Pulse Loading Research Articles

Analysis of sampling strategies for pulse loads of SARS-CoV-2: implications for wastewater-based epidemiology.

A faecal transport model was applied to a 11.3 km2 wastewater servicing area in Toronto, Ontario, Canada to explore the role that different wastewater sampling campaigns have on estimating the prevalence of SARS-CoV-2 in a population of 60,000. A stochastic wastewater and water quality model was used to evaluate the effectiveness of 11 sampling campaigns during periods of high and low COVID-19 infection among the population, tested using virtual sampling during dry-weather flow. The virtual sampling campaigns were based on the most common automatic sampler programming capabilities and widely used wastewater-based epidemiology (WBE) sampling campaigns reported in the literature. Sampling campaigns differ in weighting method (time, volume, or flow-weighted sampling), sample count, collection period, or sample time. Results suggest that grab samples should be avoided and/or that sampling campaigns with the greatest sample counts and durations are the most robust at capturing COVID-19 infection among the population. Most surprisingly, changes to the weighting method were negligible indicating that a greater number of samples, and larger sample volumes are preferred. This work suggests that investment in flow monitoring equipment for flow- or volume-weighted sampling will not improve WBE results, and that standard time based sampling is sufficient.

Cyclic pulse loads pave the road to the GRIP concept in abdominal wall reconstruction

Aim : Durable reconstruction of the abdominal wall needs to be assessed in a lifelike experimental setting and consider the reconstructed abdominal wall as a coherent compound. Our aim was to evaluate broader possibilities in preclinical testing and to deepen the understanding of the biomechanical influences. Methods : We developed a test bench that allows studying a compound under cyclic, repetitive loads. Pulse loads transmit energy to the abdominal wall repeatedly. The amount of energy is related to the load characteristics. We used porcine bellies with a round central (5 cm) defect. They were bridged in a sublay position with Cicat Dynamesh®. Further defects, located in an additional incision, were sutured in a standardized small-bite technique. We varied the number of loads, the maximum peak pressure, the pressure plateau length, and the impact area size. Results : Increasing the peak pressure by 30 mmHg lowers the durability by about 20 %. Prolonging the plateau phase led to a significant durability decrease. During the first 100 dynamic intermittent strain (DIS) impacts, the major tissue deformation and the majority of failures occur. Beyond the 425th DIS impact, about 10 % more failures occur. Increasing elongation and deformation of the tissue raise the likelihood of failure. Conclusion : If the compound does not establish a strain-stable condition during the period of plastic deformation, failure occurs. The outcome does not only depend on the reconstruction technique but also on the external influences acting on the abdominal wall compound. Considering the biomechanical reality is important for open and minimally invasive abdominal wall reconstruction.

Additional Synchronous Torque Coordinated Control for Approximate-Capacity Diesel-Storage Supply System With Low-Frequency Pulsed Load

For the power supply of low-frequency pulsed load, the approximate capacity diesel-storage supply system is difficult to dynamically synchronize the output powers of the diesel generators and energy storage converter, resulting in continuous power oscillations and unbalanced power distribution. For this, a coordinated control with additional synchronous torque is proposed to dynamically adjust the virtual torque of the energy storage converter for narrowing the angular accelerations gap with that of diesel generator, and realizing the synchronous power supply during the dynamic process. Meanwhile, the time-domain model incorporating pulsed load is established. The power oscillation mechanism is elaborated from the perspective of normalized oscillation power, and it is found that the oscillation power is mainly affected by the accelerations difference and pulsed period. Afterwards, the principle of the proposed coordinated control method is clarified, and the parameters are properly selected. Furthermore, the stability analysis with the proposed method based on Lyapunov stability theorem is elaborated. Finally, the experimental results verify the effectiveness of the proposed method.

