No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Stability Analysis of Long Composite Plates with Restrained Edges Subjected to Shear and Linearly Varying Loads
Abstract
Explicit expressions are developed for the buckling analyses of rectangular (long) plates: for ‘linearly varying axial load’ the known results for hinged supports are corrected and new results are presented for built-in and constrained edges; for ‘shear load’ new results are presented for constrained edges; for ‘uniform compression’ new results are presented when the longitudinal edges are rotationally constrained by stiffeners. The results are based on the Rayleigh—Ritz method.
... Later, researchers have expanded this methodology for orthotropic material. Plates subjected to uni-axial compression load, shear load, in-plane linearly varying load and their combination are studied by Lekhnitskii (1968), Barbero (1993), Whitney (1987), Bank and Yin (1996), Qiao et al. (2001), Kollar (2002Kollar ( , 2003, Zou (2002, 2003), Qiao and Shan (2005), Tarjan et al. (2010a), Qiao and Huo (2011). Orthotropic panels with simple boundary un-loaded edges (free, simply-supported, and clamped) are studied by Lekhnitskii (1968), Whitney (1987), and Barbero and Raftoyiannis (1993). ...
... Qiao and Zou (2003) have studied plate subjected to bending stresses with consideration of rotational restraint. However, Tarjan et al. (2010a) have reported that these equations predict infinite buckling load for a few cases, which is unrealistic. Tarjan et al. (2010a) have developed the closed-form expression for discrete plates subjected to bending, and shear stresses with consideration of rotational restraint along the un-loaded edges. ...
... However, Tarjan et al. (2010a) have reported that these equations predict infinite buckling load for a few cases, which is unrealistic. Tarjan et al. (2010a) have developed the closed-form expression for discrete plates subjected to bending, and shear stresses with consideration of rotational restraint along the un-loaded edges. Additionally, they tabulated the available equations for the orthotropic plate. ...
The lightweight superstructure is beneficial for bridges in remote areas and emergency erection. In such weight-sensitive applications, the combination of fiber-reinforced polymer (FRP) as a material and box-beams as a structural system have enormous scope. This combination offers various advantages, but as a thin-walled structure, their designs are often governed by buckling criteria. FRP box-beams lose their stability either by flange or web buckling mode. In this paper, the web buckling behavior of simply supported FRP box-beam subjected to transverse load has been studied by modeling full box-beam to consider the effect of real state of stress (stress variation in length direction) and boundary conditions (rotational restraint at web-flange junction). A parametric study by varying the sectional geometry and fiber orientation is carried out by using ANSYS software. The accuracy of the FE models was ensured by verifying them against the available results provided in the literature. With the help of developed database the influential parameters (i.e., αs, βw, δw and γ) affecting the web bucklings are identified. Design trends have been developed which will be helpful to the designers in the preliminary stage. Finally, the importance of governing parameters and design trends are demonstrated through pedestrian bridge design.
... The material properties are taken as uncertain-butbounded quantities. Long laminates under buckling loads, which have applications in aerospace industry, have been studied in a number of studies [12][13][14][15][16][17]. Local buckling of the webs of box beams can also be studied using the results based on long plates [18][19][20][21][22]. ...
... where ij D is the uncertain bending stiffness where the overhead ~ sign indicates an uncertain quantity. Equation (1) has been derived in [15,18,21] based on Rayleigh-Ritz approximation of the buckling loads of rectangular plates with simply supported and clamped boundary conditions. The results for the long plates are obtained from the corresponding results for rectangular plates by computing the buckling lengths corresponding to the lowest buckling loads. ...
... The results for the long plates are obtained from the corresponding results for rectangular plates by computing the buckling lengths corresponding to the lowest buckling loads. The resulting expressions were found to be very accurate when verified using finite element analysis [15,18,22]. [15,18,21]. ...
Elastic constants of composites are often determined with some level of uncertainty. In the present study minimum cost design problem for long cross-ply plates of hybrid construction is solved subject to material uncertainties. By using minimum amount of more expensive composite in outer layers and less expensive composite in inner layers, a more efficient and cost-effective design is obtained. The effect of hybridization is studied for uniformly and linearly varying buckling loads.
... An exact solution was also presented for buckling of simply supported symmetrical cross-ply rectangular plates under linearly varying loads [20]. The closed-form analysis of the buckling behavior of orthotropic plates with elastic clamping and edge reinforcement under uniform compressive load was presented by Weissgraeber et al. [21], and the restraint coefficient is different from those given in [9,22,23]. ...
... By simulating the buckled displacement shape function as a trigonometric series [1], the problem of buckling of plates with two opposite edges simply supported and the other two edges elastically-supported and under uniform in-plane shear was numerically solved. Using the same buckled shape function, the numerical solutions for the buckling problem of long plates under linearly varying axial load and inplane shear were obtained, and the simple expressions of local buckling loads were obtained by curve-fitting the numerical solutions [22]. The simple expressions developed were then used to predict the web buckling of composite structural shapes under various transverse loads [23]. ...
... When in the process of calculating the buckling loads, it is found that the plate buckled shape functions adopted in most of the existing studies [1,7,22] only satisfy the essential boundary conditions but not the natural (force) boundary conditions which are not necessary. Therefore, the convergence rate of the solution is slow, and it has to adopt more terms to obtain a reasonably accurate numerical solution. ...
... In recent two decades, different analytical and numerical tools, such as the Rayleigh-Ritz [4][5][6][7][8][9][10][11], Galerkin [12], finite element [13][14][15], and finite strip [16,17] methods, have been successfully used to solve the buckling problem of plates under in-plane shear loading. Based on the Rayleigh-Ritz method, the approximate formulas for the buckling of angle-ply and cross-ply laminated plates subjected to combined edge compression and shear with the four edges simply supported were presented by Kumar and Kishore [4]. ...
... Using the sine functions up to an order of eight for summation as the shape functions, both the symmetric and antisymmetric buckling modes were discussed. Numerical solutions for the local buckling of orthotropic plates under uniform compression, linearly varying compression or shear load were obtained by Tarjan et al. [11] using the Rayleigh-Ritz method. The displacement was simulated by a trigonometric series with 71 terms and 41 terms in the width and longitudinal directions, respectively. ...
