Inelastic Hinge Properties

Kinematic Hardening hysteresis model

 

Response points at the initial loading move along a trilinear skeleton curve. The unloading stiffness is identical to the elastic stiffness. It shows the tendency of strength increase with the increase in loading. This is used to model the Bauschinger effect of metallic materials. Accordingly, it is cautioned that energy dissipation may be overestimated for concrete. Due to the characteristic of the model, only the positive (+) and negative (-) symmetry is permitted for the strength reduction ratios after yielding.

Response points at the initial loading move along a trilinear skeleton curve. Response points at unloading move toward the origin and again move along the skeleton curve after reaching the opposite skeleton curve.

Response points at the initial loading move along a trilinear skeleton curve. Response points at unloading move toward the point of maximum displacement on the opposite side. If the first yielding has not occurred on the opposite side, the response points move toward the first yielding point on the skeleton curve.  

Response points at the initial loading move along a bilinear skeleton curve. Unloading stiffness is obtained from the elastic stiffness reduced by the equation below. As the deformation progresses after yielding, unloading stiffness reduces gradually.

where,

: unloading stiffness

: elastic stiffness

: yield displacement in the zone where unloading begins

: maximum displacement in the zone where unloading begins

(In the zone where yielding has not occurred, replace it with the yield displacement)

: constant for determining unloading stiffness

If the sign of loading changes in the process of unloading, response points move toward the point of maximum displacement in the zone of progressing direction. If yielding has not occurred in this zone, the response points move toward the yield point on the skeleton curve. If unloading becomes loading without changing the loading sign, the response points move along the unloading path. If the loading continually increases, loading continues on the skeleton curve again.  

 

Initial unloading state prior to yielding in the uncracked zone (for small displacement)

Initial unloading state (for large displacement) and inner loop prior to yielding in the uncracked zone

Response points at the initial loading move along a trilinear skeleton curve. The load-displacement coordinates at unloading move to the path of reaching the maximum deformation point on the opposite side due to the change of unloading stiffness once in the middle. If yielding has not occurred on the opposite side, the first yielding point is assumed to be the maximum deformation point. The first and second unloading stiffness is determined by the equations below. As the maximum deformation increases, unloading stiffness gradually decreases.

where,

KR1: first unloading stiffness

KR2: second unloading stiffness

K0: elastic stiffness

KC: first yield stiffness in the zone where the loading point is oriented due to unloading

K1: the slope between the origin and the second yield point in the zone where the loading point is oriented due to unloading

b: stiffness reduction ratio. If unloading occurs between the first and second yield points on the skeleton curve, it is fixed to 1.0.

FM+ , FM-: maximum load at each positive (+) and negative (? sides

DM+ , DM-: maximum deformation at each positive (+) and negative (? sides

Response points at the initial loading move along a trilinear skeleton curve. The unloading stiffness is determined by the location of the unloading point on the skeleton curve and whether or not the first yielding has occurred in the opposite side zone.  

If unloading occurs between the first and second yield points on the skeleton curve, load-deformation coordinates progress toward the first yield point on the skeleton curve on the opposite side. If the loading sign changes in the process, the coordinates progress toward the maximum deformation point on the skeleton curve in the zone of the progressing direction. If yielding has not occurred in this zone, the coordinates progress toward the first yield point. If the skeleton curve is encountered in the process of progressing, the coordinates progress along the skeleton curve.

If unloading takes place in the zone beyond the second yield point on the skeleton curve, the load-deformation coordinates progress following the next unloading stiffness.

 

Unloading state prior to yielding in the uncracked zone (for small displacement)	Unloading state prior to yielding in the uncracked zone (for large displacement)

 Response points at the initial loading move along a trilinear skeleton curve.  

 When the current displacement or deformation, D, exceeds D1 for the first time or the point of maximum deformation up until now

1. The point moves along the trilinear skeleton curve.

2. If unloading occurs from the straight line toward the opposite direction, the point moves toward the first yield point on the opposite side.

3. If the maximum displacement on the opposite side is in the elastic zone, the range of this elastic zone extends to the first yield point on the opposite side.

4. If the maximum deformation on the opposite side exceeds D1, the displacement in this elastic zone is defined up to the point of restoring force becoming 0.  If it passes beyond the 0 point, it moves toward the maximum deformation point on the opposite side. The stiffness at unloading from the straight line directed toward this maximum deformation uses the stiffness of unloading from the maximum deformation point on the opposite side.  

