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In this paper, cracking in the first mode (opening) is modelled for reinforced concrete beams with FRP sheets based on presenting a new method by using the principles and relations of fracture mechanics and finite element method. In this method, for modelling the relationships of determining the stress intensity factor is developed for reinforcement sheet. In the proposed method, elements of the beam are divided into two categories, including elements with and without the crack. In the elements without the crack, the relationships, equation, stiffness and mass matrices of the beam are established with considering the changes in the moment of inertia due to the reinforced FRP sheet. In the elements with the crack, a change in the cross-section of the reinforced concrete due to the crack and a discontinuity in the crack point leads to an improvement in the standard governing relationships. So that the reduction of the stiffness of the cracked element is equivalent to the change in the size of the discontinuity. Here, the variation of the stiffness of the cracked element is calculated and presented as a function of the stress intensity factor. In this approach, the simulation of the crack is done by dividing the element to two sub-elements into the two sides of the rotational spring. In which, The stiffness and mass matrices of the two sub-elements and the improved stiffness and mass matrix of the element are derived by satisfying the continuity equation at the crack point. This method is developed from a vibrating analysis. The effects of crack depth and location and the effect of crack expansion on the static and vibrational behaviour of a concrete beam are investigated. To ensure the accuracy of the proposed method, all analysis performed in Abacus software is implemented. Comparing the results of the proposed model with the results of comprehensive modelling in Abacus software is applied to verify. The comparison of the results shows that the proposed methods are suitable for the analysis of reinforced concrete structures resistant to cracking. So that it can be generalised and optimally desirable for other models. In this paper, cracking in the first mode (opening) is modelled for reinforced concrete beams with FRP sheets based on presenting a new method by using the principles and relations of fracture mechanics and finite element method. In this method, for modelling the relationships of determining the stress intensity factor is developed for reinforcement sheet. In the proposed method, elements of the beam are divided into two categories, including elements with and without the crack. In the elements without the crack, the relationships, equation, stiffness and mass matrices of the beam are established with considering the changes in the moment of inertia due to the reinforced FRP sheet. In the elements with the crack, a change in the cross-section of the reinforced concrete due to the crack and a discontinuity in the crack point leads to an improvement in the standard governing relationships. So that the reduction of the stiffness of the cracked element is equivalent to the change in the size of the discontinuity. Here, the variation of the stiffness of the cracked element is calculated and presented as a function of the stress intensity factor. In this approach, the simulation of the crack is done by dividing the element to two sub-elements into the two sides of the rotational spring.

Keywords: Vibration Analysis of Crack, Reinforced Concrete Beams, Composite Sheets, Sub-element, Rotational Spring

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