... To aid the explicit analysis of the present problem, a symbolic Maple program to compute the critical in-plane shear buckling load, skewed angle, and aspect ratio is provided in Appendix B. The explicit formulas in Eq. (12) and calculation procedures in Eq. (13) are next verified by comparing the results with the solutions obtained by the existing available methods [7,11] and numerical finite element analysis. ...
... The above solutions are based on the assumption of r = 0, and assuming the governing equation (r = 0) has the same lateral buckling mode function with the governing equation ( (Table 4). For the foundation stiffness factor k r = 0, the current solutions are compared with the previously published results (Timoshenko and Gere 1961, Bulson 1970, Nölke 1937, Leissa and Kang 2002, Kang and Leissa 2005, Weaver and Nemeth 2007, Tarján et al. 2009, Liu et al. 2014. Timoshenko and Gere (1961) presented the buckling coefficients of a simply supported plate using the energy method for α = 0, α = 1 and α = 2. Nölke (1937) illustrated the buckling coefficients of a clamped plate using the energy method for α = 1 and α = 2. Bulson (1970) summarized the buckling coefficients of square plates with both clamped and simply supported edges for α = 0, α = 1 and α = 2. ...
... The exact solutions of the plate were obtained using the power series method for α = 0, α = 1 and α = 2 Kang 2002, Kang andLeissa 2005). For α = 0 and α = 2, the buckling coefficients of the composite plate with both clamped and simply supported edges were analysed (isotropic plate was the special case) (Weaver and Nemeth 2007, Tarján et al. 2009, Liu et al. 2014. As shown in Table 4, good agreement was obtained between the current method and the previous studies. ...
An analytical method is developed for analysing the contact buckling response of infinitely long, thin corrugated plates and flat plates restrained by a Winkler tensionless foundation and subjected to linearly varying in-plane loadings, where the corrugated plates are modelled as orthotropic plates and the flat plates are modelled as isotropic plates. The critical step in the presented method is the explicit expression for the lateral buckling mode function, which is derived through using the energy method. Simply supported and clamped edges conditions on the unloaded edges are considered in this study. The acquired lateral deflection function is applied to the governing buckling equations to eliminate the lateral variable. Considering the boundary conditions and continuity conditions at the border line between the contact and non-contact zones, the buckling coefficients and the corresponding buckling modes are found. The analytical solution to the buckling coefficients is also expressed through a fitted approximate formula in terms of foundation stiffness, which is verified through previous studies and finite element (FE) method.
... 低，而且通常做成薄壁结构形式 [2][3] ，结构通常在 达到材料强度破坏之前发生局部屈曲，从而使构件 发生大变形，阻碍结构某些功能的正常发挥，或者 直接导致结构的破坏。 在复合材料型材的局部屈曲分析中，通常把构 成型材的各个板件假定为正交各向异性板，并假定 各个板件的连接处为直线 [3][4] 。确定屈曲载荷的方 法主要有以下两种 [5] ：1) 精确法，即假定结构的各 个板件同时屈曲，板件连接处满足连续条件；2) 近 似法，即把结构中各个板件单独处理，考虑相邻板 件之间的弹性约束，也称离散板分析。本文采用第 2)种方法。 1930 年，Seydel [6] 利用静力法，求得了正交各 向异性无限长板长边为简支或固支时的剪切屈曲 精确解。1971 年，Johns [7] 总结前人成果，对各向同 性和正交各向异性板的剪切屈曲问题进行了总结。 Nemeth [8][9][10] 对复合材料长板的屈曲问题进行了详 尽的分析，并进行了各种参数研究，但纵向边界均 是简支或固支的。 在均匀压力作用下， Qiao 等 [2,[11][12] 对边界为转动弹簧约束板的屈曲问题下进行了严 格的理论分析，并得到了显式解。考虑弯扭耦合效 应， Herencia 和 Weaver [13] 求得复合材料板在均匀压 力作用下的屈曲显式解。 利用 Rayleigh-Ritz 法， Qiao 和 Huo [14] 得到复合材料板长边被转动弹簧约束时 的屈曲解析解，但结果过于冗长，并且忽略了扭转 加劲肋约束的情况。基于曲线拟合技术，Tarjan 等 [15] 得到复合材料长板分别受均匀压力、纯弯曲和 均匀剪力作用下的屈曲显式解，但在均匀剪力作用 下，只考虑了线性项，忽略了高阶项；并且针对扭 转加劲肋约束的情况，由于选取的约束系数不够合 理，使求得的屈曲载荷偏高。 横向荷载作用下，高型材梁的翼缘主要承受轴 向载荷，腹板主要承受剪切载荷。依据约束板的构 造形式，必须考虑两类约束效应 [4][5]15] Fig.1 The load and boundary conditions of representative plate element ...
... 低，而且通常做成薄壁结构形式 [2][3] ，结构通常在 达到材料强度破坏之前发生局部屈曲，从而使构件 发生大变形，阻碍结构某些功能的正常发挥，或者 直接导致结构的破坏。 在复合材料型材的局部屈曲分析中，通常把构 成型材的各个板件假定为正交各向异性板，并假定 各个板件的连接处为直线 [3][4] 。确定屈曲载荷的方 法主要有以下两种 [5] ：1) 精确法，即假定结构的各 个板件同时屈曲，板件连接处满足连续条件；2) 近 似法，即把结构中各个板件单独处理，考虑相邻板 件之间的弹性约束，也称离散板分析。本文采用第 2)种方法。 1930 年，Seydel [6] 利用静力法，求得了正交各 向异性无限长板长边为简支或固支时的剪切屈曲 精确解。1971 年，Johns [7] 总结前人成果，对各向同 性和正交各向异性板的剪切屈曲问题进行了总结。 Nemeth [8][9][10] 对复合材料长板的屈曲问题进行了详 尽的分析，并进行了各种参数研究，但纵向边界均 是简支或固支的。 在均匀压力作用下， Qiao 等 [2,[11][12] 对边界为转动弹簧约束板的屈曲问题下进行了严 格的理论分析，并得到了显式解。考虑弯扭耦合效 应， Herencia 和 Weaver [13] 求得复合材料板在均匀压 力作用下的屈曲显式解。 利用 Rayleigh-Ritz 法， Qiao 和 Huo [14] 得到复合材料板长边被转动弹簧约束时 的屈曲解析解，但结果过于冗长，并且忽略了扭转 加劲肋约束的情况。基于曲线拟合技术，Tarjan 等 [15] 得到复合材料长板分别受均匀压力、纯弯曲和 均匀剪力作用下的屈曲显式解，但在均匀剪力作用 下，只考虑了线性项，忽略了高阶项；并且针对扭 转加劲肋约束的情况，由于选取的约束系数不够合 理，使求得的屈曲载荷偏高。 横向荷载作用下，高型材梁的翼缘主要承受轴 向载荷，腹板主要承受剪切载荷。依据约束板的构 造形式，必须考虑两类约束效应 [4][5]15] Fig.1 The load and boundary conditions of representative plate element ...
