Design Equations Research Articles

Numerical modelling and design of cold-formed steel battens subject to pull-out failures under bushfire conditions

Thin and high strength cold-formed steel (CFS) battens and claddings serve as non-combustible elements in roof and wall cladding systems, aligning with the recommendations of Australian bushfire standards. However, they are susceptible to significant damage from heat and strong winds experienced during bushfire events, particularly in the form of batten pull-out failures. Such failures may occur at the screw connections between cladding and battens, or battens and trusses/rafters. However, pull-out failures often occur in the thinner CFS battens, when the screw fastener pulls out of the batten’s top flange, leading to the loss of entire cladding and compromising the integrity of the building envelope. This study used finite element modelling of a range of CFS batten configurations to investigate their pull-out failures under combined wind action and bushfire conditions. Finite element models were developed to simulate the pull-out failures at elevated temperatures, validated using experimental results and used in a parametric study to investigate the effects of relevant parameters. Using the results, this study has proposed suitable design equations for predicting the pull-out capacities of CFS battens at elevated temperatures. Engineers can use them to design bushfire safe CFS cladding systems.

Numerical Investigation of the Axial Load Capacity of Cold-Formed Steel Channel Sections: Effects of Eccentricity, Section Thickness, and Column Length

Cold-formed steel channel (CFSC) sections have gained widespread adoption in building construction due to their advantageous properties, including superior energy efficiency, expedited construction timelines, environmental sustainability, material efficiency, and ease of transportation. This study presents a numerical investigation into the axial compressive behavior of CFSC section columns. A rigorously developed finite element model for CFSC sections was validated against existing experimental data from the literature. Upon validation, the model was employed for an extensive parametric analysis encompassing a dataset of 208 CFSC members. Furthermore, the efficacy of the design methodologies outlined in the AISI Specification and AS/NZS Standard were evaluated by comparing the axial load capacities obtained from the numerically generated data with the results of four previously conducted experimental tests. The findings reveal that the codified design equations, based on nominal compressive resistances determined using the current direct strength method, exhibit a conservative bias. On average, these equations underestimate the actual load capacities of CFSC section columns by approximately 11.5%. Additionally, this investigation explores the influence of eccentricity, cross-sectional dimensions, and the point-of-load application on the axial load capacity of CFSC columns. The results demonstrate that a decrease in section thickness, an increase in column length, and a higher degree of eccentricity significantly reduce the axial capacity of CFSC columns.

Behaviour and design of prestressed stayed circular concrete-filled double skin steel tubular (PS-CCFDSST) columns under axial compression

An innovative type of steel-concrete composite column, named the prestressed stayed circular concrete-filled double skin steel tubular (PS-CCFDSST) column, is introduced in this study, aiming to achieve both structural efficiency and economic advantages. Numerical simulation and theoretical analysis on the axial compressive behaviour of PS-CCFDSST columns were carried out, to understand and reveal their fundamental working mechanisms. Nonlinear finite element analysis (FEA) utilising ABAQUS was performed and validated against collected test data of CCFDSST columns and PS columns. Properly designed PS-CCFDSST columns could gain cost savings of over 30 % while maintaining identical load-bearing capacity as prestressed stayed hollow steel tube (PS-HST) columns with the same steel usage. The effects of initial pretension, nominal steel ratio, hollow ratio, column’s slenderness ratio, relative length of struts, column diameter, strut diameter and cable area on the buckling mode and buckling capacity of PS-CCFDSST columns were evaluated via parametric studies. The stiffness ratio β could be used as an index to comprehensively assess the structural efficiency of PS-CCFDSST columns, and a critical threshold of β = 2 corresponds to the transition between symmetric and anti-symmetric buckling modes. Linear calculation formulae for the theoretical buckling capacity and optimal initial pretension of PS-CCFDSST columns were derived. Finally, practical design equations for predicting the column’s buckling capacity were developed, and recommended values for the actual optimal initial pretension and other primary parameters were provided for PS-CCFDSST columns under axial compression. This study, for the first time, provides the fundamental mechanical behaviour, identification of the critical stiffness ratio for buckling mode transition, and design framework for buckling capacity of the PS-CCFDSST column, which is useful for advancing theoretical exploration and informing engineering practices for this new column type.

