The effect of cracks on the bearing capacity of reinforced concrete slabs

Cracks occur in reinforced concrete slabs for two reasons due to increased operational loads and due to manufacturing and installation defects. When cracks are detected, the question arises about the residual bearing capacity of the slab and the need of its strengthening. To solve this problem by calculation, it is necessary to take into account physical nonlinearity. An algorithm for the calculation by the finite element method is proposed. The main feature of the algorithm is the use of multilayered finite elements, which allows modeling the cracks by specifying the corresponding material characteristics of those layers which the crack passes through. A method for determining the bearing capacity of a slab with cracks after its reinforcement with composite fabrics is also considered. An example of the study of the stress-strain state of a reinforced concrete slab with cracks by the proposed method is given. The implementation of the algorithm in the PRINS program is described and the possibility of using this program for solving practical problems is discussed.


Introduction
Successes in the production of building materials [1][2][3][4][5], on the one hand, and in the technology of building production [6,7] -on the other, provide the possibility of reconstruction and repair of reinforced concrete structures without their dismantling. At that, steel reinforcement, cement-sand mixtures and composite materials based on carbon fibres are used for the strengthening of reinforced concrete structures [8][9][10]. In particular, in the repair of reinforced concrete slabs, shells, beams and columns, composite tapes and fabrics with unidirectional and bidirectional properties are used [11][12]. Recommendations for the repair and strengthening of reinforced concrete structures developed by various departments [8][9][10][11][12] imply a strength analysis of the objects under consideration both before and after the works have been performed. A feature of such studies is a change in the design model of the structure as a result of repair or reconstruction [13], as well as the presence of initial stresses and strains if reinforcement works are carried out on retention of the entire operational load or part of it. This requires the development of special software or making appropriate additions to existing programs. The method of the stress-strain state analysis of reinforced concrete slabs in which cracks arise during long-term operation and the implementation of this method in the computer program PRINS are described in this paper.

Methods
To determine the bearing capacity of reinforced concrete slabs the algorithm described in the previously published works of the author is used in the PRINS program (see, for example, [14,15]). The main features of this algorithm are as follows.
The calculation of physically nonlinear structures is carried out by the incremental/iterative method using the equation: , NL K Δu = ΔP (1) where NL K is the total nonlinear stiffness matrix, connecting the increments of the nodal forces and displacements, Δu and ΔP are the increments of the nodal displacements and the nodal forces of the finite element model, respectively.
The matrix NL K is represented as: After substitution (2) in (1), we get Equotion (2) is solved by iterative method: were iiteration number. At the end of the iterative process, the total values of displacements and stresses at the end of the loading step are found by formulas: 00 u = u + Δu; σ = σ + Δσ .
The stress increments are found by formula: where ep C is the elasto-plastic matrix of material characteristics [14 ].
At each step of loading, the stress state is analyzed and, when plastic strains and cracks occur, the stresses are corrected taking into account the adopted deformation diagrams of the materials. This requires the carrying out of the process of equilibration of the structure. The equilibrium iterations is performed in the program PRINS in accordance with the formula: where  K tangent stiffness matrix,  u nodal displacements vector,  P vector of the externally applied nodal loads,  F vector of nodal point forces that are equivalent to the element stresses, jstep number, i -iteration number.
When using formulas (1-7), the calculation is carried out with preservation of the design scheme. To solve the problem posed in this article, the algorithm implemented earlier has been updated in accordance with the flowchart shown in figure 1. Cracks are initialized in the program PRINS by specifying the corresponding characteristics of the stress-strain state of concrete layers containing cracks, and the design scheme is changed by replacing the original material package with the enhanced one.

