# RD-V: 0033 Spring (TYPE13)

Spring force as function of elongation for different stiffness formulations.

The subject of this study is to verify the behavior of the spring type beam element with different stiffness formulations, to verify the coupling between shear and bending and finally, different viscous formulations.

## Input Files

Before you begin, copy the file(s) used in this problem to your working directory.

## Model Description with Different Translational Stiffness Formulations

Units: Mg, s, mm, MPa

The model contains 5 spring elements with the same length of 100 mm and the following properties:
• Elastic linear stiffness
• K=2 N/mm
• Nonlinear elastic stiffness
• Force versus displacement function:
Elongation
Force
-20
-20
-1
-10
0
0
1
10
20
20
• Linear stiffness K= 50N/mm used for transition
• Nonlinear plastic stiffness with isotropic hardening (H=1)
• Same force versus displacement function and same linear stiffness
• Plastic behavior with H=1
• Nonlinear plastic stiffness with uncoupled hardening (H=2)
• Same force versus displacement function and same linear stiffness
• Plastic behavior with H=2
• Linear stiffness K= 50N/mm used for transition
• Same force versus displacement function and same linear stiffness
Elongation
Force
-10
-20
-5
-1
0
0
5
1
10
20
• Plastic behavior with H=6
• Linear stiffness K= 50N/mm used for transition

Imposed displacement is defined on one end of the spring element to model positive and negative (+/- 20mm) with constant velocity of 200mm/s for the 3 directions, X, Y and Z. The rotational degrees of freedom at this end of the spring are fixed to see only shear behavior.

The other end of the spring element is clamped, and the 6 degrees of freedom are fixed (/BCS).

### Results

Computation results for the force versus elongation for direction Y. The results are identical for the same loading in direction X and direction Z.
Computation results for the force versus time.
Theoretical value for the force according to time.
Table 1. Theoretical Results
Time Elastic Linear Stiffness Nonlinear Elastic Nonlinear Plastic Stiffness with Isotropic Hardening (H=1) Nonlinear Plastic Stiffness with Isotropic Hardening (H=2) Nonlinear Plastic Stiffness with Nonlinear Unloading (H=6)
t=0.1s 40 20 20 20 20
t=0.2s 0 0 -30.10526316 0 -20
t=0.3s -40 -20 -40.63157895 -20 -30.52631579
t=0.4s 0 0 50.30249307 0 9.473684211

## Model Description with Different Rotational Stiffness Formulations

Units: Mg, s, mm, MPa

The same analysis is done for each rotational degree of freedom. The model contains 5 spring elements with the same length of 100 mm and different properties.
• Elastic linear stiffness
• Nonlinear elastic stiffness
• Moment versus rotation function:
Elongation
Force
-3.141592654
-20
-.157079633
-10
0
0
.157079633
10
3.141592654
20
• Linear stiffness K= 318.3098862 Nmm/rad used for transition
• Nonlinear plastic stiffness with isotropic hardening (H=1)
• Same moment versus rotation function and same linear stiffness
• Plastic behavior with H=1
• Linear stiffness K= 318.3098862 Nmm/rad used for transition
• Nonlinear plastic stiffness with uncoupled hardening (H=2)
• Same moment versus rotation function and same linear stiffness
• Plastic behavior with H=2
• Linear stiffness K= 318.3098862 Nmm/rad used for transition
• Same force versus rotation function and same linear stiffness
Rotation
Force
-1.570796327
-20
-.785398163
-1
0
0
.785398163
1
1.570796327
20
• Plastic behavior with H=6
• Linear stiffness K= 318.3098862 Nmm/rad used for transition

Imposed rotation is defined on one end of the spring element to model rotation (+/- $\pi$ rad) with constant rotational velocity of 10. $\pi$ rad /s. The displacement Y and Z at this end of the spring are fixed to avoid shear deformation and see only bending.

The other end of the spring element is clamped.

### Results

Computation results for the moment versus rotation for the direction X. The results are identical for directions Y and Z.
Computation results for the moment versus time.
Theoretical value for the moment according to time.
Table 2. Theoretical Results
Time Elastic Linear Stiffness Nonlinear Elastic Nonlinear Plastic Stiffness with Isotropic Hardening (H=1) Nonlinear Plastic Stiffness with Isotropic Hardening (H=2) Nonlinear Plastic Stiffness with Nonlinear Unloading (H=6)
t=0.1s 40 20 20 20 20
t=0.2s 0 0 -30.10526316 0 -20
t=0.3s -40 -20 -40.63157895 -20 -30.52631579
t=0.4s 0 0 50.30249307 0 9.473684211

## Model Description for Coupling Between Bending and Shear

The model contains 2 spring elements with the same length of 100 mm and different properties:
• Elastic linear stiffness
• K= 20 N/mm
• Nonlinear elastic stiffness
• K= 5000 Nmm/rd

Imposed displacement is defined on one end of the spring element with a value of Y=+/-20mm for the first spring and Z=+/-20mm for the second spring. The rotational degrees of freedom at this end of the spring are free to see shear and bending behavior.

