Load Formula Research Articles

Multi-helix buckling of a slender compression rod constrained by a cylinder

As early as the mid-18th century, Euler proposed a critical load formula for compression rods with multiple half-sine wave modes. However, a critical load formula for compression rods constrained by cylinders with multi-helix buckling modes has not yet been proposed. In this paper, a mechanical model of multi-helix buckling of a slender compression constrained by a cylinder is established. Utilizing the dynamic relaxation method, a finite element solution strategy is presented. Its notable advantage lies in its capability to compute stable configurations of multiple helical buckling, effectively addressing challenges such as contact identification and convergence issues during computation. Furthermore, parameter analysis quantifies the influence of length, diameter, and annular gap on the critical load, while also analyzes the shear force, bending moment, and contact force after buckling. This paper presents formulas for the critical load of multi-helix buckling and the length of the multi-helix buckling modes, which is suitable for various length, diameter, and annular space gap conditions, and partially validates these results against finite element analysis.

Study on Influencing Factors and Prediction of Tunnel Floor Heave in Gently Inclined Thin-Layered Rock Mass

In recent years, the construction of new railway tunnels worldwide has become increasingly challenging due to larger cross-sections, deeper burial depths, higher in situ stress, and more complex geological conditions. During both construction and operation, some tunnels have encountered significant issues with floor heave. This paper begins by identifying the primary causes of deformation and instability in tunnel floor structures through an investigation and statistical analysis. It then examines floor heave across more than 20 railway lines, summarizing the types, generation mechanisms, and mechanical models associated with this issue. Additionally, extensive survey data indicate that tunnel floor heave is most likely to occur in gently inclined thin-layered rock masses. Therefore, using a tunnel passing through the plate suture zone in such a rock mass as a case study, numerical simulations, theoretical analyses, and on-site monitoring were conducted. This study systematically analyzed the influence of single and multiple factors, as well as the mechanical behavior of the support system, on tunnel floor heave in gently inclined thin-layered surrounding rock. Furthermore, several key models were proposed: a tunnel floor heave estimation and load formula based on a mechanical model, a dynamic relationship between surrounding rock support force and tunnel floor heave using the Nishihara model, a tunnel floor settlement estimation formula based on deformation statistics, and a tunnel floor heave energy prediction model utilizing the B-P neural network algorithm. These conclusions have been validated and widely applied in practical engineering, providing a robust theoretical foundation and technical support for future tunnel construction.

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.

Stability capacity design of shuttle-shaped restrained-buckling braces with uniform cross-section at mid-span

This paper proposes a shuttle-shaped buckling restraint brace (SS-BRB) with the uniform core and the peripheral restraint system that consists of uniform cross-section tube at its mid span and tapering sections at its two ends. This type of BRB maned as SS-BRB-UCS adopts circular steel tubes in the core and the peripheral restraint system. An isolation system is strategically positioned between the core and the peripheral restraint system to effectively transmit lateral loads. Obviously, this type of the optimizing peripheral restraint system could reduce material utilization with an economic design and beautiful appearance. First, the finite element (FE) model of the SS-BRB-UCS is established, then the elastic buckling performance investigation is carried out to determine the elastic buckling load formula through fitting techniques of numerical results, so as to obtain the formula of the restraint ratio. Additionally, an elastic-plastic stability bearing capacity analysis is conducted for the SS-BRB-UCS to assess its load-bearing capabilities, and the corresponding relationships between stability coefficient (φ) and the restraint ratio (ζ) is established to determine the lower restraint ratio (namely threshold) for static load-bearing capacity BRB design. Finally, the hysteretic responses of SS-BRB-UCSs subjected to axial compressive-tensile cyclic loads are studied numerically, and the corresponding lower limit of restraining ratio of SS-BRB-UCSs is proposed for their energy-dissipation design. Those two lower limits of restraining ratios of the SS-BRB-UCSs obtained in this study provide fundamentals for preliminary static and seismic designs of SS-BRB-UCSs.