The buckling analysis of long plates with two longitudinal edges either restrained by rotational springs or torsional stiffeners under uniform shear is presented using Rayleigh-Ritz method and numerical finite element analysis (FEA). By introducing generic non-dimensional parameters, i.e., orthotropic material parameters, restraint coefficients and buckling coefficients, the buckling formulas of long plates with two kinds of restraints are obtained using curve fitting technique. The analytical buckling formulas of long plates with two longitudinal edges restrained by torsional stiffeners can also be obtained by adapting the fitted formulas of long plates with two longitudinal edges restrained by rotational springs with a critical aspect ratio. The accuracy of the present approximate analytical formulas is validated with the numerical FEA and available solutions in the literature, and excellent agreements are reached.
... Compared to metal materials, they never rust. Beyond this, their dimensional stability allows them not to be affected by large temperature fluctuations, so no significant or permanent sectional deformations are presented [34][35][36]. ...
In the era of the climate emergency and different pandemics, systems that can provide an immediate response to housing needs are required. This paper aims to evaluate the use of fibre-reinforced plastic polymers (FRPs) to satisfy this need. In particular, a modular emergency housing system that utilises FRPs for structures and cladding is proposed, which proves adaptable to both different uses and different kinds of temporary or permanent buildings. By adapting modular emergency housing to different contexts, developing an integrated design process (IDP) and building information modelling (BIM) methodology, this research aims to provide innovations for the the architecture , engineering, and construction (AEC) sector, including FRPs, through a digitised approach, applied also to an experimental case study. A pilot unit of the modular emergency housing system, a nearly zero-energy building (nZEB), is described in detail, while laboratory tests are reported. Construction considerations confirm the sustainability and highlight the adaptability of the modular system to different housing needs conditions, justifying the possible future development of supply chain industrialisation supported by the presented methodology.
... These two solutions are applied in [28] for the local stability analysis of box-, I-, C-, Z-, and L-members. For the load cases shear and varying pressure loads, further approximate solutions can be found in [29]. A detailed investigation as well as explicit formulations regarding stiffened free edges and their minimum stiffness requirements is presented in [30,31]. ...
The couplings of unsymmetric laminated structures and the comparatively low transverse shear stiffnesses significantly influence their stability behaviour. The aim of this work is to improve the purely analytical stability analysis of unsymmetric laminated structures. This problem can be reduced to one single plate with the use of the discrete plate theory. In this study, laminates under uniaxial compressive load are considered that are simply supported at all edges and have rotational restraints on the unloaded edges (SRSR) or have a free edge (SFSR). Therefore, the present study provides explicit solutions for the buckling load in two ways. Firstly, the first explicit solution for the buckling load of unsymmetric laminates with the mentioned boundary conditions is developed, which takes the bending-extension couplings into account in a direct way. The method is an energy-based method in the form of the Rayleigh-quotient for classical laminated plate theory (CLPT), first order shear deformation theory (FSDT) and third order shear deformation theory (TSDT). Secondly, the reduced bending (RBS) method is investigated for the first time in the framework of TSDT. The comparison of unsymmetric cross-ply and angle-ply laminates shows a good agreement with finite element analysis (FEA). Due to the high computational efficiency of the presented methods, they are excellently suited for pre-design and optimisation of composite structures, with simultaneous high usability in practical application.
... Sonnenschein et al. closed-form expression for discrete plates subjected to bending and shear stresses with consideration of rotational restraint along the un-loaded edges, additionally, they tabulated the available equations for orthotropic plate. Based on these closed form equations, Tarjan et al. (2010b) have studied the web buckling of FRP box-beam under transverse loads. They compared their results with FEA results. ...
The use of Learning Management Systems as a tool for designing, sharing, monitoring, and organizing various forms of teaching and training content marked a turning point in the development of E - Learning. Major technical advancements have converted the early LMS into a sophisticated program for organizing curriculum, giving rich-content course materials, assessment, and interactive collaboration since its beginnings. With numerous current research areas dealing with various technologies connected to the LMS, the structure, operations, and implementation of the LMS will undoubtedly alter in the future. The Internet of Things is the most crucial technology that is predicted to alter many parts of the future (IoT). We present a conceptual framework for a forthcoming E Learning with Intelligent automation in this article.
We discuss how IoT will influence many aspects of the E - Learning, as well as the projected improvements and adjustments that IoT will bring to E- learning functionality. In this paper, we emphasize on the use of IoT services in the context of e-learning. We outline numerous services now provided by existing learning management systems in our framework, and show how these services will transform if they are merged with IoT services and connected to IoT modules and gadgets. We also suggest some additional services which will be achieved as a result of the integration of IoT into E-learning. We describe how each proposed E-learning. service will be implemented inside the E-learning., how it will be linked to other E-learning. services, and how it will be utilized to improve teaching and learning processes on the university campus
... Sonnenschein et al. closed-form expression for discrete plates subjected to bending and shear stresses with consideration of rotational restraint along the un-loaded edges, additionally, they tabulated the available equations for orthotropic plate. Based on these closed form equations, Tarjan et al. (2010b) have studied the web buckling of FRP box-beam under transverse loads. They compared their results with FEA results. ...
