Inelastic Material Properties

The Modified Kent & Park Concrete model allows for the consideration of confinement effects, such as the lateral confinement of reinforcing bars.

This Kent and Park (1973) concrete model, which was modified by Scott, et al. (1982), can consider the Confinement Effect due to reinforcing and is rated highly because of its clear formation and accurate analysis.

: Concrete compressive strength

: Factor, which accounts for the strength increase due to confinement

: Ratio of the volume of hoop reinforcement to the volume of concrete core measured to outside of stirrups, : Yield strength of stirrup steel

: Strain at compressive Crushing

: Strain at maximum compressive strength

Z : Strain softening slope - coefficient representing the stiffness in the concrete softening zone after compression yielding

( : Width of concrete core measured to outside of stirrups )

(S h : Center to center spacing of stirrups or hoop sets)

can be obtained by the proposed equation of Scott et al, or another value may be used at user's discretion.

In this model, the tensile strength of concrete is ignored as its influence is minimal throughout the element.

A detailed summary of the hysteresis characteristics of this model can be described as follows:

a. In the compressed state, when unloading occurs, the behavior is assumed to follow a straight line connecting the point where unloading begins and a point on the strain axis. The slope of this line decreases as the degree of compression increases, and the corresponding equation is as follows:

b. Tensile strength is ignored in this model. Therefore, under Complete Unloading or Open Crack, the stress becomes '0'(as shown in the above figure).

c. In the reloading state, the stress is assumed to be zero until reaching εp. Once a compressive strain larger than εp occurs, it is assumed that reloading follows the same straight line as during unloading. In reality, the behavior follows a nonlinear path rather than a straight line during unloading and reloading. However, the influence of such nonlinear behavior is considered to be negligible, and for the sake of efficiency and accuracy in analysis, it is sufficient to assume unloading and reloading to occur along a straight hysteresis loop, as in this model.

There is an alternative approach that uses the stiffness in the softening region, represented by E=K fc Z, beyond the compressive strength. However, since this value is negative, it can reduce the convergence efficiency in the ultimate state. Therefore, it is set to E=1.d-15 in this model.

This model is presented in Seismic Performance Comparison Investigation, P23 of Japanese Concrete Standard Specification. Softening zone after the maximum stress, residual plastic deformation and stiffness reduction effect under the condition of reloading after unloading are reflected.

f_c' : Compressive strength of concrete 

ε_peak' : Strain at maximum compressive strength

The stress-strain relationship of this hysteresis model can be described as follows:

where,

In this model, K represents the residual elastic stiffness ratio, which is used to simulate the phenomenon where the slope during unloading decreases as the compressive strain increases.

For general beam elements, the tensile strength is typically ignored. The behavioral characteristics based on the hysteresis rules can be illustrated as shown in the figure below.

As a reference, when evaluating the ultimate behavior of structures using the concrete model based on the Japanese Concrete Standard Specifications, there is a method of investigating the state based on the residual stiffness reduction factor.

This criterion determines the level of damage to the element by observing that the stiffness reduction factor during unloading decreases as the compressive strain increases. The current MIDAS software program also implements an investigation method based on this criterion.

(refer to Section Damage Check with Fiber Element function)

This model is presented in Commentary V -Seismic Design (Concrete confined by reinforcing steel, p. 161) of Japanese Roadway Specification.

The stress-strain relationship in this model is modified based on the type of seismic motion. It adequately accounts for the effects of seismic motion, confinement from reinforcement, and section geometry.

When Type I is selected                                 When Type II is selected

: Ultimate strain and the strain at maximum compressive strength are the same, and the zone of descending slope (Edes) does not exist. In this model, it is assumed that once the compressive strain reaches the maximum compressive strength, immediate crushing occurs.

: Ultimate strain is calculated by the Specification, and the zone of descending slope (Edes) is retained. 

: Young's Modulus of concrete

: Design strength of concrete

: Yield point of confining reinforcing steel

: Sectional modification coefficients

For a circular section, α=1.0, β=1.0

For trapezoidal, hollow circular and hollow trapezoidal section, α=0.2, β=0.4

: Cross-sectional area of a single confining rebar

: Spacing of lateral confinement

: Longer length of a concrete section dimension confined by main stirrups or local stirrups

: Tensile strength of concrete

: Strength of concrete confined by laterally restraining rebars

On the other hand, the volumetric ratio of transverse confinement reinforcement, denoted as Ρs, is required to be less than or equal to 0.018 according to the provided criteria.

