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 |
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Fiber Model |
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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,
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 |
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Timoshenko Beam |
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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
in which
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,
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Constitutive Relationship of Concrete |
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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).
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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: |
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Constitutive Relationship of Steel |
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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,
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 |
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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.
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Zoning Method |
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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" |
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Gauss Point |
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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. |
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Global Coordinates and Local Coordinates |
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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 coordinatesErelative position to the global coordinate can be determined as following.
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Bending Capacity of Reinforced Concrete Section |
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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. |
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Shear Capacity of Reinforced Concrete Section |
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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
in which and
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. |
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Damage Criteria |
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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.
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Nonlinear Dynamic Calculating Procedure |
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The nonlinear FEM dynamic equation of RC
structures under external forces can
be written
as
The computing flowing chart can be summarized as below.
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