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In this paper, unified shear deformation theory is used to analyze simply supported thick isotropic beams for the transverse displacement, axial bending stress, transverse shear stress and natural frequencies. This theory enables the selection of different in-plane displacement components to represent shear deformation effect. The numbers of unknow...
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Citations
... A lista de aplicações dos materiais compósitos na engenharia civil é muito vasta: Teorias de alta ordem Consideram funções de ordem superior na descrição do cisalhamento. Nessas teorias, as seções transversais não mais permanecem planas após a deformação, capturando efeitos como os de empenamento da seção [4]. ...
Esta cartilha de divulgação científica apresenta, em linguagem simples e acessível ao público geral, os avanços na modelagem matemática de materiais heterogêneos na engenharia civil. O trabalho é resultado da parceria internacional entre o Programa de Pós-graduação em Engenharia Civil (PROEC) da Universidade Federal de Sergipe (UFS) e o Instituto de Investigações em Matemática Aplicada e em Sistemas (IIMAS) da Universidade Nacional Autônoma do México (UNAM). O desenvolvimento foi fomentado pela UFS (edital Nº 13/2023 POSGRAP / CORI / AGITTE / COPES / COPGD), pelo Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq, Projeto Universal Processo No 402857/2021-6) e pela Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, PDPG Emergencial de Consolidação Estratégica dos PPGs stricto sensu acadêmicos com notas 3 e 4, Processo No 88887.711010/2022-00).
... Several formulations have been developed to describe the structural behavior of beams. The best-known and most straightforward theory is the classical Euler-Bernoulli (TEB) [1]. However, the TEB is suitable for modeling beams with a small height-to-length ratio because it does not consider shear strain. ...
Various industrial sectors require highly specialized and efficient materials for applications in fields such as the military, aeronautics, aerospace, and mechanical and civil engineering. Composite materials that meet the stringent requirements across these domains have become prominent, often serving as structural components and requiring precise mathematical modeling. Zigzag (ZZ) and Layerwise (LW) theories are commonly used for laminated-beam structural analysis. Although the LW theory provides superior accuracy, it suffers from an increase in unknowns as the number of layers grows. Conversely, the ZZ theory is less computationally intensive and less accurate. This study proposes an exponential high-order zigzag function with a unified kinematic formulation to enhance the accuracy of the ZZ theory. The results were compared with those of existing models and demonstrated excellent agreement with the reference solutions, irrespective of the layer count or slenderness index, making it a more efficient choice for laminated-beam analysis.
... Levinson [11] have presented a third-order shear deformation beam bending theory for rectangular thick beam. Ghugal [12], Sayyad [13], Ghugal and Shimpi [14], and Shimpi et al. [15] have studied refined beam theories for shear deformable rectangular beams made of isotropic, homogeneous materials. Their formulations were variationally consistent and satisfied shear stress-free conditions but failed to consider beam buckling. ...
... Several theories describe the structural behavior of beams, among which the best-known and most straightforward theory is the classical Euler-Bernoulli theory (EBT) [1]. Owing to the inability of EBT to account for shear strain, it is better suited for beams with a low height-to-length ratio. ...
Highly efficient materials and structures are becoming increasingly common in military, aeronautical, aerospace, mechanical, and civil engineering applications. Composite materials have been developed to address the need to combine two or more materials to achieve superior properties. Many structural elements, such as laminated beams, use composite materials, but an accurate mathematical model of the bending behavior is required due to the abrupt changes in material properties in the interlaminar zones. This accurate model can be achieved using zigzag theory. This theory is one of the most commonly used formulations for modeling laminated beams. This theory is an improvement of the equivalent single-layer theory as an additional term called the “zigzag function” is used to represent the variation in the axial displacement along the cross section. This paper proposes a novel high-order zigzag function in a sinusoidal format. Several higher-order beam theories are combined with the proposed functions, and their performances are compared with those of other functions in the literature. The results reveal excellent agreement between the proposed formulation and the reference solution as well as a more effective combination of zigzag functions and beam theory.
... Karama et al. [18] studied the mechanical behavior of laminated composite beams by the new multi-layered laminated composite structures model with transverse shear stress continuity. Sayyad [19,20] has carried out a comparison of various linear shear deformation theories for the free vibration analysis of thick isotropic beams. Study of the literature [9,10,19,21,22] indicates that the research work dealing with bending analysis of thick elastic beams using higher-order shear deformation theories is still in its early stage. ...