The stress-strain relationship of this model is as follows:

where,

During unloading and reloading, it is assumed that the behavior follows the elastic stiffness (as shown in the diagram below).

There is an alternative approach that uses the stiffness in the descending slope of the stress-strain curve after exceeding the compressive strength, represented by E=-Edes. However, since this value is negative and can reduce the convergence efficiency in the ultimate state, it is set to E=1.d-15 in this model.

This model is presented in Seismic Performance Comparison Investigation p.7 -Structural steel pier partially filled with concrete of Nagoya Highway Corporation.

: Compressive strength of concrete

: Strain at maximum compressive strength

: Coefficient to reflect the increase in compressive strength

: Ultimate compressive strain of concrete

: Strain at maximum tensile strength

: Strain at tensile rupture of concrete

: Ultimate tensile strain of concrete

In this model, as shown in the diagram, it is assumed that the stress remains constant until the compressive strain reaches the ultimate compressive strain, after which crushing occurs when the strain exceeds εcu.

During unloading and reloading, it is assumed to behave elastically. While there are no specific provisions in the design code regarding tensile strength, the model is implemented to allow users to input an appropriate tensile strength based on their judgment (as shown in the diagram below).

There is an alternative approach that uses the stiffness in the descending branch of the stress-strain curve after exceeding the tensile strength. However, since this value is negative and can reduce the convergence efficiency in the ultimate state, it is set to E=1.d-15 in this model.

Both tensile and compressive zones can be defined in this model. Trilinear hysteresis exists in the compression zone, which can be defined by the stress-strain relationship and the stress-stiffness reduction ratio relationship.

              Stress-strain definition method          Stress-stiffness degradation definition method

: First compression yielding of concrete

: Second compression yielding of concrete

: Stiffness of concrete after second compression yielding (required for K3 calculation)

: Strain at maximum tensile strength

: Strain at tensile rupture of concrete

: Ultimate tensile strain of concrete

: Strain at first compression yielding of concrete

: Strain at second compression yielding of concrete

: Strain after second compression yielding (required for K3 calculation)

: Strain at third compression yielding of concrete

: Initial stiffness of concrete

: Ratio of stiffness after the first yielding to the initial stiffness

: Ratio of stiffness after the second yielding to the initial stiffness

There is an alternative approach that uses the stiffness in the descending branch of the stress-strain curve after exceeding the tensile strength. However, since this value is negative and can reduce the convergence efficiency in the ultimate state, it is set to E=1.d-15 in order to preserve convergence efficiency.

When ε_c1~ε_c3 are entered in the 'σ - ε' input method, and the 'σ - α' input method is selected, K1, K2/K1, K3/K1 are automatically calculated. Also the reverse calculation is automatically done.

Characteristics

This is the concrete model confined by transverse rebar, which was suggested by Mander (1988). Transverse rebars are provided to confine core concrete, and to prevent buckling failure of main rebars and shear failure of the member. Compressive concrete confined by transverse rebar can increase strength and ductility of a member due to the confinement effect.

Mander model provides the Stress-Strain relationship of the confined concrete, regardless of the section shape. Also the Effective Confined Stress can be obtained considering longitudinal rebar area, transverse rebar area, rebar strength, and rebar type. Mander model is based on the single-axis Stress-Strain curve, suggested by Popovic(1973). However, using Mander model, Effective Confined Stress of multi-axis can be transferred to that of a single-axis.

Limitations

The Mander model can be applied to concrete structures regardless of their cross-sectional shape. However, the automatic calculation of material data and section data supports only circular and rectangular sections. For sections other than circular and rectangular, users can input the required information for automatic strength calculation to obtain the calculated strength.

Concrete Type

Unconfined Concrete : This is used to define concrete without considering the transverse confinement effect. Factors related to transverse confinement are deactivated.

Confined Concrete : This is used to define core concrete confined by transverse reinforcement. It considers the transverse confinement effect.

Unconfined Concrete Data

: Enter the compressive strength of unconfined concrete. If you wish to input it directly, select the checkbox.

: Enter the strain corresponding to the compressive strength of unconfined concrete, fco'. If you wish to input it directly, select the checkbox.

: Enter the elastic modulus of concrete. In the Mander model, it is internally calculated, but you can provide a user input if desired.

: Enter the tensile strength of concrete. You can either provide a user input or choose to neglect the tensile strength.

: It is automatically calculated based on the tensile strength.