 When the current displacement or deformation, D, exceeds D2 for the first time or the point of maximum deformation up until now

1. The point moves along the trilinear skeleton curve.

2. If unloading occurs from this straight line toward the opposite direction, the point moves from the unloading point along the straight line of the stiffness obtained from the equation below.  

,

where,

Kun2: unloading stiffness of the outer loopP1: first yield loading in the zone opposite to the unloading point

P2: second yield loading in the zone to which the unloading point belongs

D1: first yield displacement in the zone opposite to the unloading point

D2: second yield displacement in the zone to which the unloading point belongs

Dmax: maximum deformation in the zone to which the unloading point belongs

β: constant for determining unloading stiffness of the outer loop

3. If the maximum deformation point on the opposite side does not exceed D1, the range of the slope Kun2 extends up to P1 on the opposite side. If it goes beyond this P1, it directs toward the D2 point. If unloading occurs from the straight line directed toward the D2 point, it moves along the straight line of the slope Kb. If the restoring force exceeds 0, it directs toward the maximum deformation point. If unloading occurs from the straight line directed toward the maximum deformation point, it moves along the slope Kun2. If the restoring force goes beyond 0, it directs toward the maximum deformation point on the opposite side.  

4. If the loading sign changes in the process of unloading and while reloading takes place, unloading may occur before reaching the target point on the skeleton curve. Loops are formed in the process, which are all referred to as inner loops. The unloading stiffness within the inner loops is determined by the equation below.  

where,

KRI: unloading stiffness of inner loop

          Kun2: unloading stiffness of outer loop in the zone to which the start point of unloading belongs

          γ: reduction factor for unloading stiffness of inner loop

5. If the maximum deformation on the opposite side exceeds D1, the range of the slope Kun2 extends up to the point of restoring force becoming 0, and if it goes beyond 0, it directs toward the maximum deformation point. If unloading takes place from the straight line directed toward this maximum deformation point, D at this point becomes the maximum deformation point of the inner loop. And it moves along the slope Kun2 and directs toward the maximum deformation point if the restoring force exceed the 0 point. Even in the case where unloading takes place from the straight line directed toward this maximum deformation point, D becomes the maximum deformation point of the inner loop and moves along Kun2. If the restoring force goes beyond the 0 point, it directs toward the maximum deformation point.

Takeda type hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type.

 

Takeda Tetra Linear hysteresis model

Response points at the initial loading move along a tetralinear skeleton curve.

If the current displacement or deformation, D, does not exceed D3, the hysteresis rules are identical to the Original Taketa hysteresis.  

If the current displacement or deformation, D, exceeds D3, response points move along the slope K4. For unloading, response points move by the same rules as the Original Taketa hysteresis.

The Takeda tetralinear hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type.

 

Response points at the initial loading move along a trilinear skeleton curve.

If the current displacement or deformation, D, exceeds D2 for the first time or the maximum deformation point up until now, response points move along the trilinear skeleton curve. If unloading takes place from this straight line toward the opposite direction, the points move along the slope Kun2 until the point of the restoring force becoming 0. If the restoring force goes beyond the 0 point, the points move toward the maximum deformation point on the opposite side.

where,   

Kun2: unloading stiffness of outer loop

K0: elastic stiffness

D1: first yield displacement in the zone of unloading point

Dmax: maximum deformation in the zone to which the unloading point belongs

β: constant for determining unloading stiffness of the outer loop

Even in the case where unloading takes place from the straight line directed toward the maximum deformation point from the point of the 0 restoring force, the points move along the slope Kun2 until the points reach the 0 restoring force. After the point of 0 restoring force is passed, the points move toward the maximum deformation point on the opposite side.

The Modified Takeda type hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type.

Modified Takeda Tetra Linear Type hysteresis model

 Response points at the initial loading move along a tetralinear skeleton curve.

 If the current displacement or deformation, D, does not exceed D3, the hysteresis rules are identical to the Modified Taketa hysteresis.

 If the current displacement or deformation, D, exceeds D3, response points move along the slope K4. For unloading, response points move by the same rules as the Modified Taketa hysteresis.

The Modified Takeda Tetralinear type hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type.

Normal Bilinear Type hysteresis model

Response points at the initial loading move along a bilinear skeleton curve. The unloading stiffness is identical to the elastic stiffness. The Normal Bilinear type hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type.

Elastic Bilinear Type hysteresis model

Regardless of loading and unloading, response points always move along a bilinear skeleton curve without any pattern of curved loops. The Elastic Bilinear type hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type.

