Vibration Period Research Articles

Seismic reinforcement effect and vibration test of externally attached substructures in a selected Japan school building

In the long-term seismic work in Japan, an effective seismic reinforcement design system and reasonable seismic reinforcement technology have been developed, which is worth learning from. In this paper, the seismic reinforcement of a school building in Japan is taken as an example. Firstly, an overall reinforcement scheme of external prestressed precast concrete (PC) frame and steel brace is proposed. Then, based on the calculation results of seismic reinforcement and the ambient measurement before and after seismic reinforcement, the effectiveness and rationality of the seismic reinforcement method are analyzed. The results show that the seismic reinforcement method of attached substructures changes the original structural system, and solves the problems of excessive deformation and insufficient seismic performance. The reinforcement method improves the story stiffness of the structure, and the vibration period of simple harmonic motion after the implementation of seismic reinforcement is reduced by 0.845 times at most. The research results can provide reference for seismic reinforcement design and performance evaluation of existing school buildings.

A self-excited bistable oscillator with a light-powered liquid crystal elastomer

The nonlinearity in the dynamics of a liquid crystal elastomer-based systems plays an important role when designing and controlling a responsive LCE-based smart system. In this work, we proposed an LCE-based bistable oscillator and investigated its self-excited oscillation behavior. The oscillator consists of a lumped mass, connected by two lateral helical springs, and one liquid crystal elastomer fiber spring which is subjected to a steady light illumination. After inserting the dynamic model of the light-powered LCE fiber into that of the oscillator, the nonlinear vibration response is obtained by using a fourth-order Runge-Kutta method. Different from the reported mono-stable limit circle modes, the self-vibration has a snap-through oscillation mode owing to an opto-mechanical coupling where the light energy can overcome the potential well of the mechanical part. Since the light illumination energy compensates the dissipative energy loss in each nonlinear vibration period, the nonlinear vibration of the LCE-based bistable oscillator is sustainable. We discuss the influence of both the linear and nonlinear parameters on the self-excited vibration response of the bistable oscillator. The nonlinear parameters of the lateral spring, such as the stiffness ratio and the length ratio, as well as the linear parameters of the LCE fiber, including the damping ratio, spring stiffness, light intensity, light contraction coefficient, and gravity, are all taken into consideration in our analysis. Our results reveal that the self-excited vibration can be controlled by adequately tuning these linear and/or nonlinear parameters. The investigation may pave a way for promising applications such as intelligent structural design, energy harvesting, sensing, actuating, and bio-mimic/bio-inspired system using the LCE fiber.

Post-Fire Seismic Performance of Concrete-Filled Steel Tube Frame Structures Considering Soil-Structure Interaction (SSI)

Currently, reinforced thin-walled irregular steel tube concrete frame structures have been applied in engineering, but there are few researches on the seismic performance of this type of structures after fire. The seismic performance of structures after fire is generally carried out based on rigid foundation conditions. Therefore, it is of certain engineering and theoretical value to study the seismic performance considering the SSI (soil–structure interaction) in this paper. ABAQUS is employed to establish the finite element models of the reinforced thin-walled irregular steel tube concrete frame structure considering the SSI after a fire. The paper analyzes the impact of different site conditions and fire durations on the structural natural vibration period, maximum acceleration, inter-story shear force, and maximum inter-story displacement angle. The results show that the consideration of the SSI increases the basic natural vibration period of the structure by 10–30%. The softer the soil and the longer the fire duration, the more significant the increase. For harder soil, lower seismic intensity, and shorter fire duration, the acceleration assigned to the structure and foundation after considering the SSI is smaller than the results assuming a rigid foundation. The change in inter-story shear force is mainly determined by the acceleration of the structure and foundation. The inter-story displacement angle increases when considering the SSI, and the increase is more significant with softer soil, larger seismic wave acceleration amplitude, and longer fire duration.

