Validation of Numerical Modeling Techniques for Unbonded Post-Tensioned Beams under Low Velocity Impact

Prestressed concrete construction is commonly found in structures where horizontal and vertical deformation is a critical aspect of safety and design (Kang and Ibrahim Ary 2012; Lee et al. 2015; Shin et al. 2020). Developed in the late nineteenth century by Eugene Freyssinet, the use of prestressed concrete provided an economical means to help reduce material usage by aiding in the design of longer floor spans and more slender structures (Bondy and Allred 2018). The pre-compressive stresses applied to the concrete are provided by prestressing longitudinal reinforcement, referred to as tendons, which are comprised of individual or bundled hard-drawn wires and bars of high strength alloy steel. In this manner, the strong compressive properties of concrete are further utilized in conjunction with the prestressing tendon and other longitudinal reinforcement. In the 1950s, the development of unbonded tendons for post-tensioned reinforced concrete (PT-RC) design helped to extend the life span of prestressed concrete structures. As a duct is used for the insertion of the PT tendon, this allowed for additional economical aspects with the capability to replace or re-tension the tendon, common for structures such as nuclear containment facilities and bridge piers which must be regularly maintained for structural integrity.

Inherently, as the use of unbonded PT-RC construction has become widely adopted, the need to understand member behavior under all loading scenarios is important. This is especially true for dynamic loading conditions as structures such as bridge piers can be subjected to potentially large impacting masses on a recurring basis. To study these scenarios, impact tests on concrete members have been performed under low velocity impact using a drop weight apparatus, where low velocity was defined as $10\mathrm{m}/\mathrm{s}$ or less (Nghiem et al. 2021). This testing is based on the use of a hammer of significant mass dropped from a controlled height to fail the specimen in a three-point bending configuration. When impacted in such a manner, the main concern for concrete members pertains to: (1) large inertial effects that lead to increasing shear failure modes; (2) the influence of stress wave propagation effects from the impacting hammer on reaction forces; and (3) localized damage at the loading point.

Many drop weight laboratory investigations have been performed on traditional reinforced concrete (RC) type members to address both flexural- (Kishi et al. 2001; Fujikake et al. 2009; Tachibana et al. 2010; Kishi and Mikami 2012; Nghiem and Kang 2020) and shear-critical members (Kishi et al. 2002a, b; Bhatti et al. 2009; Saatci and Vecchio 2009; Nghiem and Kang 2020). Unfortunately, in comparison, little information has been previously collected on the impact behavior of unbonded PT-RC beams, as only a few experimental studies were readily available (Ishikawa et al. 1998, 2000; Wu et al. 2016). To populate this previously unavailable lab data, a recent comprehensive study was performed on both flexural- and shear-type unbonded PT-RC members (see Nghiem et al. 2021). The specimens in that study had identical longitudinal reinforcement, but varying shear reinforcement ratios were intended to be both statically flexural- and shear-critical ($V_n/P_n$ near or below 1.0), according to the ACI code. The main variables investigated included: increasing levels of effective PT force ($P_{se}$) and the application of two different impact energies per each specimen type (F = flexural, S = shear). From that study, it was concluded that: (1) shear failure mechanisms are still important and present even for statically flexural members; (2) modes of shear failure increase with increasing impact velocity; (3) the engagement of the PT tendon increases member resistance and helps restrain cracking at impact as observed with the increased values of the reaction force; and (4) increased values of initial PT forces seem to exhibit increased shear failure mechanisms in the form of increased brittle failure modes and arching mechanisms from the support point towards the impacting point (Nghiem et al. 2021).

However, experimental studies are expensive and time consuming, which lead to many limiting factors (Kang and Wallace 2008). These restrictions are especially evident in drop weight testing with physical constraints such as: (1) allowable span lengths due to testing facilities; (2) allowable drop height limited by the capacity of the testing frame and reaction floor; and (3) dimensions of the member cross-section as limited by the guiding rails used for positioning of the drop hammer. Additionally, the overall number of tested specimens capable of being tested is a large economical restraint. To further evaluate the behavior of concrete members under impact without those limitations, the use of nonlinear finite element dynamic analysis (NLFEDA) has provided additional insights. However, proper validations must be performed to prove its consistency with observed real-life phenomena.