Elastic Trilinear Type hysteresis model

Regardless of loading and unloading, response points always move along a trilinear skeleton curve without any pattern of curved loops. The Elastic Trilinear Type hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type

Elastic Tetralinear Type hysteresis model

Regardless of loading and unloading, response points always move along a tetralinear skeleton curve without any pattern of curved loops. The Elastic Tetralinear Type hysteresis model can be applied to beam element and General Link of Spring Type of Lumped Type and Distributed Type.

 

LRB Isolator Bilinear Type hysteresis model

 

 

The initial stiffness is calculated by the strain used specifically for calculating the initial stiffness. Movement on a bilinear skeleton curve is the hysteresis rule provided that the strain does not exceed Rmin.  

Qd50: yield property

Kp50: yield strength

H: layer thickness

ALF: yield stiffness ratio (1/6.5)

Rmin: strain used specifically for calculating initial stiffness

KE50: initial spring coefficient（spring coefficient at 50% deformation of the LRB device）

LRB Isolator Bilinear type hysteresis model can be applied to General Link of Spring Type only.

LRB Isolator Trilinear Type hysteresis model

 

H: layer thickness Area: contact area R_min: initial shear strain (Default 0.01) SW: bearing type 1: HDR-G12 (default) 2: HDR-G10 3: LRB-G12 4: LRB-G10 5: RB-G12 6: RB-G10 7: HDR-G8 8: HDR-S-G12 LRB Isolator Trilinear type hysteresis model can be applied to General Link of Spring Type only.

 

High Damping Rubber Isolator Type hysteresis model

 

H: layer thickness Area: contact area R_min: initial shear strain (Default 0.01) SW: bearing type 1: KL301 (default) 2: KL401 3: KL302 4: KL501 5: UHD-G6 6: HD-G8 7: TOYO 8: G=8  kgf/cm2 9: G=10  kgf/cm2 10: G=12  kgf/cm2 Gs: coefficient multiplied to shear elastic modulus (default : 1.0) Hs: coefficient multiplied to equivalent damping (default : 1.0) Us: coefficient multiplied to yield loading property coefficient (default : 1.0) HDR Damping Rubber Isolator Type hysteresis model can be applied to General Link of Spring Type only.

 

	At initial loading, response points move along the bilinear skeleton curve. When |D1| < |D|, the member is unloaded along the slope of elastic stiffness up to point A ?a point where the restoring force is 0. Once the response points reach a reversal point A during unloading, they move along the X-axis up to point B. Once response points go beyond point B, the member is reloaded along the slope of elastic stiffness until the response points reach the skeleton curve. Then the member is unloaded with the slope of elastic stiffness until the restoring force becomes 0. From a reversal point, the response points move along the Displacement Axis (X-axis). This process is repeated.  In the Slip Bilinear type hysteresis model, an initial Gap can be set.
Slip Bilinear/Tension Type hysteresis model	At initial loading, response points move along the bilinear skeleton curve. When |D1| < |D|, the member is unloaded along the slope of elastic stiffness up to point A ?a point where the  restoring force is 0, and then the response points move along the X-axis. When reloading, the response points move along the Displacement Axis (X-axis) up to point A. Once response points go beyond point A, reloading continues along the slope of elastic stiffness until the response points reach the skeleton curve.  In the Slip Bilinear/Tension type hysteresis model, only a (+) initial Gap can be set.  In the Slip Bilinear/Compression type hysteresis model, only a (-) initial Gap can be set.
Slip Bilinear/Compression Type hysteresis model
*. When the initial Gap   is entered in the Slip Bilinear type hysteresis model, plastic ratio (D/D1) is calculated by the equation .

Slip Trilinear  Type hysteresis model	At initial loading, the response points move along the trilinear skeleton curve.  When |D1| < |D|, the member is unloaded along the slope of elastic stiffness up to point A ?a point where the restoring force is 0. Once the response points reach a reversal point A during unloading, they move along the X-axis up to point B.  Once response points go beyond point B, the member is reloaded along the slope of elastic stiffness until the response points reach the skeleton curve. Then the member is unloaded with the slope of elastic stiffness until the restoring force becomes 0. From a reversal point, the response points move along the Displacement Axis (X-axis). This process is repeated.  In the Slip Trilinear type hysteresis model, an initial Gap can be set.
Slip Trilinear/Tension Type hysteresis model	At initial loading, response points move along the trilinear skeleton curve. When |D1| < |D|, the member is unloaded along the slope of elastic stiffness up to point A ?a point where the restoring force is 0, and then the response points move along the X-axis. When reloading, response points move along the Displacement Axis (X-axis) up to point A. Once the response points go beyond point A, reloading continues along the slope of elastic stiffness until the response points reach the skeleton curve.  In the Slip Trilinear/Tension type hysteresis model, only a (+) initial Gap can be set.  In the Slip Trilinear/Compression type hysteresis model, only a (-) initial Gap can be set.
Slip Trilinear/Compression Type hysteresis model
*. When the initial Gap is entered in the Slip Trilinear type hysteresis model, plastic ratio (D/D1, D/D2) is calculated by the equation .