Geçmişten günümüze kadar özellikle evlerimizde kullandığımız ve yaşam alanlarımızın
vazgeçilmez ürünü olan halıların hem tasarım hem de üretim yönünden köklü bir geçmişi
bulunmaktadır. Binlerce yıllık geçmişe sahip olan halı, dünyada giderek önem kazanmış ve
dünya ticareti içerisinde kayda değer ürün gruplarından birisi haline gelmiştir. Günümüzde
makine halıları ya dokuma tekniği kullanılarak ya da tufting metodu kullanılarak
üretilmektedir. Üretilen bu makine halılarının kalitesi için ürünün yapısal özellikleri önem arz
etmektedir. Müşterinin bir halıda bulunmasını istediği akustik yalıtım, konfor, termal yalıtım,
görünüm, dayanıklılık ve güvenlik gibi özelliklerin arka planında yer alan halının yapısal
özellikleri belirleyici niteliktedir. Makine halısının yapısal kalitesini belirleyen faktörlerin
başında ise büyük oranda hav ipliğinin yapısı gelmektedir. Yapı içerisindeki hav ipliği
incelendiğinde; sentetik liflerin sahip oldukları yüksek mukavemet ve aşınma direnci özelliği,
kolay temizlenebilme özelliği, kimyasallara karşı yüksek direnç gösterme özelliği gibi
avantajları sayesinde tekstil alanında yaygın olarak kullanıldıkları görülmektedir. Yapılan bu
çalışmada aynı üretim parametrelerinden oluşmuş farklı kesit şekillerindeki polyester iplik
kullanımının makine halıları üzerindeki etkisinin araştırılması amaçlanmıştır.
... Sonnenschein et al. closed-form expression for discrete plates subjected to bending and shear stresses with consideration of rotational restraint along the un-loaded edges, additionally, they tabulated the available equations for orthotropic plate. Based on these closed form equations, Tarjan et al. (2010b) have studied the web buckling of FRP box-beam under transverse loads. They compared their results with FEA results. ...
E-learning technology based on multimedia has had a tremendous impact on our daily
teaching and learning activities. It has centred the learning process on the student. E-learning
has become more fascinating than traditional education systems due to the usage of
multimedia technologies. It began in the 1980s and has continued to develop since then. This
innovative technology provides pupils with the benefit of studying in a fresh way. E-learning
approaches make education more engaging and fascinating. It has also had a bigger influence
on our culture and school system. It is now simple to study and teach without being fatigued.
E-learning creates virtual learning environments on the internet via teacher-student
interactions, online assessment, and interactive multimedia-based course material
dissemination. Interactive multimedia delivers various kinds of media and suitable content
delivery based on learners' learning styles, which improves learners' learning effectiveness.
This article discusses the rapidly evolving technology of "Multimedia and e-education."
Smart teaching approaches are increasingly being used in schools, as well as other institutions
and organisations. E-textbooks are electronic textbooks that are used in a learner's actual
class. The digital textbook makes advantage of the most recent smart gadget and technology.
Keywords: Multimedia, Education, E- Learning, digital classroom
... And, σ T yy , σ C yy , τ F represents the parallel traction, compression and shear failure strength of the material. In addition, to calculate the local buckling eigenvalue (λ) of the structure, an algorithm was developed, based on the method addressed by Tarjan et al. [15,16]. In this calculation, a combined request between axial, bending and transverse load was considered, in the web and in the spar flange, created by the aerodynamic forces as well as by the weight and traction produced by the electrically-driven propellers positioned on the wing. ...
The search for less polluting means of transport has been stimulating the development of batteries with higher energy densities and electric motors that are more efficient. These advances have made possible the emergence of several all-electric aircraft, which make use of multiple rotors. In this sense, there was a need to create a methodology to be followed in the conceptual design phase, focused on aerodynamic and structural issues that change in this type of aircraft. These changes, such as the lack of a Gravity Center shift and the need for a greater number of engines and rotors, alters the spar, the aerodynamic flow, the number of ribs and the stiffness of the wing itself. In this study, the engines are distributed along the wing, aiming greater traction and less variation in the angle of attack along the wingspan. An aerodynamic model was created with these requirements, based on the lift line theory, in order to calculate the flight load. Finally, using these values, employing routines of genetic algorithms, the structure was optimized, adopting the spar geometry and thickness as design variables, with the objective of minimizing weight, following the Tsai-Wu criterion and the impossibility of buckling.
... Nonetheless, as reported by Borowicz and Bank [17], assuming the simply-supported web the Kollár's equation may greatly underestimate the web buckling strength by a factor of approximately 2. In this regard, Borowicz and Bank explored the validity of web plate with two edges fixed, and a good correlation between experimental and analytical results was observed. In light of the study by Borowicz and Bank [17], Kollár's equation for fixed web plate [61] was adopted in this work, as shown in Eq. (1). ...
An arch glass fiber reinforced polymer (GFRP) I-beam with a 600 mm height was developed based on the latest curved-pultrusion technique to overcome the most critical design issues of large-scale GFRP beams, including excessive deflections and premature buckling failures. To achieve this goal, a review was first conducted regarding the complex buckling behaviors of pultruded GFRP beams, building a theoretical basis for the development of the curved beam. Then, a series of three-point bending tests for beams were conducted at full scale, in which the typical failure modes, load-carrying capacities and deflection and strain data were obtained. Compression flange delamination was found to be the dominant failure mode. The load–strain curves for flange and web plates demonstrated that the proposed beams were exempt from local buckling issues. Additionally, an analytical study and finite element modeling were carried out. Excellent agreement between experimental, analytical and numerical studies was observed. The design approach for conventional straight profiles is readily applicable to curved beams as the difference is limited. In the end, a 20-m-long full-scale GFRP pedestrian bridge was designed, constructed and tested. The great potential of the proposed curved-pultruded GFRP arch beam was successfully demonstrated.
... The discriminating between the two cross-sections is the greater vulnerability of the angular points of the square and rectangular cross-sections in relation to the manufacturing processes and the role of the matrix that mechanically constitutes the weakest component [8]. The lack of a transversal confinement triggers mechanisms for parts of the crosssection that define orthotropic plates differently bounded to each other and loaded into the plane [9][10][11]. At present, for ongoing researches and for the well-known cross section shapes, the inability to limit out-of-plane displacements -due to the low flexural and shear stiffness of pultruded FRP profiles -could be solved through cross-sections that would redraw the cross-section shapes currently available. ...
