, predicts enhanced performance in dynamic response [**10**].

**Table 1** Details of splice sleeve number 8U-X [**18**].

**Figure 2** (a) Grouted coupler and splice sleeve position position in ABC RC pier, and (b) Section A-A.

For this study, sleeve number 8U-X is considered and investigated as recommended for # 8 ASTM 706 bars used in pier for main (longitudinal) reinforcement.

The grouted couplers are strategically placed in the location where plastic hinges are expected to form at the pier-foundation connection [**14**]. In this study, splice-sleeves (Figure 1) considered from manufacturers catalogue, and grouted couplers are placed and embedded into the foundation as shown in Figure 2(a & b) . The pier section with grouted coupler is shown in Figure 2(b). Both ends of the pier are modeled as fixed supports, and the location of vehicle impact is considered at a distance 3 feet (approximately 1 m) height from the pier bottom or foundation top.

**3. ****Dynamic Impact Performance of RC Bridge Pier**

Bridge pier experiences highly dynamic impact due to seismic response, blast, and vehicular collision. This may cause health deterioration of the pier from less to severe, till collapse. From different studies and captured data from published journals [**21**], frequency of vehicular collision causing crashworthiness seems surpassing the other dynamic responses [**22**]. Vehicular impact and its catastrophic effect on the traditional reinforced concrete (RC) bridge pier has received attention. On the other hand, performance of accelerated bridge construction (ABC) using splice sleeve along with high grade concrete grouted couplers at high velocity vehicle impact are relatively unknown. In practice, couplers are generally placed at the pier-foundation connection where plastic hinges are expected to form and predicting post impacted coupler performance [**14**]. In this research, couplers are placed and embedded into foundation and not exceeding pier-foundation junction (Figure 2). Performance of the pier and the dynamic impact on it are studied for axially compressive stress and the combined stresses because of residual flexure due to impact at pier base. The coupler region must exhibit adequate stress limit over the dynamic stresses for service. Grouted couplers in plastic hinge zones must develop 150% [**15**] of the specified yield strength of the connected reinforcing bar. Stresses are computed from finite element modeling (FEM) to assess the coupler performances using high strain rate deformation [**16**]. High strain-rate deformation is considered especially as far as the impact and shocking load is concerned where the rate sensitivity leads to the high stresses causing enhanced stresses, resulting in dislocation. In addition, higher strain rate, the mobile dislocation velocity increases to accommodate the required plasticity [**23**].

*3.1 **Determination of Flexural Properties *

*3.1*

*Determination of Flexural Properties*

The representative RC pier is designed by utilizing a concrete grade of 3 ksi and longitudinal reinforcement (primary) of grade 60 steel (60 ksi tensile strength) considered from the published data [**24**], and for the shear reinforcement (transverse), grade 36 steel (36 ksi tensile strength) is used [**25**]. Sectional elevation of the RC pier and details of the pier cross-section are shown in Figure 2(a & b). End conditions are considered and utilized in pier are both ends are restrained against displacement and rotation in all directions, and unrestrained length of the pier is taken as 8.5 feet with circular cross-section throughout [**26**], as shown in Figure 3(b). The pier has primary reinforcement of 6 numbers # 8 steel re-bars throughout the foundation bottom, followed by a spirally arranged shear reinforcement with # 4 steel (grade of 36 ksi) rebar @ 2-1/2 inches pitch throughout as utilized in the representative pier [**26**]. Shear reinforcement provided in the pier conforms to the minimum shear reinforcement criteria [**27**]. In addition, the representative pier also satisfies the minimum shear reinforcement criteria for rebar diameter and vertical pitch of the spiral reinforcement [**28**].

**Figure 3** Impact Point and boundary conditions of the representative ABC bridge pier.

In this study, flexural performances of the coupler embedded at concrete foundation base is studied. Table 1 and Table 2 include computations of axial loads on pier and respective each steel rebar. DIF is shown in Table 3. Semi-trailer has been considered as a vehicle weight [**29**] for impact from the data given and as included in Table 4. Computation of the axially compressive load incurred by the RC circular bridge pier, as shown in Figure 2(b) [i.e., Sec A-A of Figure 1(a)], is given in Equation 1 [**10**]. Direct axial compression load from pier via individual reinforcing steel rebar (P_{n,s}) to coupler is transmitted and approximately evaluated by fractioning down the column axial load, using the area ratio (A_{coupler}/A_{net}), is as shown in Equation 2. Splice-sleeve details are already shown in Table 2. Computations of axial compressive load experienced by RC pier and individual coupler are shown respectively in the Table 2 and Table 3.

\[ P_{n, d}=0.85 f_{c}^{\prime}\left(A_{g}-A_{s t}\right)+A_{s t} f_{y} \tag{1} \]

\[ P_{n, s}=P_{n, d}\left(\frac{A_{ {coupler }}}{A_{ {net }}}\right) \tag{2} \]

where: *P _{n,d}* and

*P*

_{n,s}_{ }are the respective design axial compression capacities of RC pier and individual steel rebar;

*A*,

_{g}*A*,

_{st}*A*,

_{net}*A*,

_{n,s}*A*indicate gross cross-sectional area of pier, area of reinforcing steel rebar in pier section, net cross-sectional area of pier, cross-sectional area of each steel rebar, and cross-sectional area of hollow splice-sleeve; and

_{coupler}*f’*and

_{c}*f*are the compressive strength of concrete and yield strength of steel respectively.

