Non-prismatic Beams Research Articles

Shear behavior of prismatic and nonprismatic beams with trapezoidal steel web corrugations considering secondary internal force induced by external prestressing

This paper analyzes the shear behavior of statically determinate and statically indeterminate nonprismatic and prismatic beams with trapezoidal steel web corrugations (TSWCs) under external prestressing. The results indicate that for nonprismatic beams with TSWCs, the bending moments and axial forces can also induce additional shear stresses, which will lead to a redistribution of shear stresses on the sections. Moreover, for the nonprismatic statically indeterminate beams with TSWCs, the secondary bending moment and secondary axial force also affect the shear stresses distribution of the sections. This finding has not been reported previously. Therefore, it is not accurate to deem that all shear stresses on TSWCs are completely generated by shear force in the existing specifications. Then, the differences of the shear behaviors between prismatic and nonprismatic indeterminate beams with TSWCs under external prestressing are explored. Consequently, a new practical formula is derived for calculating the shear stresses on the TSWCs induced by external prestressing. The formula can be applied to not only statically determinate and statically indeterminate prismatic beam, but also statically determinate and statically indeterminate nonprismatic beams with TSWCs under external prestressing.

A Geometrically Exact Beam Finite Element for Non-Prismatic Strip Beams: The 2D Case

In this paper, a new finite element for 2D non-prismatic (curved and tapered) elastic beams with narrow rectangular cross-section is proposed. The element is geometrically exact, hence capable of handling large displacements and finite rotations, and allows for cross-section arbitrary warping through the inclusion of additional degrees of freedom (DOFs). These additional DOFs make it possible to capture the non-standard shear stress distributions that are developed in tapered members. All expressions required to implement the proposed finite element are derived and written in a simple vector/matrix form. To assess the accuracy and demonstrate the capabilities of the proposed element, several numerical examples are presented. For comparison purposes, results obtained with refined meshes of standard 2D finite elements are also shown. It is concluded that the proposed element leads to very accurate results with a small computational cost.

Investigating Static Deflection of Non-Prismatic Axially Functionally Graded Beam

In this study, the static deflection of non-prismatic axial function graded tapered beam (A-FGB) under distribution load has been analyzed using ANSYS workbench (17.2). According to a power-law model, the elastic modulus of the beam varies continuously in the axial direction of the beam. Also, the beam’s geometry, i.e., width, thickness, or both width and thickness of the beam, varies linearly in the axial direction with different values of non-uniformity parameter (1, 0.5,0, −0.5, and −0.75). The effects of martial distribution, i.e., power-law index, and non-uniformity parameter on the static deflection for A-FGB with different boundary conditions, in such free-clamped, clamped-free, and simply-supported, are studied. This research deals with functionally graded materials FGMs in more than one aspect in terms of using different boundary conditions; in addition, it studies the response of the non-prismatic beam non-uniformity parameter (α); therefore, this research studies comprehensively the deflection of the beam. The results show that the increase in power-law index causes decreasing in dimensionless deflection and its rate of change depends on the supporting types of the beam and non-uniformity parameters. The variation in both width and thickness for a free-clamped axial function–graded beam gives a significant decrease in dimensionless deflection at decreasing in non-uniformity parameter, whereas the variation in thickness for clamped-free axial function graded beam gives a significant decrease in dimensionless deflection at decreasing of non-uniformity parameter.

