Non-prismatic Beams Research Articles

Modified bar simulation method for shear lag analysis of non-prismatic composite box girders with corrugated steel webs

In this study, a modified bar simulation method is proposed for analyzing the shear lag effect of non-prismatic composite box girders with corrugated steel webs (CBGCSWs) during the elastic stage. In this theoretical method, formulas for the equivalent area of stiffening bar and the shear flow in corrugated steel webs (CSWs) are derived by properly considering the mechanical properties of non-prismatic CBGCSWs. The governing differential equations for shear lag are formulated and solved with given boundary conditions. The feasibility of the proposed method is validated by the finite element (FE) method based on a non-prismatic cantilever beam under different loading conditions. In addition, the shear lag behavior of non-prismatic cantilever CBGCSWs is compared with that of corresponding concrete box girders, and the effects of width-span ratio, girder height ratio, changing curve of girder height and loading form on the shear lag effect of non-prismatic cantilever CBGCSWs are examined based on the modified bar simulation method. Results indicate that non-prismatic cantilever CBGCSWs exhibit more significant shear lag behavior than the corresponding concrete box girders under gravity. The distribution rule of shear flow in concrete flanges, which first increase and then decrease from the cantilever end to the fixed end, causes the transformation between negative and positive shear lag effects in non-prismatic cantilever beams. Increasing the width-span ratio or girder height ratio leads to more serious shear lag effect. The effect of loading form on the shear lag behavior is significant, while the changing curve of girder height has minimal influence.

Lateral-Torsional Buckling of Singly Symmetric I-Girders with Stepped Flanges

Guidance for the lateral-torsional buckling behavior of singly-symmetric, nonprismatic (stepped flanges) sections are often absent from design specifications or are overly simplified. This paper presents simplified design procedures that estimate the buckling capacity of these complex systems that commonly face designers. The results of a robust parametric finite-element study that included 14,040 unique prismatic and nonprismatic beam sections are detailed. The study specifically examines the effects of common span-to-depth ratios, intermediate bracing schemes, degrees of monosymmetry, variable flange transitions, and moment gradients on the buckling response. A proposed weighted-average section approach as well as traditional moment gradient expressions are evaluated based on their ability to accurately approximate the finite-element solutions of these singly-symmetric and/or nonprismatic beams. The computational study and proposed design expressions focus on elastic buckling behavior, midheight loads, and stiffened web elements, such that distortional effects that may limit the beam capacity are precluded. This approach matches the assumptions used to derive the classic analytical solutions adopted in most design specifications. For a broad range of conditions, the proposed methods are shown to produce reasonable and reliable estimates of the buckling capacity obtained from the finite-element models.

Full-Range Shear Behavior of a Nonprismatic Beam with Steel Trapezoidal Corrugated Webs: Experimental Tests and FE Modeling

This paper presents the first experimental and numerical investigation of the full-range shear behavior of a nonprismatic overhanging beam with concrete slabs and steel trapezoidal corrugated webs. The results showed that the traditional calculation hypothesis that the steel trapezoidal corrugated webs resist the total vertical shear in a section was not applicable to nonprismatic members. The inclined bottom concrete slab shared a significant portion of the shear force in the region with a larger hogging moment; thus, the actual shear force experienced by the steel trapezoidal corrugated webs subsequently decreased. The dominant factor of the shear force redistribution between the concrete slabs and the steel trapezoidal corrugated webs was the Resal effect, which was also a crucial factor contributing to the differences in shear behavior between prismatic and nonprismatic beams with steel trapezoidal corrugated webs. In addition, eigenvalue buckling analyses of geometric imperfections and nonlinear buckling analyses using the Riks algorithm were carried out for a nonprismatic beam with steel trapezoidal corrugated webs considering material and geometric nonlinearities. The experiment also indicated that the cracking and crushing of the concrete slabs was another important factor affecting the ultimate bearing capacity of the test beam. A significant finding that may invalidate intuitive knowledge was that the shear buckling of the steel trapezoidal corrugated webs did not occur at the support section, although this section bore the maximum shearing force and bending moment and had the highest section depth. Because of the Resal effect, the maximum shear stress and shear buckling in the steel trapezoidal corrugated webs were located at a certain distance from the support.