Critical load for shear-induced lateral-distortional buckling in I-section steel beams considering flexural stiffness ratio

In prefabricated buildings, lateral-distortional buckling (LDB) of I-section steel beams is a common failure mode that has garnered considerable attention. However, existing studies and design methods for lateral-distortional buckling primarily focus on members subjected to in-plane flexure and do not take into account the influence of shear forces. In this paper, a study on the elastic lateral-distortional buckling behavior of I-section steel beams under in-plane shear loads was conducted. Firstly, refined finite element (FE) models of beam webs under shear loads were established and validated against theoretical formulas for different boundary conditions. Subsequently, by combining with analytical solutions and numerical results, an accurate buckling load formula for the beam web with a free bottom edge was obtained, which can be used for further study. Based on the parametric analysis results, a formula was obtained to determine the critical flexural stiffness ratio of flange to web. When the flexural stiffness ratio exceeds the critical value, the lateral-distortional buckling of I-section steel beams can be effectively avoided. Finally, a formula that incorporates both the critical flexural stiffness ratio and the span to height ratio was proposed to calculate the elastic critical lateral-distortional buckling loads of I-section steel beams solely under in-plane shear loads.

Numerical investigation of dam break flow over erodible beds with diverse substrate level variations

Abstract This study aimed to comprehensively investigate the influence of substrate level difference and material composition on dam break wave evolution over two different erodible beds. Utilizing the Volume of Fluid (VOF) method, we tracked free surface advection and reproduced wave evolution using experimental data from the literature. For model validation, a comprehensive sensitivity analysis encompassed mesh resolution, turbulence simulation methods, and bed load transport equations. The implementation of Large Eddy Simulation (LES), non-equilibrium sediment flux, and van Rijn’s (1984) bed load formula yielded higher accuracy compared to alternative approaches. The findings emphasize the significant effect of substrate level difference and material composition on dam break morphodynamic characteristics. Decreasing substrate level disparity led to reduced flow velocity, wavefront progression, free surface height, substrate erosion, and other pertinent parameters. Initial air entrapment proved substantial at the wavefront, illustrating pronounced air-water interaction along the bottom interface. The Shields parameter experienced a one-third reduction as substrate level difference quadrupled, with the highest near-bed concentration observed at the wavefront. This research provides fresh insights into the complex interplay of factors governing dam break wave propagation and morphological changes, advancing our comprehension of this intricate phenomenon.

Cable tension formulas for gravity net cage arrays in current

Due to the structural complexity and size, many gravity net cage array prototypes are modeled, simulated, and analyzed to select the potential mooring system. However, the complexity of these systems requires a significant amount of time to configure and compute the mooring dynamics in the preliminary design stage. Because the net cage array reaches an equilibrium state in a specific current condition, the cable tension is static and predictable for a given mooring system configuration. Therefore, based on a quasi-static method, load formulas that resolve cable tension are derived to reflect relationships among the cable tension, net cage array size, mooring system configuration, and current conditions. Compared to the tension obtained from numerical simulations, the cable tension predicted by the formulas has a 67% probability of error within ±5%, and a 96% probability of error within ±10%, respectively. Additionally, the formula characteristics also provide valuable information for the mooring system design. The mooring system of a 2 × 4 net cage array is designed according to the derived cable tension formulas. This research provides an efficient and rapid method for preliminary design and sizing components for a mooring system of gravity net cage array.