The historical building heritage, monumental, civil and industrial has highlighted, especially in the last decade, the high vulnerability with respect to horizontal actions. The need to design stable and strong structural systems with respect to the stresses acting in various directions has oriented research to develop lightweight technologies for structural systems, independent or ancillary, with reduced mass to mitigate the consequent applied load. In this scenario all-FRP (Fiber Reinforced Polymers) systems can be a valid alternative for new constructions and / or structural reinforcement. The scenario described above, highlights the importance of developing innovative technologies suitable for the functional and structural improvement of the existing buildings. It is necessary to define an innovative approach aimed at identifying an ad-hoc profile for the FRP pultruded profiles (currently referring to the steel shapes), which allows to capitalize the performance capabilities, limiting the main defects related to low shear stiffness of FRP pultruded material. This research analyses how the shape enhancement of the cross sections, with shape redraw, allows to increase the structural performances of FRP pultruded profiles. Experimental data of previous research on buckling behaviour of different open cross section profiles, narrow and wide flanges, have been analysed and validated by numerical approach and compared with new strengthened shapes.
... The proposed approach was based on the application of beam and plate theory in order to investigate the cross section shape and subparts relationship of pultruded profiles. The research considers the thinwalled pultruded profileswith open or closed cross-sectionas prismatic members made of flat rectangular wall segments [19][20][21]. The effect of axial and shear deformation on the vibration of bars and the vibration of plates are well-known problems [22][23][24]. ...
this paper analyses the numerical and experimental results to evaluate the dynamic properties of pultruded GFRP (Glass-Fiber Reinforced Polymers) buckled columns. The profiles are made of glass fiber reinforcement and thermosetting vinylester matrix with thin-walled open or closed cross section. The buckling phenomena of the column with fixed ends were evaluated with a non-destructive method based on experimental modal data through dynamic identification procedure.
Numerical analysis has been carried out through Finite Element models calibrated considering two consecutive stages that involve the local and global scale: i) parametric natural frequencies analysis to model the different cross sections taking into account the stiffness of the rotational constraint between the wall segments of the thin walled pultruded profiles; ii) buckling analysis to identify the inaccuracies in the specimen or in the experimental apparatus through global flexural displacements which increase continuously with the axial load.
Experimental, theoretical and numerical results were compared in order to know the wall segment effects of GFRP columns in free vibration field when affected by buckling phenomena.
The results allow to investigate the significant role that the manufacturing imperfections of pultruded material play in the structural performance of GFRP buckled columns.
... In the experimental program, 86 The averages and coefficients of variation (COV) of tests results are shown in Table 6.2. ...
An extensive experimental and analytical study investigating the stability of pultruded glass fibre-reinforced polymer (pGFRP) I-sections subject to flexure is reported. To investigate flange local buckling (FLB), specimens with flange slenderness ratios (b/2t) from 4 to 12 were tested. Lateral torsional buckling (LTB) was investigated in specimens having slenderness (Lb/r) ranging from 24 to 52. Through a comparative study with the experimental results, existing standards and the established closed-form mechanics-based equations were found to underestimate the critical FLB moment capacities of pGFRP I-sections. The finite-strip method is demonstrated to predict critical FLB moments with satisfactory accuracy. It is shown that the interaction between FLB and/or web local buckling and global LTB may lead to a reduction of critical FLB moment capacity. Such interaction tends to occur in sections having small flange slenderness ratios as well as large longitudinal lateral slenderness ratios. On the other hand, the predictions of LTB capacity were generally found to be non-conservative when compared with experimentally determined results. Such discrepancies between experimental and analytical results tend to become more significant for sections having relatively large flange slenderness ratios as well as small longitudinal lateral slenderness ratios. This underestimation of LTB moment capacity is found to result from the interaction between local and global buckling behaviours of pGFRP I-sections.
... where is the angle between the flange and the web; is the rotational restraint stiffness. However, the rotational spring restraint assumption is not always valid; Kollar [10,[12][13][14] divided the elastic restraints into two kinds based on analysis of the configurations of various sections: (1) when both of the two edges of the restrained plates are restrained, the restraints are equivalent to a rotating spring [11] and (2) when one edge of the restrained plates is free, the restraints are equivalent to torsional stiffener [15]. The former occurs when the webs and flanges of box-sections and the webs of I-, C-, and Z-sections are used as restrained plates, as shown in Figure 2(a). ...
This paper reports on approaches to estimate the critical buckling loads of thin-walled T-sections with closed-form solutions. We first develop a model using energy conservation approach under the assumption that there is no correlation between the restraint coefficient and buckling half-wavelength. Secondly, we propose a numerical approach to estimate the critical buckling conditions under the more realistic torsional stiffener constraint condition. A dimensionless parameter correlated with constraint conditions is introduced through finite element (FE) analysis and data fitting technique in the numerical approach. The critical buckling coefficient and loads can be expressed as explicit functions of the dimensionless parameter. The proposed numerical approach demonstrates higher accuracy than the approach under noncorrelation assumption. Due to the explicit expression of critical buckling loads, the numerical approach presented here can be easily used in the design, analysis, and precision manufacture of T-section webs.
... E: Modulus of elasticity (GPa). The compressive stress can be well below the material yield strength at the time of buckling the factor that determines if a column is short or long it is slenderness ratio (S) [12]: Where:- S=L/r (2) ...
This research studied the critical load of composite columns theoretical and numerical by using ANSYS14 package depended on experimental tensile properties of composite specimens. The composite specimens were prepared by hand lay-up technique made from unsaturated polyester reinforced with glass fibers with different fiber volume fraction Vf, aspect ratio (L/T), and angle of fibers for coarse and fine woven fibers. The critical load that obtained by using program (ANSYS 14) have also shown a good agreement with results that were obtained theoretically and the maximum difference was (0.7%). The results show that the maximum value of the critical load can be observed at Vf =11%, L/T = (3.5) and θ = (0 º /90 º) for fine woven fibers was (622.115N). Also its found the maximum critical load for coarse woven fibers can be observed at Vf %=8%, L/T=(3.5) and θ = (0 º /90 º) was (486.887N). Also the observed values of tensile properties and predicated values are scattered close to the (45˚)45˚45˚) line.
... One of the main failure modes of layered composites is the delamination fracture, which can occur because of machining errors, or low velocity impact during the service life [7][8][9][10][11][12][13][14][15][16]. Buckling caused by uniaxial compression is one of the critical failures for these materials [17][18][19][20][21][22][23][24]. ...