Sustainable design of reinforced concrete frames with non-prismatic beams

In this paper, sustainable design of reinforced concrete frames with non-prismatic beams is studied and relationship between optimal cost and optimal carbon dioxide emission in design of this type of frames is investigated. The objective functions are minimizing the CO2 emissions and construction cost, and the design variables are defined as the geometry of the cross sections and reinforcing bars of the beams and columns, as well as the tapered length ratio for the non-prismatic beams. The performance of five metaheuristic algorithms consisting of the particle swarm optimization, colliding bodies optimization, enhanced colliding bodies optimization, vibrating particles system and enhanced vibrating particles system is compared for solution of Example 1. The most competent algorithm is identified, and the remaining examples are optimized using the identified algorithm to minimize the cost and CO2 emissions. Numerical examples include reinforced concrete frames with prismatic beams, and frames with non-prismatic beams are optimally designed under equivalent static analysis to evaluate the objectives. Study of the results shows that compared to prismatic beams, in addition to achieving acceptable height in the floor of the structure, the use of non-prismatic beams in RC frames reduces the optimal economic costs and optimal CO2 emissions.

Structural analysis of non-prismatic beams: Critical issues, accurate stress recovery, and analytical definition of the Finite Element (FE) stiffness matrix

Non-prismatic beams are widely employed in strategic structures like bridges and sport arenas, requiring accurate analyses for a reliable and effective design. Unfortunately, features of non-prismatic beams lead their modeling to be a non-trivial task: (i) variations of both cross-section area and second moment of area impede an easy computation of analytical solutions compelling to use approximated methods; (ii) stress distributions in prismatic and non-prismatic beams are substantially different, as proved by analytical results available since the beginning of the past century; and (iii) the peculiar stress distribution in non-prismatic beams entails complicated constitutive relations, as highlighted in recent publications. Usually, commercial software does not properly account for all the features of non-prismatic beams, leading to inconsistent structural analyses, erroneous estimations of the stress distribution, and -consequently- coarse predictions of the structural element strength. The present paper proposes a strategy to effectively overcome the above-mentioned problems. We derive an accurate analytical model for 2D non-prismatic beams, able to handle the non-trivial stress distribution and the complicated constitutive relations. Thereafter, we compute both homogeneous and particular solutions using the symbolic calculus software MAPLE and we analytically define the Finite Element (FE) stiffness matrix for a planar, symmetric, linearly-tapered beam. Finally, we compare the proposed FE and SAP2000 solutions, considering several beams with different geometries, loads, and constraints. Numerical results highlight the reliability of the proposed modeling strategy, since the resulting FE consistently handles all the critical issues of non-prismatic beams with an extremely low computational cost. Conversely, SAP2000 solution remarks the need of ad hoc analysis tools and modeling strategies to be used for the design of non-prismatic structural elements.

Optimal design of reinforced concrete beams for rectangular sections with straight haunches

Objective of this research is to present a mathematical model for optimal design of rectangular cross-section beams with straight haunches under the criterion of minimum cost considering the concrete cost and reinforcing steel cost, and taking into account the equations of the regulation (ACI 318S-14). This model presents the equations for a uniformly distributed load and a concentrated load located anywhere on the beam. Two examples are developed by the proposed model, one for uniformly distributed load and another for concentrated load showing the best solution for each case. The results show the following: a) The prismatic beams for uniformly distributed load have a total cost of the 8% greater, a total volume and a total weight of the 9% greater with respect to the non-prismatic beams; b) The prismatic beams for concentrated load have a total cost, a total volume and a total weight of the 6% greater with respect to the non-prismatic beams. The main conclusions are: For a smaller b width the optimal design for both models is presented. The non-prismatic beams are more economical, also these have less volume and less weight with respect to prismatic beams.

Distribution and Properties of Shear Stress in Elastic Beams with Variable Cross Section: Theoretical Analysis and Finite Element Modelling

It was found that additional shear stress arising from the bending moment should be involved in the formulation due to the effect of variable cross section, which is quite different from the conventional method of analysis for the prismatic members. This paper focuses on the analytical formulation, properties and distribution regularity of the shear stress for the elastic tapered beams with rectangular cross-section. Currently, an analytical expression for the shear stress in elastic tapered beams is derived theoretically based on the theory of elasticity. Most notably, it is strictly proved in mathematics that the additional shear stresses arising from bending moment are self-balanced which only change the distribution of shear stress in the section but will not affect the shear stress resultant. This new finding will be beneficial to promoting the development and consummation of non-prismatic beam theory. The correctness of the theoretical formula have been verified through comparisons with finite element (FE) results of two groups with a total of 12 three-dimensional FE models. The validity of self-balanced properties of additional shear stress in tapered beams under pure bending is additionally verified by using Free Body Cuts in ABAQUS program.