_{y}**Table 2** Materials and geometric properties.

**Table 3** Design axial compressive load.

**Table 4** Computation of coupler moment [**29**].

The resulting values of axial compressive load experienced by the RC pier and partitioned load incurred by individual coupler conforming material and geometric properties (given in Table 2) using Equations 1 and 2 are shown in the Table 3. Static and dynamic moments received by the coupler because of dynamic impact are as shown in the Table 4.

*3.2 Determination of Dynamic Increase Factor (DIF)*

*3.2 Determination of Dynamic Increase Factor (DIF)*

The dynamic increase factor (DIF) is the ratio of the dynamic to static strength [**30**]. Reinforcing steel rebar being an isotropic and homogeneous material can dissipate high energy and carries out substantial impact load [**9**]. In this research, a representative RC pier specimen (Figure 2) has been considered for investigation. Vehicle weight (M) and impact velocity (V) of the semi-trailer are considered as 42,108 lbs (187.30 kN) and 100 ft/sec (30.48 m/sec) respectively [**31**]. Permissible vehicular velocity has been considered from the standardized permissible vehicular speed [**31**,**32**]. Determination of dynamic increase factor (DIF) because of vehicular impact and corresponding dynamic performance of steel re-bars are studied at quasi-static strain rate condition. Dynamic flow stress (σ_{dyn}) in steel at impact has been determined by the following Equation 3, as recommended [**2**]:

\[ \sigma_{d y n}=\sigma_{y}\left[1+\left(\frac{\overset{’}{\varepsilon} }{C}\right)^{1 / p}\right] \tag{3} \]

where: *σ _{y}* is a static flow stress of for ASTM A 706 [

**33**] using Grade 60 steel rebar and is considered as 60 ksi (420 MPa);

*C*is the material coefficient and

*p*is the strain rate parameters with the values are considered as 40 and 5 respectively [

**34**]. Quasi-static strain rate of steel re-bar (

*έ*) has been taken as 0.16 s

^{-1}for the vehicle velocity 100 ft/sec (30.48 mtr/sec) [

**31**] to assess the post impact scenario from the time of collision considering high strain rate non-linear loading, and ductile behavior as the RC pier is under the axial compression and experiencing of transverse load [

**35**]. From Equation 3, Dynamic flow stress (σ

_{dyn}) is computed as 79.8 ksi (550.2016 MPa).

Using dynamic parameter σ_{dyn} as determined from the Equation 3, ‘ξ’ can be computed by using the Equation 4 [**36**,**37**] by replacing it.

\[ \xi=0.019-0.009 *\left(\frac{\sigma_{d y n}}{60}\right) \tag{4} \]

where: *ξ* is a dynamic parameter which depends on the dynamic yield stress of steel at the strain hardening zone, and *σ _{dyn} *is the dynamic flow stress at uni-axial plastic strain rate of steel.

However, *ξ *is evaluated as 0.0172 after replacing *σ _{dyn}* (using Equation 4) as 79.8 ksi.

Dynamic Impact Factor (DIF) can be evaluated from Equation 5 by using *ξ* computed from Equation 4 [**37**,**38**].

\[ D I F=\left(\frac{έ}{10^{-4}}\right)^{\xi} \tag{5} \]

where: *DIF* is the Dynamic Impact Factor, *ξ* is dynamic parameter, and *έ *is the Quasi-static strain rate of steel re-bar.

Equation 5 yields the analytical DIF result as 1.053.

*3.3 Computation of Analytical Static and Dynamic Forces of Coupler *

*3.3 Computation of Analytical Static and Dynamic Forces of Coupler*

Static impact force (I_{S}) due to vehicular collision van be computed from Equation 6.

\[ I_{S}=\frac{W V}{t} \tag{6} \]

where: *I _{S}* is the static impact force,

*W*is the semi-trailer weight (42108 lbs or 19099.87 kg-wt);

*V*is the maximum permissible impact velocity considered as 100 ft/sec (30.48 m/sec) [

**2**] and

*t*is impact duration considered as 40 ms (milli-second) [

**39**] in this present study, and

*h*is the height of impact as shown in Figure 3.

_{I}Inserting values in Equation 6, yields I_{S} as 105270 kip-ft/sec^{2} or 142114.50 kN-m/sec^{2} (corresponding equivalent load is 3271.896 kips or 14554.12 kN) [**39**,**40**].

Corresponding static moment (M_{s}) experienced by the pier from vehicle impact can be determined by using Equation 7.

\[ M_{S}=I_{S} \cdot h_{I} \tag{7} \]

where: *M _{s}* represents the static moment incurred by RC pier, and

*h*is the height of impact considered from the foundation top.

