Theoretical Note

Fiber Model Timoshenko Beam Constitutive Relationship of Concrete Constitutive Relationship of Steel Cracking Judgment of Concrete Zoning Method Gauss Point Global Coordinates and Local Coordinates Bending Capacity of Reinforced Concrete Section Shear Capacity of Reinforced Concrete Section Damage Criteria Nonlinear Dynamic Calculating Procedure

Fiber Model

A fiber element is designed to model the 3D RC components in the RC structures. Its cross-section can be edited in a divided cell form according to the real distribution of steel and concrete in the axial direction. The section during the forced state is assumed to keep a plane, and the average strain and average stress in each cell of the section obey the corresponding constitutive relationship of concrete or steel materials. The force and moment acting on the section are obtained by integrating on the section as shown in the below. In our program only gauss point values are given. in which, ：Average sectional stress, : Section curvature in the x, y directions The section is then condensed to a point on the fiber element, which can be treated as a frame beam element in the analysis of complete structure. Timoshenko Beam theory is used to consider shear deformation which is assumed not large enough due to a presumed large shear stiffness. The geometrical nonlinear can be taken into consideration by selecting mode settings for the frame's P-delta effects

Timoshenko Beam

When the length of beams is much longer than the scale of beam section, it is enough that the beam vibration is analysed only on considering bending deformation. However when the section scale is enough large comparing to the length of beams, the shear deformation in the beams has to be taken into considering. The cross-sectional rotation angle will compose the angle by the share bending transverse-displacement and angular by the shear distortion . in which : rotating angle of section due to bending : rotating angle of section due to shear The bending moment and shear force can be expressed as the following formula. in which EI : Flexural stiffness KA: Effective shear section area G: Shear modulus By force and moment equilibrium relationship in vertical direction and around left side center, the vibration equation of Timoshenko Beam can be written as below. in which, :mass density A: cross section area

Constitutive Relationship of Concrete

The path-dependent nonlinear constitutive models of the concrete and the reinforcing bars based on the smeared crack model are applied. The unloading and the reloading processes are taken into consideration. Two state constitutive models of concrete under compression and tension are used respectively to define the relationship of average strain and average stress in each cell. The formula for these models are listed below and the following two figures are the example results of concrete stress-strain relationship computed by COM3(Fiber). 1) Compressive State in which, : average compressive stress ' : average compressive strain : fracture parameter reducing factor(=1) K : fracture factor E0 : initial stiffness() 'P : plastic strain by compressive loading 'MAX : maximum compressive strain 2) Tensile State in which, Rf :decrease factor of tensile strength(K3=) : tensile strength : cracking stain c : tension stiffness/softening factor =2.0 for plain concrete cell =0.4 for RC cell (using bar reinforcement)

The parameters of concrete such as tensile strength, compressive peak strain and cracking strain are expressed as the functions of its uni-axial compressive strength (N/mm2)which is obtained based on experiments.

Tensile strength (N/mm2): Compressive peak strain: Cracking strain:

Constitutive Relationship of Steel

The Shima's constitutive model of average stress and average strain is used for steel in the concrete. The localization of plastic strain at the crack position and the breaking action are considered in this model and a multi-linear performance can be shown. Though the yield strain obtained by this model is lower than that by bare bars before yielding, the stiffness of steel after yielding is comparable with the stiffness of bare bars. The average yielding stress is calculated by considering the average tensile performance of concrete as below. in which, :Average yield strength of steel in concrete (N/mm2) :Average tensile stress of concrete (N/mm2) :Reinforcement ratio Yield strength of bare steel (N/mm2) For the unloading and reloading operations, the Kato's model is employed for the constitutive relationship of steel for its agreement with experimental results. The Bauschinger effect can be demonstrated very well by this model too. However, due to too many parameters are used in the Kato's model, the same precise, effective multi-plastic-element numerical model is used instead. The steel is assumed to be constituted by tiny element sets and all the elements is offered the same minus strength in the plastic performance. Therefore the Bauschinger effect can be modelized. Reinforced concrete members hold higher tensile stiffness than bare steel bars even after cracking because the concrete continues to support a part of the tensile force transmitted from the reinforcing bars through the bond action. At that time,it has been verified that the average yield strength (fy')of steel bars in reinforced concrete members becomes lower than that of bare bars (fy).The decline rate is determined by the tension stiffening/softening factor(c)and effective reinforcement ratio in each element (Pe). The effective reinforcement ratio Pe is calculated by the steel area and RC Zone area which takes the bond effect of concrete and steel. For more details on the nonlinear models of RC, it can be referred to the book, 'OKAMURA, Hajime and MAEKAWA, Koichi, Nonlinear Analysis and Constitutive Models of Reinforced Concrete, Gihoudo-shuppan, 1991.5'

Cracking Judgment of Concrete

The concrete state in a fiber element is judged by the plane stress state in each cell. The crack occurrence can be judged by the following formula in the cell. for the compressive and tensile state or tensile and tensile state in which, : maximum and minimum principal stress respectively; : concrete tensile strength in one direction test; : decrease factor of concrete tensile strength.