... Sayyad [19,20] has carried out a comparison of various linear shear deformation theories for the free vibration analysis of thick isotropic beams. Study of the literature [9,10,19,21,22] indicates that the research work dealing with bending analysis of thick elastic beams using higher-order shear deformation theories is still in its early stage. Furthermore, although various technical theories for piezoelectric beams can be found in the literature [1,2,[23][24][25][26], no systematic derivation of higher-order theories for static and dynamic analysis of piezoelectric beams is available. ...
... Based on the aforementioned assumptions, the displacement field [19] and the electric potential of the piezoelectric beam are given as below: ...
This paper explores the effects of shear deformation on piezoelectric materials and structures that often serve as substrate layers of multilayer composite sensors and actuators. Based on higher-order shear elastic deformation and electric potential distribution theories, a general mathematical model is derived. Governing equations and the associated boundary conditions for a piezoelectric beam are developed using a generalized Hamilton's principle. The static and dynamic behavior of the piezoelectric structure is investigated. A bending problem in static analysis and a free vibration problem in dynamic analysis are solved. The obtained results are in very good agreement with the results of the exact two dimensional solution available in the literature.
... Most traditional SSI methods are not able to observe correctly the parameters of a structure (such as bending stiffness) when shear deformation is not negligible. SSI methods based on SMM normally use elementary beam theory, underestimating deflections and overestimating the natural frequencies since the shear deformation effect is disregarded (Sayyad 2011). Timoshenko (1921) was the first one who included shear effects into the beam theory. ...
... Nevertheless, in some structures (such as deep beams, or thin-web bridges) shear effects might play an important role. SSI methods based on SMM normally use elementary beam theory, underestimating deflections and overestimating the natural frequencies since the shear deformation effect is disregarded [16]. When shear effects are significant, most of SSI methods based on SMM are not able to observe the correct values of parameters in structures (such as bending stiffness). ...
... where comma-subscript convention in Eqs. (10) and (11) represents the differentiation with respect to the normalized coordinate and 2 2 4 1 2 2 , , 12 ...
The dimensionless equations of motion are derived based on the Timoshenko beam theory to study the transverse vibration of beams without further usage of any approximate method. The exact closed form characteristic equations are given within the validity of the Timoshenko beam theory for beams having various boundary conditions. Accurate Eigen frequency parameters are presented for a different length to height ratio for each case. The exact closed form mode shapes related to deflection, slope due to bending and stress resultants are also presented and illustrated for some cases. The modal tests are performed for beams with clamped-Free and Free-Free boundary conditions. Finally, the effect of boundary conditions, length to height ratio on the eigenvalues parameters and vibratory behavior of each distinct case are studied. Validity of the derived closed form characteristic equations are checked through comparison of numerical solutions with the available results. It is believed that in the present work, the exact closed form characteristic equations and their associated Eigen functions, except for the beams with simply supported ends, for the rest of considered cases are obtained for the first time.
... Unified shear deformation theory was used by Sayyad [10] (2011) to analyze the simply supported thick isotropic beams for transverse displacement, axial bending stress, transverse shear stress, and natural frequency. A third order shear deformation theory is developed by Dahake et.al. ...
A fifth order shear deformation theory considering transverse shear deformation effect is presented for displacement in propped cantilever beam. The considered displacement field accounts for non-linear variation corresponding to depth of beam. Governing equations and boundary conditions of the theory are obtained using the standard of virtual work. Numerical results for static deflection analysis for aspect ratio (length to depth) 2 to 50 for isotropic beam is obtained. The results of present theory are in close agreement with those of higher order shear deformation theories.
... The limitations of elementary theory of beam and first order shear deformation theory led to the development of higher order shear deformation theories. There are many higher order theories available in the literature for beams [8]. ...
... Akavci studied buckling and free vibration analysis of laminated composite plates. Sayyad [8] has carried out a comparison of various linear shear deformation theories. Study of literature [4,5,7,8,9,10] indicates that the research work dealing with bending analysis of thick beams using higher-order shear deformation theories is still in early stage. ...
... Sayyad [8] has carried out a comparison of various linear shear deformation theories. Study of literature [4,5,7,8,9,10] indicates that the research work dealing with bending analysis of thick beams using higher-order shear deformation theories is still in early stage. In the present study, linear bending problems for piezoelectric beams are studied & grouped for six high order shear deformation technical theories. ...
This paper studies the effects of shear deformation on piezoelectric materials and structures. Based on a unified displacement field for higher-order shear deformation theories, a general mathematical model; governing equations and the associated boundary conditions for a piezoelectric beam is developed; using Hamilton's principle. Shear effects due to transverse and axial loadings on a piezoelectric beam have been investigated. Static and dynamic analysis of the piezoelectric structure has been studied for a group of six higher-order shear deformation engineering theories. Problems with different boundary conditions in static analysis are solved and the obtained results are compared to available results in the literature.
