Non-prismatic Beams Research Articles

Stress Analysis of Linear Elastic Nonprismatic Concrete-Encased Beams with Corrugated Steel Webs

The basic mechanical behavior of a nonprismatic concrete-encased beam with corrugated steel webs (CSWs) was investigated through a scale model experiment and finite-element model analysis. Because of the influence of variable cross sections and the concrete encasement, results of this study contradicted the assumption that CSWs bear all of the shear force in a nonprismatic beam. The inclined bottom flange resisted a significant portion of the shear force in the section under longitudinal bending moment. Shear stress in the CSW was uniformly distributed through the cross section when the CSWs were encased in concrete. In addition, the authors made the counterintuitive finding that the average shear stress in the CSW decreased gradually from the free end to the middle support. In other words, the root section near the support did not exhibit the greatest stress concentration, even though this region had the maximum vertical shear force and the largest hogging bending moment. Last, normal strain in concrete flanges and the concrete web approximately obeyed the plane section assumption. The longitudinal bending moment was mainly balanced by the moment from coupled axial forces in the concrete flanges. In general, the CSWs encased in concrete were unable to provide sufficient bending strength.

Planar Timoshenko-like model for multilayer non-prismatic beams

This paper aims at proposing a Timoshenko-like model for planar multilayer (i.e., non-homogeneous) non-prismatic beams. The main peculiarity of multilayer non-prismatic beams is a non-trivial stress distribution within the cross-section that, therefore, needs a more careful treatment. In greater detail, the axial stress distribution is similar to the one of prismatic beams and can be determined through homogenization whereas the shear distribution is completely different from prismatic beams and depends on all the internal forces. The problem of the representation of the shear stress distribution is overcame by an accurate procedure that is devised on the basis of the Jourawsky theory. The paper demonstrates that the proposed representation of cross-section stress distribution and the rigorous procedure adopted for the derivation of constitutive, equilibrium, and compatibility equations lead to Ordinary Differential Equations that couple the axial and the shear bending problems, but allow practitioners to calculate both analytical and numerical solutions for almost arbitrary beam geometries. Specifically, the numerical examples demonstrate that the proposed beam model is able to predict displacements, internal forces, and stresses very accurately and with moderate computational costs. This is also valid for highly heterogeneous beams characterized by thin and extremely stiff layers.

Optimal vibration energy harvesting from non-prismatic axially functionally graded piezolaminated cantilever beam using genetic algorithm

Optimal vibration–based energy harvesting from the axially functionally graded non-prismatic piezolaminated cantilever beam using finite element and genetic algorithm has been proposed in this article. The functionally graded material (i.e. non-homogeneous) in the axial direction is considered, where the cross section is varied (continuously decreasing from root to tip of such cantilever beam) using a proposed power law formula. The shape variations of piezolaminated cantilever (such as linear, parabolic and cubic) have been modelled using the Euler–Bernoulli beam theory. Hamilton’s principle is used in order to derive the governing equation of motion. The governing equation is solved by considering two-noded beam element with 2 degrees of freedom at each node. The responses (such as frequency, voltage and output power) are compared between uniform and the axially functionally graded beam with arbitrary power gradient index. The effects of taper (both in the width and height directions) on output power along with frequency and voltage are analysed for axially functionally graded beam. In order to maximise the output power within the allowable limits of voltage and stresses, a real-coded genetic algorithm–based constrained optimisation technique has been proposed.

Serviceability Analysis of Non-Prismatic Timber Beams: Derivation and Validation of New and Effective Straightforward Formulas

This paper provides innovative and effective instruments for the simplified analysis of serviceability limit states for pitched, kinked, and tapered GLT beams. Specifically, formulas for the evaluation of maximal horizontal and vertical displacements are derived from a recently-proposed Timoshenko-like non-prismatic beam model. Thereafter, the paper compares the proposed serviceability analysis formulas with other ones available in literature and with highly-refined 2D FE simulations in order to demonstrate the effectiveness of the proposed instruments. The proposed formulas lead to estimations that lie mainly on the conservative side and the errors are smaller than 10% (exceptionally up to 15%) in almost all of the cases of interest for practitioners. Conversely, the accuracy of the proposed formulas decreases for thick and highly-tapered beams since the beam model behind the proposed formulas cannot tackle local effects (like stress concentrations occurring at bearing and beam apex) that significantly influence the beam behavior for such geometries. Finally, the proposed formulas are more accurate than the ones available in literature since the latter ones often provide non-conservative estimations and errors greater than 20% (up to 120%).