Zoning Method

In the real response of RC structures the crack occurrence of concrete in the plain concrete filed is different from that in steel bar peripheral field due to its adhere characteristics. The different concrete constitutive relationships, especially for the tension stiffening/softening factor C, should be applied to two fields. Therefore according to An the RC structure is divided into RC zone with an adhesive behavior and PL zone without an adhere behavior to compute respectively. In the RC zone the concrete strain-stress relationship after cracking is affected by its bond effect to steel bar and presents a tension stiffening property. In the PL zone the concrete strain-stress relationship after cracking decrease sharply in the crack normal direction because no steel bar restriction and present a tension softening property. The Zoning Method raise the precision of RC structural analysis for its distinguishing the tension stiffening or softening characteristics of concrete after cracking. In addition, the Zoning Method make the lager RC structure analysis more rapid under the same precision target for no tiny mesh division is need in the PL zone. Attentions: In general the adhere action is considered in the fields of the covering scale 2 times ( 5 times of steel diameter). Therefore the field from steel center to the covering distance is set to RC or RC (No steel). The steel-involved cells are RC and no-steel-involved cells are RC(No steel). The other fields due to no adhere action are set to Plain cells, that is PL Zone. The detailed description about Zoning Method can be referred to the paper "An, X, Maekawa, K. and Okamura, H.: Numerical Simulation of Size Effects in Shear Strength of RC Beams, Proc. Of JSCE. No. 564/V-35, pp. 297-316, 1997.5"

Gauss Point

The gauss point refers to the representing point of function for gauss numerical integral. When the element stiffness matrix is formed, the gauss point values with gauss parameters are used instead for the complicated integrating computation. The element internal forces at gauss points can be calculated easily too after global equations are solved. The detailed can refer to the related FEM books. Two gauss points of fiber element in COM3(Fiber) are defined as below.

Global Coordinates and Local Coordinates

The Fiber element section is mounted on model and the element local coordinates has to be defined relative to model's global coordinates. The local coordinates of element is defined as graph below in COM3(Fiber). The defined section is in oxy plane of local coordinates and oz axis is out of screen through the geometrical center. The local coordinates・relative position to the global coordinate can be determined as following. The origin point of local coordinates s put at i side of fiber element. The oz axis is from I to j on the fiber element. When the element is not parallel to or doesn't pile on OY axis, the oyz plane of local coordinates is created which perpendicular to the OXZ plane of global coordinates. The oy axis has a sharp degree with the OY axis and the ox axis is decided by right hand law, When the element is parallel to or piles on OY axis, the oyz plane of local coordinates is created which is parallel to OYZ plane of global coordinates. The ox axis takes the same plus direction with the OX of global coordinates and the oy axis is decided by right hand law,

Bending Capacity of Reinforced Concrete Section

It should be strengthened that the bending capacity check of Beam/column section in our program only for a reference because the different formula of bending capacity will result different results. Therefore the bending failure of sections is not made to stop program and the results are shown only in graph or comparative figure. The capacity of reinforced concrete section for resisting bending moment is defined according to the Concrete Code of JSCE.

Shear Capacity of Reinforced Concrete Section

The shear capacity can be calculated and displayed by COM3(Fiber) as a referring value. In the case that shear effects of models can be ignored or the enough shear strengthening steels are used, this shear capacity check can be neglected. But in the other cases, it has to check the shear force of the section with the given Shear Capacity. The shear capacity of columns depends on not only its cross section dimensional scale but also on the exerting axial force and moment. Of course, the materials of the section and the shear span affect the shear capacity. For the one directional, x or y, acting moment M(t), the corresponding shear capacity can be written as, V = Vc + V3 in which Vc Vc:Contribution by concrete (kN) V3 Contribution by shear reinforcement (kN) and in which and fc' :Concrete compressive strength(N/mm2) d Effective height(mm) of section a Shear Span(mm) bW Web Width(mm) p :Reinforcement Ratio in the tensile side of section Mo Moment that acts on the section with axial force and result in a zero stress on the margin of section tensile side. M(t) Acting Moment(kN-m) varying with steps. N(t) Acting Axial Forces(kN) varying with steps. The compound shear capacity can be expressed as below. It varies with the axial force and bending moment exerting on the section. For each computing step, the two directional shear forces are compound by the formula below and compared to the above calculated value(V). The shear damage can be checked.

Damage Criteria

he damage judgment of fiber element according to computation results is conducted on the strain of cell section at gauss points. Three kinds of states are defined to the section for the whole forced process according to the maximum history strain of steel and concrete. They are given as below. No Damage : The maximum steel strain is under steel yield strain. Light Damage : The maximum steel strain is more than steel yield strain(y), Damage : The maximum compressive strain of concrete is more than the assumed compressive strain level (c') and/ or the maximum tensile strain of concrete is more than the assume tensile strain level (t)

Nonlinear Dynamic Calculating Procedure

The nonlinear FEM dynamic equation of RC structures under external forces can be written as [ M ] : mass matrix. { X } : acceleration vector. [ B ] : the shape function of displacement. { F } : external force vector The Newmark-ßmethod is employed to solve the dynamic equation in the time steps and the geometrical and material nonlinear iteration use the Newton-Raphson method. The computing flowing chart can be summarized as below. Step 1: Begin iteration by setting i=0 Step 2: Begin predictor phase in which we set Step 3: Evaluate error vector Step 4: If required, form the effective stiffness matrix Step 5: Solve for displacement vector Step 6: Error corrector phase Step 7: Check convergence, if not satisfied, set i=i+1 and go to step 3, otherwise continue. Step 8: Set Step 9: Set n=n+1 and begin next time step The details of these formula refer to 'Song, C. M. and Maekawa, K, Dynamic Nonlinear Finite Element Analysis of Reinforced Concrete, Journal of The Faculty of Engineering, The university of Tokyo (B), Vol. XLI, No. 1, pp73-138, 1991'.

UC-win/COM3(Fiber)