_{I}By using simplified load fractioning method after multiplying M_{s }(Equation 7) with the area ratio (A_{coupler}/A_{net}), the static moment experienced by single coupler (M_{s,c}) can be approximately computed from Equation 8.

\[ M_{s, c}=M_{s}\left(\frac{A_{ {coupler }}}{A_{ {net }}}\right) \tag{8} \]

The corresponding approximate dynamic moment incurred by single coupler (M_{dyn,c}) is computed from Equation 9 in order for carrying out the analytical model.

\[ M_{d y n, c}=D I F . M_{s, c} \tag{9} \]

where: *M _{s,c}* and

*M*represent static and dynamic moments experienced by single coupler from vehicle impact via load transmittance, and

_{dyn,c}*DIF*represents the dynamic increase factor.

Equation 8 yields static coupler moment (M_{s,c}) as 22.257 kip-ft (30.17 kN-mtr) incurred by each coupler. The corresponding dynamic coupler moment (M_{dyn,c}) is computed from Equation 9, yields as 23.437 kip-ft, (30.18 kN-mtr), and is shown in Table 4. As the dynamic properties cannot be estimated directly due to short duration collision, it can indirectly be computed by using DIF of reinforcing steel rebar [**8**].

Computed fractioned loads for axial compression (P_{n,s}) as computed for individual steel rebar is of 3.01 kips (13.40 kN), and has been already discussed in Section 3.1. The apportioned transverse load at the free end of individual steel rebar (8 in. edge, as shown in Figure 4) developed from the static moment (M_{s}) from vehicle impact, which in furtherance incurs moment at coupler-rebar junction in pursuance of conservative analyses (Table 2, Table 3 & Table 4) yields as 22.257 kip-ft. (31.54 kN-m). This results substantial stress in the coupler-rebar junction (as shown in Figure 4).

**Figure 4** End conditions and Load model for FEM of coupler-rebar.

However, in this study, static and corresponding dynamic analyses are performed, and each result is compared to predict the material properties in demand to with the impact scenario. Dynamic properties cannot be measured directly due to short duration collision and non-linear trend, but using DIF dynamic moment (M_{dyn,c}) at coupler-rebar junction can be evaluated. This is also subjected to significant stress induced in the couple-rebar junction. However, energy dissipation occurs substantially during and immediate after the impact controlling pier behavior, and corresponding failure pattern [**8**]. The performance of the plastic failure zone as corresponding to its stiffness warrants flexural members as it governs the resistance of the post impacted RC member [**22**].

**4. Finite Element Modeling (FEM) for Grouted Coupler**

In this study, finite element modeling (FEM) has been extensively used to predict the individual coupler performance at vehicle impact. ‘ANSYS’ is used for performing static and explicit dynamic analyses to develop FE models for simulations and respective results. To develop the model, hollow cylindrical cast iron splice-sleeve (36 ksi or 248 MPa)) is used along with 6 ksi (41.36 MPa) grouting and # 8 reinforcing streel rebar embedded into the grout, are as shown in the Figure 2. For all different material’s connections, non-separable contact is considered for developing the model. The mesh size considered for the simulations followed mesh-independence is considered as 0.01 in. (0.254 mm). Steel rebar’s conforming specified yield strength (60 ksi or 420 MPa) are embedded and extended from coupler in both sides are 8 in. (20.32 cm) and 6 in. (15.24 cm) respectively, as shown in Figure 2(b). The external surface of the splice-sleeve model is considered as fixed in the peripheral surface as it is embedded and placed in the foundation concrete. The free end of the 6 in. side is also considered as fixed (as shown in Figure 4) as it is extended within the foundation and received adequate development length.

To compare the DIF’s computed from numerical simulations comprising of FE model with the analytical results, stress ratio (σ_{dyn}/σ_{static}) is considered using steel strain rate from Equation 3. The peripheral surface of the grouted coupler is considered as no lateral displacement in any directions as it is embedded in concrete, and hence considered as fixed. The equivalent fractioned forces are applied to the larger end of steel rebar of the model as shown in Figure 4. Apportioned horizontal impact (shear) and axial forces (flexure) are deployed on the model from the respective external moment and axial compressive force of pier at the free end (8 in. edge of steel bar from pier base and foundation top within pier) and shown in the Table 3 and Table 4. End conditions along with the boundary conditions of coupler-rebar model are also shown in Figure 4.

in this study material properties incorporating composite model for addressing splice sleeve and grouted coupler are developed, and post impact performance at vehicle impact has been investigated using numerical modeling through FEA. The various material properties utilized in this study are extracted from manufacturer’s data and are shown in Table 5.

**Table 5** Material properties utilized for FE Modeling.

*4.1 Meshing of Grouted Coupler*

*4.1 Meshing of Grouted Coupler*

Square mesh is considered for the entire grouted coupler and rebar model (as shown in Figure 5) along with splice-sleeve undertaken for the analyses as 0.01 in., for all elements. Mesh sizes are further re-iterated from 0.1 in., and 0.05 in. respectively to carry out the sensitivity analyses showing if any variation exist. For all three different materials and their attachments, non-separable contact has been incorporated to act as a monolithic behavior of the model under vertical axial compression and horizontal impact for modelling flexure and shear. During simulation, high frictions are developed at the contacts of all inter material surfaces. The coupler-rebar model considered in this study shows large deformation because of transmitting horizontal load while RC bridge pier experiences high strain rate velocity vehicle impact load. Figure 5 shows FE model representing the grouted coupler and rebar consisting of mesh size of 0.01 in^{2} (0.254 mm^{2}). Results are evaluated in terms of static and dynamic performances of the grouted coupler experiencing transmitted impact from RC bridge pier hit by high velocity semi-trailer.