Stress recovery of laminated non-prismatic beams under layerwise traction and body forces

Emerging manufacturing technologies, including 3D printing and additive layer manufacturing, offer scope for making slender heterogeneous structures with complex geometry. Modern applications include tapered sandwich beams employed in the aeronautical industry, wind turbine blades and concrete beams used in construction. It is noteworthy that state-of-the-art closed form solutions for stresses are often excessively simple to be representative of real laminated tapered beams. For example, centroidal variation with respect to the neutral axis is neglected, and the transverse direct stress component is disregarded. Also, non-classical terms arise due to interactions between stiffness and external load distributions. Another drawback is that the external load is assumed to react uniformly through the cross-section in classical beam formulations, which is an inaccurate assumption for slender structures loaded on only a sub-section of the entire cross-section. To address these limitations, a simple and efficient yet accurate analytical stress recovery method is presented for laminated non-prismatic beams with arbitrary cross-sectional shapes under layerwise body forces and traction loads. Moreover, closed-form solutions are deduced for rectangular cross-sections. The proposed method invokes Cauchy stress equilibrium followed by implementing appropriate interfacial boundary conditions. The main novelties comprise the 2D transverse stress field recovery considering centroidal variation with respect to the neutral axis, application of layerwise external loads, and consideration of effects where stiffness and external load distributions differ. A state of plane stress under small linear-elastic strains is assumed, for cases where beam thickness taper is restricted to 15^{circ }. The model is validated by comparison with finite element analysis and relevant analytical formulations.

Performance of reinforced concrete non-prismatic beams having multiple openings configurations

Abstract This experimental study demonstrates the gable-reinforced concrete beams’ behavior with several number of openings (six and eight) and posts’ inclination, aimed to find the strength reduction in this type of beam. The major results found are: for the openings extending over similar beam length it is better to increase the number of posts (openings), i.e., increasing opening number led to decrease in opening area, which allows us to transmit stresses and act as lever arms between the upper and the lower chords. Also, findings revealed that the inclined posts have larger loading at the mid-point relative to vertical ones. For gables with vertical posts having six and eight openings, the ultimate strength reduction was 31.5 and 25.6%, whereas it was 29 and 17.3% for those with inclined posts, respectively.

Analytical evaluation of stresses and strains in inhomogeneous non-prismatic beams undergoing large deflections

Analytical evaluation of stresses and strains in inhomogeneous non-prismatic beams undergoing large deflections

Semi-Analytical Solution for Elastoplastic Deflection of Non-Prismatic Cantilever Beams with Circular Cross-Section

A solution for the elastoplastic deflection of cantilever beams with linearly variable circular cross-section loaded by shear force at the free end, which is suitable for practical use, has not yet been developed. A semi-analytical solution for such a problem is proposed in this paper. The solution involves beams made of homogenous and isotropic materials with bilinear elastoplastic strain hardening behavior. The Bernoulli–Euler formula is used for determining the elastic deflection. However, for the plastic domain of material behavior, the differential equation of beam bending does not have a solution in closed form. Therefore, an incremental procedure for determining the curvature of the plastified region of the beam is suggested. Deflection of the cantilever beam is calculated via integration of the approximated function of the beam curvature. The proposed semi-analytical solution is validated using experimental results of the seismic energy dissipation device components which have been selected as a sample of a real engineering system. Also, validation is done via finite element analysis of six different cantilever beam models with varying geometric and material characteristics. A satisfying agreement between the proposed semi-analytical results and the subsequent experimental and numerical results is herein achieved, confirming its reliability.

Analysis of shear stresses in tapered beams under bending, shear and axial force

This paper mainly studied the analytical fomula, basic properties and distribution regularities of the shear stresses in elastic tapered beams under a combined bending moment, shear force and axial force. First, an analytical expression for calculating shear stress in elastic tapered beams was derived based on the theory of elasticity. The additional shear stresses arising from the bending moment and axial force were involved due to the effect of the variable cross-section, which is quite different from the classical formula for prismatic beams. Additionally, it was mathematically proven that the additional shear stresses arising from the bending moment and axial force were self-balanced, which changed the distribution of shear stress but did not influence the shear stress resultant. The correctness of each sub item of the analytical formula and their distribution regularities were verified through comparisons between the theoretical results and numerical simulation through 12 purposely designed three-dimensional finite element models (FEMs) under three kind of loading conditions: pure bending, shearing-bending and compressing-bending. The correctness of the self-balanced properties of additional shear stresses arising from the bending moment and axial force were also verified using Free Body Cuts in ABAQUS program. These new findings are likely to promote the development and completion of non-prismatic beam theory.

