/PROP/TYPE25 (SPR_AXI)

Description

Figure 1. Tension/Compression	Figure 2. Shear (Radial)
Figure 3. Torsion	Figure 4. Bend (Radial)

Format

Definitions

Comments

1. The spring's X direction is defined using nodes N1 and N2 of the spring.
   • If the node of the spring N3 is defined, the spring's Y direction is defined using nodes N1 and N3 of the spring. N3, N2, and N1 should not be in a line. The Z direction is:(1)
     $$\mathbf{Z}=\mathbf{X}\mathbf{\Lambda }{\mathbf{Y}}^{\prime }$$
   • If node N3 is not defined in the element input, and skew system is defined in the property input, the Z direction is:(2)
     $$\mathbf{Z}=\mathbf{X}\mathbf{\Lambda }{\mathbf{Y}}_{\mathit{skew}}$$
   • If neither node N3 nor skew system are defined in input, the Z direction is:(3)
     $$\mathbf{Z}=\mathbf{X}\mathbf{\Lambda }{\mathbf{Y}}_{\mathit{global}}$$

     
     
     Figure 5.

   • Finally, Y direction is found as:(4)
     $$\mathbf{Y}=\mathbf{Z}\mathbf{\Lambda }\mathbf{X}$$
2. In case of I_leng =0, the force in the spring is computed as:

   Linear spring:

   $\text{F}\left(\delta \right)=K_i{\delta }^i+C_i{\dot{\delta }}^i$ with $i$ =1, 2

   $\text{M}\left(\theta \right)=K_i{\theta }^i+C_i{\dot{\theta }}^i$ with $i$ = 3, 4

   Nonlinear spring:

   $\text{F}\left(\delta \right)=\text{f}\left(\frac{{\delta }^i}{Ascal{e}_i}\right)\left[A_i+B_i\ln \left|\frac{{\dot{\delta }}^i}{D_i}\right|+E_i\text{g}\left(\frac{{\dot{\delta }}^i}{F_i}\right)\right]+C_i{\dot{\delta }}^i+Hscal{e}_i\text{h}\left(\frac{{\dot{\delta }}^i}{F_i}\right)$ with i= 1, 2

   $\text{M}\left(\theta \right)=\text{f}\left(\frac{{\theta }^i}{Ascal{e}_i}\right)\left[A_i+B_i\ln \left|\frac{{\dot{\theta }}^i}{D_i}\right|+E_i\text{g}\left(\frac{{\dot{\theta }}^i}{F_i}\right)\right]+C_i{\dot{\theta }}^i+Hscal{e}_i\text{h}\left(\frac{{\dot{\theta }}^i}{F_i}\right)$ with i= 3, 4

   with $-l_0<\delta <+\infty$

   Note:

   • Here, ${\delta }^i$ (with $-l_0<{\delta }^i<+\infty$ ) is the difference between the current length $l$ and the initial length $l_0$ of the spring element for corresponding translational DOF.
   • ${\theta }^i$ is the relative angle for corresponding rotational DOF in radians.
   • For linear springs, $\text{f}\left(\delta \right)$ , $\text{g}\left(\dot{\delta }\right)$ and $\text{h}\left(\dot{\delta }\right)\hspace{0.17em}\text{f}\left(\theta \right)$ , $\text{g}\left(\dot{\theta }\right)$ , $\text{h}\left(\dot{\theta }\right)$ and are null functions and A_i, B_i, E_i, and Hscale_i are not taken into account.
   • If stiffness function $\text{f}\left(\delta \right)$ (or $\text{f}\left(\theta \right)$ ) is requested, then K is used as a slope for unloading only.
   • If K is lower than the maximum slope of function $\text{f}\left(\delta \right)$ (or $\text{f}\left(\theta \right)$ ) (K is not consistent with the maximum slope of the curve), K is set to the maximum slope of the curve.