We analyse the buckling process of composite plates with through-the-width delamination and straight crack front applying uniaxial compression. We are focusing on the mixed mode buckling case, where the non-uniform distribution of the in-plane forces controls the occurence of the buckling of the delaminated layers. For the analysis, semi-discrete finite elements will be derived based on the Lèvy-type method. The method of harmonic balance is used for taking into account the force distribution that is generally non uniform in-plane.
... E: Modulus of elasticity (Gpa). The compressive stress can be well below the material yield strength ate the time of buckling the factor that determines if a column is short or long is it is slenderness ratio (S) [10] ...
This research studied the buckling of composite columns made from glass fiber
reinforced unsaturated polyester. The composite specimens were prepared by hand
lay-up technique with different fiber volume fraction Vf, aspect ratio and angle of
fibers when reinforced with coarse and fine woven fibers. Mathematical models were
done by using statistical analysis which show the critical load of the composite
column as a function of volume fraction, fiber angle and aspect ratio.
The results show that the maximum value of the critical load can be observed at
volume fraction Vf =11%, aspect ratio L/T = (3.5) and fiber angle θ = (0º/90º) for fine
fiber was (500.72 N). Also its found the maximum critical load for coarse fiber can be
observed at Vf =8%, L/T= (3.5) and θ = (0º/90º) was (400.4 N).
... E: Modulus of elasticity (Gpa). The compressive stress can be well below the material yield strength ate the time of buckling the factor that determines if a column is short or long is it is slenderness ratio (S) [10] ...
This research studied the buckling of composite columns made from glass fiber reinforced unsaturated polyester. The composite specimens were prepared by hand lay-up technique with different fiber volume fraction V f , aspect ratio and angle of fibers when reinforced with coarse and fine woven fibers. Mathematical models were done by using statistical analysis which show the critical load of the composite column as a function of volume fraction, fiber angle and aspect ratio. The results show that the maximum value of the critical load can be observed at volume fraction V f =11%, aspect ratio L/T = (3.5) and fiber angle θ = (0 º /90 º) for fine fiber was (500.72 N). Also its found the maximum critical load for coarse fiber can be observed at V f =8%, L/T= (3.5) and θ = (0 º /90 º) was (400.4 N).
... Column fails by buckling when the axial compressive load exceeds some critical load [6] . The compressive stress can be well below the material yield strength at the time of buckling depending on the factor that determines if a column is short or long which is column slenderness ratio (S), where: S = L/r ………. ...
Experimental work and finite element techniques were used to analyze the tensile and buckling characteristics of carburized low carbon steel columns for different diameters.
The back of carburization consists of hardwood charcoal mixed with barium carbonate as the energizer. The specimens were carburized using (30%) of energizers at temperature equal to (900ºC) for 5 five hours.
The modulus of elasticity as measured experimentally was used in FEM (ANSYS) to calculate the buckling load. The results of the study showed that the Vickers micro hardness increased from (HV=210) to (HV=525), the tensile strength increased from (708 MPa) to (1065 MPa) and critical buckling load increased from (18.37 kN) to (19.6 kN) for d=8mm. Also critical buckling load increased from (5.8 kN) to (19.2 kN) as the diameter of the carburized low carbon steel column increased from 6mm to 8mm respectively for FEA analysis.
Because of their unique properties such as high strength and stiffness, lightweight, corrosion resistance, and
tailoring ability, fiber reinforced polymer beams are increasingly used in various engineering applications. Being a thin- walled member, their design is often governed by buckling. Buckling strength is influenced by various secondary effects
(shear lag, moment gradient, rotational restraint, and torsion) induced stresses. In the present study, the influence of
torsion on buckling strength is studied by changing the orientation of fiber and web geometry. Initially, the angle of twist
(θ) and buckling strength (K) were arrived using the finite element software ANSYS. Later, graphs and bar charts were used to demonstrate the effect of torsion on buckling strength. Keywords: Torsion; FRP composite; Buckling; Secondary effects; Finite element analysis
In the present paper, the shear post-buckling behavior of the FGM rectangular plates is analyzed. Furthermore, effects of the elastic foundation on the post-buckling deformation/distortion patterns and strengths are investigated. The exact energy-based 3D theory of elasticity is implemented through incorporation of the general form of Green’s strain tensor. A fully consistent 3D Hermitian element with up to the second-order mixed derivatives continuity is utilized to satisfy continuity of the stress and distortion components at the mutual nodes of the adjacent elements. The post-buckling equilibrium paths are traced using an arc-length solution scheme in conjunction with the modified Newton-Raphson method. A comprehensive sensitivity analysis is accomplished to evaluate effects of the heterogeneity index, elastic foundation stiffness, aspect ratio, and the considered two types of practical boundary conditions on the post-buckling deformation patterns, mode jumping, and equilibrium paths. Results obtained based on post-buckling observations of properly adopted reference points reveal that mode jumping and larger distortions may occur due to increasing the post-buckling loads, the elastic foundation leads to larger number of the local deformation regions, and the buckling strengths reduce remarkably when two opposite edges are clamped.
Explicit expressions are developed for the buckling analyses of rectangular plates subjected to uniaxial loads when the unloaded edges are rotationally constrained. Three types of constraints (and their combinations) are considered: rotational springs, stiffeners with torsional stiffness and stiffeners with warping stiffness. The results can be applied for the stability analysis of the webs of thin-walled I or C beams. © The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav.
Local buckling analysis of thin walled composite beams is presented, where the flanges are stiffened at their free edges. The web and the flanges are modelled by rotationally restrained long orthotropic plates. Explicit expressions are developed for the calculation of the lowest buckling load. For the stability of the flange distortional buckling, plate buckling (or local buckling) and their interaction are considered, while the stabilizing effect of the web is taken into account by elastic constraint. Web buckling is modelled with a long plate, whose edges are constrained by stiffeners with warping stiffness. The accuracy of the method is demonstrated through numerical examples.
In the present paper, shear buckling analysis of the orthotropic heterogeneous FGM plates is investigated for the first time. Moreover, influence of the Winkler-type elastic foundation is considered. The material properties are assumed to have in-plane orthotropy and transverse heterogeneity. The most accurate approach, i.e., the three-dimensional elasticity is employed instead of using the approximate plate theories. In contrast to all of the available displacement-based buckling analyses that have employed C0-continuous commercial finite element codes or semi-analytical methods, present formulations are C2-continuous due to using the proposed 3D cubic B-spline element. Results are derived based on principle of minimum potential energy and a non-linear finite element procedure utilizing a Galerkin-type 3D cubic B-spline solution algorithm. Buckling loads are detected based on a generalized geometric stiffness concept. In this regard, effects of both the prebuckling and buckling states are considered. To present a better imagination and more detailed discussions, details of the buckling mode shapes and the foundation interaction are discussed for plates with simply-supported edges. In addition, the more practical free and clamped edge conditions are also considered.