The purpose of this numerical investigation is to exhibit the capability of using NLFEDA techniques for the analysis of unbonded PT-RC beams under impact. First, a brief review was done to discuss past successful attempts at the modeling of concrete members, namely RC. This is then followed by the development of an NLFEDA model of unbonded PT-RC members. Validations were performed in two key steps: (1) a sensitivity study to determine appropriate member geometries (with respect to mesh size) and material properties; and (2) establishing the overall level of accuracy of the resulting model considering a wide range of design and loading parameters. The experimental study completed by Nghiem et al. (2021) was used for all validation purposes and was the basis for the development of the NLFEDA mesh.

The intent of this study is to present a suitable and easy to replicate method for the modeling of unbonded PT-RC beams under impact. As prior studies have relied on indirect methods in which temperature-induced shrinkage is applied to the tendon prior to loading, a direct method for the application of the pre-compressive tendon forces is performed. Additionally, past modeling has focused on members with bonded tendons; in contrast, this study focuses on addressing members with unbonded PT tendons, which, in some cases, can be more applicable for structures such as nuclear containment facilities and other high importance structures. Beyond the scope of this study, the modeling methods can potentially be used for predicting or assessing the dynamic behavior of other unbonded PT structures under other similar loading conditions for future code or guide development. Forensic investigation is one of the areas that requires a nonlinear model.

The use of specialized NLFEDA software has become increasingly popular. Seen within the previous decade, while still relatively few in number, an increasing amount of NLFEDA studies has been performed on drop weight testing of concrete members (Bhatti et al. 2009, 2011; Cotsovos 2010; Jiang et al. 2012; Adhikary et al. 2012; Jiang and Chorzepa 2015; Yi et al. 2016; Pham and Hao 2017; Zhao et al. 2017). This is due to significant advances in computational power and wider accessibility. However, one downside is that the obtained results are not readily accepted and are always susceptible to criticism without adequate justification. Commercial products, as those provided by LS-DYNA (2007), have been shown to provide accurate results. Developed by the former Livermore Software Technology Corporation (LSTC) with origins beginning in the 1970s (currently owned by ANSYS), LS-DYNA began as the first 3-D capable software for the simulation of structural elements under impact and dynamic events. Its ability to be utilized under a number of different materials and loading conditions has led to its widespread use in industries ranging from automobile, aerospace, construction, and civil engineering, to manufacturing and bioengineering applications. With respect to the low velocity impact of concrete members, its successful use has been validated by authors such as Jiang et al. (2012), Jiang and Chorzepa (2015), and Zhao et al. (2017). However, depending upon the complexity of the model, computational time can still pose a problem and balance must be made in terms of (1) ease of use, (2) required computational power, and (3) model accuracy. Sensitivity is commonly tuned to the geometric mesh size and properties of the materials used. For the modeling of large scale concrete members, two mesh sizes have been commonly incorporated: a relatively coarser mesh in the range of 20–25 mm (Bhatti et al. 2009, 2011; Jiang and Chorzepa 2015; Zhao et al. 2017) and finer meshes measuring around 10 mm (Jiang et al. 2012; Pham and Hao 2017). It was noted that anything smaller than 5 mm only added to the computational time and exhibited no large benefits in model accuracy (Pham and Hao 2017).