A novel multi-resonator honeycomb metamaterial with enhanced impact mitigation

Local resonant metamaterials show significant promise in applications involving the attenuation of impact waves. This paper introduces a honeycomb metamaterial featuring a circular distribution of multiple mass-beam resonators designed to enhance impact mitigation by broadening low frequency bandgaps. The bandgaps of unit cells with different number of mass-beam resonators (MBR) are calculated. It is found that the unit cell with circularly distributed resonators (C-MBR) has strong low frequency local resonance in all directions, thus opening a complete wide bandgap in about 500–1200Hz. Then, a method of equivalent stiffness reduction related to the corresponding mode shapes is used to research the negative effective mass density characteristic, explaining the generation of the low frequency bandgaps. Subsequently, the impact protection capacity of local resonant metamaterial is compared with that of ordinary honeycomb metamaterial through experiment and simulation. Interestingly, the maximum acceleration values under the impact load of local resonant metamaterial are about 45% lower than that of honeycomb metamaterial, and the reaction force under the blast load is about 80% reduction. Besides, by analyzing the acceleration curve waveform and transmission spectrum, the local resonant metamaterial is found to exhibit continuous short period vibration, which blocks the wave in low frequency band. Finally, the energy dissipation performance of the local resonant metamaterial explains that it converts more impact energy into the kinetic energy of the core with resonators, thereby reducing the energy transferred to the bottom panel. These results underscore the potential application of the proposed metamaterial with multi-resonators in impact mitigation and energy dissipation.

Electric-field induced droplet stable vertical vibration: Experiments and numerical simulations

In this study, stable vertical vibration of a single droplet was stimulated through Electrowetting on Dielectric (EWOD) on a Polydimethylsiloxane-coated Indium Tin Oxide glass (PDMS-ITO) surface in an oil atmosphere. The mechanism of the vibration is clarified as a self-adjusted energy earning and losing equilibrium, and a small friction at the contact line is crucial for this stable vibration. We investigated the impact of needle tip electrode height, different voltage forms (AC and DC voltages), and voltage magnitude on the droplet vibration state. As decreasing the needle tip electrode height, there is a transition of the droplet from a vibrating state to a static equilibrium state. During the vibration, the vibration period remains constant regardless of tip height, but the contact angle changes synchronously with the voltage. The electric signal in the circuit is also measured to provide solid evidence for this synchronicity. Finally, the saturation voltage of the contact angle and amplitude is about 120 V. The intrinsic connection between contact angle saturation and amplitude saturation is clarified by studying the surface energy of the droplet. A theoretical model is constructed to numerically simulate the shape and amplitude of the droplet’s vibrations.

Directionality of FIV3 ground-motion intensities during the 6 February 2023 Kahramanmaraş, Türkiye earthquake doublet

At present time, ground-motion prediction models neglect the directionality observed in horizontal components of earthquake ground motions, that is, the important changes in ground-motion intensity that occur with changes in azimuth. This study presents an investigation of the directionality of a recently proposed measure of ground-motion intensity during the 6 February 2023, Mw 7.8 Pazarcık and Mw 7.5 Elbistan earthquake doublet in the Kahramanmaraş region of Türkiye, which resulted in the collapse of more than 35,000 buildings and caused almost 60,000 fatalities. The studied intensity measure is referred to as FIV3, which has been shown to be better correlated with structural collapse than the spectral acceleration at the fundamental period of the structure. The improved intensity measure is period-dependent and is computed as the sum of the three largest incremental velocities with the same polarity obtained from the area under segments of a low-pass filtered ground acceleration time series. The following aspects are studied in this article: variation of FIV3 intensity with changes in the orientation; variation of FIV3 intensity with changes in the period of vibration; attenuation of FIV3 intensities with increasing distance; and spatial distribution of the orientation of maximum FIV3 intensity. This study is based on 231 pairs of records from the Mw 7.8 main event and 222 pairs of records from the Mw 7.5 event. Similarly to the directionality of spectral ordinates, it is found that the directionality of FIV3 intensity also increases with increasing period. Strong directionality occurred not only in the near field but up to distances as large as 400 km from the epicenter. The orientation of maximum FIV3 intensity is found to occur close to the transverse orientation, consistent with observations for the orientation of maximum spectral ordinates during strike-slip earthquakes.