The other end of the spring element is clamped, and the 6 degrees of freedom are fixed (/BCS).

### Results

The sum of the bending displacement and the displacement corresponding to the bending rotation (measured in the middle of the spring) returns the same results as the imposed displacement (error less than 0.2%).

The force from the bending moment and the shear force are similar to the reaction force.

The bending moment (measured at the middle of the spring) is also very close to the bending reaction force at the anchorage point.

## Model Description for Different Translational Viscous Formulation

Units: Mg, s, mm, MPa

The model contains 5 spring elements with the same length of 100 mm with the same stiffness and different viscous formulation:
• Nonlinear elastic stiffness
• Force versus displacement function:
Elongation
Force
-5
-20
-1
-10
0
0
1
10
5
20
• Elongation rate dependency
• Linear value
• Stiffness scale factor
• Force versus elongation rate scale factor function:
Elongation
Force
-10
-0.1
0
0
10
0.1
• Damping versus elongation rate function:
Elongation
Damping
-10
-1
0
0
10
1
• Damping versus elongation rate function with higher velocity

The imposed displacement function abscissa (time) is scaled by 0.5 to double the constant loading velocity.

Imposed displacement is defined on one end of the spring element to model tension and compression (displacement +/- 5(mm) with constant velocity of 50mm/s (with abscissa scale factor 1.0) or 100mm/s (with abscissa scale factor 0.5). The rotational degrees of freedom at this end of the spring are fixed to see only shear behavior.

The other end of the spring element is clamped.

### Results

Computation results for the force versus time for X translational direction. The results are identical for the direction Y and Z.
Theoretical value for the force according to time.
Table 3. Theoretical Results
Time Nonlinear Elastic Nonlinear Elastic + C Nonlinear Elastic + Velocity-based Force Function Scale Nonlinear Elastic + Velocity-based Damping Function (200 mm/s) Nonlinear Elastic + Velocity-based Damping Function (400 mm/s)
t=0.1s (V+) 20 25 30 25 30
t=0.1s (V-) 20 15 30 15 10
t=0.2s (V-) 0 -5 0 -5 -10
t=0.3s (V-) -20 -25 -30 -25 -30
t=0.3s (V+) -20 -15 -30 -15 -10
t=0.4s (V+) 0 5 0 5 10

## Model Description for Different Rotational Viscous Formulation

Units: Mg, s, mm, MPa

The model contains 5 spring elements with the same length of 100 mm with the same stiffness and different viscous formulation:
• Nonlinear elastic stiffness
• Moment versus rotation function:
Rotation
Moment
-3.141592654
-20
-.157079633
-10
0
0
.157079633
10
3.141592654
20
• Elongation rate dependency
• Linear value
• Stiffness scale factor
• Force versus rotation rate scale factor function:
Elongation
Force
-10
-.636619772
0
0
10
.636619772
• Damping versus rotation rate function:
Elongation
Damping
-10
-6.366197724
0
0
10
6.366197724
• Damping versus rotation rate function with higher velocity

The imposed displacement function abscissa (time) is scaled by 0.5 to double the constant loading velocity.

Imposed rotation is defined on one end of the spring element to model rotation (displacement +/- $\pi$ rd) with constant velocity of 10. $\pi$ rd/s (with abscissa scale factor 1.0) or 20. $\pi$ rd/s (with abscissa scale factor 0.5). The displacement Y and Z at this end of the spring are fixed to avoid shear deformation and see only bending.

The other end of the spring element is clamped.

### Results

Computation results for the moment versus time for X rotational direction. The results are the same for Y and Z rotational directions.
Theoretical value for the moment according to time.
Table 4. Theoretical Results
Time Nonlinear Elastic Nonlinear Elastic + C Nonlinear Elastic + Rotational Velocity-based Force Function Scale Nonlinear Elastic + Rotational Velocity-based Damping Function (200 mm/s) Nonlinear Elastic + Velocity-based Damping Function (400 mm/s)
t=0.1s (V+) 20 40 60 40 60
t=0.1s (V-) 20 0 60 0 -20
t=0.2s (V-) 0 -20 0 -20 -40
t=0.3s (V-) -20 -40 60 -40 -60
t=0.3s (V+) -20 0 60 0 20
t=0.4s (V+) 0 20 0 20 40

#### Conclusion

The Radioss computations returns results very close to the theoretical values with the expected behavior for the elastic or plastic behavior.

The coupling between the shear and bending returns the expected results.

The different option to model the translation or rotation rate effect returns the expected results.