Numerical Study on Lateral-torsional Buckling of Honeycomb Beam

A honeycomb beam is a non-prismatic wide flange beam with constant spaced hexagonal holes throughout its length. Most design codes, including Specification for Structural Steel Buildings (ANSI/AISC 360-10) do not provide provisions for lateral-torsional buckling of non-prismatic beam. In this paper, lateral-torsional buckling of honeycomb beam is studied numerically using finite element analysis (FEA). Nonlinear time history analysis is performed to simulate the behavior of the beam loaded perpendicular to its major axis gradually from zero until lateral-torsional buckling occurs. Two loading conditions are considered, namely concentrated and uniformly distributed loads. Nonlinear behavior of steel material and the presence of residual stress in the beam are considered in the analysis. The beam is assumed to be simply supported and laterally supported at both ends so that the unbraced length of the beam is the same as the beam length. Honeycomb beams with various dimensions and various unbraced length are analyzed. The lateral-torsional buckling moments obtained from finite element analyses are compared with those computed using equations in ANSI/AISC 360-10 assuming the beams were prismatic without any holes. Based on this comparison, simplified equations to predict lateral-torsional buckling moments or LTB strength of honeycomb beams under major axis bending are proposed in this paper.

Bending Capacity of Non-prismatic LVL Beams Paraserianthes Falcataria

Laminated Veneer Lumber (LVL) made from Indonesian Paraserianthes falcataria wood species has come to market since couple years ago. Paraserianthes falcataria is a typical of fast growing species having an average oven-dry specific gravity of 0.26. As an effort to develop guidance for an optimum utilization of this LVL in structural building applications such as three-hinged gable frame structures, bending test of full-scale non-prismatic LVL beams was conducted to evaluate its bending capacity as well as to observe its failure mechanism. The beams had a length of 4000 mm and a cross-section of 200 mm by 200 mm at one end and 200 mm by 400 mm at the other end. A four-point bending test configuration was adopted and distance between two supports is 3750 mm. The maximum bending load Obtained from the test was 140 kN and the beams failed due to crack opening in the tension-side beneath the applied load. The FEA was performed by ADINA and validated by the full-scale test results to gain better FE models for future use. The finite element model of the used plane stress assumption and orthotropic material. Non-linear finite element analysis was performed where the stress-strain relations in tension was assumed to be linear, while the stress-strain relation in compression was represented by a bi-linear models. Tsai-Wu failure criterion was facilitated in the FE models. Numerical analysis was able to predict the bending capacity of non-prismatic LVL Paraserianthes falcataria beam.

Application of differential transformation finite element method in aperiodic vibration of non-prismatic beam

The paper presents the combination of multi-step differential transformation method (MsDTM) and finite element method (FEM) in vibration analysis of non-prismatic beam. A new algorithm, named differential transformation finite element method (DTFEM), combines the advantages of both methods. It enables to solve a wide class problems and obtain solutions in the form of piecewise functions that are very useful in further analyses, especially in time-consuming computations of dynamic systems. In contrary to standard DTM, with the proposed method, we can successfully calculate vibrations of non-prismatic structures governed by partial differential equations with variable coefficients. The application of FEM reduces partial differential equation to system of ordinary differential equations depending on time. After applying multi-step DTM the system is converting into a set of recursive algebraic equations which can be solved in a symbolic way. The final approximate solution is composed of a sequence of single solutions obtained in every time interval.The algorithm of DTFEM was described and illustrated with an example of non-prismatic simply supported beam subjected to uniformly distributed load increasing nonlinearly in time. The results in the form of beam displacements, velocities and accelerations were compared with numerical approach based on Newmark method. Two cases of damping model were taken into account: structural damping model and external damping model represented by two viscous dampers. A very good agreement was found in both cases. Presented analyses show that DTFEM is more accurate, with the same time step, than Newmark method. Therefore, in some cases, it may be possible to obtain sufficiently accurate results faster. The main advantage of the proposed method is the solution’s form which is a function of time and not a discrete set of results.