**Figure 5** Meshing of FE Modeling for grouted coupler (a) longitudinal view, (b) top view.

*4.2 Mesh Sensitivity Study of FEM*

*4.2 Mesh Sensitivity Study of FEM*

Mesh independence analyses are performed for respective mesh sizes of 0.1 in., 0.05 in. and 0.01 in, where results from strain and total deformation are considered for both static and dynamic considerations. Mesh independence with proximity is observed from the results comprised by numerical simulations developed by using ANSYS model for both static and dynamic considerations. In the present study, simulation is performed for individual mesh size, and is considered for the optimization. Results depicting from the mesh independence study to carry out sensitivity analyses involving different mesh sizes and compare with the results of the total deformation and equivalent strain are shown in the respective Figure 6, Figure 7, and Figure 8 incorporating deformations and corresponding equivalent strain. Mesh sensitivity studies to incorporate static results are precisely shown in Figure 6, whereas the results from dynamic performance comprising deformation and strain for different mesh sizes are shown in the Figure 7 and Figure 8 respectively. Different mesh sizes for conducting sensitivity analyses show the results are within the proximity. However, in this study, 0.01 in^{2} (0.254 mm^{2}) mesh sizes are considered for static and explicit dynamics analyses using commercial software package, ANSYS. Figure 7 and Figure 8 address results for time dependent total deformations and maximum strain concerning respective mesh sizes of 0.1 in^{2}, 0.05 in^{2} and 0.01 in^{2}. Results illustrating static and dynamic analyses incur mesh independent and sensitivity study, determine total deformation and strain in the grouted coupler model embedded within the RC ABC bridge pier experiencing vehicle impact.

**Figure 6** Mesh sensitivity for total deformation and equivalent strain (Static).

**Figure 7** Mesh sensitivity for total deformation (Dynamic).

**Figure 8** Mesh sensitivity for maximum strain (Dynamic).

*4.3 Uncertainty Assessment Using Confidence Interval (CI)*

*4.3 Uncertainty Assessment Using Confidence Interval (CI)*

Confidence Interval (CI) has been utilized to capture the degree of uncertainty for assessing the numerical results evaluated from dynamic simulation using normal distribution. CI also able to evaluate the probability that a parameter falls between a pair of values around the mean. Thus, the confidence interval (CI) is utilized to assess uncertainty, and determined via using mean (μ), standard deviation (SD), confidence level (z) and sample size (N) (as shown in Table 5) and is as shown in the Equation 10 [**41**].

\[ C I=\mu \pm z \cdot \frac{S D}{\sqrt{N}} \tag{10} \]

Where: *μ* is the mean of sample size, *SD* is the standard deviation, *N* is the sample size considered as one thousand data, and *z* is the confidence or significance level considered as 98%.

CI data to capture the uncertainty is shown in Table 6.

**5****. ****Results **

*5.1 Results Using Finite Element Model (FEM)*

*5.1 Results Using Finite Element Model (FEM)*

Modeling’s in furtherance of analyzing for stress induced, governing strain and deformation are carried out by using commercial software package, ANSYS, for performing static and explicit dynamic FE simulations. Static analyses are conducted for both scaled-down axial compression and the impact experiencing by each grouted-coupler and reinforcing steel rebar. Further analyses using explicit dynamic are also undertaken and the results including stress, strain and deflection are compared with the results from statics. Substantial deformations are observed from the results of both static and dynamic analyses. However, almost the failure is identified at the junction of steel rebar and coupler, and observed from the deformations in both the directions, parallel and perpendicular to the load of 1.375 in. (34.925 mm) and 0.119 in. (3.023 mm) for static, and 2.38 in. (60.45 mm) and 0.11 in. (2.794 mm) for dynamic, respectively.

### 5.1.1 Results Showing Finite Element Model in Performing Static Analysis

Results from static analysis present considerable deformation at the steel rebar. Deformations in both X and Y directions seem uniform, as 0.12 in. (0.004 mm.) and as shown in Figure 8. High strain concentrations and significant stress are observed in the contact of grouted coupler and steel rebar as shown in the Figure 9 and Figure 11. Maximum permissible modulus of elasticity (E-modulus, i.e. considered as Maximum stress/Maximum strain) requirements from the simulation results subjected to static strain is 1.38 (Figure 10) and static stress (Figure 11) is 8.51*10^{5} psi (5.8*10^{3} MPa), whereas modulus of elasticity for steel rebar at the coupler junction demands as 6.17*10^{5} psi (7.57*10^{5} MPa), which endorses material property safe enough as material E-modulus incorporating rebar in this study has been considered as 29*10^{6} psi. Time-dependent static strain and stress are shown in Figure 12. However, maximum strain and stress concentration at coupler-rebar junctions are considered for conservative approach and safety.

