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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 91 records · Page 5
All sourcesNASA NTRSDOE OSTI
NASA NTRS1972

Metallic positive expulsion diaphragms

High-cycle life ring-reinforced hemispherical type positive expulsion diaphragm performance was demonstrated by room temperature fluid expulsion tests of 13" diameter, 8 mil thick stainless steel configurations. A maximum of eleven (11) leak-free, fluid expulsions were achieved by a 25 deg cone angle diaphragm hoop-reinforced with .110-inch cross-sectional diameter wires. This represents a 70% improvement in diaphragm reversal cycle life compared to results previously obtained. The reversal tests confirmed analytic predictions for diaphragm cycle life increases due to increasing values of diaphragm cone angle, radius to thickness ratio and material strain to necking capacity. Practical fabrication techniques were demonstrated for forming close-tolerance, thin corrugated shells and for obtaining closely controlled reinforcing ring stiffness required to maximize diaphragm cycle life. A non-destructive inspection technique for monitoring large local shell bending strains was developed.

Gleich, D.↗
DOE OSTI2026

AGR-5/6/7 Irradiation Experiment Fission Product Mass Balance

This report presents the fission product mass balance for the AGR-5/6/7 TRISO fuel irradiation experiment. The fission product inventories deposited on capsule components outside of the fuel (e.g., stainless-steel shells, graphite holders, Grafoil disks, and associated hardware) were quantified as part of the post-irradiation examination (PIE) to assess the performance of this fuel. Comparisons were made between these inventories and depletion calculations, non-destructive measurements of fuel fission product inventories in the intact fuel compacts, prior AGR experiments, and results from among each of the five distinct AGR-5/6/7 capsules. The data served as estimates of the condensable fission products released from the fuel during irradiation. Excluding Capsule 1 (which experienced accidental damage during irradiation) and Capsule 3 (which was tested at very high irradiation temperatures of >1300°C), the results indicate that the AGR-5/6/7 fuel performed comparably to fuel from earlier AGR experiments. The mass balance results were also used to estimate the number of particles with in-pile SiC failures. Subject to the assumptions made in these estimates, and excluding Capsules 1 and 3, the in-pile SiC failure rates for AGR-5/6/7 are comparable to those observed in AGR-2.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2025

Non-destructive evaluation and machine learning methods for inspection of spent nuclear fuel canisters: A state-of-the-art review

Nuclear energy is among the cleanest and most efficient energy sources currently available. The operation of nuclear power plants (NPPs) produces large amounts of high-level radioactive waste known as spent nuclear fuel (SNF). Currently, large amounts of SNF is stored in dry cask storage systems (DCSSs) for extended interim storage until a permanent disposal solution becomes available. During the extended interim storage, the DCSS, particularly the SNF canisters, may degrade and abnormal conditions may occur. Therefore, non-destructive evaluation (NDE) and machine learning (ML) approaches are necessary for inspection of SNF canisters. This paper presents a state-of-the-art review of literature by summarizing recent progress made on the applications of NDE and ML for inspection of SNF canisters. Sixteen NDE methods are examined and compared: visual inspection, ultrasonic guided waves (UGWs), laser-based approaches, acoustic emission (AE), eddy current testing (ECT), non-invasive acoustic sensing, dynamic modal testing, cosmic ray muons tomography, neutron imaging, gamma rays detection, fiber optical sensors, through-wall communications, X-ray computed tomography (CT), vibrothermography, monoenergetic photon sources, and surface acoustic wave (SAW) sensors. The technology readiness level (TRL) for each method is assessed and compared. Recent publications on ML-enhanced visual inspection, AE, non-invasive acoustic sensing, dynamic modal testing, and neutron imaging for SNF canisters are summarized and future research needs are identified. In conclusion, this review article provides a convenient reference on the state-of-the-art applications of NDE and ML methods for inspection of SNF canisters.