Performance Case Study of Concrete Bridge-Rail Post on Deck Overhang

Concrete post-and-beam bridge rails are a common bridge-rail type in the U.S. However, direct investigations of their performance on bridge-deck overhangs are rare, and current specifications do not clearly address this rail type or the design of decks supporting them. In this paper, the results of a performance case study of concrete bridge-rail posts on deck overhangs are presented. This study included (1) bogie testing of a MASH TL-4 concrete post on an instrumented deck, (2) development and calibration of a corresponding LS-DYNA model, and (3) use of the calibrated model to further describe system behavior, evaluate the effect of common design alternatives on performance, and corroborate the system performance from a full-scale crash test. The results of this study indicated that current design methods of the AASHTO LRFD Bridge Design Specifications (BDS) resulted in a significant underestimation of lateral capacity, partially because their neglect of inertial resistance. In the bogie test of the concrete post, over 40% of the peak lateral resistance of the post was attributed to inertial effects. Similarly, the calibrated LS-DYNA model indicated that the full-scale system could withstand a simplified single-unit truck (SUT) loading consisting of a 150 kip pulse load while only sustaining minor damage, despite having a BDS-predicted capacity of 73 kips. Additional findings included a characterization of deck demands, performance effects of design alternatives, such as edge distance and slab thickness, and the influence of using straight versus hooked transverse deck bars.

Very High Cycle Fatigue Data Acquisition Using High-Accuracy Ultrasonic Fatigue Testing Equipment

Abstract Fatigue testing at ultrasonic frequency is a powerful method for rapid generation of high and very high cycle fatigue data. High accuracy of cyclic loading is the basis for reproducible test results, which can be obtained with closed-loop control of vibration amplitude and resonance frequency. Self-heating of test specimens can be suppressed by intermittent (pulse-pause) loading. In the present work, the high-accuracy ultrasonic fatigue testing equipment developed at the University of Natural Resources and Life Science, Vienna (BOKU) is described, and systematic investigations are presented that demonstrate the effects of cycling frequency, specimen size, and intermittent loading. Data obtained with additively manufactured (AMed) austenitic stainless steel 316L at 19 kHz are compared with electromagnetic resonation test results measured at 90 Hz. No influence of cycling frequency on fatigue lifetimes and fatigue limit was found when specimens of comparable size were used for tests at 19 kHz and 90 Hz. In contrast, the use of as-built specimens with smaller testing volume resulted in an increase in cyclic strength, demonstrating a size effect on fatigue properties. Polishing the surfaces of AMed specimens increased the fatigue limit by a factor of two. Furthermore, ultrasonic fatigue tests with intermittent loading and four different pulse lengths as well as continuous cycling were performed. The tests were conducted with wrought aluminum alloy 7075-T651 and with specimens containing small artificial surface defects. Similar lifetimes were measured in all testing series. Moreover, the same fatigue limit is found for all pulse lengths and for continuous cycling. This demonstrates that pulsed loading is a most suitable method to avoid specimen heating at ultrasonic frequency and that there is no influence on measured fatigue properties if ultrasonic loading is appropriately controlled. Measurement artifacts could be identified and avoided by measuring and recording all load amplitudes during ultrasonic fatigue tests.

On the Problem of Arrangement of Hybrid Energy Storage Systems

. Electric energy storage systems are widely used in electric transport, power engineering and in order to provide autonomous power supply and load regulation of power systems. One of the ways to increase the technical and economic efficiency of storage devices is their hybridization, i. e. the creation of storage devices consisting of blocks of different types of batteries. The special literature contains no systematic analysis of qualitative and quantitative effects of hybridization and corresponding methodological recommendations for choosing a scheme and evaluating the effectiveness of hybridization. In the present article, this issue is considered from a theoretical and methodological standpoint, recommendations are given for the design of storage devices serving solar or low-power wind farms. A brief overview of data on the cost of buffering electricity with lithium-ion, lead-acid batteries and supercapacitors is made. A method is proposed for determining the necessity and degree of hybridization of an energy storage device based on the simplest dependencies of the storage parameters on the degree of hybridization. The notions of the coefficient of synergetic effect of hybridization and the degree of internal buffering of electricity are introduced. A quantitative-and-qualitative model for evaluating the effectiveness of hybridization is presented. A methodological approach is proposed for calculating the degree of internal recovery and evaluating the coefficient of synergetic effect of hybridization. It is shown that, in general, the adding of supercapacitor unit to lithium-ion batteries a does not lead to a reduction in the cost of buffering electricity due to the high ratio of the cost of buffering with a supercapacitor to the cost of buffering with lithium-ion batteries. At the same time, the economic feasibility of using supercapacitors to compensate for high pulse loads can be determined on basis of the analysis of the frequency spectrum of the load graph of the storage unit. The developed models and approaches can be used in the design of electrochemical energy storage systems for specified operating conditions.