An analytical study of local buckling of discrete laminated plates or panels of fiber-reinforced plastic structural shapes is presented. Two cases of composite plate analyses with two opposite edges simply supported and the other two opposite edges either both rotationally restrained or one rotationally restrained and the other free, are studied. Generic loading cases with combined linearly varying axial and in-plane shear loading are considered. A variational formulation of the Ritz method is used to establish the eigenvalue problem for the local buckling behavior, and explicit expressions for predictions of the plate buckling stress resultants, in terms of the rotationally restrained stiffness, the plate aspect ratio, and the ratios of applied stress resultants, are developed. Based on different boundary and loading conditions, simple and explicit local buckling solutions for several special cases are further reduced. Validity of the explicit solutions presented is demonstrated by a good agreement of comparisons between the present predictions and available solutions in the literature. Parametric studies are further conducted, and the influences of several parameters such as the rotationally restrained stiffness, biaxial loading stress ratio, material flexural orthotropy, and linearly varying axial loading stress gradient on the local buckling stress resultants are discussed. (C) 2013 American Society of Civil Engineers.
In the present paper, a three-dimensional elasticity approach is employed to investigate buckling of heterogeneous functionally graded plates under biaxial compression, shear, tension-compression, and shear-compression load conditions. In this regard, a formulation that employs a full compatible three-dimensional Hermitian element with 168 degrees of freedom and guarantees continuity of the strain and stress components is used. It is known that all of the available famous commercial finite element softwares and the proposed series solutions satisfy continuity conditions of the displacement rather than the stress components. Buckling occurrence is detected based on checking both the instability onset and equilibrium criteria. Results are extracted based on a Galerkin-type orthogonality. Therefore, they are more accurate than those obtained based on the traditional Ritz method. The presented three-dimensional finite element analysis and the extracted results are quite new. A vast variety of results including results of biaxial compression, compression-tension, shear, and shear-compression load cases is considered and discussed in detail.
Approximate expressions to determine the lowest buckling load of short and long composite plates are presented. The plate is subjected to uniaxial compressive load. All the four edges of the plate are elastically restrained by springs or stiffeners. Explicit expressions for the calculation of the lowest buckling load are presented as a function of the plate’s length. The derived expressions were compared to numerical solutions investigating the entire parameter range.
Pultruded fiber–reinforced plastic (FRP) composite structural shapes (beams and columns) are thin-walled open or closed sections consisting of assemblies of flat plates and commonly made of E-glass fiber and either polyester or vinylester resins. Due to high strength-to-stiffness ratio of composites and thin-walled sectional geometry of FRP shapes, buckling is the most likely mode of failure before material failure. In this paper, explicit analyses of local buckling of rectangular orthotropic composite plates with various unloaded edge boundary conditions (i.e., (1) rotationally restrained along both unloaded edges (RR), and (2) one rotationally restrained and the other free along the unloaded edges (RF)) and subjected to uniform in-plane axial action at simply-supported loaded edges are first presented. A variational formulation of the Ritz method is used to establish an eigenvalue problem, and explicit solutions of plate local buckling coefficients in term of the rotational restraint stiffness (k) are obtained. The two cases of rotationally restrained plates (i.e., the RR and RF plates) are further treated as discrete plates of closed and open sections, and by considering the effect of elastic restraints at the joint connections of flanges and webs, the local buckling of different FRP shapes under uniform axial compression is studied. The approximate expressions of the rotational restraint stiffness (k) for various common FRP sections are provided, and their application to sectional local buckling predictions is illustrated. The explicit local buckling formulas of rotationally restrained plates are validated with the exact transcendental solutions. The analytical predictions for local buckling of various FRP profiles based on the present discrete plate analysis and considering the elastic restraints of the flange–web connections are in excellent agreements with available experimental results and finite element eigenvalue analyses. A design guideline for local buckling prediction and related performance improvement is proposed. The present explicit formulation can be applied effectively to determine the local buckling capacities of composite plates with elastic restraints along the unloaded edges and can be further used to predict the local buckling strength of FRP shapes.
Closed form approximate formulas are presented for the calculation of the buckling load of rectanglar orthotropic plates with clamped and/or simply supported edges. The solution is based on the Ritz method assuming one term approximation.
Preface List of symbols 1. Introduction 2. Displacements, strains, stresses 3. Laminated composites 4. Thin plates 5. Sandwich plates 6. Beams 7. Beams with shear deformation 8. Shells 9. Finite element analysis 10. Failure criteria 11. Micromechanics Appendix A. Cross-sectional properties of thin-walled composite beams Appendix B. Buckling loads and natural frequencies of orthotropic beams with shear deformation Appendix C. Typical material properties Index.
An increase in the use of composite materials in areas of engineering has led to a greater demand for engineers versed in the design of structures made from such materials. This book offers students and engineers tools for designing practical composite structures. Among the topics of interest to the designer are stress-strain relationships for a wide range of anisotropic materials; bending, buckling, and vibration of plates; bending, torsion, buckling, and vibration of solid as well as thin walled beams; shells; hygrothermal stresses and strains; finite element formulation; and failure criteria. More than 300 illustrations, 50 fully worked problems, and material properties data sets are included. Some knowledge of composites, differential equations, and matrix algebra is helpful but not necessary, as the book is self-contained. Graduate students, researchers, and practitioners will value it for both theory and application.
Local buckling analysis of thin-walled open or closed section fiber-reinforced plastic beams is presented. In the analysis, the web is modeled as a long orthotropic plate with rotationally restrained edges. Explicit expressions were developed for the buckling analyses of rectangular (long) plates in a companion paper. These results are applied to develop explicit expressions for the calculation of the web buckling of beams with thin-walled cross sections. At last, the applicability of the method is demonstrated by numerical examples and the results are verified by finite element calculations.