Large Deflection Analysis of Prismatic Cantilever Beam Comparatively by Using Combing Method and Iterative DQM

There is no exactly analytical solution for the large deflection problem of prismatic cantilever beams under general loading conditions. In the case of considering a non-prismatic cantilever beam, the difficulty of the larger deflection problem is increased. In this study, the comparison of the Iterative Differential Quadrature Method (I-DQM) and the Combining Method (CM) was performed. Numerical solution of the large deflection problem was separately performed with both the I-DQM and the CM for prismatic cantilever beams. The obtaining results show that both of these methods gave more accurate solutions compared with a reliable semi-analytic method which was introduced by Dado and Sadder (2005). Besides, it was demonstrated that the I-DQM provided a more wide-range solution than the CM.

Free vibration of non-prismatic beam on variable Winkler elastic foundations

In this paper, the finite difference and the finite element methods are applied to evaluate natural frequencies of non-prismatic and non-homogeneous beams, with different boundary conditions and resting on variable Winkler foundation. The finite difference method is used for solving differential equation of motion, especially with variable coefficients. This technique requires a lesser computing effort and is used in situation where the exact solution is very difficult to obtain. The main idea of this method is replacing derivatives present in the free vibration equation and boundary condition equations with finite difference expressions. The natural frequencies are determined by solving the eigenvalue problem of the obtained algebraic system resulting from finite difference method. In order to illustrate the correctness and performance of the method, a comprehensive numerical example of non-prismatic beams is presented. The results are compared with the finite element results using ABAQUS software and other available numerical and analytical solutions.

Series solution of beams with variable cross-section

In structural engineering beams with non-constant cross-section or beams with variable cross-section represent a class of slender bodies, aim of practitioners’ interest due to the possibility of optimizing their geometry with respect to specific needs. Despite the advantages that engineers can obtain from their applications, non-trivial difficulties occurring in the non-prismatic beam modeling often lead to inaccurate predictions that vanish the gain of the optimization process. As a consequence, an effective non-prismatic beam modeling still represents a branch of the structural engineering of interest for the community, especially for advanced design applications in large spans elements.A straight beam of length l with variable inertia J(z) is provided in figure, subject to a generic live load condition q(z). The vertical displacement y(z) can be obtained from the solution of the differential equation of the elastic line, i.e., taking into consideration the inertia variability and neglecting, as first approximation, any shear contribution. Even if this solution is an approximate one, it is able to deal with the problem in its basic formulation.In this paper a solution for the problem stated is formulated using a series expansion of solutions, in a general load and cross section variability condition. Solution is thus obtained for a generic rectangular cross section beam with a variable height. Analytical solution is presented and evaluated using numerical evaluation of some cases of practical interest.

Numerical analysis of nonhomogeneous and nonprismatic members under generalised loadings

This research studied the problem of varying depth and varying elastic properties along flexural member lengths subjected to generalised loading, with such members considered to be nonhomogeneous and non-prismatic. The differential equation for classical thin beams was thus derived and the finite differences used in solving this equation. Fortran programs were written to solve the problem in finite differences, and three-dimensional elements were used to simulate or model the beams in finite element ABAQUS software. A parametric study of the influence of beam width, depth, load nature, and boundary conditions on deformations and internal forces was thus accomplished. The maximum variation in deformations seen between this study and previous studies was 8%. The deflection of the non-prismatic beam was reduced by 31% when the ratio of the beam moment of inertia at different sections increased by up to three times, while, the moment capacity and the right end support shear capacity were increased by 78% and 50%, respectively.

In-Situ Dynamic Response Measurement for Damage Quantification of 3D Printed ABS Cantilever Beam under Thermomechanical Load.