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
NASA NTRS1993

Nondestructive analysis and development

This final report summarizes the achievements of project #4 of the NASA/UCF Cooperative Agreement from January 1990 to December 1992. The objectives of this project are to review NASA's NDE program at Kennedy Space Center (KSC) and recommend means for enhancing the present testing capabilities through the use of improved or new technologies. During the period of the project, extensive development of a reliable nondestructive, non-contact vibration technique to determine and quantify the bond condition of the thermal protection system (TPS) tiles of the Space Shuttle Orbiter was undertaken. Experimental modal analysis (EMA) is used as a non-destructive technique for the evaluation of Space Shuttle thermal protection system (TPS) tile bond integrity. Finite element (FE) models for tile systems were developed and were used to generate their vibration characteristics (i.e. natural frequencies and mode shapes). Various TPS tile assembly configurations as well as different bond conditions were analyzed. Results of finite element analyses demonstrated a drop in natural frequencies and a change in mode shapes which correlate with both size and location of disbond. Results of experimental testing of tile panels correlated with FE results and demonstrated the feasibility of EMA as a viable technique for tile bond verification. Finally, testing performed on the Space Shuttle Columbia using a laser doppler velocimeter demonstrated the application of EMA, when combined with FE modeling, as a non-contact, non-destructive bond evaluation technique.

Moslehy, Faissal A.↗
NASA NTRS1993

Characteristics of Ferromagnetic Flux Focusing Lens in the Development of Surface/Subsurface Flaw Detector

Electromagnetic NDE techniques have in the past steered away from the use of ferromagnetic materials. Although their high permeabilities lead to increased field levels, the properties of ferrous elements in the presence of alternating magnetic fields are difficult to determine. In addition, their use leads to losses which can be minimized through the use of low conductivity ferrites. In fact, the eddy current probes which do incorporate ferromagnetic materials have focused on these losses and the shielding which can be obtained by surrounding a probe with a high permeability, conducting material. Eddy current probes enclosed in conducting and magnetic shields have been used to prevent the generated fields from interacting with materials in the vicinity of the probe, such as when testing near material boundaries. A recent invention has used ferromagnetic shielding to magnetically separate individual concentric eddy current probes in order to eliminate cross-talk between the probes so that simultaneous detection of different types of flaws at different depths can be achieved. In contrast to the previous uses of ferromagnetic materials purely as magnetic shields, an electromagnetic flaw detector recently developed at NASA Langley Research Center takes advantage of the flux focusing properties of a ferromagnetic mild steel in order to produce a simple, effective device for the non-destructive evaluation of conducting materials. The Flux Focusing Eddy Current Probe has been shown to accurately measure material thickness and fatigue damage. The straight forward flaw response of the probe makes the device ideal for rapid inspection of large structures, and has lead to its incorporation in a computer controlled search routine to locate fatigue crack tips and monitor experimental fatigue crack growth experiments.

Wincheski, Buzz↗
DOE OSTI2026

Screening and qualification methodology for SiC end plug processing methods

Deployment of SiC-ceramic-based fuel cladding for light water reactors requires a hermetic end plug–to–cladding joint that can withstand neutron irradiation during normal operation and maintain integrity during design-basis accidents. Reactor experiments have shown that some SiC composite tubes with SiC end plugs can retain hermeticity after irradiation. However, achieving consistent joint performance under irradiation remains a key challenge. Resolving this issue is essential to enable integral irradiation testing and to demonstrate fuel integrity under commercial-reactor irradiation conditions. This report aims to: (1) provide guidance for designing radiation-tolerant end plug joints for SiC cladding; (2) demonstrate experimental methods to detect processing defects that are unstable under neutron irradiation at light-water-reactor-relevant temperatures and doses; and (3) outline a step-by-step approach for designing and conducting reactor experiments to screen joining methods. The resulting data will be used to improve joint processing and to define critical defect types and sizes that must be detected and eliminated through non-destructive evaluation for quality assurance. Based on prior irradiation experiments at the High Flux Isotope Reactor, differential swelling among the cladding, bonding layer, and end plug was identified as an underlying mechanism for irradiation-induced joint degradation. Accordingly, this effect must be considered in the design of radiation-tolerant joining techniques. In this work, miniature SiC end plug joint specimens irradiated during the previous project were analyzed using X-ray computed tomography to characterize the joint microstructure. Digital volume correlation of the tomography data quantified radiation-induced microstructural changes and enabled evaluation of defect-related risks. Finally, ongoing neutron irradiation efforts using larger specimen volumes are presented. These efforts aim to statistically assess joint performance and to build a microstructure–performance (e.g., leak-tightness) dataset to inform processing improvements and quality control.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2026