Section & Confinement Rebar Type : This option is activated when Concrete Type is defined as "Confined Concrete : Core Concrete". Define the section type and rebar type of the transverse rebar. 

Section Data : You enter the dimensions of the confined section and the spacing of the transverse confinement reinforcement.

Rebar Data : You provide the necessary information for the area of the main reinforcing bars and transverse confinement reinforcement.

Confinement Effective Coefficient, Ke : The effective confinement coefficient of the selected concrete, considering the section shape, quantity of longitudinal reinforcement, quantity of confinement reinforcement, and arrangement, is determined.

The Effective Lateral Confining Stress on the Concrete

flx : Effective confining stress in the x-direction.

fly : Effective confining stress in the y-direction.

fl : Effective confining stress transferred from multi-axis to single-axis.

Confined Concrete Strength & Strain : Compressive concrete strength and strain considering confinement effect. Type your drop-down text here.

This model has been derived from the general bilinear steel model. Stiffness after compressive and tensile yielding can be freely defined. Buckling and rupturing of reinforcing steel can be considered.

: Tensile yield strength

: Compressive yield strength

: Strain at compression buckling of reinforcing steel

: Strain at rupturing of reinforcing steel after yielding

: Initial stiffness of reinforcing steel

: Stiffness of reinforcing steel after tensile yielding

: When tension becomes unloaded and reloaded in the compression zone, the E3 line limits the stiffness under compression loading.

: Stiffness of reinforcing steel after compressive yielding (By specifying a negative value, it is possible to consider a negative gradient or a descending behavior.)

: Stiffness of buckled reinforcing steel after compressive yielding

This Model can describe compressive yielding, tensile yielding, tension rupture, compression buckling, etc of reinforcing steel. The figure below shows the possible hysteresis states.  

E ach state is explained as follows:

Since tension behavior is the major cause of hysteresis in steel, the loading direction basically follows tension. During unloading, the loading direction changes from tension to compression, and vice-versa during reloading.

State 1 : It represents the elastic behavior, and the slope is E1.

1 → 2 = It refers to the transition to the yielding state, behaving with the slope of E2 during tensile yielding and the slope of E4 during compressive yielding.

State 2 : It represents the state after the yielding starts, and the slope is E2.

2 → 4 = It refers to the state of unloading or reloading after yielding.

2 → 8 = Tension is sustained after yielding, and thereafter, tension rupture is caused. The stress is always '0' after the rupture.

State 3 : Unload continuously and as a result the compression zone yields. The slope is E2. E3 should be input such that the point intersecting the line E3 and strain axis is greater than  .

3 → 4 = It refers to the state where reloading occurs after yielding, and it behaves with the slope of E1.

3 → 5 = As unloading continues, the compression strain exceeds   and compression buckling starts. The slope is E5.

State 4 : Unloading and reloading with slope E1 (elastic stiffness)

4 → 2 = As reloading continues, the state changes to tensile yielding state. Or as unloading continues, the state changes to compressive yielding state.

4 → 3 = As unloading continues, compressive yielding occurs. It can be assumed that compressive yielding occurs at the intersection with line E3.

4 → 5 = As unloading continues, compression buckling starts.  

State 5 : When compression strain exceeds buckling strain, buckling of reinforcing steel takes place. The slope is E5.

5 → 4 = Reloading continues during compression buckling.

5 → 7 = As compression is sustained, the reinforcing steel undergoes complete compression buckling. The stress becomes '0' at compression during and after State 7.

State 6 : Reload after compression buckling. Reloading will progress towards  before tensile yielding occurs. Reloading will progress towards the maximum point in the tension zone after tensile yielding occurs.    

6 → 2 = As reloading continues, tensile yielding takes place.

6 →-1 = Unload while reloading.

State 7 : Once complete compression buckling occurs, further compressive stresses cannot be generated. Although compressive stress becomes '0' it can still resist tension.

7 → 6 = Reloading occurs and progresses towards the maximum tension point.

State 8 : Once tension rupture occurs, further tensile stresses cannot be generated, and neither will compressive stresses be generated.

State -1: Unload while reloading after compression buckling (State 6). The slope is E1 (elastic stiffness).

-1 → 6 : Reloading progresses towards the maximum tension point.

-1 → 7 : Unloading and the subsequent transition to complete compression buckling.

As stated above, this model considers the various states and transitions. If the user is familiar with this model and applies the experimental parameters properly, it can be a very efficient tool. However, the user must use caution when using this model for limit states, such as tension rupture and compression buckling, as the resistance becomes '0'.