A new shear formula for tapered beamlike solids undergoing large displacements

In many engineering applications it is often necessary to determine the flow of shear stresses in the cross-sections of beamlike bodies. Taking a cue from Jourawski's well-known formula, several scholars have proposed expressions for evaluating the shear stresses in non-prismatic linear elastic beams, where longitudinal variations in the size and shape of the cross-sections produces complex stress fields. In the present paper, a new shear formula, derived using a mechanical model developed in a previous work, is presented for tapered beams subject to even large displacements and small strains. Numerical examples and comparisons with results obtained using other formulas in the literature and non-linear 3D-FEM simulations show how the new formula constitutes an important generalization of the previous ones and is able to provide particularly accurate results.

Static Shear Strength of a Non-Prismatic Beam with Transverse Openings

In this study, a predicated formula is been proposed to find the shear strength of non-prismatic beams with or without openings. It depends on the contributions of concrete shear strength considering the beam depth variation and existing openings, shear steel reinforcements and defines the critical shear section, the effect of diagonal shear reinforcement, the effect of inclined tensile steel reinforcement, and the compression chord influence. The verification of the proposed formula has been conducted on the experimental test results of 26 non-prismatic beams with or without openings at the same loading conditions. The results reflect that the predicted formula finds the shear capacity of non-prismatic beams with openings, it is conservative and can be used for designing without the strength reduction factor.

Investigation on the Bending Behavior of Tea Stalks Based on Non-Prismatic Beam with Virtual Internodes

The study aims to fully explicate the bending behavior of tea stalks under the condition of large deflection, which is crucial to improve the working performance of mechanized harvesting equipment. The mechanical model of the stalk was assumed to be a non-prismatic beam with virtual internodes that could differ from actual internodes. With the model, the stalk can be freely divided into multiple virtual internodes, whose flexural rigidities can be determined by solving an optimization problem, and deflection curves can be predicted after determining the positions of virtual nodes under given loads. Moreover, a novel method was proposed to obtain the deflection curve of the stalk based on the techniques of binocular vision and non-uniform rational B-spline (NURBS) curve fitting. The results show that R-squared values of fitted 2nd-degree NURBS curves of bending shape of tea stalks range from 0.9576 to 0.9964, with an average of 0.9797. The results indicate that flexural rigidity decreases from the bottom to the top of the tea stalk, and the deflection curve could be predicted more precisely with the model of piecewise flexural rigidities than that of average flexural rigidity. The study could be applied to the optimization design of the cutter and adaptive adjustment techniques of operational parameters for tea picking machines.

Structural Efficiency of Non-Prismatic Hollow Reinforced Concrete Beams Retrofitted with CFRP Sheets

Non-prismatic reinforced concrete (RC) beams are widely used for various practical purposes, including enhancing architectural aesthetics and increasing the overall thickness in the support area above the column, which gives high assurance to services that this will not result in the distortion of construction features and can reduce heights. The hollow sections (recess) can also be used for the maintenance of large structural sections and the safe passage of utility lines of water, gas, telecommunications, electricity, etc. They are generally used in large and complex civil engineering works like bridges. This study conducted a numerical study using the commercial finite element software ANSYS version 15 for analysing RC beams, hollow longitudinally sectioned and retrofitted with carbon fibre reinforced polymers (CFRPs), which were subjected to concentrated vertical loads. The numerical analysis results on the simulated beam models were in excellent agreements with the previous experimental test results. This convergence was confirmed by a statistical analysis, which considered the correlation coefficients, individual arithmetic means and standard deviations for all the calculated deflections of the simulated beam models. A proposed numerical simulation model with the hypotheses can be considered suitable for modelling the behaviours of simple supported non-prismatic RC beams under vertical concentrated loads. The numerical results showed that altering the cross-section from solid to hollow could reduce the load carrying capacities of the beams by up to 53% and increase the corresponding deflections by up to 40%, respectively. Using steel pipes for making recesses could enhance the loading capacity by up to 56%, increase the ductility, and reduce the corresponding deflections by up to 30%, respectively. Finally, it was found that bonding the CFRP sheets in the lower middle tensile areas of the hollow beams could improve the resistance and reduce the deformations by up to 27%. The failure patterns for all the numerical models were shear failure. The cylinder compressive strength could be used as a mechanical parameter for modelling and assessing the structural behaviours of the beam models, as its increase could improve the load carrying capacities and reduce the deflections by 30–50%.

