A novel hybrid beam element for nonlinear analysis of prestressed concrete girders

Abstract

The interaction between prestressing reinforcement and concrete material in pretensioned and post-tensioned, fully or partially prestressed concrete analysis can be accommodated totally in one-dimensional finite element mixed formulation systematically. The beam element formulation presented in this paper facilitates the nonlinear material, finite deformation as well as the cohesive behavior between prestressing tendons and concrete girders in an effective and straight forward algorithm.

The spatial L2 beam element has been enriched to incorporate the additional two degrees of freedom of the reinforcement slip/gap on the nodes of the one-dimensional mesh, corresponding to the top and bottom groups of strands/tendons prescribed in the beam section’s overlay two-dimensional mesh. The overlay mesh of the interior/exterior prestressing reinforcements gives the opportunity to extract beam section indices via a few two dimensional, e.g. Q4, elements.

The ultimate limit state of several prestressed concrete designs has been studied for unidirectional and hysteresis modes and verified using three-dimensional solid elements. The results obtained demonstrate the accuracy, stability and efficacy of the algorithm for the analysis of multi-degrees of freedom prestressed concrete structures.

Graphical abstract

Keywords

prestressed concrete / reinforcement / finite element / beam element / nonlinear analysis

1 Introduction

The author who has succeeded in creating an entire new material, “prestressed concrete (PC)”, was Eugene Freyssinet in 1928 [1,2]. Indeed, there have been some unsuccessful attempts before him, in the US and Germany, as well [3]. Failures in prestressing projects were mainly due to the loss of prestressing, resulting from a range of issues such as concrete shrinkage, creep, and the relaxation of the weaker steels of the 19th century—factors that were unclear to the original patent holders [4,5].

It was later demonstrated that the typical stress loss in steel ranges from 100 to 240 MPa, and for this loss to remain a small fraction of the initial prestress, the prestressing material needs to endure stresses between 1200 and 2000 MPa. Such high stress capabilities were not achievable with the materials used during the early developments of prestressing technology [6,7]. Michele Frizzarin’s PhD dissertation explored the link between the effective residual prestress force and the values estimated using non-destructive tests [8].

In PC, the steel reinforcement is stretched within a self-balancing system, generating tension in the steel and compression in the concrete, which optimizes the structure’s ability to withstand external forces [1]. As concrete is inherently stronger and more ductile in compression than in tension, applying precompression enhances its response to external loads. While fully prestressed concrete is attributed to Freyssinet, the development of partial prestressed concrete, which integrates prestressed and non-prestressed reinforcements allowing the concrete to handle a fraction of the tensile load, is credited to Emperger [9] and Abeles et al. [10–12].

Having overcome a range of challenges, PC has risen to the forefront of preferred materials for the material-optimized design of highway bridges and other major industrial constructions. Among these challenges are the cost of a special mold design and construction for the pretension process, higher quality materials needed in prestressing with respect to reinforced concrete (RC), demand for prestressing systems and anchorage instruments which necessitates some parts of prefabrication or composite (combined precast and cast-in-place) construction, appearance of the macro cracks on the prestressed face, durability of the steel strands and tendons, loss of prestressing due to camber and deflections of the beam and many other causes such as creep and shrinkage of the concrete, relaxation of the steel, friction losses in the conduit of the tendons, material corrosion due to aggressive environmental interaction, local bearing/spalling/bursting effects at the anchorage zone, necessity of using high strength concrete for prestressing practice, and so on [3,13,14].

On the contrary, prestressed members are more efficient in depth to at least 20%–35%, than their RC equivalents in load and span magnitudes. Thus, PC members require less concrete, and merely about 20%−35% of the reinforcement quantity of its RC alternative, as well. Therefore, for spans greater than 24–30 m, the heavy dead load of RC beams limits their efficiency, requiring the use of PC technology instead of RC arches, which are both expensive to construct and vulnerable to long-term creep and shrinkage effects [3]. A thorough reference for designing and constructing different classes of PC is available in the PCI Design Handbook [15].

However, a numerical method that efficiently and comprehensively addresses all the interactive aspects of PC analysis is scarce in the literature. The proposed numerical algorithm is expected to utilize 1 dimensional (1D) beam elements, as is commonly done by engineers in RC design, allowing for a straightforward consideration of both non-prestressed and prestressed reinforcements within the beam cross-section. It will integrate reinforcements and tendons into the analysis by compiling fiber (meshed) cross-section data into the beam’s sectional moduli, account for cohesive interactions between concrete and various reinforcement materials, address frictional contact between tendons and concrete along the beam’s length after cohesion failure, and provide several additional features that designers require when tackling comprehensive design challenges in PC; such as stirrups and tie effects, section confinement, deep beam effect on shear strength, local buckling effects of slender cross-sections and so forth.

In this paper, a 1D finite element beam element is enriched in order to reproduce the gap between prestressing reinforcement and concrete in a fibrous PC cross-section. The approach is able to be extended to higher order finite element beam elements which utilize the mixed formulation for the deflection and rotation interpolation. Accordingly, all aspects of the PC beam analysis are included in the “Beam Element-Plus” framework which takes advantage of the cross-sectional mesh besides the 1D mesh through the length of the PC beam [16].

The 21st century has seen remarkable progress in PC’s numerical analysis technology, with some key research developments outlined in the following.

El-Sheikh [17] studied the self-centering, low inelastic energy dissipation properties of the unbonded post-tensioned beam-column connections. Fanning [18] studied the nonlinear analysis of post-tensioned concrete beams using solid elements and smeared crack method in ANSYS. Vecchio et al. [19] proposed a hybrid solid-truss model for simulating shear-critical unbonded post-tensioned concrete beams. The concrete’s solid elements are constrained to the L2 truss elements representing the steel bars through “Link Elements,” which are used to model the frictional interaction between the steel tendons and the concrete.

Dall’Asta et al. [20] designed a 10 degrees of freedom (DOFs), 1D finite element algorithm specified for 2-dimensional (2D) planar analysis of exterior post-tensioning and studied the nonlinear material and geometry. The beam cross-section has been subdivided to fibers for integration for section properties through the iterations. The beam element consists of four intermediate nodes which trace the axial displacement of the external prestressing materials. In contrast to the algorithm proposed in this paper, we adopt only eight DOFs per node for full tracing of either pre-tensioning or post-tensioning, either unbonded or bonded in a 3-dimensional (3D) model of the beam.

A study by Omar et al. [21] addressed the need for improved predictions of pre-camber in PC beams by developing refined formulations based on BS 8110. Even advanced numerical analyses of PC girders often fail to provide accurate pre-camber estimates. The algorithm outlined in this work delivers a reliable and efficient calculation of this key quantity. Koshikawa et al. [22] conducted an investigation on the cyclic performance of precast bonded and unbonded post-tensioned concrete cross-type beam-column connections.

The structural performance of corroded post-tensioned bonded wires after corrosion has been investigated through empirical and numerical methods by Coronelli and colleagues [23]. Marriott et al.’s [24] research involves quasi-static and pseudo-dynamic analysis of unbonded post-tensioned rocking bridge piers incorporating exterior dampers. The shear behavior of standard-based PC designs has been comprehensively studied by Hamilton et al. [25]. The shear behavior of spliced post-tensioned girders is examined in the dissertation by Moore et al. [26,27] and by Dolan and Hamilton [28]. An external post-tensioning model for steel-concrete composite beams is developed by El-Zohairy and colleagues [29]. Various studies for post-tensioning technology, especially for exterior prestressing which have gained more attention in research since 2019 can be found in Refs. [30–61].

“Spring System Model” is proposed by Huang et al. [62,63] as a multi-scale approach for analysis of unbonded post-tensioned concrete structures and implemented by script file to ABAQUS program. Jang et al. [64], Noble et al. [65] and Tran et al. [66] investigated the use of experimental modal analysis for estimating the prestress force in grouted post-tensioned tendons. Vu et al. [67] introduced a nonlinear macro finite element modeled as a beam element with uniform average inertia for solving bending problems. The 3D finite element analysis of post-tensioned continuous beams subjected to shear loading, utilizing solid elements in ABAQUS, is carried out by Herbrand et al. [68]. A 2D analysis approach implemented in ABAQUS is demonstrated by Lee et al. [69].

Nkuako’s MSc thesis [70] offers an introduction to prestressing analysis using SAP 2000™. SAP 2000s prestressing method is based on a rigid interface between the tendons and the concrete section, with tendon nodes outside the frame section having three DOFs. Prestressing is applied at one end of the beam and propagated toward the opposite end using specific solid mechanics formulations. Furthermore, SAP 2000 does not provide any details regarding the frictional or cohesive interactions between the tendons and concrete interface. The program also fails to consider shifts in the neutral axis after the development of flexural cracks in the ultimate load-bearing phase [71].

Cornejo et al. [72] present an in-depth survey of current numerical techniques for the analysis of PC beams, proposing a serial-parallel algorithm that maintains consistency and equilibrium between 1D tendons and the surrounding 2D or 3D concrete material. Huang and Kang [73] present a nonlinear 1D finite element technique for post-tensioned concrete girder analysis, using truss elements for the tendon adjacent to the frame, similar to the method in SAP 2000. However, they include a node-to-segment frictional contact algorithm for tendon-frame interaction. Their method adds six DOFs per tendon, whereas the algorithm proposed in this paper requires only two additional DOFs per tendon bundle in three-dimensions.