Intermediate-point bracing for columns: New stiffness–buckling load formulae

According to the stiffness equation provided in the American Institute of Steel Construction (AISC) design specifications for intermediate-point braces, the required point brace stiffness increases with the number of point braces. This research aims to formulate a new stiffness design equation for intermediate-point braces that accounts for the decline in required stiffness with an increasing number of braces. The brace stiffness (β)–buckling load (Pcr) relationships for columns with point braces were investigated by performing a buckling analysis using the energy method with the Fourier series. A formula was derived, defining the β−Pcr relationships for each buckling mode of the columns, regardless of the number of point braces and whether the buckling is elastic or inelastic. The AISC stiffness equations for point braces were significantly conservative under most buckling loads. The proposed stiffness design equation requires lower brace stiffness with an increasing number of point braces, resulting in smaller brace members than the AISC design equations. Additionally, the proposed strength design equation can set relatively high strength requirements for the brace, reflecting that stiffness governs the design of the point brace. These new equations offer a more precise and efficient method for calculating required brace stiffness, providing valuable insights to improve structural design capabilities.

Efficient strategy for frequency design and bandwidth extension of curved piezoelectric ultrasonic micromachined transducers

Abstract This article proposes an efficient analytical model and strategy for designing curved piezoelectric micromachined ultrasonic transducers (curved PMUTs). The model is developed based on the Donnell-Mushtari-Vlasov (DMV) theory and the Equivalent Single Layer (ESL) method, and validated through Finite Element Analysis (FEA). Utilizing the model, we further analyze the diaphragm’s vibration modes and key design parameters. The proposed strategy is centered on 2 design equations, facilitating the rapid design of devices at any frequency through parametric sweeps. Furthermore, to minimize bandwidth loss, we employ the merging of adjacent vibration modes to broaden the bandwidth. Using the proposed method for modes merging, we have effortlessly designed devices with operating frequencies of 2.15 MHz, 6.3 MHz, 10.65 MHz, and 18.75 MHz in water. For comparison, we also designed planar PMUTs and general curved PMUTs operating around 6 MHz and 15 MHz. Compared to planar PMUTs, curved PMUTs show exceptional performance improvements in output pressure and sensitivity. Moreover, the proposed strategy for bandwidth extension results in 1.33× and 1.25× bandwidth improvements around 6 MHz and 15 MHz. The proposed design methodology is anticipated to assist engineers in designing high-performance PMUT arrays more efficiently and systematically.

Shortcut Approximation to the Coupled Equations of Heat and Electric Currents

A self-consistent approximation to the coupled equations of heat and electric current densities is developed to facilitate the integration of the relaxation time approximation of the Boltzmann transport equation in a realistic design, analysis, and optimization of energy conversion devices. By introducing two auxiliary energies in the empirical Boltzmann transport equations, the coupled equations can be simply expressed in these auxiliary energies. These auxiliary energies are then determined by optimizing both the heat current through charge neutrality equation without the presence of electric current and the carrier concentrations with the presence of electric current. The heat and the electric current densities as well as the transport properties can then be calculated. The calculated transport properties agreed reasonably well with existing experimental and computational data.

Double‐order negative group delay filtering function: A brilliant capability of the bilinear double‐order transfer function

AbstractIn this work, the capability to generate the negative group delay (NGD) phenomenon at those frequencies higher than a certain value, that is, the NGD high‐pass (HP) filtering function, of the bilinear double‐order transfer function has been demonstrated. Based on the Type‐I bilinear double‐order filter circuit, our theory has been verified by strong agreements between the formulae and their proof‐of‐concept (POC) circuit‐based simulation results. Both formula‐based and POC circuit‐based simulations give the minimum group delay of −5 ms yet the cut‐off frequency of 10 and 9 rad/s, respectively. Such slight deviation is caused by the approximation error of the bilinear double‐order impedance. By employing two orders, the bilinear double‐order transfer function has been found to be the basis transfer function for the NGD filtering function with the highest degree of freedom. Based on our design equation for the NGD filtering function, it can be seen that the distances between zero and pole and the characteristic frequency of the bilinear double‐order transfer function are governed by such characteristic frequency itself, the cut‐off frequency of the NGD filtering function, and the fractional order of the Laplacian operator. In addition, the effects of the fractional order of the Laplacian operator, the fractional order of the transfer function, and the ratio of the abovementioned distances to the characteristics of the newly found double‐order NGD filtering function have been studied in detail.

