Non-prismatic Beams Research Articles

Nonlinear Vibrations in Homogeneous Nonprismatic Timoshenko Cantilevers

Abstract Deep cantilever beams, modeled using Timoshenko beam kinematics, have numerous applications in engineering. This study deals with the nonlinear dynamic response in a nonprismatic Timoshenko beam characterized by considering the deformed configuration of the axis. The mathematical model is derived using the extended Hamilton’s principle under the condition of finite deflections and angles of rotation. The discrete model of the beam motion is constructed based on the finite difference method (FDM), whose validity is examined by comparing the results for a special case with the corresponding data obtained by commercial finite element (FE) software abaqus 2019. The natural frequencies and vibration modes of the beam are computed. These results demonstrate decreasing eigenfrequency in the beam with increasing amplitudes of nonlinear oscillations. The numerical analyses of forced vibrations of the beam show that its points oscillate in different manners depending on their relative position along the beam. Points close to the free end of the beam are subject to almost harmonic oscillations, and the free end vibrates with a frequency equal to that of the external force. When a point approaches the clamped end of the beam, it oscillates in two-frequency mode and lags in phase from the oscillations of the free end. The analytical model allows for the study of the influence of each parameter on the eigenfrequency and the dynamic response. In all cases, a strong correlation exists between the results obtained by the analytical model and ABAQUS; nonetheless, the analytical model is computationally less expensive.

Vibration analysis of an axially functionally graded material non-prismatic beam under axial thermal variation in humid environment

The vibration analysis of an axially functionally graded material non-prismatic Timoshenko beam under axial thermal variation in humid environment is carried out using the harmonic differential quadrature method. In this modeling, the length and width of the beam remains constant whereas thickness of the beam is linearly varied along the axis of the beam. The material properties are temperature dependent and are assumed to be varied continuously along the axial direction according to power law distribution. Three types of temperature variations are considered in this study, that is, uniform temperature rise, linear temperature rise, and non-linear temperature rise. The temperature of the beam remains constant under uniform temperature rise condition and it is varied linearly and nonlinearly along the length of beam for rest of the conditions. The beam is subjected to uniform moisture concentration to impose humidity. Hamiltonian’s approach is used to derive the governing equations of motion. The resultant governing equations are then solved using the harmonic differential quadrature method to obtain the natural frequencies of the axially functionally graded material non-prismatic beam. The results obtained using the harmonic differential quadrature method are compared with results obtained for special cases. The effects of thermal variation, humidity, non-homogeneity parameter, and end conditions on natural frequencies of the non-prismatic beam are reported.

Lateral buckling analysis of axially functionally graded thin-walled beam with varying cross-section under different boundary conditions

In this article, lateral-torsional stability of axially functionally graded (AFG) non-prismatic beam with doubly symmetric thin-walled cross-section under transverse loading is investigated. Material properties of FG beam with variable cross-section are supposed to vary through longitudinal direction of the constituents according to simple power-law distribution (P-FGM). Based on Vlasov’s model and using small displacements theory, the governing equilibrium equations are derived through the energy principle for AFG web and/or flanges tapered I-beams. The differential quadrature method is then selected to numerically solve the resulting system of governing fourth-order differential equations with variable coefficients. In this approach, a non-uniform mesh point distribution (Chebyshev-Gauss-Lobatto) is used for provide accuracy of solutions and convergence rate. Finally, the lateral-torsional buckling loads are calculated by solving an eigenvalue problem of the obtained algebraic system. In the case of homogenous members with variable cross-section, the outcomes of this work are compared with the available benchmarks and there is an excellent agreement. Finally, the impacts of involved parameters such as boundary conditions, load height parameter, power law index and web and flanges tapering ratios on the non-dimensional lateral buckling load are discussed. The results of this research reveal that the impacts of non-uniformity in the cross-section and gradient index play significant roles on lateral-torsional stability behavior of tapered I-beams. Furthermore, the highest buckling capacity is acquired when the transverse load is applied to the bottom flange.