In situ cryogenic characterization of proton damage in thick p-channel skipper CCDs

Skipper charge-coupled devices (CCDs) are an offshoot of standard silicon pixel detectors and are capable of performing repeated non-destructive charge measurements, enabling deeply sub-electron readout noise. This capability has opened the door to single-photon counting from the near-infrared ($\sim$1.1 $μ$m) to the soft X-ray (several keV), making these devices strong candidates for future astronomical instruments operating in the photon-starved limit. Furthermore, the p-channel architecture used to fabricate Skipper CCDs on n-type silicon has been demonstrated to have an increased hardness to the intense radiation environment of space. Building upon previous irradiation campaigns on room-temperature sensors, here we describe the first radiation-hardness tests of p-channel skipper CCDs at their cryogenic operating temperatures. We assess the performance of the floating-gate output stage and global CCD parameters (charge transfer inefficiency, dark current, hot pixels, and charge traps). We find that these devices maintain excellent performance after displacement damage doses equivalent to ${\sim}$10 years at the Earth/Sun L2 Lagrange point, demonstrating for the first time that these sensors remain radiation-hard in realistic deep-space thermal and radiation environments.

Roach, Brandon M. [Chicago U., KICP] (ORCID:000000↗
DOE OSTI2025

Correlating energy density induced residual stress, porosity, and mechanical property variations in directed energy deposition using neutron diffraction and imaging techniques

In a directed energy deposition (DED) process, the energy density deposited into the materials drives the structure and properties. The energy density is influenced by several printing parameters, such as laser power, hatch spacing, scanning speed, etc. In this study, 316L stainless steels samples were fabricated with varying hatch spacing aiming to induce variations in sample microstructure and properties. A combination of techniques including engineering neutron diffraction, neutron computed tomography, and tensile testing, were employed to correlate the structure and macroscopic properties. Porosity was predominantly observed at the bottom of the deposited materials and was effectively reduced by decreased hatch spacing. Correlatively, compressive residual stress was observed at the bottom of the specimens, while internal stress is largely determined by the hatch spacing. The residual stress in general decreases as the hatch spacing decreases. The maximum ultimate tensile strength (UTS) was found to increase from 517.4 to 528.6 MPa with a strain increased from 0.847 to 0.897 as the hatch spacing decreased from 0.45 to 0.42 mm. However, further increase in energy density by reducing the hatch spacing resulted in a significant decrease in UTS (as low as 452 MPa). This work provides new insights from bulk non-destructive techniques into correlating energy density induced residual stress, porosity, and mechanical property variations in directed energy deposition.

316L stainless steel↗
DOE OSTI2026

Self-supervised and multi-fidelity learning for extended predictive soil spectroscopy

Infrared spectroscopy is a cost-effective, non-destructive, and environmentally benign technology that is increasingly recognized as an important solution for meeting the global demand for soil data. While both near-infrared (NIR) and mid-infrared (MIR) diffuse reflectance spectroscopy enable rapid estimation of soil properties, they present a significant trade-off: NIR offers superior scalability and lower operational costs, whereas MIR provides higher analytical fidelity by capturing fundamental molecular vibrations. In this study, we propose a self-supervised, multi-fidelity learning framework designed to bridge this gap. Our approach leverages large-scale MIR spectral libraries to learn a compact, transferable latent representation, into which NIR spectra are subsequently aligned for downstream prediction. The workflow consists of pretraining a latent model on a large MIR library, adapting the representation using a smaller paired NIR–MIR dataset, and evaluating generalization on an independent external test set. Across a range of chemical and physical soil properties, we found that MIR-derived embeddings improved prediction accuracy relative to baseline models that used raw MIR inputs. Predictions derived from the spectrum conversion (NIR to MIR) task did not match the performance of the original MIR spectra but were similar or superior to predictive performance of NIR-only models, suggesting the unified spectral latent space can effectively leverage the larger and more diverse MIR dataset for prediction of soil properties not well represented in current NIR libraries.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2025

Ultrasonic Characterization of Lithium Ion Thermal Runaway Conditions for Real-time Ultrasonic Enabled BMS Integration