**Figure 9** (a) Directional deformation in X-axis; (b) Directional deformation in Y-axis.

**Figure 10** Static strain.

**Figure 11** Static stress.

**Figure12** (a) Time-dependent static strain; (b) Time-dependent static stress.

### 5.1.2 Results Showing Finite Element Model in Performing Dynamic Analysis

Results from dynamic analyses show significant deformation at steel rebar with high stress and strain concentrations at the contact of grout and steel rebar, as shown in the Figures 13 to 15. In analyses, directional deformations seem quite different in the respective directions (3.15 in. and 0.12 in. along X and Y directions as shown in the Figure 12(a) and Figure 12(b)). Maximum permissible modulus of elasticity [Maximum Dynamic Modulus = Maximum dynamic stress/Max. dynamic strain)] requirements from simulation results subjected to dynamic stress and strain are 6.25*10^{5} psi (1.82*10^{5} MPa), and 0.2 which exceeds material E-modulus incorporating rebar in this study. Using maximum dynamic stress over dynamic strain, E-modulus demand (maximum dynamic stress/maximum dynamic strain) is computed as 31.25*10^{6} psi (2.15*10^{5} MPa), whereas material E-modulus considered as 29*10^{6} psi (2.1*10^{5} MPa). However, demand of E-modulus for dynamic over material from FE analysis is computed as 1.07, whereas the numerical DIF is computed as 1.053 which commends a proximity between material and dynamic property. This result demands high E-modulus of steel rebar to perform better on impact without undergoing large deformation. However, stress concentrations and strain shown at grout are remarkably high which controls design parameters. Significant bond failure or spalling are not observed from the results albeit steel rebar and grout experience large deformation and strain. Time-dependent static strain and stress. However, for avoiding variations in strain and stress, conservative approach to determine maximum instantaneous strain and stress concentration at coupler-rebar junctions are considered for safety.

**Figure 13** (a) Directional deformation in X-axis; (b) Directional deformation in Y-axis.

**Figure 14** Dynamic strain.

**Figure 15** Dynamic stress.

Time-dependent dynamic strain and stress are also captured from utilizing explicit dynamic simulations and are shown in the Figures 16(a) & 16(b) to evaluate dynamic modulus (E_{D}). Significant plastic deformations as the result of dynamic impact load application parallel and perpendicular to it are as shown in the Figure 17(a) & Figure 17(b). Deformation that takes place parallel to the load seems significantly higher than that of the perpendicular. However, to avoid variations in deformations at different directions as a function of time, dynamic simulation to determine maximum plastic deformation is considered for being safety and as a part of conservative approach of investigating post impact material performance.

**Figure 16** (a) Time-dependent dynamic strain; (b) Time-dependent dynamic stress.

**Figure 17** (a) Deformation parallel to load; (b) Deformation perpendicular to load.

### 5.1.3 Results Showing Coupler Performing from Dynamic Analysis

Von Mises stress and corresponding strain plotted from dynamic simulations can capture the material property (dynamic modulus of elasticity) via regression analysis, come up with non-linear trend of performance function (g) of dynamic stress concentration and the corresponding dynamic strain are as shown in Figure 18.

**Figure 18** Dynamic stress and strain relationship to capture material properties.

From Figure 18, regression results depicting a non-linear trend plotted with R^{2} value of 0.99 to apprehend post impact performance of coupler is as shown in the Equation 11.

\[ g\left(\sigma_{D}, \varepsilon_{D}\right)=\sigma_{D}-4 \cdot 10^{7} \cdot \varepsilon_{D}{ }^{2}-3 \cdot 10^{7} \cdot \varepsilon_{D} \tag{11} \]

Where: *σ _{D} *is the dynamic stress concentration at the pier due to impact and

*ε*is the dynamic strain, and

_{D}*g*is the post impact performance function of coupler.

### 5.1.4 Integrity Analysis of the Model

Due to variations in static and dynamic simulations results and complexities involved to capture material modulus as a post impact performance, and exceedance of dynamic over static and material modulus, dynamic results for material modulus as demand are considered to capture post impact criteria. The mean (μ), covariance (V) and standard deviation (SD) of dynamic simulation results used to determine post impact performance integrity from dynamic stress and strain are shown in Table 7. One thousand simulations of dynamic stress and strain were developed by using random variables having incorporated stress (σ_{D}), strain (ε_{D}), and material modulus (E_{D}).

**Table 7** μ, V and SD for dynamic stress (σ_{D}), strain (ε_{D}) and modulus of material (E_{D}).

The results using ‘RAND’ function generated from random variables are shown in Figure 19, comprising integrity analyses conducted from the dynamic simulation results utilizing Table 7.

**Figure 19** Results of the integrity analysis from dynamic stress and strain.

From Figure 19, high-precision non-linear results from FE simulations are linearized and are plotted via regression analysis to precisely capture the post impact dynamic performance of coupler using integrity analyses, as expressed in the Equation 12. This equation will help to provide high accuracy results via linear relationship with R^{2} value of 0.76 from the regression result of specific impact scenario comprising post impact dynamic stress and strain concentration at coupler region.

\[ g\left(\sigma_{D}, \varepsilon_{D}\right)=\sigma_{D}-2 * 10^{7} \cdot \varepsilon_{D} \tag{12} \]

where: *σ _{D} *is the dynamic stress concentration and

*ε*is the corresponding dynamic strain at the coupler, and

_{D}*g*is the post impact performance function as expressed in terms of stress and strain.

