Non-prismatic Beams Research Articles

Investigating Static Deflection of Axial Functionally Graded Non-Prismatic Beams Using the Rayleigh Method

Investigating Static Deflection of Axial Functionally Graded Non-Prismatic Beams Using the Rayleigh Method

A Finite Difference Scheme for the Large-Deflection Analysis of Non-Prismatic Cantilever Beams

A Finite Difference Scheme for the Large-Deflection Analysis of Non-Prismatic Cantilever Beams

Shear performance and strength of reinforced concrete non-prismatic beams with basalt fiber reinforced polymer rebars

Tapered beams, which are horizontal structural elements, possess varying moments of inertia across their length. Among these, the pier cap mode is widely employed in engineering applications. This research explores the influence of incorporating basalt fiber reinforced polymer (BFRP) bars on the shear strength of eight concrete haunched beams. While two of these beams were straight beams, the others have different inclination angles. The study aimed to examine two main factors: the angles of tapered part in the compression chord and the effect of BFRP stirrups. Results from the experiment revealed a direct correlation between increasing tapered angles and the enhancement of shear capacity in haunched beams. Consequently, a comprehensive analysis was conducted using three different codes, leading to the formulation of proposed equations derived from semi-empirical methods. The analysis demonstrated a high level of accuracy in predicting shear strength with a satisfactory safety margin.

Design procedure of a shear fuse system for steel moment-resisting frames

In steel moment-resisting frames (MRFs), seismic energy is generally dissipated through the formation of flexural plastic hinges at beam ends. In ensuring such a mechanism, current code provisions limit the utilization of beams with small span-to-length ratios in MRFs. This study presents the design procedure of a novel shear fuse system that integrates replaceable shear links with non-prismatic beams to enhance the performance and repairability of MRFs with small beam span-to-length ratios. The proposed design strategically weakens the beam at its midspan by incorporating a replaceable link, prioritizing shear yielding in the link over flexural yielding in the beam outside the link, thus ensuring adequate frame ductility and protecting the critical beam-to-column connections. It can function as an efficient replaceable shear fuse system, as the damaged fuse can be substituted with a new one following a severe earthquake. In fulfilling the study’s purpose, first, several formulas are developed to provide a comprehensive design procedure. Next, example design calculations are presented for an MRF with a small beam span-to-depth ratio. Ultimately, a numerical investigation is provided to examine the proposed system’s performance compared to conventional MRFs. The results indicate that the proposed shear fuse system provides a more consistent cyclic behavior than the conventional prismatic beam design, exhibiting stable strain-hardening behavior up to the 5 % story drift ratio without noticeable strength degradation. It appropriately shifts the plastic hinges away from the beam-to-column connections while increasing the frame’s overall ductility with 1.14 times greater energy dissipation in the final loading cycles.

Lateral-torsional buckling resistance of non-prismatic and prismatic mono-symmetric I-section steel beams based on stress utilization

The lateral-torsional resistance of prismatic double-symmetric I-section beams is accurately predicted using a mechanically consistent Ayrton-Perry approach, combined with a calibrated generalized imperfection. The corresponding design formulation was recently adopted in the revised version of Eurocode 3. However, for prismatic mono-symmetric I-section beams, the General Case shall be used while for non-prismatic beams only the General Method is available. Both methods present a very large scatter and highly underestimate the lateral-torsional buckling resistance. This paper proposes an extension to the General Formulation for non-prismatic beams with arbitrary boundary conditions, partial lateral restraints, and arbitrary loading for mono-symmetric I-sections. Using an advanced numerical model calibrated with experimental test results, a large parametric study is undertaken, and its results are used to assess the available design methodologies and the proposed method. It is concluded that the General Formulation provides excellent safe-sided estimates of the LTB resistance, and it is confirmed the very poor performance of the General Case and the General Method.

Free-Vibration and Buckling Analyses of Beams using Kriging-Based Timoshenko Beam Elements with the Discrete Shear Gap Technique

A family of locking-free Kriging-based Timoshenko beam elements with a new implementation of the discrete shear gap technique was recently developed (referred to as K-beam-DSG1 elements). Performance of the K-beam-DSG1 elements has been shown to be very satisfactory in the linear static analysis of beams for a wide variety of thicknesses. This paper presents further development of the K-beam-DSG1 elements to free vibration and bifurcation buckling analyses of prismatic and non-prismatic beams. Consistent Kriging-based mass matrices are used for free vibration analysis and similarly, consistent Kriging-based geometric stiffness matrices are used for buckling analysis. The results show that for most of the cases, the K-beam-DGS1 elements yield remarkably accurate natural frequencies and critical compressive loads using a reasonable number of elements to discretize the beam. For an axially functionally graded fixed-fixed supported beam, however, the elements fail to predict the critical load accurately.