Optimal preliminary design of variable section beams criterion

The present paper discusses about optimal shape solution for a non-prismatic planar beam. The proposed model is based on the standard Timoshenko kinematics hypothesis (i.e., planar cross-section remains planar in consequence of a deformation, but it is able to rotate with respect to the beam center-line). The analytical solution for this type of beam is thus used to obtain deformations and stresses of the beam, under different constraints, when load is assumed as the sum of a generic external variable vertical one and the self-weight. The solution is obtained by numerical integration of the beam equation and constraints are posed both on deflection and maximum stress under the hypothesis of an ideal material. The section variability is, thus, described assuming a rectangular cross section with constant base and variable height which can be described in general with a trigonometric series. Other types of empty functions could also be analyzed in order to find the best strategy to get the optimal solution. Optimization is thus performed by minimizing the beam volume considering the effects of non-prismatic geometry on the beam behavior. Finally, several analytical and numerical solutions are compared with results existing in literature, evaluating the solutions’ sensibility to some key parameters like beam span, material density, maximum allowable stress and load distribution. In conclusion, the study finds a critical threshold in terms of emptying function beyond which it is not possible to neglect the arch effect and the curvature of the actual axis for every different case study described in this work. In order to achieve this goal, the relevance of beam span, emptying function level and maximum allowable stress are investigated.

An integrated method for sustainable performance-based optimal seismic design of RC frames with non-prismatic beams

In the performance-based optimal seismic design, one attempts to obtain structural design variables to meet the minimum objective function satisfying the strength-based and performance-based constraints. A limited number of studies have been conducted on the performance-based optimal seismic design of reinforced concrete frames. On the other hand, due to the importance of environmental impacts, further study is necessary for the design of RC buildings with the aim of reducing CO2 emissions. In this study, a computational procedure is developed for performance-based optimal seismic design of RC frames with prismatic and non-prismatic beams. The objective functions consist of minimizing the cost and CO2 emissions. Nonlinear pushover analysis is performed for analysis of the structures. The described procedure is applied to 4-story reinforced concrete frames and the relationship between optimal cost and optimal CO2 emissions is studied for frames with prismatic beams and frames with non-prismatic beams.

Residual Tensile Stress Estimation for Shear Strength of UHPC Nonprismatic Beams

Abstract Estimating the shear strength of Ultra-High-Performance Concrete (UHPC), with high compressive and tensile strengths, is complicated by many variables that affecting its behavior. Residual tensile stress (RTS) plays an important role in raising the efficiency of both types of resistance, especially shear strength due to the presence of steel fibers, which makes it difficult to quantify the residual tensile stress due to the different failure patterns of these fibers and the distribution mechanism within the concrete matrix. There is no study to date in assessing residual tensile stress of UHPC structural members of the variable section. Thirteen beams were selected as an experimental program to study six main variables in determining shear strength. Stirrups ratio, flexural reinforcement ratio, the volumetric fraction of steel fibers, geometry changing, existing openings along the longitudinal axis, and shear span to depth ratio. According to on Tests results, RTS is compatible with most of the global specifications.

Efficient strong Unified Formulation for stress analysis of non-prismatic beam structures

The Unified Formulation (UF) has gained attention as a powerful tool for efficient design of structural components. Due to the inherent flexibility of its kinematics representation, arbitrary shape functions can be selected in different dimensions to achieve a high-fidelity characterisation of structural response under load. Despite this merit, the classical isoparametric description of UF limits the application to prismatic structures. The weak-form anisoparametric approach adopted to overcome this limitation in a recent work by Patni et al. proves to be versatile yet computationally challenging owing to the expensive computation of its UF stiffness matrix by means of full volume integrals. We propose a strong-form anisoparametric UF (SUF) based on the Serendipity Lagrange Expansion (SLE) cross-sectional finite element and differential quadrature beam element. The main objective of the SUF is to achieve an efficient computation of the UF stiffness matrix by restricting Gauss operations to the variable cross-sections of non-prismatic structures in a discrete sense, thus eliminating the need for full volume integrals. When assessed against weak-form based UF, ABAQUS FE and analytical solutions, the static analysis of non-prismatic beam-like structures under different loads by the SUF is shown to be accurate, numerically stable, and computationally more efficient than state-of-the-art methods.