Sousa et al. [74] and da Rocha Almeida et al. [75] introduced a refined 1D finite element formulation, “Beam-Truss Model”, for composite beam sections with post-tensioned tendons, employing 10 DOFs in a 2D framework to model partial steel-concrete interaction and tendon deformation linked to frame deformation. Their method is based on Euler beam theory (EBT) and the fiber method of multiple layers for steel, concrete, and tendon cross-sections. In contrast, the finite element technique presented in this paper based on Timoshenko beam theory for deep beams uses fewer DOFs in a 3D case and treats tendons as groups, not individually fixed to cross-sections at nodes. Additionally, it considers neutral axis movements from stress softening, which Sousa’s nonlinear method overlooks. Further study with regard to Beam-Truss model is referred to Ref. [76].

Husain and Wu [77] explored and compared multiple techniques for simulating prestressing in concrete members using the LS-DYNA software environment. Brenkus’ simplified methodology [78] models post-tensioning tendons as truss elements connected to nearby concrete layers for analyzing the flexible filler interface. Yet, it fails to incorporate the cohesive and frictional contact interactions of the tendons, leading to an absence of consistent stiffness and residual force matrices.

A key aspect of prestressing is its use in bridge design, a topic that is explored in detail in Hewson’s book [79]. The design criteria and procedures for linear prestressing, i.e., prestressing of 1D elements, is reflected in several major building codes such as ACI 318, BS 8110, and AS 3600 [80]. A concise overview of partial PC behavior can be found in the work of Au and Du [81].

Ou’s [82] dissertation focused on unbonded post-tensioned precast systems for bridge columns in earthquake-prone areas, aiming to reduce prestress loss during strong seismic events while enhancing hysteretic energy dissipation. To achieve this, mild steel energy dissipation bars, continuous across segment joints, were added to the columns. The study pays particular attention to deriving hysteresis curves for PC, a well-established tradition in related research.

The strut-and-tie model, a popular analysis technique in structural mechanics, is used for simulating masonry walls, prestressing anchorage zones, and designing transverse reinforcement in deep beams. This method has been included in PC standards and is addressed in Ref. [83]. Ruiz et al. [84–85] and Huber et al. [86–87] discuss the influence of post-tensioning on the shear strength of deep beams. Lee et al. [88] investigated the shear behavior of large-scale post-tensioned girders with a small shear span-to-depth ratio.

The long-term loss of prestress in post-tensioned box girders has been evaluated by Shing et al. [89] and Kottari and Shing [90]. Researchers [91–96] examined the flexural behavior of glass fiber reinforced polymer-reinforced concrete (GFRP-RC) slabs post-tensioned with carbon fiber reinforced polymer (CFRP) tendons. A computer program created by Jayasinghe predicts the long-term performance of post-tensioned concrete beams and one-way slabs subjected to sustained loads, utilizing an analytical model of discrete structural blocks formed by flexural cracks [97]. The influence of bond conditions on the cyclic performance of post-tensioned concrete using CFRP tendons was examined by Peng et al. [98].

Composite steel-concrete girders are frequently used in the design of bridges and buildings as an alternative to homogeneous structures. The design life of these girders may be reduced due to increased design loads or corrosion effects, potentially affecting the structure’s serviceability or strength limits during maintenance. In such cases, replacement or retrofitting through stiffening or strengthening may be necessary. Exterior post-tensioning, as investigated by Uy and Craine [99], is a strong option to enhance the design life of the system.

Kim and Lee [100], Zhou and Zheng [101], Six et al. [102] and Kim and Kang [103] conducted a nonlinear analysis of unbonded post-tensioning in continuous PC beams. In his MSc thesis, Lotfy [104] analyzed precast PC wind turbine towers. In their research, Yang and Mun [105] and Kwon et al. [106] examined the flexural capacity of lightweight concrete beams with unbonded post-tensioning. Gales [107] presents a comprehensive study on the thermal-displacement response of bonded and unbonded post-tensioned thin slabs under different boundary conditions in his dissertation. Kang et al. [108] evaluates the combination of bonded and unbonded post-tensioned concrete members.

Steel tendon corrosion remains a significant concern for the performance of PC beams, especially in unbonded post-tensioned configurations. Since both steel and concrete in PC structures are expensive and cracks on the beam’s tensile face often expose the steel to environmental damage, using corrosion-resistant materials instead of steel is an attractive option. Fiber reinforced polymer (FRP) is one such alternative, although FRP-prestressed beams tend to lack ductility. Hybrid FRP rods, proposed by Liang [109], offer a pseudo-ductile solution [110,111].

By enhancing the mechanical characteristics of standard concrete, fiber reinforced concrete offers a potential solution to some of the challenges faced in PC technology. According to Johnson [112], replacing 50% of the local transverse reinforcement with just 0.5% steel fiber volume in the post-tensioned anchorage zone of bridge girders can effectively reduce the amount of transverse reinforcement needed in the tensile bursting stress zone [113]. Local post-tensioning as a method to enhance flexural performance is examined by Taoum et al. [114]. The dissertation by Gavridou [115] presents shake table testing and analytical modeling for a full-scale, four-story unbonded post-tensioned concrete wall structure, along with bonded components in the orthogonal direction. Choi et al. [116] investigate the delamination failure of post-tensioned curved walls that lack through-thickness reinforcement.

The discontinuum-based cracking particle algorithm for meshfree analysis of cohesive crack propagation in concrete structures is due to Rabczuk and Belytschko [117], and extended to three-dimensional large deformation analysis of arbitrary evolving cracks in semi-brittle environments with and without enrichments in Refs. [118–120]. Similar contributions in enrichment function application to reproduce fracture DOFs in the finite element domain is addressed by Ebrahimi et al. [121–125].

It is noteworthy that the algorithm proposed in this paper does not employ deep neural networks (DNNs), neither for data-driven modeling nor for enforcing the partial differential equation and boundary/initial conditions, as is common in physics-informed neural networks (PINNs). Nevertheless, DNN-based approaches offer several advantages in engineering applications, including mesh-free computation, the ability to handle high-dimensional problems, and the use of automatic differentiation [126–128].

The idea behind the algorithm applied in this paper is making use of the surplus DOF of the PC beam, i.e., tendon’s sliding under bond-slip law, in a consistent mechanism considered to be conjugated with the system of equilibrium to form the total statement of the virtual work principle in the structural mechanics. The additional DOFs trace the average longitudinal slip/gap between a group of prestressing reinforcements and the concrete at every single node of the topology. Hence, the cohesive interaction between top or bottom reinforcement/tendon group with concrete beam is detected using the desired interface constitutive law for bond, un-bond and debond (bond damage) during a nonlinear material and geometry equilibrium equation, “Newton-Raphson,” solution procedure.

FRP reinforcement for PC structures can be incorporated using the formulation presented in this paper; however, the numerical advantages and limitations of this algorithm, particularly concerning FRP tendons with varying tendon-concrete interfaces, will be explored in greater detail in future discussions [129].

Since the beam model treats rotation as independent from the deflection gradient, as seen in Timoshenko or deep beam models, the proposed algorithm effectively supports nonlinear analysis of circular prestressing in elements like walls, domes, arch shells, and wide or deep partially prestressed cross-sections swept along smooth curves (referred to as Sweep Shell Instances in ABAQUS). The ACI 372R standard offers recommendations for the design and construction of circular prestressing. Circular prestressing is not specifically discussed in this study.

This paper is organized as follows. Section 2 presents the material library used for the analysis examples, and Section 3 explains the proposed formulation. Section 4 is dedicated to verification, with numerical examples of PC girders of different cross-section designs and corresponding discussions. The conclusion of the study is provided in Section 5.

2 Materials library

This study utilizes two distinct types of concrete materials and a single type of mild steel for prestressing reinforcement. The concrete materials selected represent both high-strength and low-strength variants commonly employed in RC structures. The study employs the Drucker–Prager plasticity model for concrete materials in conjunction with L2-plus finite element analysis. The algorithm’s accuracy is validated using the ‘Concrete Damage Plasticity’ model from ABAQUS. The parameters for concrete and steel are listed in Table 1.

2.1 Concrete materials

The stress–strain relation for the low strength concrete used in this study is described in Fig. 1. Figure 2 presents the unidirectional constitutive relation of the high strength steel.

2.2 Stress–strain relation for steel

The effective stress–strain curve for the steel material used for prestressing rebars used in this study is depicted in Fig. 3.

As shown, the steel rebars used for prestressing in this study are not selected from high-strength steel materials. Nonetheless, the numerical algorithm presented here is suitable for use with both mild steel and high-strength steel.