Acrylonitrile butadiene styrene (ABS) offers good mechanical properties and is effective in use to make polymeric structures for industrial applications. It is one of the most common raw material used for printing structures with fused deposition modeling (FDM). However, most of its properties and behavior are known under quasi-static loading conditions. These are suitable to design ABS structures for applications that are operated under static or dead loads. Still, comprehensive research is required to determine the properties and behavior of ABS structures under dynamic loads, especially in the presence of temperature more than the ambient. The presented research was an effort mainly to provide any evidence about the structural behavior and damage resistance of ABS material if operated under dynamic load conditions coupled with relatively high-temperature values. A non-prismatic fixed-free cantilever ABS beam was used in this study. The beam specimens were manufactured with a 3D printer based on FDM. A total of 190 specimens were tested with a combination of different temperatures, initial seeded damage or crack, and crack location values. The structural dynamic response, crack propagation, crack depth quantification, and their changes due to applied temperature were investigated by using analytical, numerical, and experimental approaches. In experiments, a combination of the modal exciter and heat mats was used to apply the dynamic loads on the beam structure with different temperature values. The response measurement and crack propagation behavior were monitored with the instrumentation, including a 200× microscope, accelerometer, and a laser vibrometer. The obtained findings could be used as an in-situ damage assessment tool to predict crack depth in an ABS beam as a function of dynamic response and applied temperature.

Tapered beam–column analysis by analytical solution

By calculating the accurate solution of the governing differential equation for a non-prismatic beam–column, the exact elemental stiffness matrix will be obtained. The Euler–Bernoulli beam theory is used in the proposed formulation. This study considers the second-order effects and flexibility of the connections and supports. The effects of the ends’ flexibilities are investigated by utilising the general boundary conditions of the governing differential equation. In addition, shape factors of 2 and 4 in which the cross-section moment of inertia changes by second- and fourth-order functions through the length, respectively, are studied separately. The suggested stiffness matrix is general and can be utilised in the various analyses for a non-prismatic member belonging to a plane frame with different types of connections and restraints. It should be added that the presented stiffness matrix can be used in the finite-element method. To demonstrate the abilities of the formulation, stability analyses of columns and frames with non-prismatic members are performed. Comparing the responses with the available results shows the accuracy, efficiency and capabilities of the presented matrix.

Transverse free vibration analysis of a tapered Timoshenko beam on visco-Pasternak foundations using the interpolating matrix method

The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the effects of the Winkler coefficient, Pasternak coefficient and damping coefficient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary differential equations with variable coefficients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verified through two numerical examples. The influences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with different taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.

A static and free vibration analysis method for non-prismatic composite beams with a non-uniform flexible shear connection

Steel-concrete composite beams are widely used in practice because of their economic cross-section design. As sustainability becomes more and more important in the construction industry, the design of composite beams must be adapted to meet the requirements of the circular economy. This calls for demountability and reusability of the structural components, as well as optimized use of materials, for example by using non-prismatic beams. Linear-elastic design and the (optimized) use of demountable shear connectors are key in the design of reusable composite structures. In this paper, analytical prediction models for the elastic behaviour and the first eigenfrequency of non-prismatic composite beams with non-uniform shear connector arrangements are derived. The approach is based on 6th and 2nd order differential equations used to define matrix equations for a finite number of linearized composite beam segments. The analytical models are validated using experimental and numerical results obtained with a simply supported tapered composite beam. The analytical models are suitable for comprehensive structural analysis of non-prismatic composite beams with non-uniform shear connection.

Serviceability of non-prismatic concrete beams: Combined-interaction method

Interest in the shape optimisation of concrete members is increasing alongside the availability of fabric formwork as a relatively simple technique to cast non-prismatic concrete structures. Research has shown that up 40% of concrete can be saved when shape optimised concrete beams are cast in fabric forms. However, optimisation results in members with non-uniform cross-sections and the resulting beam is less stiff than an equivalent strength prismatic beam. Serviceability, rather than strength, may govern the design of such members and therefore understanding the serviceability behaviour (deflection and cracking) of shape optimised concrete members becomes is a critical design consideration. There are many methods which can be used to evaluate serviceability behaviour of reinforced concrete beams, including the full-interaction method, which assumes no slip between the reinforcement and the surrounding concrete, and the partial-interaction method which accounts for slip. The full-interaction method is based on a smeared crack approach and so is unsuited for the prediction of cracking behaviour. The partial-interaction method, on the other hand, assumes that cracks form through bond-stress transfer only. In the case of non-prismatic concrete beams, the cracking capacity varies along the member. Therefore, cracking can occur over extended regions (full and partial bond interaction regions) and so it can be argued that neither of these models is fully suitable for the prediction of deflections and cracking of shape-optimised concrete beams. In this paper, a novel combined-interaction method is, for the first time, presented to predict the serviceability behaviour of non-prismatic statically determinate concrete beams by simulating both full and partial bond interactions at different cracked and uncracked regions along the length of the member. In order to validate this approach, two non-prismatic simply supported beams were cast and tested. The test results for deflections, crack widths and crack spacings were in good agreement with the predicted results.