   Figure 6. Linear Spring	Figure 7. Nonlinear Elastic Spring, Hi=0
   Figure 8. Nonlinear Elastic Plastic Spring, Hi=1	Figure 9. Nonlinear Elasto-Plastic Spring with Decoupling Hardening in Tension and Compression, Hi=2
   Figure 10. Nonlinear Elastic Plastic Spring 'kinematic' Hardening, Hi=4	Figure 11. Nonlinear Elasto-Plastic Spring with Nonlinear Unloading, Hi=5
   Figure 12. Nonlinear Elasto-Plastic Spring with Isotropic Hardening and Nonlinear Unloading, Hi=6	Figure 13. Nonlinear Spring with Elastic Hystersis, Hi=7

3. If I_leng = 1, all input are per unit length:
   • Spring mass = $M\cdot l_0$ Spring stiffness = $\frac{K}{l_0}$ Spring damping = $\frac{C}{l_0}$ Spring inertia = $I\cdot l_0$

     Where, $l_0$ is the reference spring length.

   • The force in the spring is computed as:

     Linear spring:

     $\text{F}\left(\delta \right)=K_i{\epsilon }^i+C_i{\dot{\epsilon }}^i$ with $i$ =1, 2

     $\text{M}\left(\theta \right)=K_i{\theta }^i+C_i{\dot{\theta }}^i$ with $i$ =3, 4

     Nonlinear spring:

     $\text{F}\left(\delta \right)=\text{f}\left(\frac{{\epsilon }^i}{Ascal{e}_i}\right)\left[A_i+B_i\ln \left|\frac{{\dot{\epsilon }}^i}{D_i}\right|+E_i\text{g}\left(\frac{{\dot{\epsilon }}^i}{F_i}\right)\right]+C_i{\dot{\epsilon }}^i+Hscal{e}_i\text{h}\left(\frac{{\dot{\epsilon }}^i}{F_i}\right)$ with $i$ =1, 2

     $\text{M}\left(\theta \right)=\text{f}\left(\frac{{\theta }^i}{Ascal{e}_i}\right)\left[A_i+B_i\ln \left|\frac{{\dot{\theta }}^i}{D_i}\right|+E_i\text{g}\left(\frac{{\dot{\theta }}^i}{F_i}\right)\right]+C_i{\dot{\theta }}^i+Hscal{e}_i\text{h}\left(\frac{{\dot{\theta }}^i}{F_i}\right)$ with $i$ =3, 4

     Where, ${\epsilon }^i$ is the engineering strain and defined as:(5)

     $${\epsilon }^i=\frac{{\delta }^i}{l_0}$$

     Force functions are given versus engineering strain and engineering strain rate.

     Failure criteria are defined with respect to strain. Input negative/positive failure limit should be related to initial length $l_0$

4. If hardening flag is 4, hardening is kinematic. Lower and upper yield curves are the same.
5. If hardening flag is 5, residual deformation is a function of maximum displacement:

   ${\delta }_{resid}^i=\mathrm{fct}\_I{D}_{3i}\left({\delta }_{\text{max}}^i\right)$ with $i$ =1,2

   ${\theta }_{resid}^i=\mathrm{fct}\_I{D}_{3i}\left({\theta }_{\text{max}}^i\right)$ with $i$ =3,4

6. The decoupled hardening (hardening flag H_i=2) and kinematic hardening (hardening flag H_i=4) models are only valid in axial direction (tension and torsion). They are not available in radial direction (shear and bending).
7. Failure criteria:
   • If the failure criteria are uni-directional I_fail=0, the spring fails as soon as one of the criteria is met in one direction:

     $\left|\frac{{\delta }^i}{{\delta }_{\text{max}}^i}\right|\ge 1$ or $\left|\frac{{\delta }^i}{{\delta }_{\text{min}}^i}\right|\le 1$ , with ${\delta }_{\text{max}}^i$ and ${\delta }_{\text{min}}^i$ being the failure limits in direction $i$ =1, 2

     $\left|\frac{{\theta }^i}{{\theta }^i{}_{\text{max}}}\right|\ge 1$ or $\left|\frac{{\theta }^i}{{\theta }^i{}_{\text{min}}}\right|\le 1$ , with ${\theta }_{\text{max}}^i$ and ${\theta }_{\text{min}}^i$ being the failure limits in direction $i$ =3, 4

     For each direction ${\delta }_{\text{min}}^i$ (or ${\theta }_{\text{min}}^i$ ) should be negative and ${\delta }_{\text{max}}^i$ (or ${\theta }_{\text{max}}^i$ ) should be positive. If the values are zero, then no failure will be taken into account.