3 Formulation of L2-Plus beam finite element

The main concept of the algorithm is to establish consistent deformation in the prestressed beam by accounting for the bond-slip interaction between the prestressing tendons and the concrete cross-section. To achieve this, the deformation at each material point is specified as follows:

(1)\mathit{u}={\mathit{u}}^{\mathrm{c}}+⟨{\mathit{u}}^{\mathrm{g}\mathrm{a}\mathrm{p}}⟩,

where {\mathit{u}}^{\mathrm{c}} indicates the conventional deformations of the mixed L2 beam element and {\mathit{u}}^{\mathrm{g}\mathrm{a}\mathrm{p}} describes the augmented displacement of the prestressing tendons at the cross-section disclosed at each Gauss point of the element. Accordingly, the virtual work expression is written as

(2)\begin{array}{rl}& -{\int }_{\mathrm{\Omega }}^{}{\delta \mathit{\epsilon }}^{\mathrm{T}}:\mathit{\sigma }\mathrm{d}\mathrm{\Omega }+{\int }_{\mathrm{\Omega }}^{}{\delta \mathit{u}}^{\mathrm{T}}\cdot \mathit{b}\mathrm{d}\mathrm{\Omega }+{\int }_{\mathrm{\Gamma }}^{}\delta \overline{\mathit{u}}\cdot \overline{\mathit{t}}\mathrm{d}\mathrm{\Gamma }\\ & +{\int }_{{\mathrm{\Gamma }}^{\mathrm{g}\mathrm{a}\mathrm{p}}}^{}\delta {\mathit{u}}^{\mathrm{g}\mathrm{a}\mathrm{p}}\cdot {\mathit{t}}^{\mathrm{g}\mathrm{a}\mathrm{p}}\mathrm{d}{\mathrm{\Gamma }}^{\mathrm{g}\mathrm{a}\mathrm{p}}=0,\end{array}

where \{\delta \mathit{\epsilon },\delta \mathit{u},\delta \overline{\mathit{u}},\delta {\mathit{u}}^{\mathrm{g}\mathrm{a}\mathrm{p}}\} form the space of consistent deformations and \{\mathit{\sigma },\mathit{b},\overline{\mathit{t}},{\mathit{t}}^{\mathrm{g}\mathrm{a}\mathrm{p}}\} are associated to the system of equilibrium loads. The consistent deformation variables set can be decomposed as

(3)\delta \mathit{\epsilon }=\delta {\mathit{\epsilon }}^{\mathrm{c}}+⟨\delta {\mathit{\epsilon }}^{\mathrm{g}\mathrm{a}\mathrm{p}}⟩,

(4)\delta \mathit{u}=\delta {\mathit{u}}^{\mathrm{c}}+⟨\delta {\mathit{u}}^{\mathrm{g}\mathrm{a}\mathrm{p}}⟩,

(5)\delta \overline{\mathit{u}}=\delta {\overline{\mathit{u}}}^c+⟨\delta {\overline{\mathit{u}}}^{\mathrm{g}\mathrm{a}\mathrm{p}}⟩.

And the traction at the gap interface is equal to

(6){\mathit{t}}^{\mathrm{g}\mathrm{a}\mathrm{p}}=\mathit{P}\cdot {\mathit{\sigma }}^{\mathrm{g}\mathrm{a}\mathrm{p}}.

\mathit{P} is the concrete-tendon periphery at the cross-section and {\mathit{\sigma }}^{\mathrm{g}\mathrm{a}\mathrm{p}}is the cohesive traction extracted from the bond-slip relation presented in Fig. 4. The last term in Eq. (2) is present in the pre-tensioning conditions and absent in post-tensioning cases.

The prestressing tendons in the L2-Plus beam element are designed as two sets placed at the top and bottom of the cross-section. The paths of the tendons (sagging) may be defined by using a polynomial equation as a function of the local coordinate of the structural beam element, i.e., -1\le \xi \le 1, where \xi =-1,+1 indicate the two extremes of the structural beam element (Fig. 5).

An Updated-Lagrangian algorithm is employed to define the residual force at each iteration in the static analysis as

(7){\mathit{f}}_{\mathrm{e}\mathrm{x}\mathrm{t}}-{\int }_{\mathrm{\Omega }}^{}{\mathit{B}}^{\mathrm{T}}\mathit{\sigma }\mathrm{d}\mathrm{\Omega }=\left[\mathit{K}\right]\mathrm{\Delta }\overrightarrow{\mathit{U}}.

{\mathit{f}}_{\mathrm{e}\mathrm{x}\mathrm{t}} is the incremental displacement vector, \mathit{B} is the strain-displacement matrix, \left[\mathit{K}\right] is the elements stiffness matrix, \mathrm{\Delta }\overrightarrow{\mathit{U}} is the incremental displacement vector and \mathit{\sigma } at iteration i is written as

(8){\mathit{\sigma }}^i={\mathit{\sigma }}^{i-1}+{\mathit{D}}^{\mathrm{e}\mathrm{p}}\cdot \mathrm{\Delta }\mathit{\epsilon },

where {\mathit{D}}^{\mathrm{e}\mathrm{p}} is the elastoplastic stress–strain matrix of the element and \mathrm{\Delta }\mathit{\epsilon }, the incremental strain equals:

(9)\begin{array}{rl}\mathrm{\Delta }\mathit{\epsilon }=& {\left[\mathrm{\Delta }{\epsilon }_{11},\mathrm{\Delta }{\epsilon }_{22},\mathrm{\Delta }{\epsilon }_{33},\mathrm{\Delta }{\gamma }_{12},\mathrm{\Delta }{\gamma }_{13},\mathrm{\Delta }{\gamma }_{23}\right]}^{\mathrm{T}}\\ =& {\mathrm{\Delta }\mathit{\epsilon }}^{\mathrm{e}}+{\mathrm{\Delta }\mathit{\epsilon }}^{\mathrm{p}}=\mathit{B}\cdot \mathrm{\Delta }\overrightarrow{\mathit{U}},\end{array}

(10)\mathrm{\Delta }\overrightarrow{\mathit{U}}={\left[\mathrm{\Delta }U,\mathrm{\Delta }{W}_2,\mathrm{\Delta }{W}_3,\mathrm{\Delta }{\theta }_1,\mathrm{\Delta }{\theta }_2,\mathrm{\Delta }{\theta }_3,\mathrm{\Delta }{Gap}_{\mathrm{t}\mathrm{o}\mathrm{p}},\mathrm{\Delta }{Gap}_{\mathrm{b}\mathrm{o}\mathrm{t}}\right]}^{\mathrm{T}}.

This vector is introduced in Fig. 6, as well. The first six variables in \mathrm{\Delta }\overrightarrow{\mathit{U}} are the conventional variables of the mixed L2 finite element formulation. L2-Plus elements proposed in this paper, involves the last two additional variables per node, i.e., \mathrm{\Delta }{Gap}_{\mathrm{t}\mathrm{o}\mathrm{p}} and \mathrm{\Delta }{Gap}_{\mathrm{b}\mathrm{o}\mathrm{t}} for the PC Timoshenko beam elements.

The components of the incremental strain vector are defined as

(11)\mathrm{\Delta }{\epsilon }_{11}=\frac{\mathrm{\partial }{u}_1}{\mathrm{\partial }{x}_1}=\mathrm{\Delta }{\epsilon }_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}+x_3\frac{\mathrm{\partial }{\theta }_2}{\mathrm{\partial }{x}_1}-x_2\frac{\mathrm{\partial }{\theta }_3}{\mathrm{\partial }{x}_1}+⟨\mathrm{\Delta }{\epsilon }_{11}^{\mathrm{g}\mathrm{a}\mathrm{p}}⟩,

(12)\mathrm{\Delta }{\epsilon }_{22}=\frac{\mathrm{\partial }{u}_2}{\mathrm{\partial }{x}_2}=-\beta \mathrm{\Delta }{\epsilon }_{11},

(13)\mathrm{\Delta }{\epsilon }_{33}=\frac{\mathrm{\partial }{u}_3}{\mathrm{\partial }{x}_3}=-\beta \mathrm{\Delta }{\epsilon }_{11},

(14)\mathrm{\Delta }{\gamma }_{12}=\frac{\mathrm{\partial }{W}_2}{\mathrm{\partial }{x}_1}-{\theta }_3-x_3\frac{\mathrm{\partial }{\theta }_1}{\mathrm{\partial }{x}_1},

(15)\mathrm{\Delta }{\gamma }_{13}=\frac{\mathrm{\partial }{W}_3}{\mathrm{\partial }{x}_1}+{\theta }_2+x_2\frac{\mathrm{\partial }{\theta }_1}{\mathrm{\partial }{x}_1},

(16)\mathrm{\Delta }{\gamma }_{23}=0.

In the above, \left\{u_1,u_2,u_3\right\} and \left\{x_1,x_2,x_3\right\} are the sets of material point displacements and coordinates, respectively, in the local coordinate system. \beta is defined as

(17)\beta =\upsilon \left(1-SCR\right),

where \upsilon is the section Poisson’s ratio and 0\le SCR\le 1 denotes the section confinement ratio equal to zero for sections absolutely not confined and equal to unity for sections fully confined via transverse reinforcement or wrapping FRP textures.