The demand for energy storage is growing, and lithium-ion batteries are a promising technology to meet this need due to their high power/energy density, high round-trip efficiency, rapid response time, and portability. However, recent catastrophic events caused by thermal runaway have slowed their adoption, highlighting the need for an early warning system for battery failure. In this work, ultrasound is used to detect physical changes in 950 mAh batteries by identifying material property changes independent of voltage and current. [Ultrasound signal features (e.g., time of flight, maximum frequency component) were extracted as the batteries were cycled and subjected to both constant current and constant voltage overcharge and were used to develop two metrics identifying failure: a warning to detect the start of overcharge and an emergency stop (E-stop) to immediately take the battery out of service. The identification method involved locating magnitude differences of several ultrasound features compared to baseline operation considering different currents and temperatures, and the warning/stopping metrics were consistent across all experiments. For an average overcharge time of 140 minutes, the average warning was issued 124 minutes before the failure and the average E-stop was triggered 94 minutes before failure. As a test of using ultrasound for early warning detection, a battery was forced into overcharge and returned to normal cycling conditions based on the previously determined warning metric. Both the voltage profile and the ultrasound measurements returned to their baseline behavior, indicating that ultrasonic detection can not only identify battery failure before a catastrophic event, but can also provide early enough warnings such that overcharges can be detected and corrected quickly enough so the battery does not need to be decommissioned.]

25 ENERGY STORAGE↗
DOE OSTI2025

Mounted Accelerometer Frequency Response of Adhesive Products and Aluminum Frame Quick Mounts

An accelerometer mounting technique has large implications on the frequency range and accuracy of the measurement, with stiffness and the mass relative to the monitored structure as the primary concerns. The International Organization for Standardization (ISO) gives an extensive list in 5348:2021, detailing mounting methods, and provides recommendations for testing mounts that are not specifically defined. In the nuclear industry on the laboratory scale, there is a need for vibration measurements for predictive maintenance and process monitoring that are nondestructive and capable of working in high-temperature environments. Commercial adhesive products with easy application and removal were tested as nondestructive methods, while quick mounts to a commonly used aluminum frame were tested as nondestructive and have potential applicability in high-temperature environments. The sinusoidal excitation method was used, measuring frequencies from 50 Hz to 10 kHz in one-third octave band intervals, utilizing three accelerometers and comparing the results to those obtained with the stud-mounting method. Using the lowest ±3 dB threshold across each accelerometer, foam dots and poster strips were not successful, and foam tapes were accurate up to 2000 Hz, hose clamps and zip ties up to 800 Hz, and a custom 3D printed mount up to 1000 Hz. Knowing the limitations of each mounting technique allows for accurate measurements within the appropriate range.

47 - OTHER INSTRUMENTATION↗
DOE OSTI2025

Post-irradiation examination of AGR-3/4 TRISO fuel compacts using three-dimensional X-ray computed tomography

The AGR-3/4 irradiation tests combined the third and fourth planned irradiation experiments in the US Department of Energy’s Advanced Gas Reactor (AGR) testing campaign of tri-structural isotropic (TRISO) fuel compacts. In this article, we present post-irradiation examination (PIE) using X-ray computed tomography (XCT) of two unirradiated and two irradiated compacts from the AGR-3/4 irradiation tests. The irradiated compacts studied (compact 7–1 and compact 12–4) represent the upper and lower limit of burnup within the AGR-3/4 irradiation experiment. This article presents a detailed quantitative analysis on the post-irradiation structure of TRISO fuel compacts. Various quantitative parameters including shape, size, and packing of kernels, and their spatial distribution, were utilized to gain insights into the structural changes caused by irradiation. The equivalent diameter and sphericity were found to increase and decrease, respectively, in irradiated compact 7–1 due to its higher burnup. Nearest neighbor distance between fuel kernels decreased after irradiation, suggesting irradiation-induced shrinkage of graphitic matrix. Furthermore, each compact in AGR-3/4 irradiation tests contained 20 designed-to-fail (DTF) fuel particles that were meant to act as a source of fission product release to the experiment test train. Furthermore, in the present work, all DTF fuel particles in the four compacts studied were identified, and it was found that they exhibited larger kernel swelling in compact 12–4 and smaller kernel swelling in compact 7–1, compared to the driver particles.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2025