A critical aspect when performing the numerical modeling of concrete members is the consideration of how to represent member cracking and potential large brittle failure effects. How this is represented varies greatly based on the materials models used in the NLFEDA environment and is largely subjective to each researcher on how they want to interpret member cracking. A commonly applied method is “element erosion.” In this case, the erosion or removal of elements is done to remove exceedingly distorted Lagrangian elements to prevent numerical problems with respect to model convergence. When compared to real world scenarios, erosion can be seen as a way to visualize the effects of target fragmentation, spalling, scabbing of concrete, or penetration during impact. Typical methods for the assessment of element erosion are based on the following physical quantities at integration points of each element: the equivalent plastic shear strain, maximum principal strain, minimum pressure (positive in compression), damage index, and equivalent stress. Normally, the maximum principle strain has been an effective criteria for erosion criteria. Values of erosion are set at some value in reference to when the principle strain exceeds the set threshold, element deletion will occur (i.e., a value of 1.01 would mean 1% erosion over the principle strain). As erosion is a non-physical parameter, the amount of erosion is then adjusted to some measured output, such as reaction force or displacement (as based on the experimental study used for validation). As will be observed, many successful modeling attempts have been performed and the modeling methods can vary significantly from author to author. A brief review of recent modeling is provided below in chronological order.

As discussed above, commercial software LS-DYNA provides an easy-to-use method capable at delivering adequate results and was used for all numerical modeling. The approach used for the generation of the mesh for the specimens and the testing setup are provided in the following subsections. A main focus for this study is then devoted to determining the appropriate model properties, which is in reference to mesh geometries (element size) and value of material erosion for concrete elements. Using the results of Nghiem et al. (2021), validations were performed in two key steps: (1) the first step involved determining the key parameters as based on two key laboratory specimens; and (2) then the simulation of all laboratory specimens with the determined mesh and material properties to find the level of convergence of the NLFEDA model with that of the experimental results.

To determine suitable modeling parameters, a deep sensitivity study was performed based on the two mesh sizes ($\mathrm{coarse}=20\mathrm{mm}$, $\mathrm{fine}=10\mathrm{mm}$) and thresholds of element erosion for concrete (in range from 1.01 to 1.15). These two variables were the basis of the analysis, which ultimately determined the final characteristics of the model. The results were validated using two key specimens from the laboratory investigation (Nghiem et al. 2021), member ID SMPT170 and FMPT180. These specimens were chosen as ideal candidates due to the similarities in the impact energy and applied post-tensioning force; however, both belong to different specimen series, shear and flexural (S & F). Table 2 and Figs. 4 and 5 provide the specimen design and loading conditions. Additional experimental details can be found in Appendix A.

Numerical analyses were performed for both specimens, at coarse and fine meshes (20 mm & 10 mm) with values of erosion equal to 1.01, 1.02. 1.03, 1.04, 1.05, 1.075, 1.10, and 1.15. The resulting plastic strain is shown in Figs. 6 and 7 for both specimens SMPT170(N) and FMPT180(N), respectively. Note the notation “(N)” is used to designate simulated NLFEDA members in lieu of those tested in the lab. Figures refer to models tested with meshes equal to 10 mm. The resulting failure is a representative case for both mesh dimensions (20 and 10 mm). Both shear and flexural models show similar failure patterns in agreement with laboratory testing. Visually, the erosion and plastic strain can be seen as a direct form of the observed laboratory failures.

The erosion sensitivity of the NLFEDA model to shear failure is extremely noticeable, as evidenced in the simulation of SMPT170(N). For erosion equal to 1.01 and 1.02, the formation of the shear plug (as shown from distinct diagonal cracking originating from the top compressive fiber) is provided by removing elements, in which a clear triangular pattern is formed. When increasing the tolerance (erosion = 1.03, 1.04 1.05… 1.15) the threshold for deletion increases, thereby showing the failure as large strains in lieu of the deleted elements. When comparing the resulting visual failure to that of FMPT180(N), the differences are clear. This is especially notable at values of 1.01 and 1.02. However, when erosion is greater than 1.02, the two series of specimens (shear and flexural) are not as visually dissimilar.

In determining model convergence with the laboratory results, emphasis was placed on member resistance against impacting load. This was taken quantitatively as measured by the reaction force (taken at summation of the left and right supports). This was done as Zhao et al. (2017) and other authors mentioned the ability of LS-DYNA to model member parameters, such as force, much better than member deformation, especially during the initial stages of loading toward the peak response.