Lateral Torsional Buckling of Castellated Beams Analyzed using the Collapse Analysis

In designing a steel beam, one of the limit states is lateral torsional buckling. The value of the bending moment limit is called the critical moment. AISC specification only provides formulas for calculating the critical moment in prismatic beams, but not for non-prismatic beams like castellated beams. The objective of this paper is to study the lateral torsional buckling of castellated beams. The method used for this study is the collapse analysis using the finite element method, analyzed by ADINA 8.9. In the collapse analysis, a geometrically imperfect beam is loaded with an increasing load. At a certain value of the load, the lateral displacement increases progressively. The value of the maximum load is considered to be the critical load. In this study, the imperfection is taken 1/1500 of the unbraced length of the beam. The critical moment of castellated beam obtained from the finite element analysis is up to 42.31% lower than the critical moment of full beams with the same dimension calculated by using the AISC formula. Therefore, the correction factor for AISC formula to calculate the critical moment of the castellated beam is established.

The Combination of Multi-Step Differential Transformation Method and Finite Element Method in Vibration Analysis of Non-Prismatic Beam

The paper presents a new algorithm being the combination of multi-step differential transformation method (MsDTM) and finite element method (FEM) as a powerful tool for solving variety dynamic problems. The proposed algorithm, named as differential transformation finite element method (DTFEM), transforms partial differential equation into a set of recursive algebraic equations. The final form of a solution is a piecewise function which in general case may be a symbolic function. High effectiveness and accuracy of DTFEM is demonstrated on the example of forced vibrations of non-prismatic Euler–Bernoulli beam. Computed time histories of displacements, velocities and accelerations are highly consistent with results obtained by Newmark method.

Static deformation modeling and analysis of flexure hinges made of a shape memory alloy

The flexure hinge is a key element in compliant mechanisms to achieve continuous motion; however the motion range of a flexure hinge is severely restricted by the material’s allowable strain. Due to the superelasticity effect, shape memory alloys (SMAs) can undergo much larger strain than other metals; this means that they are excellent candidates for the fabrication of flexure hinges with a large motion range. In this paper, a simple static deformation modeling approach is proposed for a flexure hinge made of a SMA. The superelastic behavior of the SMA is described by Brinson’s constitutive model. The flexure hinge is considered as a non-prismatic cantilever beam associated with geometrical and material nonlinearities. Govern equations of the flexure hinge are derived and solved numerically by applying the nonlinear bending theory of the Euler–Bernoulli beam. Experimental tests show that the proposed modeling approach can predict the deformation of the flexure hinge precisely; the maximum relative error is less than 6.5%. Based on the static deformation model, the motion capacity, the stiffness characteristic and the rotational error of the flexure hinge are also investigated. The results reveal that the flexure hinge made of a SMA has great potential to construct compliant mechanisms with a large motion range.

Deformation analysis of a non-prismatic beam with corrugated steel webs in the elastic stage

Based on the concept of isoparametric mapping, a generic approach for investigating the bending and shear deformation in a non-prismatic elastic beam with corrugated steel webs (CSWs) was presented in this study. The strain field of an actual element (namely, the non-prismatic beam segments) is expressed as a function of the local coordinates in the parent element. Continually, the solution of the bending and shear deformation of the non-prismatic beam segment can be transformed into the integration of strain in the parent element under local coordinates. Since it is usually laborious to solve the displacement field by direct integration of strain in a non-prismatic beam segment using global coordinates. In addition, this study found that shear deformation cannot be ignored in the actual design of this type of structure because the thin-walled section has a small shear stiffness. In contrast, for a composite beam with CSWs, shear deformation occurs beyond bending deformation and becomes a major contributor of total deformation in some critical sections. Finally, experimental study of the mechanical properties of a non-prismatic beam with CSWs is conducted for the first time. The experimental results further demonstrate the validity of the proposed theoretical analysis methods.