An Analysis of the Effect of a Change in the Support Point Location on the Vibration of Thin-Walled Beams

This paper deals with an analysis of the free vibration of nonprismatic thin-walled beams, with a special focus on the effect of a change in the support point location on the eigenfrequencies of the systems. A change in the support point location is understood here as occurring within the same fixed cross-section of the beam where the latter is supported. The original elements of this study are a thin-walled beam model and a method of solving differential equations, not previously used by other authors. The equations describing the model used in this paper were derived using the momentless theory of shells and the Vlasov theory assumptions. The displacement equations were derived relative to an arbitrary rectilinear reference axis. In most works known to the authors the equations describing the vibration of the nonprismatic thin-walled beam do not take into account the effects due to the curvature of the axis formed by the shear centers. The recursive algorithm was used to solve the obtained differential equations with variable coefficients. The algorithm enables one to solve the analyzed displacement equations in the form of series relative to the orthogonal Gegenbauer polynomials. For special values of the parameter defining the order of the polynomials the Gegenbauer polynomials became Chebyshev or Lagrange polynomials. In the provided numerical example, the effect of a change in the support point location within the fixed cross-section is examined. It is also analyzed which of the approximation polynomials (Chebyshev or Lagrange polynomials) yield more precise results for a small approximation base. In order to verify the model and the effectiveness of the adopted solution method, the results obtained using this method are compared with the results yielded by FEM. The results obtained showed a significant effect of the position of the support point, within a fixed section, on the eigenfrequencies values of the thin-walled systems.

Stress analysis of generally asymmetric non-prismatic beams subject to arbitrary loads

Non-prismatic beams are widely employed in several engineering fields, e.g., wind turbines, rotor blades, aircraft wings, and arched bridges. While analytical solutions for variable cross-section beams are desirable, a model describing all stress components for beams with general variation of their cross-section under generalised loading remains an open and important problem to solve. To partly address this issue, we propose an analytical solution for stress recovery of untwisted, asymmetric, non-prismatic beams with smooth and continuous taper shape under general loading, considering plane stress conditions for isotropic materials undergoing small strains. The methodology follows Jourawski’s formulation, including the effect of asymmetric variable cross-section, with internal forces as known variables. We confirm the non-triviality of the stress field of non-prismatic beams, i.e., the dependency on all internal forces and beam geometry to shear and transverse stress distributions. As a particular novelty, the new formulation for transverse direct stress includes internal forces derivatives, resulting in greater accuracy than state-of-the-art models for distributed loading conditions. Also, closed-form solutions are introduced for non-prismatic and linearly tapered, generally asymmetric beams, both with rectangular cross-sections. For validation purposes, we consider three different practical beam models: a symmetric and an asymmetric, both linearly tapered, and an arched beam. The results, checked against commercial finite element analysis, show that the proposed model predicts the stress-field of non-prismatic beams under distributed loads with good levels of accuracy. Traction-free boundary condition requirements are naturally satisfied on the beam surfaces.

An efficient isogeometric beam formulation for analysis of 2D non-prismatic beams

This study presents an efficient beam formulation for static and vibration analysis of 2D non-prismatic beams with the kinematic assumptions of the Timoshenko beam theory. Beams having complex configurations can be straightforwardly constructed by using B-spline basis functions, which are used for descriptions of the beam axis and the sectional height. This feature enables analysis of free-form non-prismatic beams with strong curvatures to be performed conveniently and accurately, and is not virtually found in any existing formulations that are related to non-prismatic beams. The concept of isogeometric analysis is adopted by discretizing the kinematic unknowns, i.e., the translational displacements of the beam axis and the sectional rotation, using B-spline basis functions. A procedure for the recovery of stress components is also presented. The accuracy and performance of the developed approach are assessed through several numerical examples which cover several types of beam, e.g., sinusoidal beam, tapered beam and Tschirnhausen's beam representing free-form beams. Furthermore, convergence tests are also performed to demonstrate convergence properties of the proposed formulation.

Behaviour of reinforced concrete beams having recesses in tension and compression sides

A reinforced concrete beam with a recess is an example of a non-prismatic beam, which develops stress concentrations at the corner of the recess owing to the abrupt change in its depth. This entails appropriate detailing of internal reinforcement in order to avoid premature failure. This study presents the load transfer mechanism employing the strut-and-tie model (STM) for four non-prismatic beams with recesses in the shear span zone at both compression and tension sides, tested under a four-point loading scheme. The proposed STMs for the tested beams were developed based on both the cracking patterns near failure of experimentally tested beams and the developed stress contours based on linear static analysis. In addition, a simplified method is proposed to estimate the maximum mid-span deflection of simply supported beams with recesses. The results obtained reveal that the STM gave good predictions, with an average of about 86% of the ultimate experimental loads. Based on the results of the parametric study conducted, it was found that the most significant parameter was the recess depth from the viewpoint of stress concentration factor, cracking load and maximum crack width. Finally, recommendations and guidelines for the design of reinforced concrete beams with recesses are presented.