Calculation of ultimate bearing capacity of concrete-filled double steel tubular under compression based on the unified strength theory

Concrete-Filled Steel Tube (CFST) components are widely used in the engineering field due to their superior performance. However, the actual stress state of CFST components under axial compression is complex. As a result, calculations for CFST components are often simplified using material equivalent assumption formulas and empirical equations to determine their load-bearing capacity. These simplifications require the introduction of many coefficients and mechanical performance indicators, and the actual material principal stresses are not adequately considered. This approach weakens the completeness of the theory and adds to the burden of experimental research.The aim of this work is to develop a load-bearing capacity calculation formula for Concrete-Filled Double Steel Tubular (CFDST) components that considers the different stress states of the inner and outer steel tubes, the core concrete, and their interface interactions. This formula is designed to be convenient for engineers to understand and use. By comparing the results of this formula with experimental data collected from domestic and international scholars, it is evident that the results of this formula closely align with the collected experimental data. This validation confirms the accuracy and effectiveness of the formula developed in this paper. Furthermore, when compared with existing axial compression load formulas for CFDST, the formula presented in this paper demonstrates good accuracy. The research results provide a convenient and accurate calculation formula for the axial compression load-bearing capacity of CFDST components, bridging the gap between theoretical derivation and practical application.

Extraction, characterization and controlled release application of pectin-like/chitosan hydrogels system loaded Ciprofloxacin.

In recent decades, drug delivery applications have extensively utilized hydrogel systems based on natural polymers. Among the numerous biopolymer-based hydrogel drug delivery systems reported, a novel pectin-like substance was extracted from fig leaves and copolymerized with chitosan. The hydrogel was reformed into microspheres using glutaraldehyde (chemical cross-linker) and sodium hexametaphosphate (physical cross-linker). The extracted polysaccharide and the prepared hydrogels were characterized by FTIR, GC/MS, SEC/MALS/DRI as well as XRD, SEM, BET, and thermal analysis. SEM images revealed the formation of porous microspheres with an average size of 50 μm in diameter. Degrees of swelling in pH7 at 35°C have shown the hydrogels reached two to three times their weights. This has been reflected in their ability to load drugs or any other chemicals. The loading formula shows that hydrogels have maximum loading efficiency more than one-third of the weight of hydrogel. The antimicrobial ciprofloxacin was used as a model for loading on prepared hydrogels. The loaded hydrogels were tested for their biological activities against staphylococcus aureus (S. aureus) bacteria. The antimicrobial growth inhibition zone of the cultured (S. aureus) by ciprofloxacin-loaded hydrogel was followed, which shows controlled growth in inhibition zone sizes and for long time intervals. Results showed that the pectin-chitosan hydrogels exhibited significant antibacterial activity against gram - positive bacteria (S. aureus), with an inhibition zone of 45 mm for (CH-co-FLP)/GLU hydrogel. In vitro, the ciprofloxacin-loaded hydrogels were studied and the cumulative release of ciprofloxacin under suitable conditions was found in a controlled manner and kept release for a long time interval. Data exhibited that the cumulative release profile of ciprofloxacin from the hydrogel demonstrated sustained release over 48 hours, with a value of 6.9% released within the first 24 hours and 7.0 and 6.9% % released at the end of the study for the (CH-co-FLP)/GLU and (CH-co-FLP)/SMP hydrogels, respectively. The novel pectin-chitosan hydrogels hold the potential to enhance the quality of life for numerous patients by minimizing the need for frequent intake of chronic medications.

Exploring the effect of building openings and orientation on induced loads due to extreme flood events

Understanding the impact of building features on flood-induced loads remains a major challenge. This study experimentally investigated the effect of building openings and orientation on horizontal forces and overturning moments by unsteady extreme floods. The physical model experiments for flood impacts were conducted concerning three intensity flood events (Hi), four porosities (P) resulting by opening, and seven orientations (γ) by building rotation. Measurement of impact forces and moments on the building model was carried out to explore the flood impacts on structures under unsteady flow conditions. By introducing a reduction coefficient of maximum horizontal force ηF, the influence of the porosity and orientation on flood loads was discussed. Results showed that either the building openings or the orientation exhibit a reduction effect on the resulting force and moment, and the magnitude of the reduction is negatively related to the flood intensity. For the permeable building with oriented exposures, the reduction coefficient ηF showed a W-shaped variation trend concerning γ at certain P, exhibiting a variable behavior of force reduction and increase with changes in P and γ. Finally, the load formulas, including maximum horizontal force and maximum overturning moment, were modified by quantifying the effects of openings and orientation, supporting hydraulic engineers for the load design of building structures.