Hierarchical control of hybrid energy storage system in shipboard gas turbine power system with multiple pulsed power loads

Transient power fluctuations may occur in shipboard DC microgrids when switching a pulsed power load (PPL). However, shipboard power systems (SPSs) are isolated power grids with relatively low inertia and capacity, making them unable to respond immediately to sudden changes in pulsed loads, thereby resulting in considerable disturbances and even instability of the system. To mitigate such disturbances effectively, it is recommended to introduce energy storage devices to SPSs. In order to better leverage the buffering characteristics of energy storage devices, this paper establishes a simulation model of the SPS, which includes a micro gas turbine generator, a hybrid energy storage system (HESS), multiple PPLs and an energy management system (EMS). The HESS comprises a battery and a flywheel. Furthermore, a hierarchical coordinated control strategy for HESS in shipboard DC microgrids under multiple pulsed load conditions is proposed, which includes both an upper-layer control and a lower-layer control. The upper layer makes logical judgments through voltage difference and selects the appropriate energy storage mode for power supply. The lower layer adopts a series fuzzy logic control. To validate the effectiveness of the proposed control strategy, two scenarios of transient power mutation and total power demand exceeding the rated power of the generator were simulated by connecting different PPLs to the DC bus at different times. The results indicate that the control strategy effectively realizes the coordinated operation of HESS and gas turbine, as well as power allocation between flywheel and battery within HESS. Moreover, it reduces the frequency of charging and discharging of battery, improves battery life, and prevents the generator from running beyond the rated power.

Contribution to strengthening Bus voltage stability and power exchange balance of a decentralized DC-multi-microgrids: Performance assessment of classical, optimal, and nonlinear controllers for hybridized energy storage systems control

Stability is the primary goal in standalone power systems due to the high penetration and uncertainties of renewable energy resources, so the system reliability is affected, and the hybrid energy storage system (HESS) and diesel generator (DG) compensate for renewables' failures. This study contributes to managing and controlling DC-coupled multi-Microgrids fed by Photovoltaic, HESS, and DG resources and subjected to pulsed loading and uncertain PV outputs, besides battery SOCs and DG fuel constraints. The Dual-loop control structure compares classical Proportional Integral (CPI), Super-Twisting-Sliding-Mode-Control (ST-SMC), and Linear Quadratic Regulator with Integral Action (LQR-I) to evaluate the robustness and control performance of the HESS. For the supercapacitor (SC), a hysteresis current control (HCC) tracks its setpoint currents using a frequency decoupling-based power-split control. Matlab simulations confirmed system reliability by combining controllers with different Key Performance Indicators, where adopting the LQR-I for bus voltage regulation and the ST-SMC for HESS current control outperformed the other combinations and revealed superior system performance, improving Bus voltage regulation, DG current tracking, overall system efficiency, and loading convergence by 67.63(%), 83.63(%), 77.92(%), and 2.72(%), respectively. Instead, the basic CPI enhanced battery and SC current control with 51.55 and 63.94 (%) compared to the winner scenario.

Elasto-plastic analysis of foundations during emergency shutdown due to blast loading

Behaviour of machine foundations is investigated in this study under harmonic loading with varying frequency and pulse loading. Owing to excessive vibrations from the pulse, the machine encounters an emergency shutdown where the machine’s exciting frequency reduces to zero exponentially. The analysis is performed considering an elastic plastic single degree of freedom system with hardening and softening behaviour. Two cases are analysed depending on the time of application of pulse loading to the foundation system. For both the cases, the governing equations of motion are acquired and solved numerically using the fourth order Runge-Kutta method. Using these solutions, displacement-time graphs are obtained for a variety of parameters such as time constant, hardening/softening index, mass of machine and the foundation block, stiffness, pulse load magnitude, damping ratio and decay coefficient to study their influence on the behaviour of the machine-foundation system. The study helps in predicting the dynamic response of foundations under the emergency shutdown conditions due to pulse loading. Further, these also help in taking the decision if there exists a need for vibration barriers to have the displacements well within the permissible limits.