High precision dynamic simulations are carried out to capture the uncertainty of material performance and its behavior in terms of dynamic modulus of demand (beyond elastic limit) via regression analysis, and the results come up with a linear trend of performance function (g). The performance function (g) addresses linear trend to best capture of dynamic stress and the corresponding dynamic strain concentration at coupler-rebar junction resulted by specific vehicle impact load comprising with a little flexible R^{2} value due to large dataset.

### 5.1.5 Uncertainty Assessment Using Confidence Interval (CI)

Confidence Interval (CI) has been utilized to measure the degree of uncertainty for assessing non-linear results evaluated from dynamic simulation comprising stress (σ), strain (ε), and material modulus (E_{D}) of coupler materials resulted in during dynamic impact event. The uncertainty in the result of post impact dynamic event to capture the uncertainty involved is estimated using the confidence interval (CI) as exhibited in Table 6 is presented in Table 8. The CI results depicting uncertainties in material properties portray a substantial variation assessing post-performance behavior of the coupler material at specific impact load.

### 5.1.6 Validation of Model

To validate the model for strain variations along with the stress concentrations at the steel rebar and coupler junctions, numerical (FE) simulation results from static analyses are compared with the experimental data comprising the strain and corresponding stresses from the published journal [**10**]. The model shows a good agreement and positive coherence with the experimental results in terms of the stress - strain relationship when apportioned load is transmitted and incurred from vehicle impact event on RC bridge pier and transmits into grouted coupler. The model validation is as shown in the Figure 20.

**Figure 20** Model validation with experimental data [**10**].

**6****. ****Discussions**

In this research, an attempt has been carried out to predict the post impact performance of a RC bridge pier considering flexural response. In predicting its performance, splice-sleeve along with grouted coupler has been introduced in the bridge pier base. Stresses from vehicular impact are determined, and then compared the numerical results using FEA models for static and dynamic analyses. FEA models are generated in a conservative way where the apportioned axially compressive load along with moment are incorporated and applied via using area ratio [Load*(A_{coupler}/A_{net})], to investigate the impact scenario of splice-sleeve under axial compression. Load has been generated, and apportioned loads are applied at the free end of the steel re-bar (Table 2). Stresses in the pier-base due (maximum stress concentration at the coupler end) to vehicular impact are analyzed for maximum stress concentration, strain compatibility, and deformations. Stresses resulted due to impact and the dynamic amplification effect draw an insightful correlation between DIF’s computed analytically (1.053) and numerically from the FE simulation (1.07) using the ratio of E-moduli (Dynamic material modulus/Static material modulus). FEM analyses present a little conservative result correlating proximity to the realistic material performance. Due to variations in the results of static and dynamic simulations, and complexities involved to capture material modulus as a post impact performance and exceedance of dynamic over static and material modulus, dynamic results for material modulus as demand are considered. The following observations perceived from this research are as shown:

- The study depicts a little conservative result due to the boundary conditions deployed, and hence seems more realistic in design as far as dynamic effect is concerned. No further mesh refinement is necessary to predict split or spalling. Results shown from the exact solution while experiencing DIF of 1.053 (i.e., 5.3%), whereas material properties with a DIF of 1.07 (i.e., 7%) in terms of dynamic modulus of elasticity demands a little more conservative of 1.6% increment. FEM simulations are used to determine and validate DIF computed analytically. DIF computed by using analytical method and numerical simulations are quite similar, with a 1.6% difference. This indicates high stress concentration in rebar-coupler junction, and from steel deformation seems critical in rebar due to conservative modeling.
- Maximum stresses developed at the contact of steel rebar and coupler zone. This seems almost failure followed by large deformations in dynamic performance governing the control of material property. On the other hand, significant deformations are also observed along both the axes without being failure in steel rebar for static performance.
- The time dependent static strain with stress results is predicted from simulations and are plotted by incorporating high strain rate loading. Dynamic performances of the steel and concrete composite system and its post impact behavior are further assessed and as shown in the Figure 17(a & b). Non-linear stress and strain concentration during specific vehicle impact is plotted and as shown in the Figure 18, and the results are expressed in Equation 18. Integrity analysis of dynamic stress-strain resulted from simulation has been undertaken via regression analyses. Results from regression analyses (as shown in Figure 19) are capable to capture material property while undergoing substantial deformation. From confidence interval (CI) the material property warrants about dynamic impact event as the dynamic demand (E
_{D}) of coupler material exceeding the modulus of elasticity (E). - Validation of model from the results of static stress and strain are compared with the experimental data and is as shown in Figure 20. This study also represents a positive correlation of the model compared with the published data from articles extracted from the experimental results. Risk analysis has been conducted using confidence interval and the results are shown in Table 5.