Ground-Motion Intensity Measures for the Seismic Response of the Roof-Isolated Large-Span Structure

Ground-motion intensity measures (IMs), which quantify and describe the characteristics of earthquake ground motion, are of utmost importance in the assessment of seismic risk and the design of resilient structures with large spans. The appropriate selection of a ground-motion IM is crucial in establishing a reliable and robust correlation between seismic hazards and structural demands. The current study presents a novel ground-motion IM that incorporates the influence of multiple vibration modes and period elongation resulting from isolation based on the velocity spectrum. A comprehensive study has been conducted to examine the efficiency of 37 different ground-motion IMs on a roof-isolated large-span structure with engineering demand parameters (EDPs), using far-field ground-motion data. The initial examination of the proposed intensity measure involves a planar lumped-mass model. Subsequently, a numerical model of a large-span roof-isolated structure, specifically the Beijing Workers’ Stadium, is constructed and examined. The results suggest that the proposed intensity measure (IM) demonstrates satisfactory adequacy and achieves optimal efficiency when considering three different engineering demand parameters among 37 other ground-motion intensity measures.

Shaking-Table Test and Finite Element Simulation of a Novel Friction Energy-Dissipating Braced Frame

To enhance the effect of seismic mitigation in medium-sized buildings, this study introduced a novel friction damper within a braced frame, forming a friction energy-dissipating braced frame (FDBF). The seismic reduction mechanism of the FDBF was examined, and its performance was evaluated through shaking-table tests and finite element simulations. The hysteresis performance of the novel damper was assessed through low-cycle repeated loading tests, which yielded predominantly rectangular and full hysteresis curves, exemplifying robust energy dissipation capacity. The shaking-table tests of the FDBF indicated significant modifications in the dynamic characteristics of the original frame structure, which notably reduced the natural vibration period and enhanced the damping. Additionally, the FDBF remarkably reduced both acceleration and displacement responses during seismic excitation. Optimizing the orientation of the energy dissipation brace significantly improved seismic reduction efficiency. A dynamic time history analysis, employing finite element software, was conducted on the FDBF equipped with a friction energy dissipation brace at each level. Comparative analysis with both the moment-resistant frame and ordinary braced frame revealed that the FDBF substantially lowered the peak acceleration at the apex of the structure, achieving a reduction rate of 40–50%. Under both design and rare earthquake conditions, the FDBF demonstrated superior seismic mitigation capabilities, especially under rare earthquakes. Future studies should investigate various structural types with energy dissipation braces at different levels to identify the most efficient layout for the novel friction energy dissipation brace, thereby guiding relevant engineering practices.

Validation of numerical results of complex seismic analysis through simple analytics

The paper analyzes the application of the numerical findings of the program, which is becoming increasingly difficult for civil and structural design. Since, as in many other countries, the verification of design using a software model is now required by current Italian codes as well. Given this, the structural engineer must provide a technical report using a licensed software tool, attached to other project documents to get the Seismic Authorization at the local Civil Engineering Department offices. Following a brief explanation of structural analysis methodologies, this study presents a criterion for assessing the applicability of numerical findings obtained using any structural software. Three case studies of this criterion are shown to demonstrate how to check them using simple manual calculations: (i) the normal stress in RC columns subjected to gravity loads; (ii) the periods of vibration, participating masses, and seismic base shear derived from dynamic modal analysis; and (iii) the main parameters characterizing the pushover curves of existing buildings. Finally, this work underlines the significance of confirming the application of numerical results obtained by software in civil and structural design. The offered criteria and scenarios exhibit realistic techniques to ensure accuracy and reliability in structural performance assessment, according to the structural requirements imposed by current codes in Italy and similar countries.

Investigation on the free vibration characteristics of vertical quasi-zero stiffness isolation system considering the disc spring's loading position

This study investigates a quasi-zero stiffness (QZS) isolation system designed for vertical vibration control in isolation structures. The QZS characteristics are achieved by connecting a negative stiffness disc spring in parallel with the linear positive stiffness spiral spring. Initially, the force-displacement relationship of the disc spring is established, taking into account the loading position, and validated through static experiments. Subsequently, the free vibration equations of the QZS system are derived. The Jacobi elliptic functions and the Newton-harmonic balance (NHB) method are then employed to study its free vibration characteristics without considering damping. The results demonstrate that the natural vibration period of the QZS system is related to the nondimensional amplitude and the height of the loaded inner cone. Furthermore, the second-order analytical approximate solution obtained via the NHB method exhibits remarkable accuracy. Finally, the critical damping ratio of the damped QZS isolation system is investigated using a numerical approach. The findings indicate that the critical damping ratio is only correlated with the nondimensional amplitude. This study addresses the gaps in existing research and provide recommendations for the parametric design of vertical QZS isolation system.