Nondimensional Shape Optimization of Nonprismatic Beams with Sinusoidal Lateral Profile

Nondimensional Shape Optimization of Nonprismatic Beams with Sinusoidal Lateral Profile

Analytical theory of composite beams with CSWs using the state space method

Based on the zig-zag theoretical model of corrugated steel web beam (CSW), the governing equations in the form of state equations are derived and their analytical solutions are obtained for different loads, variable cross-sections and boundary conditions. Through numerical examples, it is verified that the bending moment, deflection and stresses predicted by the present analytical method have a good agreement with the results of the three-dimensional finite element model. The applicability of the present method for non-prismatic continuous beams with CSWs under different loads is also demonstrated. Meanwhile, it is found that the quasi-plane assumption is still valid in non-prismatic continuous beams with CSWs. Through quantitative analysis of the flexural stresses in the top and bottom slabs, it is confirmed that the zig-zag theory in this study can well reflect the accordion effect of CSWs. Because of the inclined bottom slabs in the non-prismatic beams with CSWs, the Resal effect on the shear stresses in the CSWs and bottom slabs, which is caused by the vertical component of the axial force in the bottom slab, is discussed. A refined shear formulation is developed for more accurate shear stresses in the CSWs and bottom slabs, which provides fundamental theoretical support for the strength prediction of the non-prismatic beam with CSWs.

A two-parameter multiple shooting method and its application to the natural vibrations of non-prismatic multi-segment beams

A two-parameter multiple shooting method and its application to the natural vibrations of non-prismatic multi-segment beams

Free Vibration Analysis of Non-Prismatic Beam under Clamped and Simply Supported Boundary Conditions

Free Vibration Analysis of Non-Prismatic Beam under Clamped and Simply Supported Boundary Conditions

Heterogeneous Beam Element for Multiscale Modeling of Non-prismatic Composite Beam-like Structures

Anisotropy, heterogeneity, and geometry make full system-level three-dimensional (3D) finite element analysis (FEA) simulation of real composite beam-like structures, such as wind blades, impossible due to limitations on computing resources. A popular approach is to cut the structure into multiple cross-sections, from which the beam properties based on the Euler-Bernoulli or Timoshenko beam models are extracted using two-dimensional (2D) cross-sectional analyses. These properties are then utilized in beam elements for system-level analyses. However, the mechanics behind 2D cross-sectional analyses are based on prismatic beams, limiting the accuracy of this approach on non-prismatic structures, especially in predicting stresses. In this work, we present a new multiscale method based on Mechanics of Structure Genome to analyze beam-like non-prismatic composite structures. A beam-like structure is homogenized into a series of 3-node Heterogeneous Beam Elements (HBE) with effective beam element stiffness matrices, which are then used as input for one-dimensional (1D) beam analysis using Abaqus User Element subroutine (UEL). Using the macroscopic beam analysis results as input, we can also perform dehomogenization to predict the stresses and strains in the original structure. We use three examples, a prismatic composite beam, an isotropic homogeneous tapered beam, and a composite tapered beam, to demonstrate the capability of HBE and show its advantages over the traditional cross-sectional analysis approach. HBE can capture macroscopic behavior and detailed stresses due to non-prismatic geometry. HBE provides a new concept to model non-prismatic composite beams by modeling heterogeneous beam-like structures as equivalent beam elements rather than beam properties of material points of the reference line.

A Geometrically Exact Beam Finite Element for Non-Prismatic Strip Beams: The Spatial Case

A new beam finite element for non-prismatic (tapered and curved) beams with thin rectangular cross-section (“strip” beams) is proposed in this paper. The element is geometrically exact and constitutes the extension, to the large 3D displacement and rotation setting, of those proposed by the author for the 2D [R. Gonçalves, Int. J. Struct. Stab. Dyn. 23 (2022) 2350037] and linearized lateral-torsional buckling [R. Gonçalves, Int. J. Struct. Stab. Dyn. 23(12) (2023) 2350139] cases. The element incorporates torsion-related warping, as well as arbitrary mid-line warping, and allows nonlinear tapered geometries, arbitrary initially curved configurations (including pre-twist) and eccentric loads (loads offset from the centroid). All expressions required to implement the proposed element are given in a straightforward vector/matrix format. Several numerical tests are presented to show that the element has a very good predictive capability and a low computational cost, when compared with solutions obtained with refined shell finite element meshes, even for members with a high taper ratio and initial curvature.