Experimental and numerical studies on pulse shaping techniques used in split Hopkinson pressure bar for testing concrete material

Split Hopkinson pressure bar (SHPB) is commonly used to characterise materials under high strain rates. However, conventional SHPB tests on brittle materials have encountered several experimental challenges for high strain rate loading. In relatively brittle materials like concrete, the deformation of the specimen is very small when subjected to impact loading; hence, it is very difficult to obtain the prerequisites of valid SHPB tests such as dynamic equilibrium and constant strain rate in the specimen. To overcome these issues, the current study presents the importance of the pulse shaper approach in SHPB application for the dynamic characterisation of concrete material. Selection of the appropriate dimension of pulse shaper assists in facilitating dynamic stress equilibrium and constant strain rate in the specimen. In the present study, copper pulse shapers are used for the evaluation of concrete under high strain rate loading using an SHPB set-up. Parameters such as the effect of dimensions (diameter and thickness) of pulse shapers on loading pulses, dynamic equilibrium, constant strain rate and material responses are studied. Experimental results reveal the prediction of suitable pulse shapers for 50–200/s strain rates. In addition, numerical simulation is also performed and results are validated with the experimental data.

Impact of microstructure on the performance of WRe10 conversion layers for stationary and rotating anodes

The focal track of high-performance rotating anodes withstands extreme temperatures, thermal shock, and thermal fatigue during the operation of X-ray tubes. Overloading the focal track manifests itself in undesirable operational instabilities and reduced X-ray output of the X-ray tube.In the present work a novel test system was developed to test stationary anodes, exhibiting WRe10 conversion layers with different microstructures, by applying pulsed electron beam loading. The samples were exposed to a maximum temperature of 2350 °C and showed surface damage features compared to rotating anodes. The measured reduction in X-ray output was up to 35% due to target damage while keeping consistent filtration and a 100 kV tube voltage.A columnar grain microstructure led to a significant increase in performance due to the early formation of stable and orientated crack networks. In contrast, pancake-shaped grains stacked parallel to the surface caused local overheating due to the formation of cracks parallel to the surface and the resulting local thermal isolation.

Weapon target assignment strategy for shipboard power system considering high power pulse loads integration

In the current naval combat environment, the instantaneous high-energy nonlinear high-power pulse load is gradually developed into a core equipment and facilities for military ships, and the nonlinear and transient impact characteristics of high-power pulse load are obvious. In this paper, the weapon–target assignment (WTA) modeling and solving problem are studied considering the pulse loads integration. Firstly, the multi-objective optimization method is used in this paper, and the optimization objective function is defined as maximizing the attack ability of high power pulse load in a short time and minimizing the total survival probability of incoming targets. The WTA model under multiple constraints is established. Then, genetic algorithm (GA) is used to solve the problem, taking into account the characteristics of impulsive load and WTA problem. Finally, simulation cases are constructed for experimental analysis. The simulation results show that the proposed model and algorithm can obtain an effective solution to the WTA problem with impulsive load, which has certain significance of reference.

Phase change heat transfer in a vertical metal foam-phase change material thermal energy storage heat dissipator

A metallic foam heat dissipator for cooling electronic components was addressed. A heat dissipator is a partitioned aluminum container loaded along with aluminum metallic foam and saturated with paraffin wax. A heat flux at a surface contains a basic uniform flux and the step transient raise, which should be managed by a heat dissipator and a Phase Change Material (PCM). The regulating equations for a melting/solidification transfer of heat & momentum transport in a heat dissipator were instituted into a structure of partial differential equations. Then, the vital monitoring equations were converted into a general dimensionless type and solved by the Finite Element Method. A mesh adjustment technique & automated time-step control was employed to control the accuracy & convergence of the result automatically. An adaptation technique controls the mesh resolution at the melting/solidification interface. The dimensionless temperature of fusion is a vital factor in the control of the surface temperature and heat dissipator efficiency. Considering a fixed amount of material for walls, a heat dissipator with thick sidewalls and thin top and bottom walls results in slightly better thermal performance. Using a PCM heat sink could reduce the heated surface temperature by >175 % during the pulse load.