Non-destructive electrochemical diagnosis of failure mechanisms in aqueous zinc batteries

The early detection of secondary reactions that affect the life and performance of zinc manganese oxide batteries requires a shift from conventional time-consuming and often destructive procedures to rapid lifetime-predictive techniques. In this work, an electrochemical approach is employed to elucidate independent signatures for four common types of failure mechanisms in zinc manganese dioxide (Zn||MnO2) batteries—namely, the loss of zinc inventory, the loss of active material at the cathode, electrolyte depletion, and increased cell impedance. Our findings, specific to coin cell configurations, reveal that each induced failure mechanism can be distinctively modeled and identified based on responses from the rest voltage and columbic-efficiency data for prompt detection. For instance, electrolyte depletion response manifests a distinctive abrupt (>80 %) decrease in columbic efficiency (CE) and charge-rest voltage (Vc) while the discharge-rest voltage remained constant at ~1.3 V. Furthermore, electrolyte rejuvenation of the cell increased the CE to >95 % and restored Vc from ~0.3 to >1.7 V. Recovery experiments and reference performance tests demonstrated consistency between electrochemical descriptors and their associated failure mechanisms. Further, the outcomes of this work provide valuable insights and data models for some of the dominant failure mechanisms present in zinc manganese battery chemistries, which are beneficial to accelerated early-lifetime diagnosis and advancement of Zn batteries development.

25 ENERGY STORAGE↗
DOE OSTI2026

MIND-MAC: Multi-Level In-memory Quasi Non-Destructive MAC Operation in Compact 2T-nC FeRAM for Efficient DNN Accelerator

We present MIND-MAC, a compact 2T-nC FeRAM architecture that performs multi-level, quasi-non-destructive in-memory multiply–accumulate (MAC) for deep neural networks. By exploiting voltage-controlled partial domain switching in MFM capacitors and read-transistor amplification, the cell stores multi-bit weights and gates bit-serial inputs to produce an accumulated current on shared lines. We combine TCAD-extracted parasitics with experimentally calibrated ferroelectric models in SPICE to validate device-/circuit-level behavior, and validate multi-level sensing and QNRO with measurements on a fabricated 2T-3C test vehicle. An analytical system model maps MIND-MAC to a 6-GB main-memory in-memory compute (IMC) architecture and benchmarks VGG13 inference in 61.08 ms at 964.99 mJ. Results indicate high density, reduced rewrite overhead, and energy efficiency, positioning 2T-nC FeRAM as a promising IMC candidate for next-generation AI hardware.

36 MATERIALS SCIENCE↗
DOE OSTI2026

Machine learning enables reconstruction of past fire regimes from charcoal-derived fire intensity and fuel composition