Results
To illustrate the developed technique, a square symply supported reinforced concrete slab was investigated with the following initial data: slab size 8x8 m, slab thickness 25 cm; concrete of class B30, normative compressive strength of 22 MPa, normative tension strength of 1.75 MPa, elastic modulus Е = 3.25х10 7 MPa, Poisson's ratio μ = 0.2; reinforcement of class A500, standard resistance 500 MPa, Poisson's ratio μ = 0.3; the reinforcement was made on the load q = 20 kPa, while 4 rods with a diameter of 14 mm per linear meter of slab length were taken (spacing of reinforcement equal to 250 mm) ; to strengthen the slab, a bidirectional composite fabric with a tensile strength of 4.9 GPa, elastic modulus of 245 GPa and a residual strain of 1.8% was used ; the slab was divided in thickness into 32 layers; two packages of materials -the original (package number 1) and Input of the initial data The analysis of the structure by the step-iteration method to achieve the design load Initialization of crack (cracks) Is the construction strengthened?
Change of the design scheme in accordance with the strengthening adopted Continuation of the calculation by the stepiteration method until the destruction of structure

No
Yes repair (package number 2) were used in the calculation.The characteristics of layers are shown in Tables 1 and 2. The angle α in the tables 1 and 2 is the steel reinforcement orientation angle. The layers 1 and 32 are fictitious. Fictitious layers are introduced to enable the output of the stresses and cracks on the lower and upper surfaces in the postprocessor. The layers 30 and 31 in the package № 1are also fictitious with the fictitious material having a modulus of elasticity several orders of magnitude lower than the modulus of elasticity of concrete and steel reinforcement. These layers have almost no effect on the operation of the slab in its original state. They are needed for the transition from one calculation scheme to another.
The design scheme of the slab with the numbering of elements is shown in figure 2. At the first stage, the slab was calculated by a step-iteration method up to destruction according to the PRINS program. The maximum value of the deflection under load q = 20 kPa was equal to 2.04 cm, which is less than the allowable value of / 200 800 / 200 4 L  cm. Figure 3 shows the first cracks on the lower surface under a load of 8 kPa, and Figure  4 shows cracks with a nominal load of 20 kPa.  As can be seen from figures 3 and 4, two mutually perpendicular cracks occur in the central region of the plate, and their directions are parallel to the outer contour of the plate. In the remaining regions, one crack arises at each point, and its direction depends on the ratio between the shear and normal stresses. Closer to the corners of the slab, the direction of the cracks approaches 45 degrees with respect to the external contour. The crack opening at a nominal load of 20 kPa was found taking into account the recommendations of building acts [16] according to the formula: The coefficient s  is assumed to be one if the condition is met: where , crc ult a  the maximum allowable width of crack opening.
Provided the safety of the reinforcement is maintained, the value In figure 5 the load-displacemen curve is shown. As can be seen from figure 5, the loadbearing capacity of the reinforced concrete slab with the given characteristics is equal to 74 kPa. When the slab was reinforced at the stage of operation with the two layers of composite fabric, the behavior of the structure was described by the load-displacement curve shown in figure 6. It is seen from Figure 6, that when the structure is strengthened at the operation stage, the displacement gradients decrease, which is explained by an increase in the structure rigidity. The ultimate load have increased with the above strengthening to 78.6 kPa.  Characteristics of cracks are given in table 3. The angle between the normal to the crack and the local x axis of the element is indicated by the letter β.The calculation according to the PRINS program showed that when the cracks discribed above occurred, the ultimate load decreased to a value of 56 kPa (figure 8). The crack formation on the bottom surface with the initial and initiated cracks under the load of 20 kPa is shown in figure 9  The result of the calculation of the slab with initiated cracks after reinforcement with two layers of composite fabric is shown in figure 10 as a load-desplacement curve. The maximum load have increased to 67.2 kPa.

Conclusion
The algorithm developed in this article and implemented in the PRINS program makes it possible to calculate reinforced concrete slabs with internal and external reinforcement, taking into account the actual loading history, up to destruction.This is especially important when the works related to the repair or reconstruction of structures are carrying out without their dismantling. The PRINS program allows us to estimate the bearing capacity of reinforced concrete slabs in their initial state, to find the ultimate factor of safety in the event of defects in the form of cracks and to select the necessary reinforcement by calculation. The use of the PRINS program at the stage of development of a project for the reconstruction or repair of reinforced concrete structures will give to civil engineers the opportunity to choose the strategy for future works more sensible and to avoid unnecessary waste of time and material resources.