\mathrm{\Delta }{\epsilon }_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}=\mathrm{\partial }U/\mathrm{\partial }{x}_1 is the axial strain increment which is approximated prior to other strain increment components in order to relocate the local coordinate axes of the cross-section at the current integration point of the L2-Plus beam element. The local unit vectors {\overrightarrow{e}}_2 and {\overrightarrow{e}}_3 are determined based on the \mathrm{\Delta }{\sigma }_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}} calculated from \mathrm{\Delta }{\epsilon }_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}} by using the following procedure:

(18)\mathrm{\Delta }{x}_2=-{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{x}_2{\mathrm{\Delta }\mathit{\sigma }}_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}\mathrm{d}A/{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{\mathrm{\Delta }\mathit{\sigma }}_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}\mathrm{d}A,

(19)\mathrm{\Delta }{x}_3=-{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{x}_3{\mathrm{\Delta }\mathit{\sigma }}_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}\mathrm{d}A/{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{\mathrm{\Delta }\mathit{\sigma }}_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}\mathrm{d}A,

(20){\overline{x}}_2^i={\overline{x}}_2^{i-1}+\mathrm{\Delta }{x}_2,

(21){\overline{x}}_3^i={\overline{x}}_3^{i-1}+\mathrm{\Delta }{x}_3,

(22)I_{22}={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{{\overline{x}}_3}^2\mathrm{d}A,

(23)I_{33}={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{{\overline{x}}_2}^2\mathrm{d}A,

(24)I_{23}={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{\overline{x}}_2{\overline{x}}_3\mathrm{d}A,

(25)\vartheta =\frac{1}{2}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\frac{2I_{23}}{I_{33}-I_{22}},

(26){\left[\begin{array}{c}{x}_2\\ x_3\end{array}\right]}^i=\left[\begin{array}{cc}\mathrm{c}\mathrm{o}\mathrm{s}\vartheta & -\mathrm{s}\mathrm{i}\mathrm{n}\vartheta \\ \mathrm{s}\mathrm{i}\mathrm{n}\vartheta & \mathrm{c}\mathrm{o}\mathrm{s}\vartheta \end{array}\right]{\left[\begin{array}{c}{\overline{x}}_2\\ {\overline{x}}_3\end{array}\right]}^i.

Consequently, the strain-displacement matrix associated to node i is defined as

(27){\mathit{B}}^{\left(i-node\right)}={\left[B_{11},B_{22},B_{23},B_{12},B_{13},B_{23}\right]}^{\mathrm{T}}.

The arrays B_{ij} are defined as follows:

(28){\mathit{B}}_{11}={\mathit{B}}_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}+{\mathit{B}}_{11}^{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{l}}+⟨{\mathit{B}}_{11}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}⟩+⟨{\mathit{B}}_{11}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}⟩,

(29){\mathit{B}}_{22}={\mathit{B}}_{33}=-\beta {\mathit{B}}_{11},

(30){\mathit{B}}_{12}={\left[\begin{array}{ccccccc}0& {\displaystyle \frac{\mathrm{\partial }{\overline{N}}_1^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}+{\overline{\overline{N}}}_1^{\left(i-node\right)}& 0& {-x}_3{\displaystyle \frac{\mathrm{\partial }{N}^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& 0& -{\displaystyle \frac{\mathrm{\partial }{\overline{N}}_2^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}-{\overline{\overline{N}}}_2^{\left(i-node\right)}& O_{1\times 2}\end{array}\right]}^{\mathrm{T}},

(31){\mathit{B}}_{13}={\left[\begin{array}{ccccc}{\mathit{O}}_{1\times 2}& {\displaystyle \frac{\mathrm{\partial }{\overline{N}}_1^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}+{\overline{\overline{N}}}_1^{\left(i-node\right)}& x_2{\displaystyle \frac{\mathrm{\partial }{N}^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& {\displaystyle \frac{\mathrm{\partial }{\overline{N}}_2^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}+{\overline{\overline{N}}}_2^{\left(i-node\right)}& {\mathit{O}}_{1\times 3}\end{array}\right]}^{\mathrm{T}},

(32){\mathit{B}}_{23}={\mathit{O}}_{8\times 1}.

The ingredient variables in the aforementioned equations are defined as below:

(33){\mathit{B}}_{11}^{\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{a}\mathrm{l}}={\left[\begin{array}{cc}{\displaystyle \frac{\mathrm{\partial }{N}^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& {\mathit{O}}_{1\times 7}\end{array}\right]}^{\mathrm{T}},

(34){\mathit{B}}_{11}^{\mathrm{f}\mathrm{l}\mathrm{e}\mathrm{x}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{l}}={\left[\begin{array}{ccccccc}0& x_2{\displaystyle \frac{\mathrm{\partial }{\overline{\overline{N}}}_1^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& x_3{\displaystyle \frac{\mathrm{\partial }{\overline{\overline{N}}}_1^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& 0& x_3{\displaystyle \frac{\mathrm{\partial }{\overline{\overline{N}}}_2^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& {-x}_2{\displaystyle \frac{\mathrm{\partial }{\overline{\overline{N}}}_2^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& {\mathit{O}}_{1\times 2}\end{array}\right]}^{\mathrm{T}},

(35){\mathit{B}}_{11}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}={\left[\begin{array}{ccc}{\mathit{O}}_{1\times 6}& {\displaystyle \frac{\mathrm{\partial }{N}^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}& 0\end{array}\right]}^{\mathrm{T}},

(36){\mathit{B}}_{11}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}={\left[\begin{array}{ccc}{\mathit{O}}_{1\times 6}& 0& {\displaystyle \frac{\mathrm{\partial }{N}^{\left(i-node\right)}}{\mathrm{\partial }{x}_1}}\end{array}\right]}^{\mathrm{T}}.

{\mathit{O}}_{n\times m} denotes the zero matrix of size n\times m. N^{\left(i-node\right)} indicates the standard interpolation function of the L2 beam element written as follows:

(37)N^{\left(1\right)}\left(\xi \right)=\frac{1-\xi }{2},

(38)N^{\left(2\right)}\left(\xi \right)=\frac{1+\xi }{2}.

{\overline{N}}_{j=1,2}^{\left(i-node\right)} and {\overline{\overline{N}}}_{j=1,2}^{\left(i-node\right)} are the Hermit functions of the L2 element for the interpolation of deflections and rotations, respectively.

(39){\overline{N}}_1^{\left(1\right)}\left(\xi \right)=\frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\left\{\frac{J^2}{6}{\xi }^3-\left(\frac{J^2}{2}+{\beta }_{\mathrm{T}}\right)\xi +\left(\frac{J^2}{3}+{\beta }_{\mathrm{T}}\right)\right\},

(40)\begin{array}{rl}{\overline{N}}_2^{\left(1\right)}\left(\xi \right)=& \frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\{-\frac{J^3}{6}{\xi }^3+\frac{J}{2}\left(\frac{J^2}{3}+{\beta }_{\mathrm{T}}\right){\xi }^2\\ & +\frac{J^3}{6}\xi -\frac{J}{2}\left(\frac{J^2}{3}+{\beta }_{\mathrm{T}}\right)\},\end{array}

(41){\overline{N}}_1^{\left(2\right)}\left(\xi \right)=\frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\left\{-\frac{J^2}{6}{\xi }^3+\left(\frac{J^2}{2}+{\beta }_{\mathrm{T}}\right)\xi +\left(\frac{J^2}{3}+{\beta }_{\mathrm{T}}\right)\right\},

(42)\begin{array}{rl}{\overline{N}}_2^{\left(2\right)}\left(\xi \right)=& \frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\{-\frac{J^3}{6}{\xi }^3-\frac{J}{2}\left(\frac{J^2}{3}+{\beta }_T\right){\xi }^2\\ & +\frac{J^3}{6}\xi +\frac{J}{2}\left(\frac{J^2}{3}+{\beta }_{\mathrm{T}}\right)\},\end{array}

(43){\overline{\overline{N}}}_1^{\left(1\right)}\left(\xi \right)=\frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\left\{\frac{J}{2}\left(1-{\xi }^2\right)\right\},

(44){\overline{\overline{N}}}_2^{\left(1\right)}\left(\xi \right)=\frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\left\{\frac{J^2}{2}{\xi }^2-\left(\frac{J^2}{3}+{\beta }_{\mathrm{T}}\right)\xi -\left(\frac{J^2}{6}-{\beta }_{\mathrm{T}}\right)\right\},

(45){\overline{\overline{N}}}_1^{\left(2\right)}\left(\xi \right)=\frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\left\{-\frac{J}{2}\left(1-{\xi }^2\right)\right\},

(46){\overline{\overline{N}}}_2^{\left(2\right)}\left(\xi \right)=\frac{1}{2\left(J^2/3+{\beta }_{\mathrm{T}}\right)}\left\{\frac{J^2}{2}{\xi }^2+\left(\frac{J^2}{3}+{\beta }_{\mathrm{T}}\right)\xi -\left(\frac{J^2}{6}-{\beta }_{\mathrm{T}}\right)\right\}.