 

 

(1) Overview of Hysteresis

Multi-Linear Elastic Type Hysteresis is nonlinear but elastic. The force-displacement relationship of the skeleton curve is defined by a multi-linear curve. Irrespective of loading and unloading, no hysteresis loop is generated in Multi-Linear Elastic Type, and the force-displacement relationship exists only on the skeleton curve.

The curve can be symmetrically or unsymmetrically defined. The types of corresponding elements include lumped hinge, distributed hinge, spring and truss elements.

 

(2) Definition of Skeleton Curve

Force-Displacement Curve

The skeleton curve is defined by the force-displacement relationship defined by the user. The following restrictions apply to defining the force-displacement curve:

l Force-Displacement Curve has no limitation on the number of data.

l The initial value must be set to (0,0).

l No identical values can be used for Displacements, and the force-displacement data are arranged in reference to the displacements.

l The signs of force and displacement must be the same at all times.

l A negative slope is not permitted in Force-Displacement Curve except for the final value. As such, the forces must gradually increase on the positive side and decrease on the negative side except for the last points on the curve. No fluctuation is permitted.

 

(3)   Rules for Hysteresis of Multi-Linear Elastic Type

The rules for hysteresis of Multi-Linear Elastic Type are identical to those of Elastic Tetralinear Type.

(1) Overview of Hysteresis

Multi-Linear Plastic Kinematic Type Hysteresis is defined on multi-linear skeleton curves based on the kinematic hardening rules. The curve can be symmetrically or unsymmetrically defined. The types of corresponding elements include lumped hinge, distributed hinge, spring and truss elements.

 

(2) Definition of Skeleton Curve

Force-Displacement Curve

The skeleton curve is defined by the force-displacement relationship defined by the user. The following restrictions apply to defining the force-displacement curve:

Force-Displacement Curve has no limitation on the number of data.

l At least one data point must be defined on both the positive and negative sides, and the numbers of data on the positive and negative sides must be identical.

l The initial value must be set to (0,0).

l No identical values can be used for Displacements, and the force-displacement data are arranged in reference to the displacements.

l The signs of force and displacement must be the same at all times.

l A negative slope is not permitted in Force-Displacement Curve. As such, the forces must gradually increase on the positive side and decrease on the negative side. No fluctuation is permitted.

 

(3)   Rules for Hysteresis of Multi-Linear Plastic Kinematic Type

1. In the case of , the hysteresis curve for Multi-Linear Plastic Kinematic Type follows the conventional kinematic hardening rules.

2. In the case of  , when the force is unloaded on the skeleton curve, the unloading takes place backward at a slope of K0 by the magnitude of P1(+) or P1(-) (Rule:1). It is then directed towards the point of unloading by the magnitude of the first

yielding displacement, D1(-) or D1(+), on the opposite side until the restoring force becomes 0 (Rule:2). Once the restoring force exceeds 0, the kinematic hardening rules apply.

(1) Overview of Hysteresis

Multi-Linear Plastic Takeda Type Hysteresis is a multi-linear stiffness degradation model. The curve can be symmetrically or unsymmetrically defined. The types of corresponding elements include lumped hinge, distributed hinge, spring and truss elements.

Multi-Linear Plastic Takeda Hysteresis Model

(2) Definition of Skeleton Curve

Force-Displacement Curve

The skeleton curve is defined by the force-displacement relationship defined by the user. The following restrictions apply to defining the force-displacement curve:

l Force-Displacement Curve has no limitation on the number of data.

l At least one data point must be defined on both the positive and negative sides, and the numbers of data on the positive and negative sides must be identical.

l The initial value must be set to (0,0).

l No identical values can be used for Displacements, and the force-displacement data are arranged in reference to the displacements.

l The signs of force and displacement must be the same at all times.

l A negative slope is not permitted in Force-Displacement Curve. As such, the forces must gradually increase on the positive side and decrease on the negative side. No fluctuation is permitted.