Deflection Analysis of Non Prismatic Beam with Trapezoidal Sectorial Section with Uniformly Perpendicular Loading Condition

Non-prismatic beams are extensively used for many engineering application. Due to varying section, deflection analysis becomes very complex. Loading condition also makes the whole theory complex. This paper shows analytical analysis for the deflection of trapezoidal sectorial section with uniformly perpendicular loading condition. The method illustrated is comparatively easy and can be applied for the similar other sections as well. Analytical method is validated using Finite Element Analysis using Creo Simulation which shows good amount of match with analytical result.

Geometric nonlinear finite element and genetic algorithm based optimal vibration energy harvesting from nonprismatic axially functionally graded piezolaminated beam

The present article deals with optimal vibration energy harvesting from an axially functionally graded (FG) non-prismatic piezolaminated beam using genetic algorithm (GA). Geometric nonlinear based finite element (FE) formulation using Newmark method in conjunction with Newton-Raphson method is formulated to solve the obtained governing equation. A two noded beam element with two degrees of freedom (DOF) at each node is used in the present FE formulation. The FG material (i.e. non-homogeneity) in the axial direction is considered which varies (continuously decreasing from root to tip of such cantilever beam) using a proposed power law formula. The various cross section profiles (such as linear, parabolic and cubic) are modelled using the Euler-Bernoulli beam theory. The effects of nonlinearity on the responses (such as displacement, voltage and output power) are deliberated with arbitrary power gradient index. The effects of tapers (both width and height in length directions) on the output power and voltage are analysed. A real-coded genetic algorithm based constrained optimization technique is also proposed to determine the best possible design variables for optimal power harvesting within the allowable limits of ultimate stress of beam and breakdown voltage of PZT sensor.

Equivalent computational models and deflection calculation methods of box girders with corrugated steel webs

This study found that it is improper to use equivalent concrete webs (ECWs) to replace corrugated steel webs (CSWs) in the MIDAS Civil. To reflect the mechanical performance difference of CSWs in different directions, a more precise theoretical model, the assimilated orthotropic plate (AOP), is presented to simulate CSWs. 3D Finite element analysis demonstrates that the AOP model can yield a more precise description of CSWs compared with the ECW model in MIDAS. In addition, this paper presents a unified calculation method for bending deformation suitable for the analysis of beams with equal-area or variable-area cross-sections, in which the non-prismatic beam is made equivalent to the prismatic beam by introducing the concept of equivalent inertia moment. Additionally, the formula of shear deformation in a non-prismatic box girder with CSWs is also derived with Timoshenko beam theory. These derived formulas for non-prismatic box girder with CSWs agree well with the numerical and measured data.

Serviceability of Fabric-Formed Concrete Structures

Fabric form-work is a newly developed technique to cast concrete structures with a great advantage of saving concrete material of up to 40%. This technique is particularly associated with optimized concrete structures that usually have smaller cross-section dimensions than equivalent prismatic members. However, this can make the structural system produced from these members prone to smaller serviceability safety margins. Therefore, it is very important to understand serviceability issue of non-prismatic concrete structures. In this paper, an analytical computer-based model to optimize concrete beams and to predict load-deflection behaviour of both prismatic and non-prismatic concrete beams is presented. The model was developed based on the method of sectional analysis and integration of curvatures. Results from the analytical model were compared to load-deflection behaviour of a number of beams with different geometric and material properties from other researchers. The results of the comparison show that, the analytical program can accurately predict load-deflection response of concrete beams with medium reinforcement ratios. However, it over-estimates deflection values for lightly reinforced specimens. Finally, the analytical program acceptably predicted load-deflection behaviour of on-prismatic concrete beams.