Enhancement of underwater shock wave by coupling electrical explosion and al-powder-suspension combustion: Facing advanced reservoir stimulation technology

The development of fossil energy requires eco-friendly and high-efficient approaches, especially in contemporarily low-carbon scenarios. Using the high-power pulsed discharge to induce fractures has drawn increasingly academic and industrial attention in reservoir stimulation. Therein, the electrical wire explosion is commonly adopted as the pulsed discharge load to transfer electrical energy to mechanical one (shock wave and plasma bubble) efficiently. To strengthen mechanical loadings, a discharge capsule with a metallic wire inside Al-powder suspension has been designed and tested. Experiments were performed with a μs-timescale pulsed power source with stored energy from 150 to 750 J, and the electrical explosion were diagnosed via high-speed photography, electrophysical, and mechanical measurements. The experiment shows that there is an optimal concentration of Al-powder suspension in order to obtain the maximum peak pressure of shock wave for a given discharge condition. Too high or low concentrations of the aluminum powder will cause a lower peak pressure. This paper shows that at the stored energy of 500 J, when the concentration of Al-powder suspension is 20 g/L, the peak pressure can reach 4.81 ± 0.27 MPa, which is 2.01 times as large as the shock wave generated by the electrical explosion under the same conditions in pure water. Through the electrophysical measurement, it is found that the aluminum powder almost does not affect the phase transition of the wire. However, in the subsequent plasma formation, the gaseous reaction products make the discharge channel more complex and in a higher pressure density state, which is shown by the increase of channel resistance. Through the backlit images taken by the high-speed camera, it is obvious that two discrete strong shock waves can be seen in the Al-powder-suspension environment.

Sliding mode control for pulsed load power supply converters in DC shipboard microgrids

Pulsed power load (PPL) is a special load type in shipboard microgrids (SMGs), which consists of the generation module, energy storage system, and various types of loads. Having a reliable power supply to shipboard loads is a challenge as the SMG operates in islanded mode in most cases. Particularly, the PPLs require high transient power transfer with fast dynamics and strong robustness. Conventional solution to supply for the PPL is based on proportional-integral (PI) control, which can be used by linearizing the system around the equilibrium operation point. However, for a pulsed power supply (PPS) system, the load demand drastically changes in a short time, usually in millisecond level, making the operating point changes when the pulsed power is triggered or terminated. To supply the PPL with fast dynamics and robustness, an improved PPS control method is presented in this paper. By adopting a nonlinear sliding mode control (SMC) method, fast voltage regulation and robust pulse power tracking can be achieved. In the PPS, the PPL power demand is divided into two terms: one is the average power that is supplied by the SMG and the other is the fast pulsed power that is supplied by the storage capacitor. The size and cost of the storage capacitor are reduced as it is intentionally driven to a deep discharge. The PPS system configuration and coordination principle, SMC controllers, and sizing of passive elements in the PPS are analyzed in detail. The effectiveness of the presented PPS is verified by simulation results.

An experimental study of the saturated impulse for metal plates under slamming

Engineering structures can suffer severe local structural damage when subjected to extreme dynamic pulse loadings such as slamming impact. However, there is paucity of information on how the loading pulses affect the large plastic deformations. This paper presents information on an experimental investigation into the dynamic pressure pulse generated by slamming and the plastic response of metal plate. In our experiments, plates made of aluminum and steel alloy, were dropped from various heights ranging from 0.1 m to 1.3 m, onto the calm water in a tank. Based on the experimental results, the effects of structural material and water impact velocity on the slamming pressures pulse are investigated in detail. The results indicate that not only the peak pressure but also the pressure pulse duration play important roles in governing the dynamic response of the impacted structures. It is shown that the maximum strain and deflection at the plate center increase almost linearly with the drop height of the drop model. In view of the lack of previous experimental evidence for the saturation phenomenon of structures under slamming, we compared the dynamic response at the center of the plate with the experimentally measured slamming pressure under various drop heights, so that we have evidently verified the saturation phenomenon of flat plates for the first time under slamming in drop tests. It is also revealed that the dynamic response of the flat plates under slamming becomes more prone to the saturation phenomenon with the increase of drop height.

Field Evaluation of Piezoelectric Energy Harvesters on Bridge Structure

This study aims to develop and evaluate vibration-based piezoelectric energy harvesters for generating power from a bridge structure. New designs of multiple-degree-of-freedom (DOF) cantilevers were proposed and evaluated in a laboratory and on a full-scale bridge. It was found that all cantilever designs showed potential of generating 35 V voltage outputs under a simple sinusoidal vibration scenario in the laboratory. Field testing results showed that the match between the vibration frequencies of bridge structure and the resonant frequencies of cantilevers significantly affected the voltage output from the piezoelectric energy harvester under moving tire loads. Through adding more DOF on the same cantilever, the voltage attenuation from peaks generated by the cantilever turned to be less significant after each load passing, leading to greater energy outputs in some cases. With adjusting the mass combination in the 3-DOF cantilever design, the voltage output and energy production reached 11.1 V and 58.2 μJ under one single loading pulse, respectively, which was higher than 9.2 V and 14.9 μJ obtained from the best scenario of 1-DOF cantilevers. The study findings indicate the potential of developing multi-band piezoelectric energy harvesters for harvesting energy from bridge vibrations.