Background Fire is a foundational ecological process that shapes ecosystem structure, diversity, and resilience. Quantifying paleofire regime attributes such as frequency, severity, and intensity is essential for understanding the historical range of variability in fire behavior and its ecological effects. While frequency and severity are often reconstructed in paleofire studies, quantitative reconstructions of fire intensity remain limited. Recent work has shown that maximum pyrolysis temperature—a proxy for fire intensity—and plant species type can be inferred from charcoal using transmission Fourier-transform infrared (FTIR) spectroscopy. However, the sample preparation for transmission FTIR is destructive and time-consuming, limiting application and reuse of materials for other analyses. We evaluated reflectance FTIR spectroscopy as a non-destructive alternative for reconstructing combustion temperature and plant species from laboratory-generated charcoal. We also examined the influence of contrasting airflow environments (ambient air versus nitrogen-rich) on pyrolysis temperature and plant species reconstruction prediction accuracies and compared predictive performance between a novel, neural network–based deep learning model with the traditional modern analogue technique (MAT) using k-nearest neighbor functions. As proof of concept, we apply our enhanced methodology to ancient charcoal to demonstrate applicability at improving long-term fire regime reconstructions and the ability to link paleofire records with contemporary fire ecology. Results Our analysis shows that transmission and reflectance FTIR spectra yield comparable spectral profiles. However, sample preparation for reflectance FTIR is minimal and non-destructive, unlike transmission FTIR which is destructive. We demonstrate that oxygen environments improved reconstruction accuracy relative to nitrogen-rich conditions. Finally, our deep learning neural network (DL) achieved testing accuracies of 98.7% for temperature and 96.2% for species identification, outperforming MAT’s k-NN approach (89.8% and 65.9%, respectively). A Shapley importance analysis identified 5 key spectral regions that greatly influenced the model’s temperature or species categorization. When applied to ancient charcoal, our results show historic fires from the most recent past primarily burned at low intensities (400–500 °C), reflective of natural fire regimes in ponderosa pine forests. Our results corroborate charcoal morphology data that suggests all ancient charcoal originated from burned woody plant types. Conclusions By combining reflectance FTIR spectroscopy with a deep learning approach, we provide the first accuracies high enough to confidently identify both species and temperature from laboratory-produced charcoal, improving quantitative reconstructions of fire intensity and fuel composition from paleofire records. This opens a wide range of research into the link between fire and larger drivers (i.e., climate or human) and greater ecological understanding of fire regimes beyond that of burn scars or recent observations. These methodological improvements have direct relevance for fire management by improving interpretation of historical fire behavior, informing fuel–fire relationships, and providing a scalable analytical framework applicable to both long-term ecological studies and contemporary fire science.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1989

Onorbit electron beam welding experiment definition

The proposed experiment design calls for six panels to be welded, each having unique characteristics selected to yield specific results and information. The experiment is completely automated and the concept necessitated the design of a new, miniaturized, self-contained electron beam (EB) welding system, for which purpose a separate IR and D was funded by the contractor, Martin Marietta Corporation. Since future tasks beyond the proposed experiment might call for astronauts to perform hand-held EB gun repairs or for the gun to be interfaced with a dexterous robot such as the planned flight telerobotic servicer (FTS), the EB gun is designed to be dismountable from the automated system. In the experiment design, two separate, identical sets of weld panels will be welded, one on earth in a vacuum chamber and the other onorbit in the aft cargo bay of an orbiter. Since the main objective of the experiment is to demonstrate that high quality welds can be achieved under onorbit conditions, the welds produced will be subjected to a wide range of discriminating non-destructive Q.C. procedures and destructive physical tests. However, advantage will be taken of the availability of a fairly large quantity of welded material in the two series of welded specimens to widen the circle of investigative talent by providing material to academic and scientific institutions for examination.

William H. Hooper↗
NASA NTRS1991

Quantitative nondestructive evaluation of materials and structures

An experimental investigation was undertaken to quantify damage tolerance and resistance in composite materials impacted using the drop-weight method. Tests were conducted on laminates of several different carbon-fiber composite systems, such as epoxies, modified epoxies, and amorphous and semicrystalline thermoplastics. Impacted composite specimens were examined using destructive and non-destructive techniques to establish the characteristic damage states. Specifically, optical microscopy, ultrasonic, and scanning electron microscopy techniques were used to identify impact induced damage mechanisms. Damage propagation during post impact compression was also studied.

Smith, Barry T.↗
NASA NTRS1993

A combined experimental and finite element study to predict the failure mechanisms in SiC coated carbon/carbon composites at room and elevated temperatures under flexural loading

Response of quasi-isotropic laminates of SiC coated Carbon/Carbon (C/C) composites have been investigated under flexural loading at various temperatures. Variation of load-deflection behavior with temperatures are studied. Increase in flexural strength and stiffness are observed with the rise in temperature. Extensive analyses through Optical Microscope (OM) and Non-Destructive Evaluation (NDE) have been performed to understand the failure mechanisms. Damage zone is found only within the neighborhood of the loading plane. Isoparametric layered shell elements developed on the basis of the first order shear deformation theory have been used to model the thin laminates of C/C under flexural loading. Large deformation behavior has been considered in the finite element analysis to account for the non-linearities encountered during the actual test. Data generated using finite element analysis are presented to corroborate the experimental findings, and a comparison in respect of displacement and stress-strain behavior are given to check the accuracy of the finite element analysis. Reasonable correlation between the experimental and finite element results have been established.

Mahfuz, Hassan↗