{\beta }_{\mathrm{T}} is the deep beam index defined as below:

(47){\beta }_{\mathrm{T}}=\{\begin{array}{ccc}E{I}_{22}/k_{2}AG,& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}& \text{'}{2}^{\prime }\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n},\\ E{I}_{33}/k_{3}AG,& \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{i}\mathrm{n}& \text{'}{3}^{\prime }\mathrm{d}\mathrm{i}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}.\end{array}

E is Young’s modulus. k_{i=2,3} are shear coefficients of the section in the i direction, G is the shear modulus of the material, A is the cross-section area and I_{22}, I_{33} are section moduli in “2” and “3” directions, respectively. J denotes the Jacobean determinant introduced as

(48)\mathit{J}=\frac{\mathrm{d}{x}_1}{\mathrm{d}\xi }=\left[\begin{array}{cc}{\displaystyle \frac{\mathrm{\partial }{N}^{\left(1\right)}}{\mathrm{\partial }\xi }}& {\displaystyle \frac{\mathrm{\partial }{N}^{\left(2\right)}}{\mathrm{\partial }\xi }}\end{array}\right]\left[\begin{array}{ccc}{X}_1& Y_1& Z_1\\ X_2& Y_2& Z_2\end{array}\right]{\left[{\overrightarrow{\mathit{e}}}_1\right]}_{3\times 1}.

By applying the Hermite shape functions described earlier, the fourth-order equilibrium equation is satisfied. The use of two Gauss points for integrating the flexural and shear terms in the stiffness matrix and residual force vector prevents locking issues and promotes effective energy norm convergence.

The displacement variation at any material point within the cross-section is determined as follows:

(49)\delta \mathit{u}={\left[\delta u_1,\delta u_2,\delta u_3\right]}^T=\left[\mathit{N}\right]\delta \overrightarrow{\mathit{U}},

(50){\left[\mathit{N}\right]}^{i-node=1,2}=\left[\begin{array}{cccccccc}{N}^{\left(i-node\right)}& x_2{\overline{\overline{N}}}_1^{\left(i-node\right)}& x_3{\overline{\overline{N}}}_1^{\left(i-node\right)}& 0& x_3{\overline{\overline{N}}}_2^{\left(i-node\right)}& {-x}_2{\overline{\overline{N}}}_2^{\left(i-node\right)}& 0& 0\\ 0& {\overline{N}}_1^{\left(i-node\right)}& 0& 0& 0& -{\overline{N}}_2^{\left(i-node\right)}& 0& 0\\ 0& 0& {\overline{N}}_1^{\left(i-node\right)}& 0& {\overline{N}}_2^{\left(i-node\right)}& 0& 0& 0\end{array}\right]+⟨{\mathit{N}}_{\left(i-node\right)}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}⟩+⟨{\mathit{N}}_{\left(i-node\right)}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}⟩,

(51){\mathit{N}}_{\left(i-node\right)}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}=\left[\begin{array}{ccc}{\mathit{O}}_{1\times 6}& N^{\left(i-node\right)}& 0\\ {\mathit{O}}_{2\times 6}& {\mathit{O}}_{2\times 1}& {\mathit{O}}_{2\times 1}\end{array}\right],

(52){\mathit{N}}_{\left(i-node\right)}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}=\left[\begin{array}{cc}{\mathit{O}}_{1\times 7}& N^{\left(i-node\right)}\\ {\mathit{O}}_{2\times 7}& 0\end{array}\right].

The section force vector is given as follows (Fig. 6):

(53)\overrightarrow{\mathit{F}}={\left[\begin{array}{cccccccc}N& Q_2& Q_3& T_1& M_2& M_3& F_{\mathrm{t}\mathrm{o}\mathrm{p}}& F_{\mathrm{b}\mathrm{o}\mathrm{t}}\end{array}\right]}^{\mathrm{T}},

(54)N={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{\sigma }_{11}\mathrm{d}A,

(55)Q_2={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{\sigma }_{22}\mathrm{d}A,

(56)Q_3={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{\sigma }_{33}\mathrm{d}A,

(57)T_1={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}\left(x_2{\tau }_{13}-x_3{\tau }_{12}\right)\mathrm{d}A,

(58)M_2={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{x}_3{\sigma }_{11}\mathrm{d}A,

(59)M_3={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{-x}_2{\sigma }_{11}\mathrm{d}A,

(60)F_{\mathrm{t}\mathrm{o}\mathrm{p}}={\int }_{A_{\mathrm{t}\mathrm{o}\mathrm{p}}}^{}{\sigma }_{11}\mathrm{d}A,

(61)F_{\mathrm{b}\mathrm{o}\mathrm{t}}={\int }_{A_{\mathrm{b}\mathrm{o}\mathrm{t}}}^{}{\sigma }_{11}\mathrm{d}A.

3.1 Transformation to global coordinate system

The transformation matrix is used to map the section vectors to the global (nodal) coordinate system as follows (Fig. 6):

(62)\begin{array}{rl}{\overrightarrow{\mathit{U}}}^{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}=& {\left[\mathit{T}\right]}^{\mathrm{T}}\overrightarrow{\mathit{U}}\\ =& {\left[\begin{array}{cccccccc}{U}_X& U_Y& U_Z& {\mathrm{\Theta }}_X& {\mathrm{\Theta }}_Y& {\mathrm{\Theta }}_Z& {GAP}_{\mathrm{t}\mathrm{o}\mathrm{p}}& {GAP}_{\mathrm{b}\mathrm{o}\mathrm{t}}\end{array}\right]}^{\mathrm{T}},\end{array}

(63)\begin{array}{rl}{\overrightarrow{\mathit{F}}}^{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}=& {\left[\mathit{T}\right]}^{\mathrm{T}}\overrightarrow{\mathit{F}}\\ =& {\left[\begin{array}{cccccccc}{F}_X& F_Y& F_Z& T_X& M_Y& M_Z& F_{\mathrm{t}\mathrm{o}\mathrm{p}}& F_{\mathrm{b}\mathrm{o}\mathrm{t}}\end{array}\right]}^{\mathrm{T}}.\end{array}

The transformation matrix for L2-Plus element is defined as,

(64)\begin{array}{rl}& \left[\mathit{T}\right]\\ =& \left[\begin{array}{ccc}{\left[\begin{array}{ccc}{\overrightarrow{e}}_1& {\overrightarrow{e}}_2& {\overrightarrow{e}}_3\end{array}\right]}_{3\times 3}& {\mathit{O}}_{3\times 3}& {\mathit{O}}_{3\times 2}\\ {\mathit{O}}_{3\times 3}& {\left[\begin{array}{ccc}{\overrightarrow{e}}_1& {\overrightarrow{e}}_2& {\overrightarrow{e}}_3\end{array}\right]}_{3\times 3}& {\mathit{O}}_{3\times 2}\\ {\mathit{O}}_{2\times 3}& {\mathit{O}}_{2\times 3}& \left({\overrightarrow{e}}_1\cdot {\overrightarrow{e}}_g\right){\mathrm{e}\mathrm{y}\mathrm{e}}_{2\times 2}\end{array}\right].\end{array}

Finally, the internal force vector and consistent elemental stiffness matrix are written as follows in the global coordinate system:

(65)\begin{array}{rl}{\mathit{f}}_{\mathrm{i}\mathrm{n}\mathrm{t}}=& {\int }_{l_{\mathrm{b}}}^{}([\{[\begin{array}{cc}{t}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}& {\mathit{O}}_{1\times 2}\end{array}]{\mathit{N}}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}\\ & +[\begin{array}{cc}{t}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}& {\mathit{O}}_{1\times 2}\end{array}]{\mathit{N}}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}\}[\mathit{T}]{]}^{\mathrm{T}}\\ & +{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{(B[\mathit{T}])}^{\mathrm{T}}\sigma \mathrm{d}A)\mathrm{d}l,\end{array}

(66){\mathit{f}}_{\mathrm{e}\mathrm{x}\mathrm{t}}-{\mathit{f}}_{\mathrm{i}\mathrm{n}\mathrm{t}}={\mathit{K}}^{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}\mathrm{\Delta }{\overrightarrow{\mathit{U}}}^{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}},

(67)\begin{array}{rl}{\mathit{K}}^{\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{b}\mathrm{a}\mathrm{l}}=& {\int }_{l_{\mathrm{b}}}({\left({\mathit{N}}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}\left[\mathit{T}\right]\right)}^{\mathrm{T}}{P}^{\mathrm{t}\mathrm{o}\mathrm{p}}{k}^{\mathrm{t}\mathrm{o}\mathrm{p}}{\mathit{N}}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{t}\mathrm{o}\mathrm{p}}\left[\mathit{T}\right]\\ & +{\left({\mathit{N}}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}\left[\mathit{T}\right]\right)}^{\mathrm{T}}{P}^{\mathrm{b}\mathrm{o}\mathrm{t}}{k}^{\mathrm{b}\mathrm{o}\mathrm{t}}{\mathit{N}}^{\mathrm{g}\mathrm{a}\mathrm{p}-\mathrm{b}\mathrm{o}\mathrm{t}}\left[\mathit{T}\right]\\ & +{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}{\left(B\left[\mathit{T}\right]\right)}^{\mathrm{T}}{D}^{\mathrm{e}\mathrm{p}}B\left[\mathit{T}\right]\mathrm{d}A)\mathrm{d}l.\end{array}

3.2 Elastoplastic Cauchy stress tensor

To compute the residual force and consistent stiffness matrix from the above formulations, the incremental stress \mathrm{\Delta }\mathit{\sigma } and elastoplastic stress–strain matrix {\mathit{D}}^{\mathrm{e}\mathrm{p}} must be determined. The stress increment for each iteration is formulated as follows:

(68)\mathrm{\Delta }\mathit{\sigma }=D^{\mathrm{e}}\mathrm{\Delta }{\mathit{\epsilon }}^{\mathrm{e}}=[{\mathrm{\Delta }\sigma }_{11},{\mathrm{\Delta }\sigma }_{22},{\mathrm{\Delta }\sigma }_{33},{\mathrm{\Delta }\tau }_{12},{\mathrm{\Delta }\tau }_{13},{\mathrm{\Delta }\tau }_{23}],

(69)\begin{array}{rl}{D}^{\mathrm{e}}=& \frac{E}{\left(1+\nu \right)\left(1-2\nu \right)}\\ & \cdot \left[\begin{array}{cccc}1-\nu & \nu & \nu & {\mathit{O}}_{1\times 3}\\ \nu & 1-\nu & \nu & {\mathit{O}}_{1\times 3}\\ \nu & \nu & 1-\nu & {\mathit{O}}_{1\times 3}\\ {\mathit{O}}_{3\times 1}& {\mathit{O}}_{3\times 1}& {\mathit{O}}_{1\times 3}& \left(0.5-\nu \right){\mathrm{e}\mathrm{y}\mathrm{e}}_{3\times 3}\end{array}\right].\end{array}

Assuming elastic shear behavior for the deep beam, the incremental shear components of the stress tensor are expressed as

(70)\mathrm{\Delta }{\tau }_{12}=k_{2}G\mathrm{\Delta }{\gamma }_{12},

(71)\mathrm{\Delta }{\tau }_{13}=k_{3}G\mathrm{\Delta }{\gamma }_{13}.

k_{j=2,3} is the shear coefficient in the j axis direction. Nonetheless, \mathrm{\Delta }{\tau }_{12} and \mathrm{\Delta }{\tau }_{13} play a role in determining the elastoplastic stress state at every point within the cross-section. With the strain increment \mathrm{\Delta }\mathit{\epsilon } identified, the initial stage of calculating \mathrm{\Delta }\mathit{\sigma } assumes an elastic behavior, implying \mathrm{\Delta }\mathit{\epsilon }=\mathrm{\Delta }{\mathit{\epsilon }}^{\mathit{e}}.

After calculation of the updated Cauchy stress \mathit{\sigma }, the Yield criterion is applied to determine if the inequality F\left(\sigma ,\kappa \right)\le 0 holds (elastic) or not (plastic). If the criterion is violated, then we have the following additional constraint:

(72)\mathrm{d}F\left(\sigma ,\kappa \right)=0,

\kappa is the stress hardening parameter. From the equation above, the elastoplastic stress–strain matrix is derived as

(73)D^{\mathrm{e}\mathrm{p}}=\frac{A}{A+{\mathit{a}}^{\mathrm{T}}{D}^e\mathit{a}}{D}^{\mathrm{e}},

(74)A=-\frac{\mathrm{\partial }F}{\mathrm{\partial }\kappa }\frac{\mathrm{\partial }\kappa }{\mathrm{\partial }{\epsilon }^{\mathrm{p}}}\frac{\mathrm{\partial }F}{\mathrm{\partial }\sigma },

(75)a={\left[\frac{\mathrm{\partial }F}{\mathrm{\partial }\mathit{\sigma }}\right]}^{\mathrm{T}}.

The plastic criterion for ductile materials, i.e., von-Mises is expressed as

(76)F\left(\sigma ,\kappa \right)=\sqrt{J_2^{{}^{\prime }}}-\frac{{\sigma }_Y\left(\kappa \right)}{\sqrt{3}}\le 0,

(77)J_2^{\prime }=\frac{1}{2}{\sigma }_{ij}^{\prime }{\sigma }_{ij}^{\prime },

where {\sigma }_{ij}^{\prime } is deviatoric stress tensor. For concrete materials, the Drucker–Prager yield criterion is used, written as

(78)F\left(\sigma ,\kappa \right)=\sqrt{J_2^{\prime }}+\frac{2\mathrm{s}\mathrm{i}\mathrm{n}\varphi }{\sqrt{3}\left(3-\mathrm{s}\mathrm{i}\mathrm{n}\varphi \right)}{J}_1-\frac{6c\mathrm{c}\mathrm{o}\mathrm{s}\varphi }{\sqrt{3}\left(3-\mathrm{s}\mathrm{i}\mathrm{n}\varphi \right)}\le 0,

where, c and \varphi are cohesion and angle of internal friction, respectively.

3.3 Multiscale topology description (overlay mesh)

Since the prestressing reinforcement area is negligible relative to the beam cross-section, generating a fibrous mesh for the beam section is expected to produce elements of two distinct scales. This characteristic renders the L2-Plus algorithm computationally inefficient. Replacing a compatible non-uniform triangular or quadrilateral mesh with an ‘overlay mesh’ improves the algorithm’s computational cost efficiency.

This method involves meshing the tendons or reinforcement cross-sections independently from the concrete cross-section of the beam. As a result, the contribution of the prestressing materials is accounted for twice: once in the steel and once in the background mesh of the concrete. In nonlinear static procedures, the contribution of the latter should be removed from the calculations of internal force and the consistent stiffness matrix. A schematic of the ordinary mesh versus the overlay mesh is presented in Fig. 5.

This research observes that an overlay mesh decreases the mesh size by a factor of two to ten, especially when the tendons’ distribution in the beam cross-section is highly entropic.

3.4 Nonlinear static algorithm for prestressed concrete beams

The updated-Lagrangian algorithm for the analysis of PC beam is described in Fig. 7.

4 Verification and numerical examples

In this section, the proposed algorithm is validated through analytical calculations for an elastic PC beam and the ABAQUS standard program. Subsequently, the proposed L2-Plus element and algorithm are applied to various PC beam sections. In all numerical examples, the deep beam formulation is applied to a 4000 mm beam with one end pinned and the other end on a roller support. A three-point load test is performed on the beams under unidirectional and cyclic loading scenarios.

The analytical approach assumes elastic and bonded behavior of the beam. For elastoplastic behavior, the Drucker–Prager plasticity model in the proposed L2-Plus algorithm has been validated against the “concrete damage plasticity” algorithm in ABAQUS.

As previously noted, this study uses mild steel prestressing bars instead of high-strength steel. However, both high-strength and medium-strength concrete are considered for the PC beam analysis.

4.1 Verification example

Shown in Fig. 8 is the section considered for verification. This figure displays the beam dimensions, tendon details, and the overlay mesh of the cross-section.

In the following analysis, the top and bottom tendon sets are stretched from both ends with varying amounts of prestressing. The beam is then subjected to both unilateral and bilateral loading to examine the effect on its flexural load-bearing capacity under a three-point bending test.

4.1.1 Analytical calculations for elastic prestressed concrete beam with top and bottom prestressing reinforcements using Euler−Bernoulli theory

Figure 9 outlines the components of the cross-section equilibrium in the elastic mode.

The equilibrium between the tensile stress in the top and bottom tendons and the compressive forces in the concrete is expressed as

(79)\begin{array}{rl}N=& {\int }_{A_{\mathrm{c}\mathrm{s}}}^{}\sigma \mathrm{d}A={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}E\epsilon \mathrm{d}A\\ =& {\int }_{A_{\mathrm{c}\mathrm{s}}}^{}E\left(\frac{\mathrm{d}g}{\mathrm{d}x}-\mathrm{z}\frac{{\mathrm{d}}^{2}y}{{\mathrm{d}x}^2}\right)\mathrm{d}A+F_{\mathrm{t}}+F_{\mathrm{b}}=0.\end{array}

The moment equilibrium at the section is expressed as

(80)\begin{array}{rl}M=& -{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}\sigma \mathrm{z}\mathrm{d}A=-{\int }_{A_{\mathrm{c}\mathrm{s}}}^{}E\left(\frac{\mathrm{d}g}{\mathrm{d}x}-\mathrm{z}\frac{{\mathrm{d}}^{2}y}{{\mathrm{d}x}^2}\right)\mathrm{z}\mathrm{d}A\\ & -F_{\mathrm{t}}{\mathrm{z}}_{\mathrm{t}}-F_{\mathrm{b}}{\mathrm{z}}_{\mathrm{b}}=\frac{1}{2}Px.\end{array}

In the equations above, \mathrm{d}g/\mathrm{d}x denotes the derivative of the tendons’ slip along the length of the beam and F_{\mathrm{t}}, F_{\mathrm{b}} indicate the prestress loss due to elastic strain in the beam. The expression for \mathrm{d}g/\mathrm{d}x on the top and bottom prestressing tendons is given as

(81){\int }_0^{l_{\mathrm{b}}/2}{g}_{\mathrm{t}}^{\prime }\mathrm{d}x=g_{0\mathrm{t}},

(82){\int }_0^{l_{\mathrm{b}}/2}{g}_{\mathrm{b}}^{\prime }\mathrm{d}x=g_{0\mathrm{b}}.

Through the expansion of the equilibrium equation for section forces, we derive:

(83){\int }_{A_{\mathrm{t}}}^{}E{g}_{\mathrm{t}}^{\prime }\mathrm{d}A+{\int }_{A_{\mathrm{b}}}^{}E{g}_{\mathrm{b}}^{\prime }\mathrm{d}A-\left({\int }_{A_{\mathrm{c}\mathrm{s}}}^{}E\mathrm{z}\mathrm{d}A\right)\frac{{\mathrm{d}}^{2}y}{{\mathrm{d}x}^2}=-F_{\mathrm{t}}-F_{\mathrm{b}}.