In-Plane Failure Mechanism and Strength Design of Plate-Tube-Connected Circular Steel Arches

The in-plane elastoplastic failure mechanism of plate-tube-connected steel circular arches with inverted triangular cross sections is investigated in this study by using theoretical derivation and numerical simulation. First, the in-plane elastic buckling load formula of the arch under full-span uniform radial load (FSURL) is presented. Then, the limited conditions of avoiding the connecting plate and chord local failure before global elastic instability are derived. Lastly, the elastic–plastic failure mechanisms of arches are studied under FSURL, full-span uniform vertical load (FSUVL), and half-span uniform vertical load (HSUVL). It is found that the arch will experience global failure, chord local failure, combined connecting plate and chord failure, and connecting plate local failure under FSUVL and HSUVL. The failure mode is mainly related to the stiffness of the connecting plate. The corresponding design formulas are proposed for the global failure of arches and local failure of the chord. The proposed formulas and FE results are in good agreement.

Compressive behaviour of stiffened thin-walled CFDST columns with large hollow ratio

Concrete-filled double skin steel tubular columns with large hollow ratio (LHR-CFDST) have been commonly used in recent years. With the increasing size of the engineering structures, one efficient way to decrease the consumption of steel is to increase the diameter-to-thickness ratio. Few studies have been conducted to report on thin-walled LHR-CFDST columns. At the same time, local buckling may occur in these structures in the absence of stiffening measures. Based on this, an experimental and numerical analysis was carried out for the stiffened thin-walled LHR-CFDST columns under axial compression in this paper. Five thin-walled LHR-CFDST columns with different stiffening measures (ribs, studs) were tested. The failure modes, load versus shortening curves, local buckling behaviours and ductility were investigated. The test results indicates that the stiffened measures effectively delayed the appearance of local buckling and increased the peak load. Besides, the finite element models validated by the test results were applied to the subsequent mechanical mechanism analysis of the stiffened thin-walled LHR-CFDST. It can be seen that the stiffened measures enhance the confinement of the inner and outer steel tubes on the concrete and greatly improve the ultimate strength of the concrete. Finally, the predicted peak load formulae in the existing design specifications were modified by combining the characteristics of the specimens in this paper. The peak load values obtained by the improved formulae agreed well with the test results.

In‐plane stability and shear deformation analysis of the H‐beam hollow arch

SummaryH‐shaped circular arc is a relatively novel type of open‐web steel arch, and currently, no reports have been published concerning its in‐plane stability. In this paper, the elastic and elastic–plastic in‐plane stability of the H‐shaped hollow circular arch is studied by theoretical deduction combined with numerical simulation. First, the overall shear rigidity of the H‐shaped circular arch is calculated, and the elastic buckling load formula of the arch is proposed and verified considering double shear deformation under full‐span radial and uniform loading. The overall elastic buckling load deduced in this paper is reasonable according to the finite element analysis. The results indicate that the influence of shear deformation on the overall elastic buckling load of the arch decreases with the increase of the span length. The arch‐bearing capacity is the largest when the rise‐span ratio is 0.25. Second, the restriction conditions necessary for avoiding local buckling of the chordal web before integral buckling of the H‐shaped steel hollow circular arch are analyzed. Finally, the elastic–plastic failure mechanism of the H‐shaped arch under full‐span radial and uniform loading is examined, and the formula for determining the ultimate bearing capacity that is achievable before failure under full‐span radial and uniform loading is proposed. ANSYS analysis shows that under the radial uniform loading, the chordal bars will yield near 1/4L and 3/4L, and ultimately, the structural failure of the lower chord occurs in the vicinity of 1/4L. The formulas presented in this paper agree well with the results obtained from the finite element analysis and can be used as a reference for engineering applications.

