Concerning the design and strengthening of reinforced concrete (RC) structures, an appropriate performance level should be provided for the RC structures in serviceability limit state (SLS) conditions. For this purpose, the total deflection of a structure, resulting from flexural and shear deformations, should be limited to cover the requirements of SLS due to deflection.

The current study aims to develop a novel simplified analytical model using the force method (also known as flexibility method) to predict the response of RC structures in terms of total deflections by considering the contribution of flexural and shear deformations up to the failure of these structures. In this regard, the current study proposes a new strategy to evaluate the influence of flexural cracks on the shear stiffness degradation of RC structure during the post-cracking stage. The applicability of the developed analytical model is not limited to statically determinate RC elements, since the force method principles were used to extend its use to statically indeterminate RC structures. The good predictive performance of the proposed model is appraised by predicting the force-deflection response registered in the experimental programs composed of determinate and indeterminate RC beams and slabs.

The objective of the research is to develop a novel simplified analytical model based on the flexibility (force) method to predict the deflections of statically determinate and indeterminate RC structures up to their failure, which can be in bending or in shear. The main highlights of this model are:

- The capability of predicting the deflections of both statically determinate and indeterminate structures
- Prediction of RC structures responses in terms of total deflections by considering the contribution of flexural and shear deformations up to the failure of these structures
- Proposing a new strategy to evaluate the influence of flexural cracks on the shear stiffness degradation of RC structure during the post-cracking stage

The implementation of the proposed model to predict the total deflection (including the flexural and shear deformation) of statically determinate and indeterminate RC structures using the flexibility method is described in the flowchart exposed in Fig.1. In this algorithm, in the first block, the initial values of the accumulative variables of the formulations are defined. After the definition of initial values, a loop of displacement increments is executed up to an assumed maximum deflection. In each increment of the displacement, the bending moments are updated in block (2). Then, each term of the flexibility matrix is determined in block (3) by evaluating the contribution of all the elements the structure is decomposed (nel), and considering the flexural and shear deformation. In the next step, the incremental force vector is obtained in block (4). Then, the total force vector is updated in block (5). After updating the total deflection in block (6), a new step of incremental deflections is executed if the maximum deflections were not yet attained, otherwise this incremental loading process is ended. Each term of flexibility matrix (f_ij) is obtained by applying the principal of virtual work resulting:

where fij is the displacement at coordinate i (in the direction of Fi) due to the application of a real unit load at coordinate j (Fj = 1) on the released structure (see Fig.2(a)). By applying Fj = 1, Nj, Mj, Vj and Tj are the internal axial force, bending moment, shear force and torsional moment, respectively. Besides, by applying a unit virtual load Fi = 1 at coordinate i on the released structure, following internal forces Ni, Mi, Vi and Ti are produced at any section. In Eq. (1) EA, EI, GA* and GJ are the axial, flexural, shear and torsional stiffnesses, respectively.

*Fig.1 - Algorithm to drive the force-deflection relationship*

*Fig.2 - (a) Physical meaning of the terms of the flexibility matrix, based on the displacements for each equilibrium configuration, (b) Three stages of shear deformational behavior of RC element, (c) Distribution of internal shears in beam with web reinforcement, (d) Schematic representation of longitudinal strain distribution for assisting on the determination of the shear retention factor*

In a 3D frame bar, two bending moments and two shear forces can develop in correspondence to the principal axis of the cross section, but for the present version of the proposed model, a 2D bar is assumed, so the torsional term is not considered, and only one bending component and one shear force is considered for the flexibility terms of bending and shear. Furthermore the axial deformation is also neglected (term (a) in Eq. (1)), since the target type of RC elements are those mainly submitted to bending and shear forces.

The flexural deflections of a RC structure (due to bending moment) are estimated considering the tangential flexural stiffness of the cross section obtained from the corresponding moment-curvature relationship of the section. The moment-curvature relationship (M - X) of a cross section representative of a generic element was determined using the sectional analysis software DOCROS (Design Of CROss Sections(. DOCROS assumes that a plane section remains plane after deformation and perfect bond exists between distinct materials. According to this sectional analysis software, a cross section is divided in layers that can be composed by materials with the constitutive models available in DOCROS.

The shear deflections of a RC structure (due to shear force) is determined by considering the tangential shear stiffness of the cross section during the loading process. For this purpose, the shear behavior of a RC structure is assumed simulated by a diagram representing the pre-cracking, post-cracking and post-diagonal cracking stages, delimited by the following points: concrete crack initiation (point (cr)); diagonal crack initiation (point (dcr)); and ultimate shear capacity (point (us)) (see Fig 2(b)).

According to the experimental evidence, prior to flexural cracking (pre-cracking stage) the shear force applied on the cross section is carried exclusively by the uncracked concrete, Vcz (Fig. 2(c)). Since the flexural cracking and the initiation of the diagonal cracking, the external shear force is resisted by the uncracked concrete (Vcz), the vertical component (Vay) of the crack shear stress transfer capacity (Va, also known as aggregate interlock shear resisting mechanism), and the dowel shear effect carried by the tensile longitudinal steel reinforcement (Vd) (Fig. 2(c)). After diagonal cracking and before the yield initiation of stirrups (post-diagonal cracking stage), a portion of the applied shear force is resisted by the web reinforcement (Vs) (Fig. 2(c)).

Furthermore, the current study proposes a new strategy to evaluate the influence of flexural cracks on the shear stiffness degradation of RC structure during the post-cracking stage. In this regard, the contribution of concrete in the cracked tension zone for determining the shear stiffness was considered using a retention factor (beta ), which reduces the elastic shear modulus (beta*Ge) (Fig 2(d)).

The good predictive performance of the proposed model is appraised by predicting the force-deflection response registered in the experimental programs composed of determinate and indeterminate RC beams and slabs (Fig 3).

*Fig.3 - Analytical prediction of total deflection which is composed of flexural and shear deflection*