Experimental Investigation and Damage Identification of High-Pile Wharf Framed Bents under Horizontal Impact Loads.

This study investigates damage characteristics, dynamic structural performance changes, and quantitative damage assessment of high-pile wharf framed bents exposed to horizontal impact loads. Through extensive testing of wharf framed bents under such loads, a damage identification approach based on stiffness, natural vibration period, and acceleration data derived from experiments is presented. The findings reveal that under horizontal impact loads, framed bents initially exhibit tensile damage and leaning piles, followed by short straight piles. Additionally, structural damage results in a reduced self-oscillation frequency and an increased amplitude decay rate. Both stiffness-based and cycle-based damage indicators effectively track the cumulative damage progression of the structure. However, the cycle-based damage indicators demonstrate superior stability and accuracy, while acceleration-based indicators precisely identify the moment of damage mutation. This research contributes to enhancing local components, implementing damage identification methods, and advancing health monitoring practices in high-pile wharf projects, aligning with the standards of scientific publications in the field.

Spatio-temporal distribution of aerodynamic forces and their associated vortex drift patterns around a closed-box girder during torsional vortex-induced vibration

Torsional vortex-induced vibrations (VIVs) pose potential risks to prototype bridges with increasing spans and lower damping levels. However, the underlying mechanism remains unclear. Wind tunnel tests revealed that torsional VIVs occurred at an initial attack angle of + 3°. To unravel the VIV mechanism throughout the lock-in range, encompassing the ascent stage, amplitude extremum, and descent stage, the spatio-temporal distribution of aerodynamic forces around the bridge girder was analyzed. The results demonstrated the presence of traveling pressure waves with approximately 2.5 wavelengths and three uniformly distributed separated vortices over the upper side throughout the lock-in range. The primary source of these travelling pressure waves stemmed from the vortex drift patterns, with the aerodynamic wave travelling velocity equating the vortex drift velocity. The formation of separated vortices, known as the “double-vortex model”, occurred at the windward barrier on the upper side and windward maintenance rail on the lower side. These vortices directly imposed travelling pressure waves on the girder, thereby inducing torsional VIVs. Among them, the former vortices contributed more to the VIVs than the latter. Therefore, the formation of separated vortices and their related aerodynamic waves in the windward edge with the 2nd SVM/AWM modes predominantly governed the occurrence of torsional VIVs. Furthermore, both the ratio of the wave travelling time to the vibration period and the mean value of the wave travelling velocity ratio exhibited an inverse relationship with the VIV responses. However, the actions exerted by the vortex drift patterns were proportional to the VIV responses on both the upper and lower sides. Therefore, there existed evolutionary characteristics of aerodynamic waves and their related vortex drift patterns throughout the lock-in range, which was responsible for the “lock-in” phenomena.

Nonlinear periodic response of viscoelastic laminated composite plates using shooting technique

The nonlinear periodic response of viscoelastic laminated composite plates subjected to harmonic excitation is carried out in time domain using the generalized Maxwell model in the form of a Boltzmann integral. The integral form of the viscoelastic constitutive relation is converted to the incremental form for the finite element formulation based on the Reissner–Mindlin plate theory incorporating von Karman geometric nonlinearity. The recursive relations are developed to compute the current time-step solution using only the previous time-step solution. The periodic response is obtained using shooting technique coupled with Newmark’s time integration and arc length continuation method. The implementation of the shooting technique for the Boltzmann Integral based viscoelasticity done for the first time in this study involves derivation of a number of additional recursive relations and initial conditions at the beginning of each shooting cycle. The nonlinear periodic vibration characteristics such as frequency response, damping factor, steady-state response history, phase plane plots, and frequency spectra are presented for viscoelastic plates with different boundary conditions and lamination schemes. Further, the influence of relaxation time and number of Maxwell elements on the frequency response characteristics are investigated. It is seen that the shooting technique is capable of predicting periodic responses of viscoelastic dynamical systems directly from the second-order equations of motion and is more efficient than the direct time integration method. The nonlinear response of viscoelastic plates predicted using the equivalent elastic model with Rayleigh proportional damping (having same linear frequency response) is found to be significantly different from the one predicted using viscoelastic constitutive model.