Performance of structural insulation panels under eccentric axial and racking loads with varying opening ratios

Structural insulation panels, which are actively applied in the housing market in North America and Europe by utilizing wood, an eco-friendly material, as the main material, are composed of OSB (Oriented Strand Board), EPS (Expanded Polystyrene), and SPF (Spruce-Pine-Fir), which is used as a connection between the panels. Structural insulated panels are used as a load-bearing wall to resist external forces without installing separate columns. The production of structural insulation panels consists of applying adhesive to both sides of EPS, which acts as an insulation material, and pressing wood boards (OSB), which exert strength as a structural material. The OSB on the outside resists compressive and tensile forces, while the EPS on the inside resists shear forces. The joints and openings of the structural insulation panels are reinforced with SPF (Spruce-Pine-Fir) structural wood. When structural insulated panels are used as an exterior wall structure, the installation of openings is essential. Based on the opening size that is popularly used in wooden structures, the performance evaluation of structural insulation panels under eccentric axial load and repeated lateral load due to the reduction of the cross-sectional area by installing the opening was carried out, and the behavior analysis through variable analysis and comparison analysis with the design equation was carried out.

A computationally efficient approach of tuned mass damper design for a nuclear cabinet based on two-step machine learning and optimization methods

Enhancing nuclear power plant (NPP) safety is demanded because of the recent beyond-design-basis earthquake near a NPP. Therefore, research on improving the seismic performance of the electrical cabinet, which ensures the safe operation of NPPs, is needed. In this paper, a tuned mass damper (TMD) is employed to control the seismic response of cabinet. To design the TMD, we employ existing design equations or perform numerical model–based optimization. However, limitations, such as inconsistencies with targeted control of the load and structure, the possibility of converging a local solution, and the high cost of numerical analysis. Therefore, this paper proposes a two-step machine learning and optimization method. Such an approach is utilized to find the optimal global design solution and reduce numerical analysis costs. Each step involves the design of experiment (DOE), response surface, and optimization. Notably, range setting in the DOE accounts for the difference between each step. In the first step, the sampling range is widened to determine the relationship between the design variables and the cabinet's response, and in the second step, the sampling range is narrowed depending on the result of the first step. Consequently, the proposed method reduced the cabinet's response by 35.4 % on average and numerical analysis cost declined by 1/3.

Improvement of field falling-head test and determination of hydraulic conductivity using Darcy’s equation

This study presents a novel permeameter design for field falling-head tests to determine vertical hydraulic conductivity (K) using Darcy's equation. The design features an open-ended standpipe with two ports for simultaneous measurement of hydraulic heads at both ends of the sediment column, enabling direct estimation of flow rate (q) and hydraulic gradient (i). Flow rate is calculated by differentiating the best-fit curve for water level change above the sediment, and the hydraulic gradient is derived from the head difference between the ports. Laboratory and field tests consistently demonstrated a strong linear relationship between q and i (R2 > 0.999), validating the applicability of Darcy's equation for this new permeameter design. The K values obtained using the proposed method matched those obtained using the Hvorslev equation method. Furthermore, the design allows for continuous measurement of heads after the falling-head permeameter test, facilitating the collection of time series data for the hydraulic gradient. When combined with a pre-determined K value, this enables calculation of time series for the seepage rate across the surface water/sediment interface. To demonstrate this capability, preliminary tests were conducted using commercially-available pressure transducers to monitor heads and obtain seepage time series. The results of these tests are also presented.