Table 3 shows the peak reaction values for specimens SMPT170(N) and FMPT180(N) at mesh sizes of both 20 and 10 mm, respectively, and at all cases of erosion (1.01 to 1.15). For the shear model of SMPT170(N), the obtained values at 20 mm mesh were unconservative with values overcalculated when compared to the laboratory results. The resulting maximum reaction force for SMPT170 was equal to 773 kN. For NLFEDA results, this value was in a range of 766 kN at high erosion ($=1.15$) and around 850 kN ($\mathrm{erosion}=1.02$ and 1.03), with resulting values approximately 10% over the lab-obtained value. The amount of error greater than the lab value is given to the right peak value shown in Table 3. No direct trend was observed when looking at the reaction force quantitatively. The same unconservative behavior was also observed for FMPT180(N), with a range of values from around 900 to 920 kN, compared to the laboratory result of 813 kN. The same amount of error was also observed, around 10% for all erosion values.

In contrast, the use of 10 mm typical mesh dimension provided conservative calculations with results in good approximation with the laboratory study. For SMPT170(N), values were in agreement with $-10\%$ to $-20\%$ error with larger errors more associated with high values of erosion ($>1.04$). The same trends are observed from flexural specimen FMPT180(N) with values in the range of $-5\%$ and $-25\%$ errors. The lower values of error were associated with lower erosion thresholds with values of 1.01 and 1.02.

In considering visually observed failures, the use of an erosion threshold equal to 1.02 appeared to be the ideal choice for this particular drop weight test setup. Visually, a threshold of 1.02 showed differences of failure in the two specimens. However, for specimen SMPT170(N), a large number of elements were deleted. This behavior was characteristic of the lab testing, in which high brittle failure occurred. This was in the form of an immediate shear plug at the contact of the hammer, followed by brittle failures at the top and bottom fiber of the left span as detailed by Nghiem et al. (2021). Please note that the use of low erosion may be atypical for members under cyclic loadings due to repeated openings and closings of the cracks. The immediate erosion for this specific case is suitable, as member failure was relatively instantaneous as noted by Nghiem et al. (2021). With a mesh equal to 10 mm, quantitatively, the use of erosion at 1.02 was able to produce conservative calculations with errors of $-15\%$ and 5% for SMPT170(N) and FMPT180(N), respectively (see Table 3). As noted by previous authors, further reduction in the mesh size would most likely not lead to more positive results and only increase calculation time and space required for data storage.

To access the full characteristics of the model under the governing mesh size and erosion parameters, the NLFEDA model was run for the full matrix of laboratory testing (Nghiem et al. 2021). Convergence of the NLFEDA is determined below, in which the range of errors associated with a wider range of measured parameters is evaluated. Calibration of the erosion was done relative to the peak reaction of the beam. In addition to the two laboratory specimens, SMPT170 and FMPT180, the full testing matrix expanded on the design parameters with increasing levels of initial PT force and impacting energies. Additional details on the testing matrix are given in Appendix A.

There are few examples for the numerical modeling of unbonded PT-RC members under low velocity drop weight impact. The study done here was to substantiate the use of readily available commercial software and material models, provided here in the use of LS-DYNA. From the results, the use of LS-DYNA provided a practical method of NLFEDA modeling of unbonded PT-RC members, both at replicating the prestressing force provided by the tendon and also the impacting conditions for both shear- and flexural-critical members. Qualitatively, all measured parameters, such as impact and reaction forces, PT forces, and member deformation. were in good agreement with those of the experimental values, with only minor differences in the distribution. Additionally, all major failure modes were observed in the form of element erosion, noting the occurrence of shear failure common to concrete members under this particular loading. However, some differences were noted with the peak response of all parameters with a broad range of errors dependent upon various design or loading conditions.