Unloading Stiffness Parameter, β   

 The stiffness at unloading on the (+) and (-) sides is computed as follows. When β=0 , the unloading stiffness becomes the same as the elastic stiffness.

 

where, D1(+), D1(-) : Yielding displacements on (+) & (-) sides

D1max(+), D1max(-): Maximum displacements on (+) & (-) sides (Replace with the yielding displacement when yielding has not occurred.)

β: Unloading stiffness parameter (0≤ β ≤1)

 

(3)   Rules for Hysteresis of Multi-Linear Elastic Type

1.   In the case of │Dmax│<D1, the curve becomes linear elastic, which retains the elastic slope, K₀, passing though the origin.

2.     When D first exceeds D1(+) or exceeds the maximum D up to the present, the curve follows the skeleton curve.

3.     When the force is unloaded at the state, D1(+) < D or D<D1(-), the curve follows the unloading stiffness at a slope of Kr(+) or Kr(-).

4.     D moves towards the D_max on the opposite side when the sign of the force changes in the process of unloading. If the opposite side has not yielded, the yielding point becomes the maximum displacement.

(1) Overview of Hysteresis

Multi-Linear Plastic Pivot Type Hysteresis (Pivot Hysteresis hereafter) is a multi-linear stiffness degradation model proposed by R. K. Dowell, F. Seible & E. L. Wilson(1998). Pivot Hysteresis uses multiple pivot points to control the nonlinear relationship of stress-strain or moment-rotation of reinforced concrete members. Thus, this model can accurately depict the stiffness degradation and the pinching effect when unloading takes place.

The curve can be symmetrically or unsymmetrically defined. The types of corresponding elements include lumped hinge, distributed hinge, spring and truss elements.

(2) Definition of Skeleton Curve

n Force-Displacement Curve

The skeleton curve is defined by the force-displacement relationship defined by the user. The following restrictions apply to defining the force-displacement curve:

l Force-Displacement Curve has no limitation on the number of data.

l At least one data point must be defined on both the positive and negative sides, and the numbers of data on the positive and negative sides must be identical.

l The initial value must be set to (0,0).

l No identical values can be used for Displacements, and the force-displacement data are arranged in reference to the displacements.  

l The signs of force and displacement must be the same at all times.

l A negative slope is not permitted in Force-Displacement Curve except for the final value. As such, the forces must gradually increase on the positive side and decrease on the negative side except for the last points on the curve. No fluctuation is permitted.

n Primary Pivot Point

The Primary Pivot Points, P₁ and P₃ represent the points towards which the unloading curves are oriented in the Q₁ and Q₃ zones. The Primary Pivot Points, P₁ and P₃ control the degradation of the unloading stiffness caused by the change in deformation or displacement. P₁ and P₃ are located along the extended lines of the initial stiffness on the (+) and (-) sides, which are defined by the yield strengths, Fy⁽⁺⁾ and Fy^(-) and Scale Factors, α1 and α2.

α1	: Scale Factor used to define the pivot point, P1 when unloading from the Q1 side (α1 ≥1)
α2	: Scale Factor used to define the pivot point, P3 when unloading from the Q3 side (α2≥1)

The locations of the Primary Pivot Points, P₁ and P₃ move to P₁^* and P₃^* after yielding respectively, whenever the maximum displacement point is renewed by the Initial Stiffness Softening Factor, η. However, when η=0, the locations of the Primary Pivot Points, P₁ and P₃ remain unchanged.

Primary Pivot Point

n Pinching Pivot Point

 

The Pinching Pivot Points, PP₂ and PP₄ represent the points towards which the unloading curves are oriented in the Q1 and Q3 zones after the restoring force exceeds 0. PP₂ and PP₄ are located on the skeleton curve in the elastic zone on the (+) and (-) sides, which are defined by the yield strengths of the initial stiffness, Fy(+) and Fy(-) and Scale Factors, β1 and β2.

β1	: Scale Factor used to define the pivot point, PP2 when loading on the Q2 side (0< ≤1)
β2	: Scale Factor used to define the pivot point, PP4 when loading on the Q4 side (0< ≤1)

The locations of the Pinching Pivot Points, PP₂ and PP₄ after yielding will move to PP₂^*and PP₄^* respectively, whenever the maximum displacement point is renewed by the Initial Stiffness Softening Factor, η. However, when η =0, the Pinching Pivot Points, PP₂ and PP₄ remain unchanged.