Multi-objective optimization of a type of ellipse-parabola shaped superelastic flexure hinge

Abstract. Flexure hinges made of superelastic materials is a promising candidate to enhance the movability of compliant mechanisms. In this paper, we focus on the multi-objective optimization of a type of ellipse-parabola shaped superelastic flexure hinge. The objective is to determine a set of optimal geometric parameters that maximizes the motion range and the relative compliance of the flexure hinge and minimizes the relative rotation error during the deformation as well. Firstly, the paper presents a new type of ellipse-parabola shaped flexure hinge which is constructed by an ellipse arc and a parabola curve. Then, the static responses of superelastic flexure hinges are solved via non-prismatic beam elements derived by the co-rotational approach. Finite element analysis (FEA) and experiment tests are performed to verify the modeling method. Finally, a multi-objective optimization is performed and the Pareto frontier is found via the NSGA-II algorithm.

A novel stiffness/flexibility-based method for Euler–Bernoulli/Timoshenko beams with multiple discontinuities and singularities

Nonprismatic beams have vast and various applications in mechanical and structural systems; thus, much research is dedicated to develop well-performed stiffness matrices for beams with different forms of section changes and singularities. This paper introduces stiffness matrices by the use of flexibility and stiffness methods. For this purpose, the spring model of the beam element is introduced. This model provides an innovative physical interpretation for the beam element. It can also be used to develop finite elements for both Euler–Bernoulli and Timoshenko beams with different combinations of tapering, singularity and discontinuity. In this model, beam sections are represented by some appropriate virtual springs; their connection in series/parallel gives the flexibility/stiffness matrix of the beam element. The obtained matrices are in general forms and applicable to different beam conditions.

Non-prismatic beams: A simple and effective Timoshenko-like model

The present paper discusses simple compatibility, equilibrium, and constitutive equations for a non-prismatic planar beam. Specifically, the proposed model is based on standard Timoshenko kinematics (i.e., planar cross-section remain planar in consequence of a deformation, but can rotate with respect to the beam center-line). An initial discussion of a 2D elastic problem highlights that the boundary equilibrium deeply influences the cross-section stress distribution and all unknown fields are represented with respect to global Cartesian coordinates. A simple beam model (i.e. a set of Ordinary Differential Equations (ODEs)) is derived, describing accurately the effects of non-prismatic geometry on the beam behavior and motivating equation’s terms with both physical and mathematical arguments. Finally, several analytical and numerical solutions are compared with results existing in literature. The main conclusions can be summarized as follows. (i) The stress distribution within the cross-section is not trivial as in prismatic beams, in particular the shear stress distribution depends on all generalized stresses and on the beam geometry. (ii) The derivation of simplified constitutive relations highlights a strong dependence of each generalized deformation on all the generalized stresses. (iii) Axial and shear-bending problems are strictly coupled. (iv) The beam model is naturally expressed as an explicit system of six first order ODEs. (v) The ODEs solution can be obtained through the iterative integration of the right hand side term of each equation. (vi) The proposed simple model predicts the real behavior of non-prismatic beams with a good accuracy, reasonable for the most of practical applications.

Finite Element Based Vibration Analysis of a Nonprismatic Timoshenko Beam with Transverse Open Crack

The present day structures and machineries are designed based on optimizing of multi-objectives such as maximum strength, maximum life, minimum weight and minimum cost. Due to this they are flexible and allow having a very high level of stresses. This leads to development of cracks in their elements. Many engineering structures may have structural defects such as cracks due to long-term service. So it is very much essential to know the property of structures and its response in various cases. The present article deals with finite element based vibration analysis of a nonprismatic cracked beam. The beam is modeled using the Timoshenko beam theory. The governing equation of motion is derived by the Hamilton's principle. In order to solve the governing equation two noded beam element with two degrees of freedom (DOF) per node is considered. In this work the effect of structural damping is also incorporated in the finite element model. The dynamic analysis is carried out by using state space model in time domain.

Finite Element Based Modeling of a Piezolaminated Tapered Beam for Voltage Generation

The present article deals with finite element (FE) based modeling of a nonprismatic piezolaminated cantilever beam for voltage generation. The beam is modeled using the Euler-Bernoulli beam formulation. The governing equation of motion is derived by using the Hamilton's principle. In order to solve the governing equation two noded beam element with two degrees of freedom (DOF) per node is considered. In this work the effect of structural damping is incorporated in the finite element model. The effects of taper (both in width and height direction) on output voltage are discussed as well.