A Trilinear Model for the Load-Slip Behavior of Headed Stud Shear Connectors.

Headed stud shear connectors are most broadly applied in various composite structures. There exist plenty of empirical formulae for load-slip curves. However, most of them are fitting formulae in particular forms. Due to the lack of physical model support, fitting empirical formulae apply only to cases with similar parameters to the tests. Therefore, this paper analyzes the load-slip curves of existing headed stud connectors, proposes three stages of slip deformation in the shear connectors and the corresponding trilinear model, and presents the analytical formulae for the stiffness and strength of headed stud shear connectors. Firstly, we model the headed studs and surrounding concrete as beams on the foundation model, derive the equivalent shear stiffness equations for headed studs, and establish the load-slip behaviors for the first two stages. Then, the connectors' shear stiffness and shear strength in the third stage are derived based on the head stud's plastic deformation characteristics and failure mode. Finally, the numerical results are presented and verified with the existing test results, showing that the trilinear model is conceptually straightforward, easy to apply, and has sufficient accuracy.

Testing Bed Load Transport Formulas: A Case Study Of The Lower Amur Using Bed Load Yield Data Obtained With Multi-Beam Echo-Sounders (Mbes)

The development of bed load calculation methods directly depends on the reliability of the measurement data. The most reliable measurement data remains the data obtained by the volumetric method when observing the filling of reservoirs, borrows, ditches etc. Nevertheless these data are the rarest. In this paper on the base of the data obtained when observing the process of filling of a ditch across the Amur River a comparison of a number of bed load calculation methods is performed. The observations were carried out with a multi-beam echo-sounder during summer floods of 2018, from 21st of July to 22nd of August. Over this time 5 surveys were performed, that allows to have 4 calculation periods for determining bed load yield. The total number of the measurements at different calculation verticals is 108. These data are used for verification of 80 bed load formulas. Four methodological approaches are considered: bed form approach, critical velocity approach, critical water discharge approach and regression approach. The bed form approach has shown the greatest accuracy: 17 formulas out of 26 gave the error less than 60%. For the other 56 methods which were considered only 5 formulas showed the error less than 60%, all of them correspond to the critical velocity approach.

EXPERIMENTAL REPORT ON GLASS PANELS AGAINST STATIC AND REPEATED IN-PLANE COMPRESSION LOADING

Primary load-bearing members used in glass structures, such as columns consisting of glass panels, are frequently subjected to in-plane loading. Its allowable stress for safety design appears to be a significant subject. By analyzing the results of static (without repeated loading) and cycling compression loading tests, the authors explore the mechanical features and failure mechanism of the tempered glass panels subjective to in-plane compression loading, including its ultimate stress and buckling load. In addition, a statistical method based on the experimental results is adopted to evaluate the allowable compressive stress. The experimental results show that the ultimate loads of specimens with high slenderness ratios satisfy Euler's critical load formula. In contrast, the ultimate loads of specimens with low thickness ratios distribute in a certain variance range. For the cycling loading test, after 30 times of repeated loading with maximum stress equivalent to 2/3 of the ultimate static stress, the average strength is almost the same or bigger than that of static loading tests. No fatigue fracture due to repeated loading was found for the tempered glass panels. Consequently, the allowable stress for in-plane compressive loading can be formulated based on Euler's critical load formula for a glass panel with a bigger slenderness ratio. The formulation for the allowable compressive stress of a glass panel with a smaller slenderness ratio is promoted using statistical analysis with a 95 percent confidence level for the safety design.