Optimized Interaction Equations for More Efficient Design of CFS Channels under Combined Compression and Biaxial Bending

Optimized Interaction Equations for More Efficient Design of CFS Channels under Combined Compression and Biaxial Bending

Fourier-Based Function Generation of Four-Bar Linkages With an Improved Sampling Points Adjustment and Sylvester's Dialytic Elimination Method

Abstract Four-bar linkages are critical fundamental elements of many mechanical systems, and their design synthesis is often mathematically complicated with iterative numerical solutions. Analytical methods based on Fourier coefficients can circumvent these difficulties but have difficulties with sampling points adjustment and solutions of the design equations in previous studies. In this paper, an improved Fourier-based analytical synthesis method is presented, which transforms the function generation synthesis of planar four-bar linkages into a problem of solving design equations. Calculation of the Fourier coefficients is discussed, including the discretization of the prescribed function and an improved sampling points adjustment method. It is shown that the Fourier coefficients can be computed efficiently and accurately by discretizing the prescribed function with a small number of sampling points. The proposed sampling adjustment method overcomes the difficulty of easily resulting in non-Grashof solutions by considering the complete period of the prescribed function. An improved Sylvester's dialytic elimination method is presented to solve design equations. The method reduces the computation time and avoids cumbersome procedures without generating additional invalid solutions. Several examples are presented to demonstrate the advantages of the proposed synthesis method, which is easy-understanding and efficient, and yields more accurate solutions than available synthesis methods.

Designing Corrugated Board Properties to Mitigate Puncturing Impacts Occurring in E‐Commerce or Other Demanding Parcel Handling Distribution Chains

ABSTRACTThe growth of e‐commerce trade has led to more parcel shipments, causing an increase in corrugated board box packaging. Individually shipped boxes often sustain more damage, like crushing or puncturing, compared to pallet‐transported boxes. In contrast to the known relationships between containerboard properties and box compressive strength, there is no standard method for designing corrugated board against puncturing, which can result in damages, customer dissatisfaction and costly returns. A hole is typically seen as a severe compromise to package integrity, whereas some crushed boxes can be acceptable for nonfragile items. Addressing this, this research has focused on developing design equations linking containerboard and corrugated board properties to resistance against puncturing. These equations give the energy loss of an impacting object that would cause a hole and define the energy absorbance limit of the material. The equations enable material design aimed at minimizing puncturing risks based on specific distribution chain conditions, like fall heights and velocities.

Design of a compact MOSFET model suitable for geometric programming

ABSTRACT The design of analog integrated circuits is a demanding task that involves many constraints and objectives. Also, the transistor models employed must consider high-order physics effects to achieve accurate solutions. Geometric Programming (GP) allows finding the global minimum in optimisation problems, but requires design equations to be in monomial or posynomial form. This work proposes a simplified model inspired on the Advanced Compact MOSFET model tailored to be used in GP problems. The proposed model considers short-channel effects and it is valid in the saturation region with all inversion levels. The equations are presented, first showing the transistor current-voltage characteristic and then the associated capacitances. The validity of the equations is proven by their capacity to fit to the simulated data, achieving a mean error of 0.4% for the main NMOS characteristic. After that, the optimal design of two basic amplifier stages is performed to demonstrate the model usability with GP. The predicted voltage gain and bandwidth of the designed amplifiers are compared with simulations, presenting mean errors of 24.7% and 31.7%, respectively. These errors are low when viewed from a logarithmic perspective and considering the wide design space covered. As GP optimisation guarantees the global minimum location and does not require the usage of a circuit simulator in the loop, the proposed GP-friendly compact model can enable fast and accurate optimisation of analog integrated circuits.