Effect of a “Slight” Curvature of the Axis of a Thin-Walled Nonprismatic Beam on its Free Vibration

This paper presents an analysis of the effect of a “slight” curvature of the axis of a thin-walled beam on its eigenfrequencies and forms. By “slight” we understand such a beam curvature at which no theory of curved beams (arches, horizontally curved girders), but classical beam theory is applied. The subject of the analysis is the open nonprismatic thin-walled beams with any geometrical parameters, and prismatic beams. The thin-walled beam model used in the analysis was derived on the basis of the momentless theory of plates, using the Vlasov assumptions. The free vibrations of the beams were analysed. Calculations were performed for four groups of beams. Each of the groups comprised two beams with the same configuration of flanges and identical geometrical cross section characteristics, but differing in web geometry. One of the beams had a symmetrical web (a beam with a rectilinear axis in the web plane), whereas the web of the other beam was asymmetrical (a beam with its axis curved in the web plane). The algorithm based on Chebyshev polynomials was used to solve the variable coefficient equations describing the model’s vibration. According to this algorithm, solutions are sought in the form of Chebyshev series and the closed analytical recurrence formulas generated by the algorithm are used to calculate the coefficients of the equations. It has been shown that even a “slight” curvature of the beam’s axis has a significant effect on its eigenfrequencies and forms, as confirmed by a comparison of the results with the ones obtained using Finite Element Method (FEM).

Residual post fire strength of non-prismatic perforated beams

The main aim of this study is to assess the performance and residual strength of post-fire non-prismatic reinforced concrete beams (NPRC) with and without openings. To do this, nine beams were cast and divided into three major groupings. These groups were classified based on the degrees of heating exposure temperature chosen (ambient, 400, and 700°C), with each group containing three non-prismatic beams (solid, 8 trapezoidal openings, and 8 circular openings). Experimentally, given the same beam geometry, increasing burning temperature caused degradation in NPRC beams, which was reflected in increased mid-span deflection throughout the fire exposure period and also residual deflection after cooling. But on the other hand, the issue with existing openings was exacerbated. The burned NPRC beams were then gradually cooled down by leaving them at ambient temperature in the laboratory, and the beams were loaded until failure to examine the effect of burning temperature degree on the residual ultimate load-carrying capacity of each beam by comparing them to unburned reference beams. It was found, increasing the exposure temperature leads to a reduction in ultimate strength about (5.7 and 10.84%) for solid NPRC beams exposed to 400 and 700°C, respectively related to unburned one, (21.13 -32.8) % for NPRC beams with eight trapezoidal openings, and (10.5 - 12.8) % for those having 8 circular openings. At higher loading stage the longitudinal compressive strain of Group ambient in mid-span of solid beams reach 2700 με, while the others with openings exhibit divergent strain higher than that, it’s about 3300 με meanwhile, the lower chord main reinforcements have been pass beyond yielding stress. Exposure to high temperatures reduces rafters’ stiffness causing a reduction in load carrying capacity, companion with premature failure consequently reduce the strain at the ultimate stage.