Optimal strain-deflection analyses of a wedge edged beam for enhanced vibration energy scavenging in patch-like harvester

Recently, non-prismatic beams have been considered as perfect substitutes to replace uniform cantilever beam structures in harvester design. This is due to their tendencies to provide uniform strain distribution, which enhances piezoelectric (PZT) charge generation when coupled with the beams. As the voltage produced by the PZT depends largely on the geometrical shape of the substrate, thus there is need for formulations to analyze the behavior of the shim prior to practical implementation. In this paper, analytical and finite element models were developed to study deflection and strain distribution of a non-prismatic beam termed, wedge edged cantilever (WEC) for patch-like harvester application. The analyses offer possibility for maximum utilization of strain location along the beam span for efficient harvester design. Vibration energy harvesting characteristics of the WEC beam with partially covered PZT material forming three configurations namely: Config1, Config2 and Config3 were studied. Both numerical and experimental results show that Confg3 with macro fiber composite (MFC) as the PZT material covering the determined optimal strain position yielded 2 to 3 times more peak output voltage than Config1 and Config2 with the MFC bonded at non-optimal positions.

Broadband dynamic elastic moduli of honeycomb lattice materials: A generalized analytical approach

A generic analytical framework is proposed to obtain the dynamic elastic moduli of lattice materials under steady-state vibration conditions. The dynamic deformation behaviour of the individual beam elements of a lattice is distinct from the behaviour under a static condition. This leads to a completely different global deformation pattern of the lattice material and subsequently opens up a tremendous opportunity to modulate amplitude and phase of the elastic properties of lattices as a function of the ambient vibration. The dynamic stiffness approach proposed in this article precisely captures the sub-wavelength scale dynamics of the periodic network of beams in a lattice material using a single beam-like member. Here the dynamic stiffness matrix of a damped beam element based on the Timoshenko beam theory along with axial stretching is coupled with the unit cell-based approach to derive the most general closed-form analytical formulae for the elastic moduli of lattice materials across the whole frequency range. It is systematically shown how the general expressions of dynamic elastic moduli can be reduced to different special cases by neglecting axial and shear deformations under dynamic as well as classical static conditions. The significance of developing the dynamic stiffness approach compared to conventional dynamic finite element approach is highlighted by presenting detailed analytical derivations and representative numerical results. Further, it is shown how the analytical framework can be readily extended to lattices with non-prismatic beam elements with any spatial variation in geometry and intrinsic material properties. In general, research activities in the field of lattice metamaterials dealing with elastic properties revolve around intuitively designing the microstructural geometry of the lattice structure. Here we propose to couple the physics of deformation as a function of vibrating frequency along with the conventional approach of designing microstructural geometry to expand the effective design space significantly. The stretching-enriched physics of deformation in the lattice materials in addition to the bending and shear deformations under dynamic conditions lead to complex-valued elastic moduli due to the presence of damping in the constituent material. The amplitude, as well as the phase of effective elastic properties of lattice materials, can be quantified using the proposed approach. The dependence of Poisson's ratio on the intrinsic material physics in case of a geometrically regular lattice is found to be in contrary to the common notion that Poisson's ratios of perfectly periodic lattices are only the function of microstructural geometry. The generic analytical approach for analysing the elastic moduli is applicable to any form of two- or three-dimensional lattices, and any profile of the constituent beam-like elements (different cross-sections as well as spatially varying geometry and intrinsic material properties) through a wide range of frequency band. The closed-form expressions of elastic moduli derived in this article can be viewed as the broadband dynamic generalisation of the well-established classical expressions of elastic moduli under static loading, essentially adding a new exploitable dimension in the metamaterials research in terms of dynamics of the intrinsic material.