In a similar manner, expanding the moment equilibrium equation yields:

(84)\begin{array}{rl}& {\int }_{A_{\mathrm{t}}}^{}E\mathrm{z}{g}_t^{\prime }\mathrm{d}A+{\int }_{A_{\mathrm{b}}}^{}E\mathrm{z}{g}_{\mathrm{b}}^{\prime }\mathrm{d}A-\left({\int }_{A_{\mathrm{c}\mathrm{s}}}E{\mathrm{z}}^2\mathrm{d}A\right)\frac{{\mathrm{d}}^{2}y}{{\mathrm{d}x}^2}\\ =& -\frac{1}{2}Px-F_{\mathrm{t}}-F_{\mathrm{b}}.\end{array}

Since {\int }_{A_{\mathrm{c}\mathrm{s}}}E\mathrm{z}\mathrm{d}A=0, Eq. (83) simplifies to

(85)E_{\mathrm{s}}{A}_{\mathrm{t}}{g}_{\mathrm{t}}^{\prime }+E_{\mathrm{s}}{A}_{\mathrm{b}}{g}_{\mathrm{b}}^{\prime }=-F_{\mathrm{t}}-F_{\mathrm{b}}.

Averaging g_{\mathrm{t}}^{\prime } and g_{\mathrm{b}}^{\prime } from Eq. (82), gives us:

(86)g_{\mathrm{t}}^{\prime }=\frac{2g_{0\mathrm{t}}}{l},

(87)g_{\mathrm{b}}^{\prime }=\frac{2g_{0\mathrm{b}}}{l}.

From Eqs. (85)−(87), we extract the values of F_{\mathrm{t}} and F_{\mathrm{b}} and replace in the moment’s equilibrium of Eq. (84) to have:

(88)F_{\mathrm{t}}=-\frac{2g_{0\mathrm{t}}}{l}{c}_{1\mathrm{t}},

(89)F_{\mathrm{b}}=-\frac{2g_{0\mathrm{b}}}{l}{c}_{1\mathrm{b}},

where c_{1\mathrm{t}} and c_{1\mathrm{b}} are defined as

(90)c_{1\mathrm{t}}=E_{\mathrm{s}}{A}_{\mathrm{t}},

(91)c_{1\mathrm{b}}=E_{\mathrm{s}}{A}_{\mathrm{b}}.

At this stage, the moment equilibrium equation contains all the variables except {\mathrm{d}}^{2}y/{\mathrm{d}x}^2. The equation becomes as follows:

(92)\begin{array}{rl}& g_{\mathrm{t}}^{\prime }{\int }_{A_{\mathrm{t}}}^{}E\mathrm{z}\mathrm{d}A+g_{\mathrm{b}}^{\prime }{\int }_{A_{\mathrm{b}}}^{}E\mathrm{z}\mathrm{d}A-\left({\int }_{A_{\mathrm{c}\mathrm{s}}}^{}E{\mathrm{z}}^2\mathrm{d}A\right)\frac{{\mathrm{d}}^{2}y}{{\mathrm{d}x}^2}\\ =& -\frac{1}{2}Px+\frac{2g_{0\mathrm{t}}}{l}{c}_{1\mathrm{t}}{\mathrm{z}}_{\mathrm{t}}+\frac{2g_{0\mathrm{b}}}{l}{c}_{1\mathrm{b}}{\mathrm{z}}_{\mathrm{b}}.\end{array}

We define the parameters c_{2\mathrm{t}} and c_{2\mathrm{b}} and c_3 as follows:

(93)c_{2\mathrm{t}}={\int }_{A_{\mathrm{t}}}^{}E\mathrm{z}\mathrm{d}A,

(94)c_{2\mathrm{b}}=-{\int }_{A_{\mathrm{b}}}^{}E\mathrm{z}\mathrm{d}A,

(95)c_3={\int }_{A_{\mathrm{c}\mathrm{s}}}^{}E{\mathrm{z}}^2\mathrm{d}A.

Consequently, the equation can be written as follows:

(96)c_3\frac{{\mathrm{d}}^{2}y}{{\mathrm{d}x}^2}=\frac{1}{2}Px+\frac{2g_{0\mathrm{t}}}{l}\left(c_{2\mathrm{t}}-c_{1\mathrm{t}}{.\mathrm{z}}_{\mathrm{t}}\right)-\frac{2g_{0\mathrm{b}}}{l}\left(c_{2\mathrm{b}}+c_{1\mathrm{b}}{\mathrm{z}}_{\mathrm{b}}\right).

The following boundary conditions apply to the second-order differential equation above:

(97)\{\begin{array}{rl}& {y|}_{x=0}=0,\\ & {\frac{\mathrm{d}y}{\mathrm{d}x}|}_{x=l/2}=0.\end{array}

By solving the above equation, the deflection at the mid-span of the beam is derived as follows:

(98)c_3{y|}_{x=l/2}=\frac{l}{4}\left[g_{0\mathrm{b}}\left(c_{2\mathrm{b}}+c_{1\mathrm{b}}{.\mathrm{z}}_{\mathrm{b}}\right)-g_{0\mathrm{t}}\left(c_{2\mathrm{t}}-c_{1\mathrm{t}}{.\mathrm{z}}_{\mathrm{t}}\right)\right]-\frac{1}{48}P{l}^3.

The first term in the equation above represents the camber of the prestressed beam, while the second term corresponds to the slope of the load–deflection curve from the three-point bending test. Figure 10 illustrates the analytical load–deflection curves for various prestressing conditions.

As shown in Fig. 10, increasing the pre-tensioning gap results in a greater camber for the PC beam. In the following the results extracted by ABAQUS program are investigated.

4.1.2 Elastic and elastoplastic analysis of prestressed concrete beam via ABAQUS program

A three-dimensional solid model of the beam, consisting of 41013 nodes and 31120 C3D8R elements, has been created in ABAQUS software. The configuration of the meshed parts is presented in Fig. 11.

Three steps have been defined for the analysis of pre-tensioned beam in ABAQUS. In the first step, the top and bottom tendons have been stretched respectively 1 and 2mm from both ends of the beam. In the second step, the mid-span section has been restrained against vertical movement and the tendons are in contact with the surrounding concrete with cohesive parameters of \left[{\delta }_0,{\delta }_{\mathrm{t}},{\sigma }_{\mathrm{b}}\right]=\left[0.1\mathrm{m}\mathrm{m},2.0\mathrm{m}\mathrm{m},1.0\mathrm{M}\mathrm{P}\mathrm{a}\right]. In the last step, the mid-span section has been displaced in the vertical direction.

ABAQUS results have been extracted for both elastic and elastoplastic material behavior of the beam. The elastoplastic results of Mises stress and equivalent plastic strain at ultimate condition for bond-slip properties of \left[{\delta }_0,{\delta }_{\mathrm{t}},{\sigma }_{\mathrm{b}}\right]=\left[0.1\mathrm{m}\mathrm{m},2.0\mathrm{m}\mathrm{m},1.0\mathrm{M}\mathrm{P}\mathrm{a}\right] and pre-tensioning gaps of 1.0 mm top and 2.0 mm bottom are displayed in Figs. 12 and 13.

The Mises stress contour for the top and bottom pre-tensioning reinforcements has been illustrated in Fig. 14. The load–deflection curve of the beam and the loads in the mid-span and end-span of the top and bottom tendons are sketched in Fig. 15. As shown in the figure, the elastic deformation prestressing loss are different for top and bottom tendons and the tendons are carrying more loads at the end-span in contrast to the mid-span due to bond-slip behavior through the tendon’s length around the mis-span.

Load deflection results by ABAQUS is demonstrated in Fig. 16. As shown in the figure, Load–deflection curves in ABAQUS does not show the initial jump in the load due to existing camber on the mid-span of the beam. The beam reacts to the camber smoothly to the extent that the load–deflection curve observed never seems the expected bilinear curve.

4.1.3 Analysis of prestressed concrete beam via proposed L2-Plus elements

By using L2-Plus element proposed with overlay mesh the components seven and eight of the nodal DOFs are loaded to produce a certain measure of prestressing. In this first step the section components are not stress free and only the prepressing components are stressed. In the second step the mid-span is restrained and the beam redistributes the tension in the prestressing tendons to the whole section fibers simultaneous with pushing down the mid-span section of the beam.

The prestressing gap and bond-slip parameters are the same as the three-dimensional solid model in ABAQUS. The beam is loaded to the extent that the load–deflection curve reaches the yield point. The contour of tendon debonding at the ultimate capacity of the Timoshenko beam is described in Fig. 17 for top tendons and in Fig. 18 for bottom tendons.

The {\sigma }_{11} stress component is depicted in the slice view of Fig. 19. It is observed that at the limit state a part of the beam in the middle span of the beam the section works in tension. However, the sections near the two ends of the beam are working completely in compression.

The load transfer in the top and bottom prestressing tendons are shown in Fig. 20. In comparison to the results of ABAQUS, the load in the individual bars of top and bottom of the end section are not the coincident absolutely. Bottom reinforcements which have been stretched 2.0 mm in comparison to the top reinforcement which are pre-tensioned only 1.0 mm hold more internal tensile loads. However, the ratio of the tensile load in the bottom reinforcement are not twice that of top reinforcement. The prestressing loss is observed in Fig. 20 as well. However, the prestress loss is not as sharp as seen in Fig. 15 corresponding to the ABAQUS results.