An elastic stability analysis is given for an aluminium/carbon fibre flange of a beam or column subjected to a uniform in-plane load. In the theory, coupling between bending and stretching is avoided by using ‘reduced bending rigidity’ terms and this leads to a relatively simple formulation of the problem. Consideration is given to the degree of restraint offered by the web, which is shown to lie between simply supported and fully restrained conditions. Theoretical results are supported by tests on fabricated column specimens which show a decrease in the buckling load as the percentage of reinforcing material is increased.
The local buckling analysis of fiber reinforced plastic (FRP) composite open and closed thin-walled section beams and columns is presented. Explicit expressions are developed for axially loaded and for bent box-, I-, C-, Z-, and L-members. These simple, explicit formulas contain the width and the bending stiffnesses (D11, D22, D12, D66) of each wall segment. The utility of the method is demonstrated by numerical examples.
An explicit elastic stability analysis for local buckling of fiber-reinforced plastic (FRP) structural shapes is presented. Flange of pultruded FRP wide-flange sections is modeled as a discrete panel with elastic restraint at one unloaded edge and free at the other unloaded edge [restrained-free (RF) condition] and subjected to uniform distributed axial in-plane force along simply supported loaded edges. By considering a linear combination of simply supported-free and clamped-free boundary displacement fields as an interpolation function of the RF buckling, a variational formulation of the Ritz method is used to establish an eigenvalue problem, and a flange critical local buckling coefficient is determined. An explicit solution is obtained for local compressive buckling strength of orthotropic panels with the RF condition and is expressed in term of the coefficient of elastic restraint based on the flexibility of flange-web connection. The explicit predictions are in good agreements with experimental data. exact transcendental solutions, and finite-element analyses for local buckling of FRP wide-flange columns. The formulation developed in this paper is the first attempt in the literature for explicit buckling analysis of orthotropic plates with RF condition and can facilitate the local buckling analysis and design of open FRP structural profiles (e.g., I and channel shapes).
An analytical study of local buckling of rectangular composite plates rotationally restrained elastically along unloaded edges and subjected to nonuniform in-plane axial action at simply supported loaded edges is presented. A variational formulation of the Ritz method is used to establish an eigenvalue problem, and by using combined harmonic and polynomial buckling deformation functions, which satisfy all the restrained boundary conditions, the explicit solution of plate local buckling coefficients is obtained. The explicit formulas for local buckling strength of orthotropic plates are simplified to the cases of isotropic plates, which are consistent with classical solutions. The elastically rotationally restrained plates are further treated as discrete plates or panels of fillet-reinforced plastic (FRP) box shapes, and by considering the effect of elastic restraints at the joint connections of flanges and webs, the local buckling strength of FRP box shapes is predicted. The theoretical predictions are in good agreement with transcendental solutions and finite-element eigenvalue analyses for local buckling of FRP box columns. The present explicit formulation can be applied to determine local buckling capacities of composite plates with elastic restraints along the unloaded edges and can be further used to predict the local buckling strength of FRP shapes.
A variational formulation of the Ritz method is used to establish an eigenvalue problem for the local buckling behavior of composite plates elastically restrained along their four edges (the RRRR plates) and subjected to compression along one axis, and the explicit solution in terms of the rotational restraint stiffness (k) is presented. Based on the different boundary and loading conditions, the explicit local buckling solution for the rotationally restrained plates is further simplified to several special cases (e.g., the SSSS, SSCC, CCSS, SSRR, RRSS, CCRR, and RRCC plates) with a combination of simply supported (S), clamped (C), and/or rotationally restrained (R) edge conditions. The unique deformation shape function combining the harmonic and polynomial functions is proposed by considering the effect of elastic rotational restraint stiffness (k) along the four edges of the orthotropic plate. The explicit local buckling solutions of the RRSS and SSRR cases are validated with the exact transcendental solutions. A parametric study is conducted to evaluate the influences of the rotational restraint stiffness (k) and the aspect ratio (γ) on the local buckling stress resultants of various rotationally restrained plates. The applicability of the explicit solutions of restrained composite plates is illustrated in the discrete plate analysis of two composite structures: short thin-walled composite columns and honeycomb sandwich cores. The local buckling strength values of plate panels in short FRP box columns and core walls between the top and bottom face sheets of sandwich are predicted, and they are in excellent agreement with the numerical finite element solutions and experimental results. The simplicity of the “explicit” local buckling solution of elastically restrained composite plates facilitates analysis, design, and optimization of composite structures.
Closed form approximate formulas are presented for the calculation of the buckling load of rectangular orthotropic plates with clamped and/or simply supported edges. The solution is based on the Ritz method assuming one term approximation.
An analytical study of local buckling of discrete laminated plates or panels of fiber-reinforced plastic (FRP) structural shapes is presented. Flanges of pultruded FRP shapes are modeled as discrete panels subjected to uniform axial in-plane loads. Two cases of composite plate analyses with different boundary con-ditions and elastic restraints on the unloaded edges are presented. By solving two transcendental equations simultaneously, the critical buckling stress resultant and the critical value of the number of buckled waves over the plate aspect ratio are obtained. Using this new solution technique and regression analysis, simplified ex-pressions for predictions of plate buckling stress resultants are efficiently formulated in terms of coefficients of boundary elastic restraints. The effects of restraint at the flange-web connection are considered, and explicit expressions for the coefficients of restraint for I-and box-sections are given; it is shown that actual cases lie between simply supported and fully restrained (clamped) conditions. The theoretical predictions show good agreement with experimental data and finite-element eigenvalue analyses for local buckling of FRP columns. In a similar manner, web plate elements of FRP shapes under in-plane shear loads are modeled with and without elastic restraints provided by the flange panels. The present formulation can be applied to several cases to determine local buckling capacities of laminated plates with elastic restraints along the unloaded edges and can be further used to predict the local buckling strength of FRP shapes, such as columns and beams.
Composites for Construction
- L C Bank
Bank, L. C. (2006). Composites for Construction, J. Wiley & Sons, Hobken.
Buckling of Metal Structures
- F Bleich
Bleich, F. (1952). Buckling of Metal Structures, McGraw-Hill, New York.
Anisotropic Plates, Gordon and Breach Science Publishers
- S G Lekhnitskii
Lekhnitskii, S. G. (1968). Anisotropic Plates, Gordon and Breach Science Publishers, New York.
- Bank, L.C.