Examples of hysteresis modeling

RIFLEX has several options for modeling of hysteresis. The following is a description of allowed combinations of input and expected behaviour.
Allowed combinations of cross-section input is described in the user manual as shown in table below. The examples below apply to cross sections of type advanced axisymmetric pipe.

Allowed combinations of RIFLEX input:

Case	IEJ	IPRESS	Allowed IMF-values	Data required
0	0	0	0	None
1a	1	0	0	EI, GA_s
1b	1	0	1	EI, MF
2	1	1	0	Not implemented
3	N	0	0 1	CURV (I), I=1,N BMOM (I), I=1,N
4	N1	N2	0	Not implemented

The test model is a 10 m cantilever beam which is fixed in all degrees of freedom at end 1, while a lateral time-varying force of 500N is applied at the free end. Results are plotted for the cross-section 2m from the fixed end.

1. Model 1, Hysteresis-InternalFrictionMoment, (IEJ=1, IMF=1)

This case corresponds to Case 1b in the RIFLEX user manual.
The following input is used for the cantilever beam:

Model 1

1.00E+05

10*EI (calculated by RIFLEX)

1.00E+06

Internal friction moment

2.00E+03

Generated hysteresis is selected in SIMA for the hysteresis option in the cross section input while a constant bending stiffness is applied.
The initial stiffness is equal to 10*EI (calculated by RIFLEX). When the moment passes the internal friction moment the stiffness is reduced to EI. A prescribed oscillating force resulting in a moment with amplitude 4.0kNm gives the response shown in the figure below.

Figure 1. Results for Model 1

2. Model 2 Hysteresis_Hardening, IEJ>1, IMF=1, 0.0<=HARPAR<=1.0

This will generate an elasto-plastic material model with isotropic/kinematic hardening, depending on the hardening parameter. Hardening parameter = 0.0 will result in isotropic hardening while 1.0 will give kinematic hardening.
The following input is used for the cantilever beam:

Model 2
Moment (Nm)	Curvature (1/m)
0	0
2500	0.02
5000	0.06

The plots below show base moment vs. curvature for isotropic and kinematic hardening.

Figure 2. Results for isotropic hardening

Figure 3. Results for kinematic hardening

3. Example of shear stiffness contribution

The effect of shear stiffness contribution may be included by adjusting the shear stiffness. The shear stiffness GAs is the product of the shear module and the shear area of the cross-section. If the shear stiffness is set to 0.0, the shear stiffness contribution is not included (equivalent to setting a very high shear stiffness).
The effect of including shear deformation can be shown using a simple cantilever beam with a vertical tip force at the free end.

Input for shear deformation test case:

\( EI (Nm^2) \)

5.29E+07

F (N)

1.00E+05

L (m)

0.6

Resulting tip deflection:

Gas (N)

Tip deflection (m)

2.32E+08

3.95E-04

1.36E-04

In this case the shear deformation represents about 65% of the total deformation when shear stiffness is included (deflection increases).