Frequency Restoration of Grid-Forming Inverters in Pulse Load and Plug-in Events

This article presents a restoration technique using grid-forming inverters in an islanded microgrid during pulse load and plug-in events. In a microgrid powered by droop-controlled inverters, frequency is the variable that is accessible to all inverters for adjusting their power contributions. The problem is that the microgrid frequency deviates from its nominal value after a load change. The presented method enables the inverters to restore the frequency to the nominal value and regulate their bus voltage amplitudes. The frequency and voltage restorations are performed without communication while achieving the desired power-sharing between grid-forming inverters. The restoration is activated after detecting any active or reactive power changes. In this article, the dynamic model of an inverter equipped with enabled restoration paths is developed to verify the stability of the inverter controller under various conditions. The frequency and voltage restorations are examined during pulse load and plug-in events. This study is performed using a laboratory-scale islanded microgrid powered by two 208 V, 60 Hz, 5 kVA inverters.

Optimization of Hybrid Energy Storage System Sizing With Considering Energy Management Strategy for High-Power Pulsed Load in Aircraft

High-power pulsed load (HPPL) such as airborne laser weapon and radar poses a severe challenge to aircraft electrical power system. Since peak power cannot be satisfied only by generator, hybrid energy storage system (HESS) should be well sized to match their energetic performances versus instantaneous power requirement and improve reliability of electrical power system. To deal with these challenges, a sizing optimization for HESS with considering energy management strategy (EMS) based on virtual-droop control in aircraft is proposed in this paper. By analyzing the spectrum characteristics of HPPL profile, splitting frequency between different types of energy storage devices is specially analyzed and designed to give full play to HESS. HESS weight, loss and parameters in EMS are taken into account in this multi-objective optimization based on Non-dominated Sorting Genetic Algorithms II (NSGA-II). Series of Pareto fronts can be obtained through the optimization process, and the most suitable solutions of different HESS combinations are selected by a decision-making means which combines subjective and objective weighting methods. Finally, optimization results of four different combinations are compared. Simulation results are obtained in MATLAB/Simulink environment to prove the rationality and applicability of the optimal sizing tool. The research has resulted in a solution of HESS and EMS optimization for HPPL used in aircraft.

Elastic effects on the dynamic plastic deflection of pulse-loaded beams

Beams under intense dynamic pulse loading may experience large elastic-plastic deformation. Despite numerous studies have been conducted based on the rigid-perfectly plastic idealization of material, it is of great interest in both theoretical modeling and practical application to characterize and evaluate the effect of elastic deformation on the dynamic behavior of structures. This paper is aimed to examine the range of applicability of rigid-plastic theoretical solutions on large deflection of beams in response to dynamic pulse loading. The energy ratio R and the characteristic time ratio are first precisely defined by referring to the characteristics of a “plastic string”, which represents the ultimate status of beams under very large deformation. Based on the recently published complete solutions on the dynamic response of fully-clamped and simply-supported rigid-plastic beams subjected to uniformly distributed rectangular pressure pulse, the discrepancies in final deflection and in saturated impulse of the rigid-plastic theoretical predictions from the elastic-plastic simulation results are evaluated and analyzed. It is revealed that if the energy ratio R>5 (with no limitation to the pulse duration), the rigid-plastic solutions can provide reliable predictions on final deflection and saturated impulse for the usage in engineering design. Then, the effects of the key dimensionless parameters related to the energy ratio R (e.g., the structural and material parameters as well as the loading one) on the discrepancies in final deflection and in saturated impulse are thoroughly examined. Finally, a unified dimensionless stiffness is identified to characterize the combined effect of structural and material parameters. For the cases of ζ≤1.581, which represent most of the practically used beams, 1/(2R) and 1/(3R) could provide upper bounds to the absolute value of the discrepancy in final deflection for fully-clamped and simply-supported beams, respectively.