Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

Fogang, Valentin

doi:10.20944/preprints202111.0327.v1

Cited by 3 publications

(5 citation statements)

References 7 publications

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“…Fogang [14] derived equations for the buckling analysis of isotropic plates, one of which was as follows (32a)   The results are in agreement with the structural behavior of the deep beam. The slender beam ( = 1/10; 1/15; 1/25; 1/35) can be regarded as a portal frame with supports connected with a tie.…”

Section: Buckling Of Deep Beamsmentioning

confidence: 59%

“…The FDA of the axial/shearing stresses are formulated using Equations (9d-f) and (12a-b) as follows (14) (15) The number of additional nodes associated to an edge node corresponds to the number of boundary conditions at the node of interest: therefore, three additional nodes are introduced at the angle node k and two additional nodes at the other edge nodes (e; ee; s; ss …).…”

Section: Displacement Potential Function Fda At An Interior Nodementioning

confidence: 99%

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Two-Dimensional Stress Analysis of Isotropic Deep Beams Using the Finite Difference Method

Fogang¹

2022

Preprint

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This paper presents an approach to the two-dimensional analysis of elastic isotropic deep beams using the finite difference method (FDM). Deep beams are subjected to in-plane loading and present a shear span to height ratio of less than 2.50; consequently, Euler-Bernoulli beam theory and Timoshenko beam theory do not apply. Deep beams analysis is generally conducted using numerical methods such as the finite element method and to a lesser extent the FDM; the strut-and-tie model and the stress field method are also widely utilized. Analytical approaches usually make use of the Airy stress function, where stresses are formulated in terms of the stress function; however, the exact solution of this function satisfying all of the boundary conditions can hardly be found, even for simple cases. In this paper, deep beams were analyzed using the FDM. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected nodes of the structure. Therefore, the deep beam was discretized with a two-dimensional grid, and additional nodes were introduced at the boundaries and at positions of discontinuity (openings, brutal change of material properties, non-uniform grid spacing), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at boundary nodes and satisfy the boundary and continuity conditions. An Airy stress function approach and a displacement potential function approach were considered in this study whereby strong formulations of equations (equilibrium, kinematic, and constitutive) were set. Stress and stability analyses were carried out with this model; furthermore, deep beams of varying stiffness, layered beams, and beams having openings were analyzed. For slender beams, the results obtained with the Airy stress function approach showed good agreement with those of the Euler-Bernoulli beam theory, and for deep beams, the shapes of stress distributions were in good agreement with a proper understanding of the behavior of structures. On the other hand, the displacement potential function approach delivered unsatisfactory results, probably due to the use of an inefficient equation solver; a more powerful tool will be needed in future research for this purpose.

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Section: Buckling Of Deep Beamsmentioning

confidence: 59%

Section: Displacement Potential Function Fda At An Interior Nodementioning

confidence: 99%

Two-Dimensional Stress Analysis of Isotropic Deep Beams Using the Finite Difference Method

Fogang¹

2022

Preprint

Self Cite

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show abstract

“…The FDA of the axial/shearing stresses are formulated using Equations (9d-f) and (12a-b) as follows (14) (15)…”

Section: Displacement Potential Function Fda At An Interior Nodementioning

confidence: 99%

“…Fogang [14] derived equations for the buckling analysis of isotropic plates, one of which was as follows Therefore, the axial/shearing forces per unit length at any node are determined according to the present study. Thereafter the buckling analysis of the deep beam, in fact a plate buckling analysis, is carried out using Fogang [14].…”

Section: Buckling Of Deep Beamsmentioning

confidence: 99%

Two-Dimensional Stress Analysis of Isotropic Deep Beams Using the Finite Difference Method

Fogang¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

This paper presents an approach to the two-dimensional analysis of elastic isotropic deep beams using the finite difference method (FDM). Deep beams are subjected to in-plane loading and present a shear span to height ratio less than 2.50; consequently, Euler-Bernoulli beam theory and Timoshenko beam theory do not apply. Deep beams analysis is generally conducted using numerical methods such as the finite element method and to a lesser extent the FDM; the strut-and-tie model and the stress field method are also widely utilized. Analytical approaches usually make use of the Airy stress function, where stresses are formulated in terms of the stress function; however, the exact solution of this function satisfying all of the boundary conditions can hardly be found, even for simple cases. In this paper, deep beams were analyzed using the FDM. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected nodes of the structure. Therefore, the deep beam was discretized with a two-dimensional grid, and additional nodes were introduced at the boundaries and at positions of discontinuity (openings, brutal change of material properties, non-uniform grid spacing), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at boundary nodes and to satisfy the boundary and continuity conditions. An Airy stress function approach and a displacement potential function approach were considered in this study whereby strong formulations of equations (equilibrium, kinematic, and constitutive) were set. Stress and stability analyses were carried out with this model; furthermore, deep beams of varying stiffness, layered beams, and beams having openings were analyzed. For slender beams, the results obtained with the Airy stress function approach showed good agreement with those of Euler-Bernoulli beam theory, and for deep beams, the shapes of stress distributions were in good agreement with a proper understanding of the behavior of structures. On the other hand, the displacement potential function approach delivered unsatisfactory results, probably due to the use of an inefficient equation solver; a more powerful tool will be needed in future research for this purpose.

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“…Application of the plate theory used in calculating the structure of the storage tank using the Kirchoff-Love theory (David et al, 2020;Fogang, 2021). The tank structure's plate is conceptualized as a thin shell consisting of various components, including the annular plate, shell plate, and roof plate.…”

Section: Introductionmentioning

confidence: 99%

Design of a 5000 Kl Crude Palm Oil (Cpo) Tank With Steel Plates

Atmadja,

Sutandar,

Aryanto

2024

JTS

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The study is centered on designing a 5000 KL Crude Palm Oil (CPO) Tank within Indonesia's palm oil industry, ensuring adherence to the API 650 13th Edition standard to effectively withstand operational and environmental loads. Plate theory, specifically the Kirchoff-Love theory, is employed to model structural components such as annular, shell, and roof plates, which must endure various loads including dead, internal (hydrostatic and pressure), external (wind, pressure, snow), live, and seismic loads as specified by API guidelines. Stress analysis is conducted to ensure compliance with material limits, utilizing ASTM A573M Grade 485 steel for plates and standard compressive strength for foundations.The research involves detailed phases encompassing data collection, preliminary design, structural modeling, loading analysis, stress assessment, weld design, and foundation planning utilizing SAP2000 software. Calculated stresses on tank plates remain well below API limits, confirming structural robustness. Welding details for butt and corner joints are carefully planned for durability, while deep pilings and pile caps provide stability against ground movements.By integrating empirical data and theoretical calculations, the study presents a comprehensive plan for constructing a dependable and cost-effective palm oil storage facility, thereby enhancing safety and efficiency within Indonesia's palm oil sector.

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Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

Cited by 3 publications

References 7 publications

Two-Dimensional Stress Analysis of Isotropic Deep Beams Using the Finite Difference Method

Two-Dimensional Stress Analysis of Isotropic Deep Beams Using the Finite Difference Method

Two-Dimensional Stress Analysis of Isotropic Deep Beams Using the Finite Difference Method

Design of a 5000 Kl Crude Palm Oil (Cpo) Tank With Steel Plates

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