Deformability of non-prismatic prestressed concrete beams with multiple openings of different configurations

Abstract This work presents experimental research using draped prestressed steel strands to improve the load-carrying capacity of prestressed concrete non-prismatic beams with multiple openings of various designs. The short-term deflection of non-prismatic prestressed concrete beams (NPCBs) flexural members under static loading were used to evaluate this improvement. Six simply supported (NPCBs) beams, five beams with openings, and one solid specimen used as a reference beam were all tested as part of the experiment. All of the beams were subjected to a monotonic midpoint load test. The configuration of the opening (quadrilateral or circular), as well as the depth of the chords, were the variables studied in this study. In comparison to a solid beam, experimental results show that beams with openings have a lower load-carrying capacity not exceeding (2.3–10.6%) and higher mid-span deflection through all loading stages of elastic, service, and ultimate loads (14–73%), (19–44%), and (31–55%), respectively. Furthermore, specimens with circular openings had stiffer behaviour under load than those with quadrilateral openings. Beams with quadrilateral openings and inclined posts, on the other hand, were stiffer than beams with quadrilateral openings and vertical posts.

Serviceability of Post-fire RC Rafters with Openings of Different Sizes and Shapes

This study deals with the serviceability of reinforced concrete solid and perforated rafters with openings of different shapes and sizes based on an experimental study that includes 12 post-fire non-prismatic reinforced concrete beams (solid and perforated). Three groups were formed based on heating temperature (room temperature, 400 °C, and 700 °C), each group consisting of four rafters (solid, rafters with 6 and 8 trapezoidal openings, and rafter with eight circular openings) under static loading. A developed unified calculation technique for deflection and crack widths under static loading at the service stage has been provided, which comprises non-prismatic beams with or without opening exposed to flexure concentrated force. Two approaches were used to compute the deflection: The first attempt was conducted by using the moment of inertia for solid non-prismatic beam and reduced for those with openings by the ratio of residual rafter self-weight. The second was performed by using the moment of inertia of transformed cracked sections depending on the segmental rafter method. The crack width was determined using the ACI code's equation. The analytical and experimental results were evaluated and found to be in good agreement.

Analytical plane-stress recovery of non-prismatic beams under partial cross-sectional loads and surface forces

High levels of strength- and stiffness-to-mass ratio can be achieved in slender structures by lengthwise tailoring of their cross-sectional areas. During use, a non-prismatic beam element can be subject to surface forces or loads acting on only a part of their cross-section. Practical examples involve tapered aircraft wings, wind turbine and helicopter rotor blades under fluid pressure and shear forces; arched beams in bridges subject to vehicular traction forces and tensile stresses in tendons of prestressed concrete. Presently, beam theories generalise the external loads to the entire cross-sectional area. However, this technique does not accurately describe surface-load boundary conditions and beams under partial cross-sectional loads. Hence, an efficient analytical plane-stress recovery methodology is introduced in the present study that generalises the external load to a specific sub-area of the cross-section of homogeneous non-prismatic beams with one plane of symmetry. As a result, the transverse stress components become piecewise distributions, i.e. non-smooth but continuous in the thickness direction. Additional novelties include the boundary equilibrium recovered considering applied surface loads and terms up to second-order derivatives of the internal forces to define the transverse stress field. Closed-form solutions for the specific case of non-prismatic beams with a rectangular cross-section loaded both on top and bottom surfaces are presented. For validation purposes, different numerical examples are modelled with results compared to solid-like finite element analyses as well with relevant analytical theories. The results show that the developed formulation predicts the stress field in non-prismatic beams under surface forces and non-uniform loads applied to a part of the cross-sectional area with goods levels of accuracy. The error associated with the proposed method escalates with the taper angle, such that a 10° taper angle could result in a 6% error at the surfaces and reduced values for interior zones, while the analytical state-of-the-art models were not able to predict the transverse stresses correctly.