The differences in the mechanical performance of prismatic and non-prismatic beams with corrugated steel webs: A comparative research

The differences in the mechanical performance of prismatic and non-prismatic beams with concrete flanges and corrugated steel webs (CSWs) are numerically investigated in this paper with respect to their behaviour in bending, shear and shear buckling. With regard to the bending behaviour, it is found that the quasi-plane assumption is still valid for the non-prismatic beams with CSWs, similar to the prismatic beams with CSWs, at their elastic stage and their bending stiffness are mainly provided by the concrete flanges. This is attributed to the accordion effect of such webs. In terms of the shear behaviour, it is found that the inclined bottom flange shares a significant portion of the shear force in the non-prismatic beam with CSW, while the effective shear force carried by the CSW is greatly reduced; this phenomenon is called the Resal effect which is the most important reason leading to the shear performance difference between the prismatic and non-prismatic beams with CSWs. Thus, the traditional calculation assumption that the CSWs bear all the vertical shear force in the prismatic beams with CSWs is no longer applicable in calculating the shear stress in the non-prismatic cases. Additionally, it is noticed from two numerical cases, simulating the prismatic and non-prismatic beams with CSWs under end loading, that the average shear stress in the CSW of the non-prismatic beam decreases gradually from the free-end to the fixed-end with the increase of the hogging bending moment, while the shear stress in the CSW remains nearly constant in the prismatic beam. The numerical results, as well, show that the root cross-section, although it bears the maximum bending moment and shear force, is not the critical shear section of the CSW of the non-prismatic beam. Finally, considering the shear buckling behaviour, it is found that the shear buckling stress of the non-prismatic with CSW is much greater than that of an equal-weight prismatic beam with CSW through eigenvalue buckling analyses.

Bi-directional functionally graded thin-walled non-prismatic Euler beams of generic open/closed cross section Part I: Theoretical formulations

This paper presents a detailed treatment of the formulation for buckling and free vibration analysis of Bi-Directional Functionally Graded (BDFG) Thin-Walled non-prismatic beams of generic open/closed cross section. The theory developed is limited to small strains, moderate deflections and small rotations. It is based on the assumption that the shear strain on the mid-surface contour of the cross section of the member is neglected. Using the membrane theory of shells rigorous expressions for strains are obtained by which the effect of nonlinear tapering is considered. The material properties are assumed to be graded both in the longitudinal and depth-wise directions of the plate segment of the thin-walled beams. The governing equations are developed defining the displacements with reference to any arbitrary point (say geometric centre) and get away with the usual shear centre and centroid. For computer solutions for classical buckling and vibration analyses, the Finite Element Method is used. Simpson’s rule of numerical integration is adopted for the computation of axial, coupled and bending rigidities as well as inertial properties of the plate segment and the Gaussian Quadrature of numerical integration is used for the computation of flexural stiffness, geometric stiffness and mass matrices of an element over the length of the beam. Critical buckling loads and the natural frequencies and their corresponding buckled and mode shapes for various examples are obtained by solving as an eigen-value problem and compared with the published results in the companion paper.

Bi-directional functionally graded thin-walled non-prismatic Euler beams of generic open/closed cross section Part II: Static, stability and free vibration studies

In the current study, deflections, buckling loads, natural frequencies and the corresponding mode shapes and forms for BDFG thin-walled tapered beams of generic open/closed/open-closed sections are examined using the finite element method. A tapered thin-walled beam finite element with the seven degrees of freedom at each node is investigated. A rigorous expression for strains based on the membrane theory of shells is considered. The flexural stiffness matrix, geometric stiffness matrix and the consistent mass matrices are developed in a companion paper. Using the present theory, one is able to investigate various torsional, flexural static and dynamic systems. Examples are presented and comparisons are made with the existing available literature. With the modifications of axial, coupled and bending rigidities one will be able to investigate composite beams with fibre reinforced composite laminates.

Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

Abstract This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).