In Fig. 21, the load–deflection of the proposed method is demonstrated. As shown in the figure, L2-Plus elements reproduce the camber effect on the three-point flexural capacity curve similar to the analytical curve.

4.1.4 Comparison of the results of analytical, ABAQUS and L2-Plus approaches

Due to numerous causes of prestress loss in PC technology, one of the important cases of design is to merely anchor the reinforcements rather than prestressing. In this research it is indicated that the non-prestressed but anchored tendons’ results resemble more to the prestressing results rather than non-prestressed and not anchored reinforcements of RC technology.

In Fig. 22, the elastic and elastoplastic curves for three different methods are shown for non-prestressed but anchored PC state. The results of the analytical, ABAQUS and L2-Plus elements look very similar to each other. However, L2-Plus beam element show a more conservative results in comparison to the other alternatives.

For 0.25 mm top/0.5 mm bottom prestressing the results of the three alternatives are compared in Fig. 23.

Once more, the analytical and ABAQUS results are close to the results reproduced by L2-Plus elements. However, for large amounts of prestressing gap values the results are not matching so much. Figure 24 illustrates the results for 1.0 mm top/2.0 mm bottom prestressing regime.

The last case of pre-tensioning the PC beam shown in Fig. 11 deals with 5.0 mm top/10.0 mm bottom pre-tensioning analysis. Figure 25 compares the results generated by our three alternative methods. While the slops of the elastic line are identical, the three methods do not agree on the camber moment at the beginning of the three-point loading test.

4.2 Numerical examples for the analysis of different prestressed concrete girders by using L2-Plus element

In this section, the proposed L2-Plus element is utilized for the analysis of different pre-tensioned and post-tensioned beam sections. From the verification analyses it is known that the L2-Plus element’s results match the results by ABAQUS for relatively small prestressing gap values.

However, the results of L2-Plus elements are more effective in terms of computational cost and user-friendliness. Because, due to local element distortion and complexities provided by automatic mesh generation in ABAQUS, many engineering pre-tensioned and even post-tensioned beam cross-sections fail to be simulated efficiently in ABAQUS or being calculated by analytical formulations. Nevertheless, every section with every tendon’s arrangement can be simulated by L2-Plus elements in companion with the proposed elastoplastic prestressing algorithm.

4.2.1 Prestressed T-beam

A typical PC T-beam design is shown in Fig. 26. This beam is considered to analyze and extract different facts regarding the L2-Plus analysis algorithm. In Fig. 27, two meshes of the beam is introduced. One compatible mesh and one un-compatible “overlay” mesh. The compatible mesh consisted of 252 elements while the overlay mesh incorporates only 64 Q4 elements. This opportunity is provided by meshing the tendons’ section and the concrete section independently. In the overlay mesh the accuracy of the numerical integration and not the geometry governs the number of elements.

By using the ordinary mesh, the T-Beam introduced above is loaded under three-point bending conditions. The top reinforcements are non-prestressed while the bottom tendons are prestressed with different pre-tensioning and post-tensioning gap values from both ends of the 4000 mm length beam. For the pre-tensioning cases the bond-slip properties are considered \left[{\delta }_0,{\delta }_{\mathrm{t}},{\sigma }_{\mathrm{b}}\right] = [0.1\mathrm{m}\mathrm{m},2.0\mathrm{m}\mathrm{m},0.3\mathrm{M}\mathrm{P}\mathrm{a}].

The results of the three-point bending test indicates that, pre-tensioning load capacity is greater than post-tensioning capacity, while the ultimate load bearing of the beam in flexure is independent of pre-tensioning or post-tensioning and the amount of gap applied to the beam end-spans.

One other important result is that, the anchorage of the prestressing tendon which is the last step of prestressing technology is more important than the prestressing gap. Because, the anchorage itself produces the most of the additional flexural strength of the beam.

The considered T-Beam is analyzed with uplifting load on the mid-span in Fig. 28. In this figure as well, the ordinary or compatible mesh has been applied which is not necessary in the sense of the accuracy of the results.

Once more, we notice that for the ultimate limit the effect of pre-tensioning or post-tensioning disappears and the only fact that dominate in the ultimate capacity of the beam is “ANCHORAGE”.

4.2.1.1 Tendon profile effect on three-point bending test load–deflection curve

Studying the effect of curved tendon paths on the load–deflection curve is carried out using the T-beam considered in this section (Fig. 29). The first series of T-beams has not any prestressing tendons nor any reinforcement anchorage. The results are demonstrated in Fig. 30. Excepting the ultimate limit part of the load–deflection curve, the results of sagging curves are identical to that of non-sagging curves.

The effect of the 1.0 mm pre-tensioned tendons’ sagging on the load–deflection curve is observable from Fig. 31. The effect of sagging on 1.0 mm post-tensioned tendons’ result is reflected in Fig. 32. The effect of sagging on 5.0 mm post-tensioned tendons’ load–deflection curve is given in Fig. 33. The figure illustrated the negative effect of highly sagging tendon-curves on the three-point bending test results.

4.2.1.2 Hysteresis curves for different prestressing regimes

Cyclic loading test illustrates important issues on the energy absorbing and ductility of the structural elements as well as structural systems. The area surrounded by the richest hysteresis curve under three-point bending test implies the potential of the structural detail to maintain and improve the integrity of the structure by effective load redistribution through the material. Figure 34 illustrates the hysteresis curves for different pre-tensioning and post-tensioning measures of the T-beam PC section. As shown in Fig. 34, The broad hysteresis curves indicate effective energy dissipation during seismic excitations, a characteristic feature of the pre-tensioned and post-tensioned girders with moderate prestressing gap values.

4.2.2 Prestressed box girders

The box section shown in Fig. 35 is considered for analysis with L2-Plus element in this section. The configuration of the box girder with associated overlay mesh is displayed in Fig. 35. The total number of the Q4 elements used for discretization of the box girder section is 133.

The box girder has been analyzed under three-point bending text, the results of which is outlined in Fig. 36. As shown in this figure, the pre-tensioned cases show more service load capacity in comparison with relative post-tensioned case. However, as mentioned before, the ultimate capacity of the girder is governed by anchorage of the tensons rather than pre-tensioning or post-tensioning and the amount of prestressing gap.

4.2.2.1 Bridge box girder under cyclic loading and various prestressing methods

The hysteresis curves for different prestressing methods, various prestressing tendon gaps and different bond-slip parameters are analyzed using the L2-Plus element proposed in this paper (Fig. 37).

4.2.3 Prestressed I-beam with interior tendons

The unsymmetrical I section with interior prestressing tendons and associated overlay mesh are depicted in Fig. 38. By using overlay mesh, the number of Q4 element for discretization of the cross-section reduced to 82 elements. The I beam has been pushed down in accord to three-point testing method and the results presented in Fig. 39 are emerged. The bond-slip parameters for the pre-tensioned items in the figure have been adopted as \left[{\delta }_0,{\delta }_{\mathrm{t}},{\sigma }_{\mathrm{b}}\right]=\left[0.1\mathrm{m}\mathrm{m},2.0\mathrm{m}\mathrm{m},1.0\mathrm{M}\mathrm{P}\mathrm{a}\right].

4.2.4 Prestressed I-beam with external tendons

The prestressed unsymmetrical I beam with external tendons and its relative overlay mesh could be seen in Fig. 40. The overlay mesh facilitates discretizing the section with only 90 Q4 elements while the ordinary compatible mesh is not able to provide an optimum mesh size with the same quadrature accuracy. The beam has been analyzed using multiple prestressing arrangements, the results of which is provided in Fig. 41.

5 Conclusions

In this paper, a novel beam element based on multi-fiber approach has been developed to cover several aspects of pre-tensioned and post-tensioned concrete beams’ analysis and design. To optimize the efficiency of the cross-sectional mesh in this fibrous description of the beam section, an overlay mesh has been applied which reduces the computation cost multiple times and makes the proposed L2-Plus element formulation feasible for engineering applications.

The beam element proposed have been validated and verified by analytical formulations and ABAQUS program which indicated that in the range of finite deformation there is a profound agreement between the results of L2-Plus Timoshenko beam and three-dimensional solid model in ABAQUS environment.

Several pre-tensioning, post-tensioning, anchorage and non-prestressed reinforcements have studied by considering a diverse set of PC sections and different bond-slip properties as well as external prestressing items.

It is shown that, pre-tensioning provides a relatively enhanced load–deflection curve in serviceability limit state for the intermediate prestressing gap ranges. In the ultimate limit, all prestressing properties reduces to a simple anchorage condition which is still effective in contrast to ordinary reinforced non-PC condition. Then in the absence of a mild prestressing gap, it is recommended to provide a simple “anchorage” for the tendons at the two ends of the girder.

Hysteresis curves are derived from some of the cross sections frequently used in bridge design. These curves indicate the significance of the mild pre-tensioning fairly before the ultimate capacity of the beam. Thus, prestressing can be adapted to seismic and cyclic applications as well.

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