Shear design method for non-prismatic concrete beams reinforced using W-FRP

Flexible fabric formworks and bespoke Wound Fibre-Reinforced Polymer (W-FRP) cages have been proposed to address the difficulties of constructing geometrically optimised concrete structures with minimised material consumptions. However, the existing codes of practice for FRP reinforced concrete structures are not able to provide accurate shear behaviour predictions, resulting in potential premature anchorage failure. This paper presents a revised shear design approach for non-prismatic beams reinforced with W-FRP based on the Modified Compression Field Theory (MCFT), which considers the non-prismatic geometries, geometrical nonlinearity and internal load redistributions. The revised shear design approach is calibrated against previous fabric-formed T beam tests and parametric analyses are conducted to minimise the material usage of the previous T-beams. The validity of the shear design approach is further confirmed with a new T-beam test. The new test shows that the T beam designed with the MCFT based approach failed in the expected flexural failure mode with an ultimate capacity of 245 kN applied load. The predictions can effectively model the bar force development of the inclined flexural reinforcement with a Normalised Root Mean Square Error (NRSME) of 4.4%. The proposed shear design method improved the prediction accuracy of flexural bar force at ultimate capacity by 40% compared to the codified shear design method. By addressing the shear contributions of inclined flexural reinforcement and required anchorage strength, further applications of this novel flexibly formed beams with optimised geometries are expected to achieve material consumption reduction and hence lower carbon emissions from concrete structures.

Strengthening of Reinforced Concrete Stepped Beam Using Cotton Belt (New Technique)

The stepped beam is a type of non-prismatic beams that extend the stepped joint by stress collection and require adequate detail for this joint to prevent early failure. This research demonstrates an experimental study aimed at the behaviour of reinforced concrete stepped beams. The status issue is the premature failure of the stepped joint. The experimental program consists of eight concrete beams. All beams are tested under a four-point load. Three variables are studied in this work, the first of which is compressive strength, which is normal strength of 40 MPa and high strength of 80 MPa. The second variable is the details of the steel reinforcement at the stepped joint and the third variable is the strengthening with a cotton belt. Models achieved increases ranging from (26.3-37.4) kN, (1.88-3.95) mm, and (31.75-116.882) kN.mm in the ultimate capacity, deflection, and energy absorption respectively. Whereas it showed a decrease ranging (14.38-27.68) in the initial stiffness for the compression strength variable. As for the rebar detail variable, it showed an increase ranging between (39-138) kN, (7.5-28.29) mm, (244-3099.619) kN.mm, and (22.285-31.425) kN/mm, for the maximum load, deflection, energy absorption, and initial stiffness, respectively. On the other hand, the strengthened beams achieved an increase in the maximum load by 120% with a slight decrease in stiffness by 3.5% concerning an unstrengthened beam.

Strong internal resonance in a nonlinear, asymmetric microbeam resonator

Exploiting nonlinear characteristics in micro/nanosystems has been a subject of increasing interest in the last decade. Among others, vigorous intermodal coupling through internal resonance (IR) has drawn much attention because it can suggest new strategies to steer energy within a micro/nanomechanical resonator. However, a challenge in utilizing IR in practical applications is imposing the required frequency commensurability between vibrational modes of a nonlinear micro/nanoresonator. Here, we experimentally and analytically investigate the 1:2 and 2:1 IR in a clamped–clamped beam resonator to provide insights into the detailed mechanism of IR. It is demonstrated that the intermodal coupling between the second and third flexural modes in an asymmetric structure (e.g., nonprismatic beam) provides an optimal condition to easily implement a strong IR with high energy transfer to the internally resonated mode. In this case, the quadratic coupling between these flexural modes, originating from the stretching effect, is the dominant nonlinear mechanism over other types of geometric nonlinearity. The design strategies proposed in this paper can be integrated into a typical micro/nanoelectromechanical system (M/NEMS) via a simple modification of the geometric parameters of resonators, and thus, we expect this study to stimulate further research and boost paradigm-shifting applications exploring the various benefits of IR in micro/nanosystems.

Unified calculation formula for predicting the shear stresses in prismatic and non-prismatic beams with corrugated steel webs

The existing shear design theories for hybrid beams with corrugated steel webs (CSWs) used in China and Japan were originally proposed for only prismatic hybrid beams with CSWs. The traditional calculation method assumes that the bending stiffness and axial stiffness of a beam with CSWs are provided by the top and bottom concrete flanges, while the shear stiffness is contributed entirely by the CSWs. However, due to the influence of the Resal effect on a non-prismatic hybrid beam with CSWs, the calculated shear stress of the CSWs will be overpredicted if the shear capacities of the top and bottom concrete flanges are ignored (especially for the case of an inclined bottom flange). These traditional assumptions may no longer be applicable to non-prismatic structural members. Considering the Resal effect and accordion effect of non-prismatic beams with CSWs, this study proposed a practical unified calculation formula suitable for calculating the shear stresses of prismatic and non-prismatic hybrid beams with CSWs. The accuracy and adaptability of this unified calculation formula were verified through prismatic and non-prismatic beam model tests as well as finite element simulations. Moreover, the authors quantitatively discussed the calculation errors of the traditional calculation assumptions for non-prismatic beams with CSWs. In addition, a finite element parameter analysis was performed on a non-prismatic beam with CSWs near the support to clarify the applicable conditions of the unified calculation formula. The numerical results show that when the inclination angle of the bottom flange varies within the range of 0° to 20°, the calculation error of the proposed unified calculation formula is controlled to within 15%, exhibiting good adaptability to changes in the inclination angle. The unified calculation formula can still predict the shear stresses in tapered CSWs with an error of less than 20% when the sum of the thicknesses of the top and bottom concrete flanges does not exceed 20% of the total beam height. It is expected that this study can further develop the shear stress calculation theory for non-prismatic beams with CSWs, thus providing a reference for the reasonable design of tapered CSWs.