   • If the failure criteria is multi-directional I_fail=1, the spring fails when the following criteria is fulfilled:(6)
     $${\sum _{i=1,2}{\alpha }_i\left(\frac{{\delta }^i}{{\delta }^i{}_{fail}}\right)}^{{\beta }_i}+{\sum _{i=3,4}{\alpha }_i\left(\frac{{\theta }^i}{{\theta }^i{}_{fail}}\right)}^{{\beta }_i}\ge 1$$

     For "old" displacement formulation (I_fail =0), the coefficients ${\alpha }_i$ and $\beta$ _i are equal to 1.0 and 2.0, respectively.

     New formulation ( I_fail2 > 0) allows to model velocity dependent failure limit for translational DOF:(7)

     $${\delta }^i{}_{fail}=\{\begin{array}{c}{\delta }_{\text{max}}^i+c_i\cdot {\left|\frac{v^i}{v_0}\right|}^{ni},if\left({\delta }^i>0\right)\\ {\delta }_{\text{min}}^i-c_i\cdot {\left|\frac{v^i}{v_0}\right|}^{ni},if\left({\delta }^i\le 0\right)\end{array}$$

     Where, ${\delta }_{\text{min}}^i$ or ${\delta }_{\text{max}}^i$ is the static failure limit in translational directions (Lines 5 and 8), and ${\nu }_0$ is the reference velocity.

   • Force and energy criteria are activated with I_fail2=2 or 3:(8)
     $${\delta }^i{}_{fail}={\delta }_{\text{max}}^i+c_i\cdot {\left|\frac{v^i}{v_0}\right|}^{ni},if\left({\delta }^i>0\right)$$

     In this case the displacement values are replaced by positive failure force or failure energy values.

     New formulation (I_fail2 =1) allows you to model velocity dependent failure limit for rotational DOF:(9)

     $${\theta }^i{}_{fail}=\{\begin{array}{c}{\theta }_{\text{max}}^i+c_i\cdot {\left|\frac{{\omega }^i}{{\omega }_0}\right|}^{ni},if\left({\theta }^i>0\right)\\ {\theta }_{\text{min}}^i-c_i\cdot {\left|\frac{{\omega }^i}{{\omega }_0}\right|}^{ni},if\left({\theta }^i\le 0\right)\end{array}$$

     Where, ${\theta }_{\text{min}}^i$ or ${\theta }_{\text{max}}^i$ is the static failure limit in rotational direction (Lines 11 and 14), and $\omega$ ₀ is the reference velocity.

     Moment and energy criteria are activated with I_fail2=2 or 3:(10)

     $${\theta }^i{}_{fail}={\theta }_{\text{max}}^i+c_i\cdot {\left|\frac{{\omega }^i}{{\omega }_0}\right|}^{ni},if\left({\theta }^i>0\right)$$

     In this case the rotation values are replaced by positive failure moment or failure energy values.

8. Spring activated and/or deactivated by sensor:
   • If sens_ID ≠ 0 and I_sflag = 0, the spring element is activated by the sens_ID.
   • If sens_ID ≠ 0 and I_sflag = 1, the spring element is deactivated by the sens_ID.
   • Spring elements with sensor activation or deactivation are mainly used for the pretension model.
   • If a sensor is used for activating or deactivating a spring, the reference length of the spring at sensor activation (or deactivation) is equal to the nodal distance at time =0.