**7. Conclusions**

In reinforced concrete (RC) structures, piers are usually the most vulnerable members to collisions due to its exposed face and its slender behavior. In particular, the desired characteristics and the associated impact performance levels of a structure during a vehicle collision are not well defined. Therefore, there has been a need to assess the accountability of the existing structures against such collisions, and proffer solutions to limit such susceptibility and enhancing its performance level to safely withstand the impact force. A specific method to assess the material capacity and demand of the coupler to its dynamic post impact performance is studied in this research. Based on the comparison of analytical studies, FEM simulation results, and its validation with the experimental results published in journals, the following conclusions can be drawn:

- Studies are conducted for comparing analytical and numerical analyses using FEM comprising static and dynamic force models to assess material moduli in demand to withstand high strain rate dynamic impact and are compared with the respective static material modulus. To assess material performance, dynamic material modulus is compared with the static material modulus to compute DIF in demand, and to come up with a good understanding and agreement on material calibration utilizing approximate evaluation of the transmitted loads.
- Performance based studies of the impacted coupler are investigated and presented for short duration impact where steel strain rates play significant role for the deformation. Steel strain rate which is a function of DIF directly influences the resulting stress in the coupler-rebar junction as shown in the Figure 18 and Figure 19 can help providing a fair decision about material viability of the coupler at impact, and hence resolving and controlling material performance at impact.
- This research is also an attempt to investigate the material requirement for enhancing resistance of 7 to 10% strength upgradation in material modulus is recommended for the cast-iron of splice-sleeve and steel rebar to withstand vehicle impact safely without failure. This study instills an insightful idea and realistic correlation to accomplish material properties that can be safely predicted for the essential criteria and useful calibration as well.
- The integrity study utilizing simulation results will help to detect high accuracy results at coupler from vehicle impact scenario. This will also provide improved information on material behavior and post impact performance for future calibration.
- Risk analysis conducted using CI provides a comprehensible understanding for capturing the uncertainty in using present coupler materials to withstand specific dynamic impact. Materials need to be upgraded for safely withstanding the dynamic effect.

However, high precision experimental studies involving various geometries, material properties, different boundary conditions, and different impact scenarios are recommended before considering accelerated bridge constructed (ABC) piers for widespread use.

Conversion Chart for the US Customary to the Equivalent SI Units:

**Symbols**

**Acknowledgments**

This publication was supported by NMB Splice Sleeve, North America, Inc. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of NMB Splice Sleeve.

**Author Contributions**

Suman Roy: Conceptualized and developed the theoretical formalism, numerical models (FEM) and simulations, validation, performed the analytic calculations, writing draft and editing.

**Funding**

NMB Splice-Sleeve – North America, USA.

**Competing Interests**

I, the author of the manuscript declare that I have no competing interests.