Pinching Pivot Point

n Initial Stiffness Softening Factor : η

η is an initial stiffness softening factor used to control the initial stiffness degradation after yielding. After yielding, the Primary Pivot Points, P₁ and P₃ are relocated to P₁^*and P₃^*, which are located on the lines extended from the maximum displacement points on the (+) and (-) sides respectively. P₁^*and P₃^* are defined by Fy⁽⁺⁾ and Fy^(-), Scale Factors, and , and the initial stiffness softening factor, η.

In addition, the Pinching Pivot Points,PP₂ and PP₄ move to PP₂^* and PP₄^* respectively. PP2* (or PP4*) is defined by the intersection point of the straight line passing through P1* and the origin (or P3* and the origin) and the straight line connecting PP2 (or PP4) to the maximum displacement point on (-) side (or (+) side).

 

n Renewal of Scale Factors, β1 and β2

The Pinching Pivot Point Scale Factors, β1 and β2 are renewed after yielding under the conditions below.

Initial Stiffness Softening Factor

Renewal of Scale Factors, β1 and β2

 

 

(3)   Rules for Hysteresis of Multi-Linear Plastic Pivot Type

1.   In the case of │Dmax│<D1, the curve becomes linear elastic, which retains the elastic slope, K_o passing the origin. (Rule: 0)

2.   i) The curve follows the skeleton curve when the displacement exceeds for the first time. (Rule: 1)

 ii) When unloading takes place on this straight line, the curve is directed towards P₁ or P_3. (Rule: 2)

 iii) In the case of reloading before the restoring force reaches 0, the curve continues to follow the same unloading straight line. (Rule: 3) If it reaches the skeleton curve, it follows along the skeleton curve. (Rule: 4)

iv) When the restoring force exceeds 0, the curve is directed towards PP₂ or PP₄. (Rule: 5)

v) When PP₂ or PP₄ is exceeded and yielding has not occurred, the curve moves along the straight line of the elastic slope. (Rule: 6) When yielding takes place due to large deformation, the curve moves along the skeleton curve. (Rule: 7)

3.   i) When unloading takes place on the skeleton curve after both sides have yielded, the curve moves towards P₁ or P_3.

 (Rule: 8) However, it is directed towards the renewed P₁^* or P₃^* if η is not equal to 0.

ii) If the restoring force exceeds 0, the curve is directed towards PP₂ or PP₄. However, it is directed towards the renewed PP₂^*or PP₄^* if η is not equal to 0. (Rule: 9)  

iii) If unloading takes place before reaching PP₂ or PP₄, the curve moves along the straight line passing through the unloading point and P4 (or P2). (Rule:10) If reloading takes place before the restoring force reaches back to 0, the curve moves back towards P3 (or P1). (Rule: 11)

iv) If the restoring force exceeds 0, the curve moves along the line connecting the point of zero restoring force to P3 (or P1). (Rule: 12) When the curve intersects with a line connecting PP₂ (or PP₄) and (or ), it is directed towards  (or ). (Rule: 13)

P-M Multi-Curve Type allows for bilinear and multilinear plasticity. Strain hardening can be isotropic, kinematic or mixed. The moment-curvature relationship can either be symmetric or non-symmetric with respect to the sign of the curvature. Whether it is symmetric or non-symmetric, entire moment-curvature curve should be entered. The first data point corresponds to negative rupture and the last data point corresponds to positive rupture. One data point – the zero point – must be at the origin. A different number of data points can be used for the positive and negative sections of the curve.

Bending moment and curvature relationship can depend on the axial force and the dependence can be different in tension and in compression. The input for the bending moment-curvature curve consists of bending moment-curvature curves for different levels of axial force. Note that the number of axial forces must be two at least.

To obtain the bending moment-curvature curve for a level of axial force not input, interpolation is used. This interpolation is performed, not on the bending moment-curvature curves, but on the bending moment-plastic curvature curves; the bending moment-plastic curvature curves are automatically calculated from the bending moment-curvature curves.

The moment-curvature relationship can either be symmetric or non-symmetric. Whether it is symmetric or non-symmetric, entire moment-curvature curve should be entered.

Strain hardening can be isotropic, kinematic or mixed.

Each moment component will follow 1-D von-mises model.

  

Under the specific axial force, the yield moment can be defined as the followings.