Optimization of thin-walled beam structures: Monolithic versus staggered solution schemes

This contribution compares the performance of a monolithic and two staggered numerical update schemes for a coupled shape and topology optimization method for thin-walled beam structures. In order to limit the computational time of the optimization method, the thin-walled beam structures are modeled as 2.5D configurations. In this approach, 1D beam elements are used to simulate the beam response in the longitudinal direction, while the cross-sectional properties of the beam elements are calculated from additional 2D finite element method analyses. The sensitivities with respect to the shape and topology design variables are derived in closed form in order to take advantage of a computationally efficient, gradient-based optimization algorithm. The numerical examples concern basic circular and square thin-walled cantilever beam structures, as well as a more complex, non-prismatic thin-walled beam structure representative of a rotor blade used in a horizontal-axis wind turbine. For each numerical example the computational time and the solution computed by the monolithic and staggered update schemes are compared, which provides clear insight into the numerical efficiency of the solution procedure and the uniqueness of the computational result. Although the three different update schemes for the cases examined result in comparable optimized design concepts, the computational efficiency and the specific minimal compliance value found turn out to be rather sensitive to the algorithmic details of the numerical update scheme applied. Since the monolithic update scheme typically navigates a larger design space than the two staggered update schemes, for most cases examined it provides the lowest structural compliance. In the specific case whereby shape optimization (virtually) has no influence on the final, optimized beam configuration, the structural compliance computed by the monolithic scheme may relate to a local minimum that is less optimal compared to the value calculated by a staggered update scheme.

Minimum Cost for Reinforced Concrete Rectangular Beams with Parabolic Haunches

El objetivo de esta investigacion es presentar un modelo matematico para el diseno optimo de trabes de seccion transversal rectangular con cartelas parabolicas bajo el criterio de costo minimo tomando en cuenta el costo del concreto y costo de acero de refuerzo, y considerando las ecuaciones del reglamento (ACI 318S-19). Este modelo presenta las ecuaciones para dos tipos de cargas: Carga uniformemente distribuida y Carga concentrada localizada en cualquier parte de la trabe. Dos ejemplos han sido desarrollados por el modelo propuesto para ambos casos. Los resultados muestran que las trabes prismaticas para los dos casos tienen un costo total, un volumen total y un peso total mayor con respecto a las trabes no prismaticas. Por lo tanto, las trabes no prismaticas son mas economicas, tienen menos volumen y tambien tienen menos peso con respecto a las trabes prismaticas.

Improving the concrete sections after removing intermediate support of RC continuous non-prismatic beam

An experimental study was conducted to investigate the performance of the 2-span continuous reinforced concrete beams of different section depth after removing the middle support by adding steel fibers or steel plates. The beams were loaded monotonically with two-point loads. One continuous beam and six simply supported beams of non-prismatic section were tested using two different content of steel fibers and three different locations of steel plates welded to the reinforcement. The test results reveal that using inclined steel plates at the region of changing of cross section thickness at the middle support increase the load capacity of the beam significantly up to 75% of the continuous beam capacity, and a significant warning before failure is shown. Adding steel fibers to the concrete has less influence on the capacity of the beam. The failure mode of the beams with no middle support is the same, but with different values of deflection. The inclined steel plate again is the most effective way to decrease the deflection because of the increased stiffness of the cross section. To achieve the same capacity of the continuous beam after removing the middle support, it is recommended to use horizontal steel plates welded to the reinforcement at the region of the middle support extended within one fourth the length of each span of different thickness to avoid the stress concentration resulted from the large deflection at that region.