Behavior of non-prismatic RC beams with conventional steel and green GFRP rebars for sustainable infrastructure

This study presents an experimental and finite element analysis of reinforced concrete beams with solid, hollow, prismatic, or non-prismatic sections. In the first part, a total of six beams were tested under four-point monotonic bending. The test matrix was designed to provide a comparison of structural behavior between prismatic solid and hollow section beams, prismatic solid and non-prismatic solid section beams, and prismatic hollow and non-prismatic hollow section beams. The intensity of shear was maximum in the case of prismatic section beams. The inclusion of a tapered section lowered the demand for shear. In the second part, Nonlinear Finite Element Modeling was performed by using ATENA. The adopted modeling strategy resulted in close agreement with experimental crack patterns at ultimate failure. However, the ultimate failure loads predicted by nonlinear modeling were generally higher than their corresponding experimental results. Whereas in the last part, the developed models were further extended to investigate the effect of the strength of concrete and ratio of longitudinal steel bars on the ultimate load-carrying capacity and cracking behavior of the reinforced concrete beams with solid, hollow, prismatic, or non-prismatic sections. The ultimate loads for each beam predicted by the model were found to be in close agreement with experimental results. Nonlinear modeling was further extended to assess the effects of concrete strength and longitudinal reinforcement ratio on failure patterns and ultimate loads. The parametric study involved beams reinforced with glass fiber-reinforced polymer (GFRP) bars against shear and flexural failure. In terms of ultimate load capacities, diagonal cracking, and flexural cracking, beams strengthened with GFRP bars demonstrated comparable performance to the beams strengthened with steel bars.

Geometric nonlinear analysis of slender layered non-prismatic beams with interlayer slip

This paper presents a novel model that, for the first time, simultaneously captures the effects of variable cross-section and interlayer slip on the geometrically nonlinear response of slender symmetrically layered beams within the framework of a beam theory. While the central layer has a constant cross-section along its length, the height of the two outer layers can vary arbitrarily. A nonlinear axial strain–displacement relationship captures the influence of moderately large deflections for immovably supported beams. The kinematic hypothesis of the Euler–Bernoulli theory is applied layerwise. The relationship between the interlaminar shear tractions and the interlayer slips is assumed to be linear. The Galerkin method is used to solve the derived initial–boundary value problem. Several application examples demonstrate the influence of the interlayer slip, the variable cross-section, and the geometrically nonlinear formulation of the equations of motion on the static and dynamic response. The results of comparative analyses with a numerically much more computational expensive finite element model discretized by plane stress continuum elements are in excellent agreement with the outcomes of the presented theory.

Exact structural-property matrices and Timoshenko’s shape functions for nonprismatic frame elements defined by noncentroidal axes

When one is attempting to model framed structures and adopts beam-like finite elements formulated with reference to their centroidal axis—which avoids the interaction between axial and flexural effects—one must, in many practical situations, take into account geometrically complex curved-line finite elements. This fact, which can be quite disadvantageous, is typically called for when solving physically nonlinear problems or modeling nonprismatic beams having cross sections varying asymmetrically around their straight centerlines. This paper proposes then a flexibility-type method based on the principle of virtual forces (PVF) to obtain, according to the Timoshenko beam theory (TBT), exact expressions for the structural properties of nonprismatic frame elements described with reference to noncentroidal axes. The main advantage of this method is that it allows, through the use of straight-line segments, the modeling of complex variable-rigidity frame elements. To interpolate the variable sectional rigidities over the elements, one adopts polynomials of different orders. In addition, this new method allows one to obtain the exact Timoshenko’s shape functions (TSFs) for nonprismatic frame elements with noncentroidal axes. Such shape functions are needed to evaluate deformation-dependent structural properties, such as geometric stiffness. We validate the robustness of the proposed formulation by solving in-plane tapered beams possessing thin-walled monosymmetric cross sections. Using analytical and highly accurate 3D ANSYS responses, this work determines and compares responses described in terms of displacements, rotations, internal forces, and stresses.

Load-Bearing Performance of Non-Prismatic RC Beams Wrapped with Carbon FRP Composites

