Fiber Material Properties

Stress-strain relationships between reinforcing steel and concrete are defined to carry out pushover analysis using Fiber Elements. Each model is unique based on the proponents and specifications.

From the Main Menu select Pushover > Properties > Hinge Properties  >  Define Inelastic material Properties.

Each beam element's cross-section is divided into small fibers, and each fiber cell within the cross-section retains a linear and nonlinear Stress-Strain relationship.  

The properties of fiber materials are defined.

: Enter or add a new fiber material property.

: Modify or check a fiber material property previously defined.

: Delete a fiber material property previously defined.

Name: Name of a fiber element model to be defined

Material Type: Steel or concrete for which a hysteresis model will be defined

Hysteresis Model: Select a hysteresis model to define a fiber element out of 4 reinforcing steel models and 5 concrete models.  

Skeleton Curve: Specify the property values defining the hysteretic behavior of the fiber material following the diagram.

Concrete

• Kent & Park Model

  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.

   

  

  Inelastic Material Model

   

  

   

   

  Note
  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 concrete tensile strength is ignored since it has little effect on the entire member.

   

  Hysteresis characteristics of this model are as follows:

   

  a. When a member is unloaded under compression, its behavior is represented by the line connecting   and ;

   where  is the strain at the beginning of unloading and  is a certain point on the strain axis. As the compression increases, the slope of this line becomes flatter. The relationship is defined as:

   

   

  

   

   

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

  c. When reloaded, the stress values before attaining  shall be taken as zero Once the compressive strain exceeds , reloading is applied along the line where the unloading occurred. In reality, Unloading and Reloading cannot be applied linearly, since it is actually a nonlinear application. However since there is not much difference between linear and nonlinear behavior, the results will be accurate enough even if linear Unloading and Reloading is assumed

• Japan Concrete Standard Specification 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.

   

   

  

   

   

  : Compressive strength of concrete cylinder

   

  ε'peak: Strain at maximum compressive strength

• Japan Roadway Specification Model

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

   

   

  

   

  Type I Selection         

   

                                                        

  

   

  Type II Selection

   

   

  Earthquake Type I: Ultimate strain and the strain at maximum compressive strength are the same, and the zone of descending slope (Edes) does not exist.  

   

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

   

  Ec: Young's Modulus of concrete

   

  σck: Design strength of concrete

   

  σsy: Yield point of confining reinforcing steel

   

  α, β: Sectional modification coefficients

   

  Note
  For a circular section, α=1.0, β=1.0
  For trapezoidal, hollow circular and hollow trapezoidal section, α=0.2, β=0.4

   

  Ah: Cross-sectional area of a single confining rebar

   

  s: Spacing of lateral confinement

   

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

   

  σbt: Tensile strength of concrete

   

  σcc: Strength of concrete confined by laterally restraining rebars

• Nagoya Highway Corporation Model

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

   

    

  

   

   

  σck: Compressive strength of concrete cylinder

   

  εcc : Strain at maximum compressive strength

   

  K: Coefficient to reflect the increase in compressive strength

   

  εcu: Ultimate compressive strain of concrete

   

  εt0: Strain at maximum tensile strength

   

  εt1: Strain at tensile rupture of concrete

   

  εtu: Ultimate tensile strain of concrete

   

• Trilinear Concrete 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

   

   

  

   

  Stress-Stiffness reduction ratio Definition

   

   

  σc1: First compression yielding of concrete

   

  σc2: Second compression yielding of concrete

   

  σc3: Stiffness of concrete after second compression yielding (required for K3 calculation)

   

  ε: Strain at maximum tensile strength

   

  εt1: Strain at tensile rupture of concrete

   

  εtu: Ultimate tensile strain of concrete

   

  εc1: Strain at first compression yielding of concrete

   

  εc2: Strain at second compression yielding of concrete

   

  εc3: Strain after second compression yielding (required for K3 calculation)

   

  εcu: Strain at third compression yielding of concrete

   

  K1: Initial stiffness of concrete

   

  K2/K1: Ratio of stiffness after the first yielding to the initial stiffness

   

  K3/K1: Ratio of stiffness after the second yielding to the initial stiffness

   

  Note
  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.

   

• Mander Model

  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.

   

   

  

   

  Stress-Strain Definition

   

   

  Concrete Type

   

  Unconfined Concrete : Select to define cover concrete. Confinement effect will not be considered and confinement related options will be inactivated.

   

  Confined Concrete : Select to define core concrete confined by transverse rebar.

   

   

  Unconfined Concrete Data

   

  

   

  Stress-Stiffness Reduction Ratio Definition

   

   

  fco': Unconfined concrete strength. Select this option to directly define the value of concrete strength.

   

  εco : Unconfined concrete strain. Select this option to directly define the value of concrete strain.

   

  Ec : Elastic modulus of concrete. When "Mander et al." is selected, it is automatically calculated. When "User Input" is selected, the user can directly enter the value.

