Transformation of multi-spring model of plate body with bulk structure method

Constitutive relationship of shear springs For reinforced concrete beam-column members, even if the shear capacity exceeds the shear resistance corresponding to bending yield, shear yielding may be induced by bending yield and does not guarantee the elasticity of the members. Shear behavior. Due to the complexity of shear properties and the lack of experimental data, current shear hysteresis models are rare. The author uses a two-line origin pointing model to describe the shear hysteresis performance of the component <11>, as shown.

Shear spring elastic section stiffness (before shear yielding), the equivalent elastic shear stiffness can be approximated, ie, Ke=GAl(1) where: G is the elastic shear modulus of concrete, and A is the component. The effective cross-sectional area, l is the initial length of the spring group; is the non-uniformity of the section shear stress. The stiffness after shear yielding was taken as 001 Ke. The yield strength of the spring is determined by equation (2) or equation (3). <12> Vy=07ftbh0+1.25fyvAsvsh0(2)Vy=1.75+1ftbh0+fyvAsvsh0+0N(3) The constitutive relation of the shear spring Symbols can be found in the concrete specification. In the actual calculation and analysis, if considering the effect of the axial force, the estimated value of the axial force N of the component must first be given.

The shear limit deformation vu of the shear spring can be determined according to formula (4) <13>vu=vuh0(4) where: vu is the ultimate tensile strain of the stirrup; h0 is the effective height of the cross section of the member in the direction of the shear spring. The force-displacement relationship of the spring group and the motion equation of the unit are as shown. When the unit i-1, i is slightly displaced, the deformation vector dsl={dxldyll}T of the spring group l, that is, the relative displacement of the unit, can be expressed by 5) Obtained: unit displacement and spring group deformation relationship dxl=ui-ui-1dyl=vi-vi-1-i-1li-1/2-ili/2l=ii-1(5) where: dxl, dyl, l is the axial, shear, and corner deformation of the spring group l; ui-1, vi-1, i-1, ui, vi, and i are the displacements of i-1 and i, respectively. The deformation of each spring in the axial spring group can be in the formula (6): eACl, j, eASl, j is the deformation of the jth concrete and the steel spring; yACl, j, yASl, j are the cross-sectional position coordinates of the spring, mc And ms are the number of concrete springs and rebar springs respectively; Al is the cross-sectional position matrix of the axial spring group.

According to the deformation, loading history and constitutive relationship of each concrete spring and steel spring, the restoring force vector fAl={fACl, 1fACl, mcfASl, 1fASl, ms}T of the axial spring can be obtained. According to formula (7), the resultant force vector FAl={NAl0MAl}T of the axial spring group can be obtained.

The deformation of FAl=ATlfAl(7) is the shear deformation of the spring group. According to its constitutive relation and loading history, the restoring force fsl of the shear spring is obtained, that is, i-1, i acts on the interface between the two. Mutual shear binding force Vsl. Combining the resultant force vector of the axial spring group with the restoring force of the shear spring is the resultant force vector Fsl={NAlVslMAl}T of the spring group l, that is, the mutual restraining force of i-1, i due to the relative displacement.

According to formula (8), the shear binding force acting on the cell interface is equivalently translated to the centroid of the element, and the binding force vector Fci acting on the i centroid when i-1, i is relatively displaced is obtained, i-1={Ni , i-1Vi, j-1Mi, i-1}TNi, i-1=NAlVi, i-1=FslMi, i-1=MAl+Fslli/2(8) The constraint resultant force vector acting on the unit i is Fci =Fci,i-1+Fci,i+1(9) sums the restraining force acting on i with the main power (applied load) Fai, and the resultant force vector acting on i is Fi=Fai+Fci={NiViMi }T. According to Newton's second law of motion and considering the damping force, the equation of motion of element i is miu+cuiu=Nimivi+cvivi=Vi(i=1,2,,n)Ii+ci=Mi(10) where mi is The mass of i, Ii is the moment of inertia of i relative to its centroid, ui, vi, i, ui, vi, i are the translational speed, angular velocity, translational acceleration and angular acceleration of i, cui, cvi, ci are i Damping coefficient in the direction of motion.

The numerical solution method and the numerical solution method of the calculation parameter model and the determination method of the calculation parameters are the same as those in the literature <9>. The numerical calculation results show that the improved multi-spring model of the rod segment can better simulate the general load of reinforced concrete columns. Bending and bending coupling performance and shear nonlinearity. Due to space limitations, the author did not give a verification example of the model.

Conclusion The author has improved the multi-spring model of the segment. By introducing a shear spring, the shear force between the units is transmitted, thereby perfecting the multi-spring model of the rod segment. The constitutive relation of the shear spring adopts a two-line origin pointing model. The improved multi-spring model of the rod segment can better describe the nonlinear performance of RC beam-column members under the combined action of compression, bending and shear loads, and lays a foundation for the application of discrete element method in reinforced concrete structure analysis.

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