Practical Calculation Of Geometrical Nonlinear And Material Nonlinear Of Reinforced Concrete Structures

The stability problem of reinforced concrete structures is a geometrically nonlinear problem,that is,development of load and deformation is no longer limited to linear relations.Two types of methods are usually used to calculate the geometric and material nonlinearity of reinforced concrete structures.One is the direct method,which obtain the internal force by finite element software;The other is the indirect method,which calculates the second-order effect magnification factor(the magnification factor of internal force and deformation),and indirectly obtains the second-order internal force by amplifying the first-order internal force,which is also used by the current specifications("Specifications for Design of Concrete Structures",etc.)method,Finally,the second-order internal force is less than the first-order resistance to calculate the strength and reinforcement.The above calculation methods have some problems: whether the finite element results are correct;how to consider the second-order superposition principle when it failure;how to obtain the secondorder internal force from the first-order internal force;how to convert the calculation of the entire structure into a member;how to get it practical calculation methods;How to consider the error caused by the separate calculation of the resistance and load.These are the components of the basic theoretical research on the geometry and material nonlinearity of reinforced concrete structures,and they are also the content that needs to be supplemented and improved in the current code.The process of structure to members and then to cross section for research is Adopted: At the structural level,this paper proposes a new calculation model-"springswaying column model" to calculate the critical bearing capacity of the structure.The critical force is connected with stiffness by the expanded structure,and turns the transcendental equation that calculated the critical force into an algebraic equation,which can consider the mutual support between the columns of the same story and the support between the story,and make up for the length coefficient method in the calculation.At the member level,the member level is transformed into the section level through the assumption of the curvature of the model column.At the section level,the calculation is carried out through the process of strain-stress-first order bearing capacity-second order bearing capacity,and the strain method is used in the calculation.Compared with the traditional stress method,its advantage is that the complete constitutive relationship of the material can be considered,and the load and the resistance can be coupled to calculate.At the same time,the strain method in this paper does not require equivalent rectangular stress conversion,and the calculation formula is an algebraic equation without iteration.The main research contents are:(1)The spring-swinging column model is used to define the concepts of the internal and external stiffness of the structure,and the calculation formulas for the internal and external stiffness of the separation column are further deduced.The critical equation of the frame established by the K-P model(K stands for spring and P stands for Pendulum)is an algebraic equation,and the solution is very simple.The core of the K-P model is not to establish the critical equation on the frame itself(on the original structure),but to move the load on the frame to the additional swinging column(on the extended structure)to establish the critical equation.This greatly simplifies the solution of the critical force for the overall stability of the frame.The solution idea of this method is clear.The calculation formula is an algebraic equation,which can take into account the mutual support effect between the columns of the same story and different story,which makes up for the shortcomings of the length coefficient method.(2)The solution of the critical force of the frame structure is converted into the calculation of the displacement of the structure through the deflection method,and the formula for calculating the length coefficient of the single-story and multi-story frame is deduced.Finally,the critical force of each column of the frame can be obtained according to the Euler formula.This method takes the entire frame structure as the object to study,and can well consider the interaction of the columns in the frame.The derived formula has very good applicability for single-story or multi-story,single-span or multi-span or asymmetrical loads,and the frame parameters are not uniform.(3)Based on the complete constitutive relationship between concrete and reinforcement,all strains of concrete and reinforcement can be used without the need for equivalent rectangular stress conversion.The strain method for the first-order bearing capacity of rectangular section is derived.The strain is used to calculate the stress,and then It is a method of calculating stress from strain and then calculating internal force;when calculating the second-order bearing capacity,the numerical integration method and the model column method are used respectively to obtain the load-deflection curve and the axial force-bending moment correlation curve of the rectangular section column.Finally,based in the limit state,a simplified formula for the ultimate curvature of the section is deduced,and a simplified formula and a nomogram for the calculation of the second-order bearing capacity are obtained.The reinforcement can be calculated according to the external load,and the bearing capacity of the member can be estimated according to the section parameters,which can be applied in practice engineering and verify the correctness of the finite element software calculation results.(4)The calculation formula of the circular cross section is deduced by the analytical method,and a practical calculation chart(nomogram)is obtained based on this.In the derivation,in accordance with the relevant provisions of the constitutive relationship between concrete and reinforcement in the "Specifications for Design of Concrete Structures",5 ultimate strain regions are divided for short columns and slender columns respectively,and the stress and internal force can be determined by the strain,and the second order bearing capacity is derived.Due to the nondimensional derivation,the formula in this paper can be applied to different reinforcement grades and all concrete strength grades below C50.For concrete members with strength greater than C50,only the constitutive relationship in the text needs to be replaced,and it can be applied in engineering and verify the correctness of the finite element software calculation results.(5)The strain method is used to calculate the bidirectional eccentric load bearing capacity of the rectangular section.The strain method starts from the strain and obtains the ultimate strain diagram through the constitutive relationship between concrete and reinforcement given in the "Specifications for Design of Concrete Structures",that is,the strain is known,and then through the process of strain-stressforce to calculate bearing capacity,which avoids the problem of iterative solution.In the formula derivation,the resultant moment and the corresponding inclined section are used to solve this problem,which avoids the superposition problem.The formula derivation and the production of the nomogram are all in dimensionless form,which has the characteristics of strong applicability and fast and convenient calculation.It is a very good practical calculation tool for the calculation of the reinforcement of the rectangle section subjected to axial loads and biaxial bending.