Nevertheless, the observed differences of the NLFEDA modeling with that of the experimental agreed with those, as noted by previous authors. These details include the: (1) ability to model flexural specimens more accurately than shear specimens; and (2) difficulty at modeling deformation, especially during the post-peak phase. Both of these attributes are associated with the non-continuum problem as stated by Zhao et al. (2017), in which it is noted that the model basically begins to be separated into two or more distinct pieces upon large shear failures. This problem is difficult to avoid, as one key element of the lab study was the extent of shear failure especially for members without transverse reinforcement and the increased brittle nature observed at higher levels of prestress. This led to a particular importance in capturing these characteristics. By introducing low thresholds of element erosion ($=1.02$) to represent the brittle nature of the unbonded PT-RC members under impact, conservative results were obtained, and the interaction of the PT tendon with the surrounding concrete in presence of large failures was met with similar convergence between both shear and flexural members when compared to the experimental testing.

A critical modeled aspect was the loss of residual PT tendon forces after impact. This was one observation of the lab study (Nghiem et al. 2021), in which the brittle nature of shear members directly led to a loss in the ability to maintain PT force after impact. Per Nghiem et al. (2021), it was noted that: (1) increases in impact velocity increased shear failure mechanisms which directly led to inabilities of shear members to maintain prestress; and (2) increases in the value of initial PT potentially led to increased brittle failure mechanisms thereby increasing the potential inability to maintain prestress after impact. For this study, the use of erosion to model the loss of strength with respect to the ability of the member to maintain prestress after impact was on the conservative side. This was observed in the modeling of members SLPT172(N), SMPT118(N), and SMPT170(N), in which erosion of elements contained within the impacting zone (shear plug) leads to the failure of the surrounding concrete to hold PT forces, thereby resulting in PT forces after impact being less than the initial values. Additionally, as noted in the lab setting, this behavior seems to also be attributed to increased levels of prestress. For the modeling of flexural members, this was observed for FHPT200(N), a member with relatively higher levels of initial PT forces and impacted at a higher impact velocity. Per the NLFEDA model, this interaction between the PT tendon and the surrounding concrete in the presence of large failures is a crucial aspect even for flexural members that should be studied more thoroughly.

The use of LS-DYNA for NLFEDA modeling of unbonded PT-RC members under low velocity drop weight impact was validated under a broad range of design and loading conditions. When considering the lab results of Nghiem et al. (2021), a model with a typical mesh size of 10 mm and low thresholds for element erosion ($=1.02$) of concrete were suitable at providing conservative and practical results. Calculations made per NLFEDA were able to visually show important failures during the loading process and maintain a broad range of validation with respect to impact and reaction forces and member deformation. Importance was placed on the ability to capture the distinct failure modes of shear- and flexural-critical members, with shear models (SL and SM) being able to reflect the extreme brittle nature observed in the experimental testing through the large number of eroded elements in the form of a shear plug at the loading point. Conversely, for flexural models (FM and FH), erosion was less severe but still showed aspects of shear failure combined with flexural failure modes throughout the loading process. Additionally, the interaction between the prestress as applied from the PT tendon was well modeled in the presence of large shear failures of the concrete. With respect to the observed phenomenon and inertial effects of low velocity impact loading, this study substantiated the use of commercial software and material models. The modeling performed in this study can provide beneficial and additional methods that can easily replicate the behavior of unbonded members under low velocity impact. These modeling tools, coupled with experimental studies, can be helpful in expanding the range of understanding of the behavior of such members by capturing data not easily obtained in a lab setting, either due to testing complexity or economic constraints.

All data, models, and code generated or used during the study appear in the published article.

This work was supported by the Nuclear Safety Research Program through the Korea Foundation Of Nuclear Safety (KoFONS) using the financial resource granted by the Nuclear Safety and Security Commission (NSSC) of the Republic of Korea (No. 2003007). Partial support by the Institute of Construction and Environmental Engineering at Seoul National University is also appreciated. The views expressed are those of the authors and do not necessarily represent those of the sponsors.