   

  ft': Tensile strength of concrete. When "Neglect Tensile Strength" is selected, tensile strength of concrete is ignored. When "Mander et al." is selected, it is automatically calculated. When "User Input" is selected, the user can directly enter the value.

   

  εt: Tensile strain of concrete. It is automatically calculated based on the defined 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.  

   

   

  : Define the core concrete dimension and confinement rebar spacing, etc.

   

   

  : Define the longitudinal rebar and confinement rebar related data.

   

   

  Confinement Effective Coefficient, Ke

   

  Confinement effective coefficient of concrete that considers section size, rebar area, and rebar type.

   

   

  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

   

Steel

• Menegotto-Pinto Model

  This is the steel model of Menegotto and Pinto, which was modified by Filippou et al.

   

   

  

   

   

  fy: Yield strength of reinforcing steel

   

  E: Modulus of elasticity

   

  b: Stiffness reduction factor after yielding

   

  Ro, a1, a2: Coefficients for shape index (R) for steel fiber constitutive model

   

  This Stress-Strain hysteresis model is outlined as follows:

   

  

   

  The above formula represents a curved transition from the elastic range to the yielding range.

   

  and  are normalized values and are calculated as follows:

   

   

   

  This model has two asymptotes as shown below. One asymptote has a slope of elastic stiffness and the other has a slope of yielding stiffness.  is a point intersected by two asymptotes.  is the point where the last unloading occurred. While unloading and reloading, these points are updated, so that the transition curve is affected.

   

   

   

  R is the value that can affect the shape of the transition curve and represents the Bauschinger effect. The value of R is determined as follows:

   

   

   

  Coefficients  are determined from experimental hysteresis results. MIDAS uses 18.5, 0.15 and 20, respectively, as default values. These default values are suggested in the original reference (Menegotto and Pinto,1973).

   

  is a shape coefficient and updated at every unloading.

• Bilinear Model

  This model represents a general symmetric Bilinear model for reinforcing steel.

   

   

  

   

   

  fy: Yield strength of reinforcing steel

   

  E1: Initial stiffness of reinforcing steel

   

  E2/E1: Ratio of stiffness after yielding to the initial stiffness

  As shown in the figure below, the model behaves elastically when it is unloaded and reloaded after yielding.

   

  

   

• Asymmetrical Bilinear Steel Model

  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.

   

   

  

   

   

   

  σy: Tensile yield strength

  σcy: Compressive yield strength

   

  ε1: Strain at compression buckling of reinforcing steel

   

  ε2: Strain at rupturing of reinforcing steel after yielding

   

  E1: Initial stiffness of reinforcing steel

   

  E2: Stiffness of reinforcing steel after tensile yielding

   

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

  under compression loading.

   

  E4: Stiffness of reinforcing steel after compressive yielding

   

  (A negative slope can be specified by entering a negative value.)

   

  E5: 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.  

   

  

   

  Each 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 = Transition to yielding state. The slope is E2 under tensile yielding and E4 under compressive yielding.

   

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

  2 → 4 = Unload and reload after yielding.

  2 → 8 = Tension is sustained after yielding, and thereafter, tension rupture is caused. The stress is always ??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 = Reload. The slope is 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 ??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 ?? 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 ?: 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 zero.   

• Trilinear Steel Model

  This model represents a Trilinear model of three slopes.  The hysteresis can be defined by the stress-strain relationship and the stress-stiffness reduction ratio relationship.

  When unloading and reloading, the model behaves elastically.

   

   

  

   

  Stress-Strain Definition     

   

                                                 

  

   

  Stress-Stiffness reduction ratio Definition

   

   

  σ1y: First yield strength in tension

   

  σ2y: Second yield strength in tension

   

  σ3y: Stiffness after second tensile yielding (required for K3 calculation)

   

  σ'1y: First yield strength in compression

   

  σ'2y: Second yield strength in compression

   

  σ'3y: Stiffness after second compressive yielding (required for K5 calculation)

   

  ε1y: Strain at first yielding in tension

   

  ε2y: Strain at second yielding in tension

   

  ε3y: Strain after second tensile yielding (required for K3 calculation)

   

  ε'1y: Strain at first yielding in compression

   

  ε'2y: Strain at second yielding in compression

   

  ε'3y: Strain after second compressive yielding (required for K5 calculation)

   

  K: Initial stiffness of reinforcing steel

   

  K2/K1: Ratio of stiffness after first tensile yielding to the initial stiffness

   

  K3/K1: Ratio of stiffness after second tensile yielding to the initial stiffness

   

  K4/K1: Ratio of stiffness after first compressive yielding to the initial stiffness

   

  K5/K1: Ratio of stiffness after second compressive yielding to the initial stiffness

   

  Note
  When ε1y~ε'3y are entered in the 'σ - ε' input method, and the 'σ - α' input method is selected, K1, K2/K1, K3/K1, etc. are automatically calculated. Also the reverse calculation is automatically done.