Displacement prediction equations for seismic design of single friction pendulum base‐isolated structures

AbstractWe propose a set of equations for preliminary risk‐targeted seismic design of structures base‐isolated using single friction pendulum (SFP) bearings. The equations offer statistical estimates of the maximum displacement of the superstructure relative to the base and the maximum displacement of the isolated base response quantities of interest (RQIs), given ground motion intensity measures and the essential dynamic properties of the SFP bearing base‐isolated structure. The set of proposed equations enables preliminary seismic design of SFP base‐isolated structures using the mean annual frequency of exceeding displacement‐defined limit states related to the two RQIs. To develop the design equations, we define and use a two‐degree‐of‐freedom (2DOF) surrogate model that features a few essential dynamic parameters of the SFP bearing base‐isolated structure. We first verify that the 2DOF surrogate model is accurate enough to represent the base and superstructure displacements compared to the complete multiple‐degree‐of‐freedom model of the prototype SFP bearing base‐isolated structure. Next, we perform more than 5 million non‐linear dynamical analyses of 4900 distinct 2DOF surrogates subjected to 210 different recorded ground motions scaled five times. We fit the response data of each 2DOF surrogate to a linear model and use two different modeling techniques to derive the design equations. One model is based on Polynomial Chaos Expansion (PCE), and the other model is based on Linear Regression (LR). The PCE model is more accurate than the LR model, with the trade‐off regarding its simplicity. Nevertheless, both the PCE and the LR models are accurate enough to estimate the design quantities of interest of the SFP bearing base‐isolated structure for preliminary design purposes. Lastly, we exemplify the use of the proposed design equations and we show that the estimate of base displacement is conservative in the cases when the superstructure of the SPF bearing base‐isolated structure yields.

A two‐step sizing method for multistage Op Amps based on behavioral initial sizing followed by Spice‐in‐the‐loop refining

AbstractAnalog integrated circuit sizing is a laborious process that requires many times of iteration with Spice simulations. Even by applying the method, it still requires iterations and test assignment of device values (and biasings) in each iteration until reaching a satisfactory sizing result. When facing a design of multiple‐stage circuits (such as multiple‐stage operational amplifier [Op Amp]), sizing becomes more challenging due to the requirement on pole‐zero placement in order to achieve a better high‐frequency performance. Simulation‐in‐the‐loop method has been the dominating means taken by a great many of existing circuit sizer, but they suffer from intolerable runtime while lacking the possibility of offering insight or design knowledge acquisition. On the other hand, behavioral‐level synthesis methods, although intuitive and fast, suffer from accuracy loss due to the adoption of simplified device models and significantly condensed design equations. However, a proper combination of these two types of methods could lead to a lucrative research territory, yet it demands a methodological development for efficient deployment in practice, namely, implementation easy, less runtime, and guaranteed sizing quality. In this paper, we propose a two‐step method that takes the advantage of behavioral synthesis (in the first step) that is capable of fast and broader coverage of design space then makes correction (in the second step) on sizing refining by Spice simulation. Due to the profiling knowledge acquired on the design space in the first step, it can avoid blindness of the optimization landscape that is faced by many simulation‐centric methods which could easily get trapped in local optima. Conducted experiments over eight three‐stage Op Amps with different compensation configurations, including nested Miller capacitor (NMC), NMC with feedforward path (NMCF), NMC with feedforward path and nulling resistor (NMCFNR), NMC with nulling resistor (NMCNR), double pole‐zero cancellation (DPZC), transconductance with capacitances feedback compensation (TCFC), impedance adapting compensation (IAC), active feedback frequency compensation (AFFC), have validated that the proposed method can successfully generate qualified sizing results, offering an opportunity to compare fairly what merit a specific compensation strategy could possibly have.

Steel truss reinforced concrete composite walls: Numerical investigation and design consideration

Conventional design practices for reinforced concrete (RC) structural walls in the lower stories of super high-rise buildings involve employing large thickness and dense reinforcement to ensure satisfactory seismic performance. This is necessary to withstand the considerable gravity and seismic forces caused by the superstructure. However, this approach brings a design challenge in engineering practice because excessively thick RC walls not only occupy useable space but also magnify earthquake forces due to the increased self-weight. For this reason, steel truss RC (STRC) composite walls have been developed to address the above-mentioned shortcomings of conventional designs. This paper elucidates the working mechanism of STRC walls under cyclic loading, indicating that these two diagonal embedded steel elements mainly exhibit truss behavior. The working mechanism is further verified through computational models developed in OpenSees with high accuracy and efficiency. Moreover, the design provisions in the current specification are evaluated in terms of the flexural strength and shear strength of STRC walls. Results show that the flexural strength of the design equations for STRC walls in the current specification is underestimated because the contribution of steel trusses is not considered. The findings of this study offer valuable insights into the engineering application of STRC walls.