Post-Fire Behavior of Non-Prismatic Beams with Multiple Rectangular Openings Monotonically Loaded

The main objective of this paper is to study the behavior of Non-Prismatic Reinforced Concrete (NPRC) beams with and without rectangular openings either when exposed to fire or not. The experimental program involves casting and testing 9 NPRC beams divided into 3 main groups. These groups were categorized according to heating temperature (ambient temperature, 400°C, and 700°C), with each group containing 3 NPRC beams (solid beams and beams with 6 and 8 trapezoidal openings). For beams with similar geometry, increasing the burning temperature results in their deterioration as reflected in their increasing mid-span deflection throughout the fire exposure period and their residual deflection after cooling. Meanwhile, the existing openings situation was compounded. The burned NPRC beams were left to gradually cool down under ambient laboratory conditions, and afterward, they were loaded until failure. The influence of temperature on the residual ultimate load-carrying capacity of each beam was studied by comparing these beams with unburned reference beams. Increasing exposure temperature reduces the ultimate strength of solid NPRC beams exposed to temperatures of 400°C and 700°C by about 5.7% and 10.84% respectively. Meanwhile, NPRC beams with trapezoidal openings showed ultimate strength reductions of 21.13% and 32.8% (for beams with 8 openings) and 28% and 34.4% (for beams with 6 openings) under the same burning conditions. The excessive mid-span deflections for these three types of beams were 2%–30.8%, 1.33%–21.8%, and 1.5%–17.4% under the same burning conditions.

Buckling and Prestressed Vibrations of Periodic Nonprismatic Beams Using Integral Equation Approach

This paper presents the use of Fredholm–Volterra integro-differential equation approach for determining the critical buckling load and natural frequencies of periodically supported nonprismatic Euler–Bernoulli beams. Neumann series solution is used to solve the integro-differential equation by using the orthogonal Chebyshev polynomials, to avoid ill-conditioning. Gauss quadrature is used to perform various needed integrals. Critical buckling load, natural frequencies, and the corresponding mode shapes are compared with those available in the literature, those obtained using the principle of virtual work, and the ones obtained using NASTRAN®. The effect of variation of taper ratio, both in width and height of a rectangular cross section, on the nondimensional critical buckling load and the nondimensional natural frequency, is also investigated. A singularity is observed near the taper ratio of unity, which impacts the accuracy of both the finite element and the principle of virtual work results. The integral equation approach converges with a lesser number of terms than the principle of virtual work, especially for higher modes for beams approaching the taper ratio of unity. Benchmark examples for free vibration of a multiple-bay beam and of inflated tube are also studied, and the results are compared with those presented in the literature.

Analytic and numeric analysis for deformation of non-prismatic beams resting on elastic foundations

BackgroundThe buckling load as well as the natural frequency under axial load for non-prismatic beam is a changeling problem. Determination of buckling load, natural frequency, and elastic deflection is very important in civil applications. The current paper used both perturbation method (PM), analytic method, and differential quadrature method (DQM), numerical method, to find buckling load and natural frequency with different end supports. The deflection of the beam resting on an elastic foundation under transverse distributed and axial loads is also obtained. Both PM and DQM are used for non-prismatic beams with rectangular and circular cross sections in the vibration analysis. The comparisons of results obtained from both PM and DQM showed perfect agreement with analytical solution for uniform beams with different end supports. The PM and DQM succeeded powerfully for investigating the buckling load as well as the natural frequency for non-prismatic beam.ResultsThe percentage of relative error between DQM and PM doesn’t exceed than 5% if the gradient of rectangular section height and the gradient of circular section radius are less than 0.6. As the gradient of height and radius increase, the maximum deflection decreases and the location of maximum deflection displaced toward the smaller moment of inertia.ConclusionsThe PM has not been used for solving the problem of non-prismatic beams resting on elastic foundations subjected to transverse distributed and axial loads. The current research proved the good ability of PM as an analytical solution for a complicated problem and defined its range of accuracy as compared to DQM. Also, it introduced accurate empirical formulae to find both natural frequency and buckling load of non-prismatic beams. These empirical formulae represent a good achievement in vibration analysis of non-prismatic beams.