Structural stability analysis and deformation control of constraint-anchorage support system in soft rock mass tunnel

The primary support of the soft rock mass tunnel is prone to lose its bearing capacity in advance due to the buckling instability. To solve it, the buckling critical load formula was derived in combination with structural stability theory. It is found that the stability can be improved by strengthening or increasing the constraint conditions for the structure. Consequently, a constraint-anchorage support system was proposed inspired by this conclusion. It provides effective constraints for the primary support by anchoring the pipes in deep rock mass with grouting high-performance material, and connecting with the primary support by I-beam connector. Reasonable support parameters were presented by analyzing the influence of support parameters on the deformation and stress of the tunnel structure. Finally, this support system was applied to the actual the tunnel construction. It was found that the stability of the primary support was improved and the tunnel deformation was effectively controlled.

Comparative study on overall and local static performance of steel open‐web girders

SummaryTo quantify the mechanical properties of five types of steel open‐web girders that have been tested or applied in practical engineering, a systematic study of the overall and local mechanical properties of steel open‐web girders based on the same dimensions and material conditions was presented. An overall study of the quasi‐intersection beam method for shear deformation was introduced for 8 × 8 grid open‐web girders to derive the determinants of the load distribution factor. Based on the load distribution factors, the essence and formula of any shear key node domain (SKND) were derived theoretically. The reliability of the numerical model was verified by tests on the H‐type unidirectional SKNDs, which led to a comparative load–displacement study of unidirectional and bidirectional SKNDs. Quantitative results of the overall flexural capacity and SKND load capacity of the H‐type, TH‐type, TS‐type, T‐type, and DT‐type open‐web girders were obtained. The maximum shear deformation in the elastic phase of the five open‐web girders accounted for 39.04% to 54.76% of the overall deformation. In this study, the adjustment factors of the modeling analysis based on the current practical design of equivalent dense‐ribbed solid beams (DRSBs) were revised. A method of SKND's design load capacity under the action of bidirectional forces was put forward for the first time.

Prediction of transverse stability of composite micro-curvature plates

This article is focused on the instability of marine composite micro-curvature plates subjected to transverse load, which is different from the axial loading case. The jump instability mechanism of orthogonal laminated plates has been analyzed, and transverse stability is predicted through the modified formula. Based on the classical elastic stability theory, suitable transverse stability analysis models for cylindrical and spherical shells are proposed. The instability load of the plate is predicted theoretically. The influential factors of the instability load are methodically analyzed based on an orthogonal experiment. Finally, according to the correlation analysis results of design variables, the calculation formula of the instability load is appropriately modified by considering the chord length of the plate. The obtained results show that the thickness has the most significant effect on the instability load, followed by the radius and the chord length. The instability load formula based on the elastic stability theory does not consider the chord length, which leads to low accurate results for the case of unclosed structures such as spherical crown plates. A comparison study confirms that the predicted results by the modified formula are generally closer to the experimentally observed data for the unclosed micro-curvature plate. In the present investigation, the load-bearing capacity of plates is evaluated in conjunction with the theory, nonlinear numerical simulation, and experimental research. Further, it can be applied to the theoretical prediction of the instability load for composite micro-curvature plates.

Targeted Model Error Determination for Limit Load Formulas for Axial Surface Defect in a Pressurized Pipe

Abstract There are dozens of formulas in literature and standards for residual strength calculation of the pressurized ductile pipes with axial rectangular defect. Their accuracy is usually characterized by the so-called “model error” notion. The latter is established as the averaged deviance of experimental data from the calculated results. In fact, the model error determined in such way is partly related to the validity and range of the chosen experimental data. The idea of our work is to introduce the notion of “targeted model error.” It is expected to characterize the accuracy of given formula for the particular defect only with given ratio of its dimensions, which is named as the “point of interest.” So, every experiment can give the contribution to the targeted model error according to its weight on the point of interest. These weights are subjectively established based on the difference between the dimensionless residual strengths calculated for the point of interest and for the real defect dimensions of the considered experiment. Practical examples of determination of the targeted model errors are performed for the ratios of the defect depth to wall thickness equal to 0.5, 0.6, 0.7, and 0.8 for three well-known formulas—ASME, PCORRC, and O-formula.