**References**

- Hauksson E, Kanamori H, Stock J, Cormier MH, Legg M. Active Pacific North America Plate boundary tectonics as evidenced by seismicity in the oceanic lithosphere offshore Baja California, Mexico. Geophys J Int. 2014; 196: 1619-1630. [CrossRef]
- Feyerabend M. Hard transverse impacts on steel beams and reinforced concrete beams. Karlsruhe: University of Karlsruhe; 1988.
- Sharma H, Gardoni P, Hurlebaus S. Performance-based probabilistic capacity models and fragility estimates for RC columns subject to vehicle collision. Comput Aided Civ Inf. 2015; 30: 555-569. [CrossRef]
- Roy S, Unobe ID, Sorensen AD. Investigation of the performance of grouted couplers in vehicle impacted reinforced concrete ABC bridge piers. Adv Bridge Eng. 2022; 3: 18. [CrossRef]
- Roy S, Unobe ID, Sorensen A. Damage characterization and resilience optimization of reinforced concrete bridge piers under vehicle impact. Adv Bridge Eng. 2022; 3: 16. [CrossRef]
- Arias-Acosta JG, Sanders DH. Evaluation of the seismic performance of circular and interlocking RC bridge columns under bidirectional interlocking RC bridge columns under bidirectional shake table loading. Proceedings of the 15th World Conference on Earthquake Engineering; 2012 September 24-28; Lisbon, Portugal. Kalyanpur: Indian Institute of Technology Kanpur.
- Pantelides CP, Ameli MJ, Parks J, Brown DN. Seismic evaluation of grouted splice sleeve connections for precast RC bridge piers in ABC. Taylorsville: Utah Department of Transportation; 2014.
- Kowalsky MJ. Deformation limit states for circular reinforced concrete bridge columns. J Struct Eng. 2000; 126: 869-878. [CrossRef]
- Roy S, Sorensen A. Energy based model of vehicle impacted reinforced bridge piers accounting for concrete contribution to resilience. In: International Probabilistic Workshop. Cham: Springer; 2021. pp. 301-315. [CrossRef]
- Ameli MJ, Pantelides CP. Seismic analysis of precast concrete bridge columns connected with grouted splice sleeve connectors. J Struct Eng. 2017; 143: 04016176. [CrossRef]
- Thomas RJ, Steel K, Sorensen AD. Reliability analysis of circular reinforced concrete columns subject to sequential vehicular impact and blast loading. Eng Struct. 2018; 168: 838-851. [CrossRef]
- Ameli MJ, Brown DN, Parks JE, Pantelides CP. Seismic column-to-footing connections using grouted splice sleeves. ACI Struct J. 2016; 113: 1021-1030. [CrossRef]
- Jacob GC, Fellers JF, Starbuck JM, Simunovic S. Crashworthiness of automotive composite material systems. J Appl Polym Sci. 2004; 92: 3218-3225. [CrossRef]
- Ebrahimpour A, Earles BE, Maskey S, Tangarife M, Sorensen AD. Seismic performance of columns with grouted couplers in Idaho accelerated bridge construction applications. Boise: Idaho. Transportation Dept.; 2016. [CrossRef]
- Tazarv M, Saiidi MS. Seismic design of bridge columns incorporating mechanical bar splices in plastic hinge regions. Eng Struct. 2016; 124: 507-520. [CrossRef]
- Ebrahimpour A, Earles BE, Maskey S, Tangarife M, Sorensen AD. Seismic performance of columns with grouted couplers in Idaho accelerated bridge construction applications. Boise: Idaho. Transportation Dept.; 2016. [CrossRef]
- Zhou D, Li R, Wang J, Guo C. Study on impact behavior and impact force of bridge pier subjected to vehicle collision. Shock Vib. 2017; 2017: 7085392. [CrossRef]
- ICC-ES Report. FSR, FTX, fire-retardant-treated wood shakes and shingles. Los Angeles: ICC Evaluation Service; 2016; ESR-1410.
- ICC-ES Report. ICC-ES evaluation report. Los Angeles: ICC Evaluation Service; 2014; ESR-3433.
- For P, By A. Guidelines for preparing UDOT research reports. Available from: https://rppm.org/documents/guidelines-for-preparing-udot-research-reports/.
- El-Tawil S, Severino E, Fonseca P. Vehicle collision with bridge piers. J Bridge Eng. 2005; 10: 345-353. [CrossRef]
- Zhao X, Wu YF, Leung AY, Lam HF. Plastic hinge length in reinforced concrete flexural members. Procedia Eng. 2011; 14: 1266-1274. [CrossRef]
- Gray III GT. High-strain-rate deformation: Mechanical behavior and deformation substructures induced. Annu Rev Mater Sci. 2012; 42: 285-303. [CrossRef]
- Roy S, Sorensen A. A reliability-based crack propagation model for reinforced concrete bridge piers subject to vehicle impact. In: International Probabilistic Workshop. Cham: Springer; 2021. pp. 95-108. [CrossRef]
- Holt J, Garcia U, Waters S, Monopolis C, Zhu A, Bayrak O, et al. Concrete bridge shear load rating, synthesis report. Washington: Federal Highway Administration, Office of Infrastructure; 2018.
- Grouted Splice Sleeve Connectors for ABC Bridge Joints in High-Seismic Regions–Transportation Blog. [cited date 2020 July 13]. Available from: https://blog.udot.utah.gov/2014/09/grouted-splice-sleeve-connectors-for-abc-bridge-joints-in-high-seismic-regions/.
- American Concrete Institute. ACI 318-11: Building code requirements for structural concrete. Farmington Hills, MI: American Concrete Institute; 2011.
- Furlong RW. Design for shear. 2014. Available from: https://by.genie.uottawa.ca/~murat/Chapter%202%20-%20SHEAR%20DESIGN%20SP%2017%20-%2009-07.pdf.
- AFDC. Vehicle weight classes & categories. Washington: Alternative Fuels Data Centre, U.S. Department of Energy; 2018.
- Chopra AK. Dynamics of structures, theory and applications to earthquake engineering. Upper Saddle River: Pearson-Prentice Hall; 2001.
- Wikipedia. Speed limits in the United States–Wikipedia [Internet]. Wikipedia; 2022 [cited date 2020 April 24]. Available from: https://en.wikipedia.org/wiki/Speed_limits_in_the_United_States.
- Skszek SL. Actual speeds on the roads compared to the posted limits. Arizona. Dept. of Transportation; 2004; FHWA-AZ-04-551.
- ASTM. Standard specification for low-alloy steel deformed and plain bars for concrete. West Conshohocken: ASTM International, United States of America; 2015; A706/A706M−15.
- Sharma H, Hurlebaus S, Gardoni P. Performance-based response evaluation of reinforced concrete columns subject to vehicle impact. Int J Impact Eng. 2012; 43: 52-62. [CrossRef]
- Tavio T, Tata A. Predicting nonlinear behavior and stress-strain relationship of rectangular confined reinforced concrete columns with ANSYS. Civ Eng Dimens. 2009; 11: 23-31.
- Malvar LJ. Review of static and dynamic properties of steel reinforcing bars. Mater J. 1998; 95: 609-616. [CrossRef]
- Malvar LJ, Crawford JE. Dynamic increase factors for steel reinforcing bars. 28th DDESB Seminar; 1998 August; Orlando, FL, USA.
- Roy S, Unobe I, Sorensen AD. Vehicle-impact damage of reinforced concrete bridge piers: A state-of-the art review. J Perform Constr Facil. 2021; 35: 03121001. [CrossRef]
- Cowper GR, Symonds PS. Strain-hardening and strain-rate effects in the impact loading of cantilever beams. Providence: Brown Univ Providence Ri; 1957. pp. 1-46. [CrossRef]
- Buth CE, Brackin MS, Williams WF, Fry GT. Collision loads on bridge piers: Phase 2, report of guidelines for designing bridge piers and abutments for vehicle collisions. Austin: Texas Transportation Institute; 2011.
- Crosby D, Cheremisinoff NP. Practical statistics for engineers and scientists. Technometrics. 1988; 30: 234. [CrossRef]