Prospects for mathematical modeling in mining system development: accounting for global oscillations and seismic waves

The article discusses the potential of mathematical modeling in understanding the impact of vibrations and seismic waves, aiming at enhancing the sustainability of systems within the mining industry. It explores the dynamic response of a tall, elastic structure with a uniform cross-section and a fixed cylindrical fluid reservoir, subject to various complex boundary conditions. The study delves into the vibrational behavior of the structure when exposed to seismic and harmonic forces, calculating frequency, vibration periods, and deriving formulas for stress, tension, deformation, bending moments, and shear forces in different parts of the structure through both theoretical and experimental approaches. Additionally, the article derives the differential equation for the free oscillation of a tall hydraulic structure in pure bending with an incorporated mass load under appropriate boundary conditions, identifying specific vibration frequencies and periods. The forced vibration scenario is also examined, focusing on the structure's foundation movement due to external harmonic forces. Numerical computation technology is utilized to analyze the change laws of principal quantities that describe both free and forced vibrational movements of the hydraulic structure, showcasing the applicability of these models in predicting and mitigating the effects of seismic activities on mining infrastructure.

Characterization of landslide and measurement of ground vibration period in the Salado-Paccha sector using GNSS equipment and refraction seismic testing

A landslide is a soils movement that depending on its level can be a very risky event, causing from material losses to human lives in some occasions. It can occur naturally or by anthropogenic means, the main causes are usually high intensity rainfall, soils with poor geomechanical characteristics, considerable seismic movements, erosion, lack of vegetation cover, among others. Paccha is a parish where there are several active landslides that put its inhabitants and their material goods at risk. As a result, this research analyzes one of them with UTM17s, 728845,38 E 9679536,46 N coordinates, located in the El Salado sector, for its study it is intended to characterize it, measuring its displacement and velocity through a monitoring of 36 days in 3 data collection campaigns with GNSS equipment using the RTK method, in addition a seismic refraction test is used to measure the period of vibration of the soil at the site. The results obtained indicate that the soil in the sector is clay and silt of high plasticity, the slip is extremely slow, semi-deep, with lateral propagation and rotational geometry. The displacement between points is 4,14 cm while the velocity is 0,43 m/year, finally the vibration period corresponds to a value of 0,13s and the shear wave velocity Vs is 178,85, which is categorized as a type E soil.

Experimental and Numerical Investigation of a Real Four‐Story Building with an Open Ground Floor in a Seismic Region for Proper Strengthening

Many existing buildings fail to meet the requirements of current earthquake resistance codes. Therefore, it is necessary to strengthen and/or upgrade such buildings. Strengthening and/or upgrading include adding new floors, parts, elements, etc. Modern codes usually do not reflect upgrading of existing buildings. These buildings may be damaged and require maintenance and repairs, which may change the original material and structural properties. Design of such buildings is based on dominant vibration periods obtained by empirical dependences according to the building codes or from dynamic numerical analysis using available software. In both cases, the results usually do not consider the present state of the structure, influence of nonstructural elements, and possible differences in material properties in various elements of the building due to construction problems, concrete creep, etc. Such evaluation can be done just based on dynamic analysis, experimental and numeric, whereas the first one is essential. Therefore, the present research is aimed at numerical and experimental investigation of an existing real multistory residential reinforced concrete building before its strengthening and/or upgrading according to the requirements of modern seismic codes. The present study is a first stage of the research. It is focused on obtaining the dynamic parameters of the building before its strengthening and upgrading. Experimental values of natural vibration periods and damping ratios are compared with those obtained numerically. Following these values, the design spectrum of the seismic code is updated and the building is designed using this updated spectrum. The findings of this study form a basis for the second research stage that will be aimed at monitoring and analyzing of this building at each stage of its strengthening and upgrading. This approach iaimed at improving the available techniques for strengthening and upgrading the building. It will also be the basis for further research in this direction.

Natural frequencies of bars with variable cross-sections

This paper presents the possibility of determining the natural frequency of flexural bars treated as Bernoulli-Euler beams using the Rayleigh method. It is assumed that the shape of the bar axis during vibration is the same as the shape of the deflection line of this bar under continuous load. By comparing the potential energy in the deflected position with the kinetic energy in the undeflected position, it is possible to determine the frequency and period of natural vibration as a function of Young's modulus and material density, as well as the parameters describing the geometric shape of the bar. Examples of solutions for truncated cone and truncated wedge-shaped bars are shown, as well as a solids of revolution with the generatrix described by linear and curvilinear (parabolic, exponential) forms. It applies to both solid and hollow bars, and different types of fixing the ends of the bar. The accuracy of the obtained results (compared with the literature and with the results of FEM calculations) is sufficient for practical applications. The authors also showed the possibility of extending the method to higher frequencies of vibration.