Web crippling design of cold‐formed ultra‐high strength steel lipped channels under ITF loading: A numerical parametric investigation

AbstractIn recent years, there has been a compelling need to adopt cold‐formed ultra‐high‐strength steel (CFUSS) in the construction industry owing to its numerous advantages, such as a higher strength‐to‐weight ratio, flexibility in achieving desired shapes, and adaptability over longer spans. Among the various applications, CFUSS lipped channel sections are commonly used as purlins and joists in steel structural systems. However, these sections are susceptible to different failure modes, particularly web crippling, which presents significant challenges. Currently, the current design rules lack specific guidelines for estimating the web crippling capacity of CFUSS sections. To address this crucial gap, the present study focuses on a comprehensive numerical investigation of the web crippling response of CFUSS lipped channel sections under interior‐two‐flange (ITF) loading conditions. Finite element (FE) models were developed using the ABAQUS package, verified against published test data, and subsequently used in an extensive parametric study. The ultimate web crippling capacity obtained from the parametric study was used to evaluate the accuracy of the current design equations in various design standards. The findings revealed that the existing design equations inadequately predicted the ultimate web crippling capacity of CFUSS lipped channel sections subjected to the ITF loading condition. Consequently, a modified design equation is proposed, utilizing the same approach as the current design standards, and a new direct strength method (DSM) approach is developed and verified through reliability analysis. The proposed modified design equations offer promising solutions to ensure safer and more reliable design practices for CFUSS structures in the construction industry.

Calibration of resistance factors for seismic design of masonry infilled frames using the random finite element method

This paper presents a reliability analysis of concrete masonry infill walls with an aim to examine the efficacy of the resistance factor specified in the infill design under seismic loading. A numerical model for masonry infilled reinforced concrete frames was developed and encoded in OpenSees for this study. The model was validated using experimental data. Two features were involved in this reliability study, including 1) that the effect of spatially varying masonry compressive strength f′m on the lateral resistance of masonry infills was considered using the Random Finite Element Method, and 2) that seismic load with different return periods was considered to provide a more complete treatment of a reliability analysis. The study selected six Canadian cities with varying levels of seismic intensity for this analysis. By calculating the probability of failure of infill frames subjected to earthquake loading at different locations, the paper showed that the current resistance factor of 0.6 in the infill design equation suggested by the Canadian masonry design standard is conservative. To achieve a reliability index of 3, this factor could be increased to 0.65 for all cities studied. For cities with low seismicity such as Toronto, the results suggest that an even higher resistance factor is justified.

An experimental study of cold-formed steel connections with screwed clip angle under tension and shear loads

This paper aims to investigate the performance of typical cold-formed steel (CFS) beam-to-beam connections between floor bearers and floor joists through a series of full-scale experiments. The investigated connections consisted of a lipped channel (floor joist), an unlipped channel (floor bearer), a thin-walled clip angle (connector), and a self-drilling screw (fastener). In this study, customised T-shaped and H-shaped CFS connection specimens were tested to examine the performance under tension and shear loads, respectively. Two sizes of self-drilling screws, 10 G and 14 G, were used with the selected load bearing clip angle. The fabricated connection specimens were subjected to displacement-controlled tension and shear loads. The experiments uncovered the ultimate connection strengths, load-displacement relationships, and failure mechanisms of the designed CFS connections under the applied loads. Moreover, the experimental tensile and shear strengths of the connections were compared with the strengths predicted by the standard design equations. It was found that the screw pullout on the anchored leg of the clip angle was the main mode of failure under the tensile load. On the other hand, two different failure modes—fracturing of 10 G screws or shear buckling of the cantilevered leg of the clip angle with 14 G screws—were observed in the H-shaped specimen under the applied shear load. Among the designed connection specimens, the maximum tensile load capacity was 30.86 kN and the maximum shear load capacity 88.15 kN. The standard design calculations predicted screw pullout failure under tension, while the shear bucking failure mode of the clip angle under shear load was overlooked.