This study investigated the influence of CFRP composite wrapping techniques on the load–deflection and strain relationships of non-prismatic RC beams. A total of twelve non-prismatic beams with and without openings were tested in the present study. The length of the non-prismatic section was also varied to assess the effect on the behavior and load capacity of non-prismatic beams. The strengthening of beams was performed by using carbon fiber-reinforced polymer (CFRP) composites in the form of individual strips or full wraps. The linear variable differential transducers and strain gauges were installed at the steel bars to observe the load–deflection and strain responses of non-prismatic RC beams, respectively. The cracking behavior of unstrengthened beams was accompanied by excessive flexural and shear cracks. The influence of CFRP strips and full wraps was primarily observed in solid section beams without shear cracks, resulting in enhanced performance. In contrast, hollow section strengthened beams exhibited minor shear cracks alongside the primary flexural cracks within the constant moment region. The absence of shear cracks was reflected in the load–deflection curves of strengthened beams, which demonstrated a ductile behavior. The strengthened beams demonstrated 40% to 70% higher peak loads than control beams, whereas the ultimate deflection was increased up to 524.87% compared to that of the control beams. The improvement in the peak load was more prominent as the length of the non-prismatic section increased. A better improvement in ductility was achieved for the case of CFRP strips in the case of short non-prismatic lengths, whereas the efficiency of CFRP strips was reduced as the length of the non-prismatic section increased. Moreover, the load–strain capacity of CFRP-strengthened non-prismatic RC beams was higher than the control beams.

Exact Stiffness Matrix for Nonprismatic Beams with Parabolic Varying Depth

Exact Stiffness Matrix for Nonprismatic Beams with Parabolic Varying Depth

Lateral-Torsional Buckling of Non-Prismatic I-Beams Using FEM Approach

Essential steel components with variable cross-section are fabricated from welded plates, which are primarily employed for the growth of the construction industry in beams according to the stress and stiffness requirements of the structure. Lateral-torsional buckling, in which the beam experiences non-uniform twisting and buckling about its weaker axis, is one of the most common failure modes. This dissertation focuses primarily on the lateral-torsional buckle of non-prismatic I-beams. The development of differential equations for deformation analysis of the non-prismatic beam. ANSYS results and deformation equation results are compared to validate the methodology. Using ANSYS, the lateral torsional buckling of a non-prismatic I-beam section with uniformly distributed load is analyzed. In finite element analyses, the solid element approach is used to determine the lateral buckling load for various cross-sections (β = 0.1 to 1.0) by analyzing their behavior. In addition, a stiffener is used to prevent lateral buckling, and the results are compared to a model without stiffeners.

Shear performance of prestressed composite box beams with ultra-high-performance concrete and corrugated steel webs under different loading conditions

Composite box beams with concrete flanges and corrugated steel webs (CSWs) have been widely adopted in bridge engineering owing to their light weight and ability to prevent web cracking. By replacing conventional concrete flanges with those made of ultra-high-performance concrete (UHPC), a novel non-prismatic prestressed CSW-UHPC composite box beam was designed and proposed in this study to achieve lighter weight, longer span, and more rapid construction for highway bridges. Owing to differences in geometric dimensions and material properties, the shear states in the proposed novel prestressed CSW-UHPC composite box beam may differ from those of conventional beams. In this study, a formula for calculating the shear stress in a non-prismatic beam with CSWs was derived, and a simplified formula was suggested. Additionally, a large-scale non-prismatic prestressed CSW-UHPC composite box beam was designed and tested to investigate the typical shear stress distributions during different construction stages. To simulate the states during construction, six loading conditions were considered by varying the loading positions and boundary conditions. The shear stress distributions along the depth and longitudinal direction of the specimen were obtained and agreed well with the results obtained from the simplified formula and finite element model (FEM). As the inclined bottom flanges contributed a considerable proportion to the shear resistance, the shear bearing ratio of the CSWs varied under different loading conditions, which correlated with the combined effect of internal forces. The effective shear force or shear bearing ratio of the CSWs varied depending on the consistency of the shear force directions. Finally, several design considerations were proposed for the design of novel non-prismatic prestressed CSW-UHPC composite box beams.

A Geometrically Exact Beam Finite Element for Non-Prismatic Strip Beams: Linearized Lateral-Torsional Stability

This paper presents a new geometrically exact beam finite element able to perform, accurately and efficiently, linear stability analyses of curved and tapered elastic strip beams (beams with thin rectangular cross-section) susceptible to lateral-torsional bucking. This element constitutes a non-trivial extension, to the spatial (3D) case, of that previously reported by the author for the in-plane (2D) case in Gonçalves [R. Gonçalves, A geometrically exact beam finite element for non-prismatic strip beams: The 2D case, Int. J. Struct. Stab. Dyn. (2022) 2350037]. To allow capturing accurately torsion effects, a torsion-related warping function is derived and included as an additional cross-section degree-of-freedom (DOF). Since the element is developed using a geometrically exact framework, complex effects such as load height and out-of-plane flexural-torsional effects are straightforwardly included. For implementation purposes, all fundamental expressions are provided in simple vector/matrix forms. A set of numerical examples is presented and discussed, to show that the proposed element provides very accurate solutions with a small DOF number, even for heavily curved and tapered members.