Estimation of the fundamental period of vibration of the dental clinics and the dental classroom of the Catholic University of Cuenca using environmental vibration recordings

In the Faculty of Dentistry of the Catholic University of Cuenca, located in the city of Cuenca, province of Azuay. The lecture hall and the clinics of the faculty are built. These structures were built during 2018 and 2019. Due to the seismic events occurred in recent years, monitoring was carried out to obtain the characterization of the dynamic behaviour. The two structures are formed by a structural system based on structural steel frames (beam-column), the structure for laboratory use has metal columns filled with concrete, while the structure for classroom use has metal columns without concrete filling. The main objective was to obtain the vibration periods experimentally from the recording of environmental vibrations and also to obtain them analytically with mathematical models elaborated in structural analysis programs and environmental vibration tests were performed. The results show that building 1, composed by the dentistry clinic, obtained a vibration period of 0.41 s in X and 0.33 s in Y (Analytical Model); and, in the environmental vibration test it showed a period of 0.46 s in X and, 0.30 in Y. Building 2, composed of the dental faculty classroom, obtained periods of 0.53 s in X and 0.29 s in Y (Analytical Model). While in the environmental vibration studies they gave values of 0.52 s in X and 0.26 in Y.

Site Specific Non-Linear Ground Response Analysis and Soil Structure Interaction Study of Newly Constructed Dharahara Tower

This study investigates the seismic performance of the recently reconstructed Dharahara tower, a historically significant monument in Nepal that has experienced damage in past earthquakes and was completely destroyed in the recent Gorkha earthquake. Ground Response Analysis (GRA) and Soil-Structure Interaction (SSI) investigations are conducted to assess its structural behavior. GRA includes free field and structure-influenced analyses, with subsequent comparative assessment. Additionally, 3D non-linear finite element analysis is employed to derive equivalent spring constants representing soil and foundation characteristics, which are incorporated into the SSI analysis. These computed spring constants are used to model support conditions, allowing for an evaluation of the super-structural response. The study also employs non-linear dynamic analysis to compare structural responses between fix-based and spring-based models. The results indicate that the presence of the structure significantly influences surface wave motion amplification in GRA, resulting in peak ground accelerations (PGAs) that exhibit de-amplification in the free field and amplification in the presence of the structure. Furthermore, the introduction of an equivalent spring system in the soil-structure model changes the system's vibration period and damping characteristics, leading to enhanced dynamic response compared to the fixed-base model.

Artificial neural network-based prediction model of elastic floor response spectra incorporating dynamic primary-secondary structure interaction

The evaluation of the Floor Response Spectrum (FRS) holds paramount significance in assessing the seismic behavior of secondary structures. Precise FRS prediction empowers engineers to make informed decisions concerning structural design, retrofitting, and safety precautions. This study aims to scrutinize the impact of dynamic interaction between primary and secondary structures on FRS. Both the elastic primary structure (PS) and elastic secondary structure (SS) employ a single-degree-of-freedom (SDOF) system. Governing motion equations for both coupled (with dynamic interaction) and uncoupled (without dynamic interaction) systems are formulated and solved numerically. The study investigates how variations in the vibration period of PS (Tp), tuning ratio (Tr), mass ratio (μ), and damping ratio (ξs) of SS influence FRS. The FRS impact remains minimal at μ = 0.001 (0.1%); however, with increasing mass ratio, PS-SS dynamic interaction significantly affects SS's spectral acceleration response. Coupled analysis is crucial only for secondary structures tuned to the primary structure's vibration period (0.8≤Tr≤1.2). This study utilizes two-layer feed-forward Artificial Neural Networks (ANNs) for FRS prediction. The Levenberg-Marquardt (LM) backpropagation (BP) algorithm trains the network using a comprehensive dataset. In summary, it is evident that the ANNs, once trained, enable accurate prediction of the FRS, exhibiting a R2 of 99%. Additionally, a design expression is formulated utilizing the ANN model and subsequently compared with the existing formulation.