Load application

In this study, a combination of axial (pre-loading) and impact load were applied on the model.

In order to combine the impact load which is a transient load, and the compressive axial load which is rather static, in a single model, different possible numerical approaches are available in LS-DYNA.

One method is to use the implicit LS-DYNA solver for the pre-loading and the explicit solver for the transient impact loading. The implicit-explicit switching functionality in LS-DYNA can be conducted within one single input deck. The other method is to use solely the explicit LS-DYNA solver for the pre-loading and impact loading. To avoid oscillations, the pre-loading should be applied in a “ramp up” manner from zero to its final value and then extended beyond the termination time for stability. This method requires iteration to find a “ramp up” duration which minimises the response fluctuation, hence it is less efficient. Another option to prevent oscillation is to apply the displacement-controlled pre-loading within a sufficiently long time interval, making this method less computationally efficient as well. However, this approach is more stable. Alternatively, the explicit dynamic relaxation (DR) feature (CONTROL_DYNAMIC_RELAXATION) can be used to pre-load the system before onset of transient loading. In explicit DR, the computed nodal velocities are reduced each time step by

Chapter 5: Development and Validation of Numerical Model 155 the dynamic relation factor (default=0.995). Therefore, the solution undergoes a form of

damping during DR phase. The distortional kinetic energy (KE) is monitored. When this KE has been sufficiently reduced (i.e., the “convergence factor” has become sufficiently small), the dynamic relaxation analysis stops and the current state becomes the initial state of the transient analysis. This option is very straightforward and efficient.

In this study, the explicit analysis with dynamic relaxation option was used due to its convenience, simplicity and efficiency. The axial load was applied as nodal forces to the springs. Dynamic relaxation was invoked by setting parameter SIDR in the LS-DYNA define (load) curve function to 2. By completion of the dynamic relaxation phase, the model was stabilised and the axial stress in the column acted as initial conditions for the impact analysis.

Figure 5-6 shows the convergence behaviour (in terms of kinetic energy) for the dynamic relaxation run when the axial load was 400 kN.

Upon completion of dynamic relaxation phase, the transient impact load was applied on the model. In order to apply the impact load, the impact head was assigned with an initial velocity in the global Y direction that corresponded to the initial velocity of the impact head immediately before impacting the CFDST column in the experiments (i.e., 7.8 m/sec).

Figure 5-6: The convergence behaviour (in terms of kinetic energy) for the dynamic relaxation run in the numerical model simulating test series 4 (axial load=400 kN)

156 Chapter 5: Development and Validation of Numerical Model 5.2.7 Hourglassing energy

With the exception of triangular shells and tetrahedral solids, any under-integrated (single integration point) shell, thick shell or solid formulation will undergo hourglass modes which are nonphysical, zero energy modes of deformation that produce zero strain and no stress.

These undesirable hourglass modes will often have periods which are much shorter than that of the structure’s natural period, and can hence have a significant effect on the system’s behaviour. Therefore, it is important to inhibit the hourglass mode by implementing a suitable hourglass control algorithm which internally calculates and applies counteracting forces. The two main categories of hourglass control are stiffness and viscous forms, where elastic springs or viscous dampeners are selectively added to the element to prevent the formation of the anomalous modes. The hourglass deformation modes are orthogonal to elemental strains, and hence the work performed by hourglass control is not calculated in the global energy balance, which can lead to energy loss of system. Work performed by hourglass control can be measured within LS-DYNA and is considered insignificant if it is less than 5% of the total deformation energy (Hallquist, 2007).

It should be noted that the hourglass concerns can be entirely eliminated by switching to element formulations with fully integrated or selectively reduced (S/R) integration; however, it may involve a number of downsides including relatively high cost, instability in large deformation applications (negative volumes are much more likely), and the tendency to “shear lock” and thus behave too stiffly in applications where the element shape is poor (Hallquist, 2007). Refining the existing mesh without altering parameters can be used alternatively to control hourglassing. Nevertheless, this option is computationally expensive and may be complicated and time consuming. Furthermore, it may not adequately eliminate hourglassing.

The hourglass resisting force f_iα^k can be calculated using Equation 5-35:

f_iα^k = a_hh_iαΓ_αk Equation 5-35 where, ℎ_𝑖𝛼 depends on nodal coordinates, 𝛤_𝛼𝑘 depends on nodal velocity and ah can be calculated using

Equation 5-36,

a_h= Q_hgρ𝑣_𝑒^{2⁄ c3}₄^s Equation 5-36

Chapter 5: Development and Validation of Numerical Model 157 in which Qhg is a user defined constant, ve is the element volume, ρ is the density, and cs is the

material sound speed. A detailed description of each parameter can be found in (Hallquist, 2006).

In this study, the hourglass formulation 4 (Flanagan-Belytschko) with hourglass coefficient 0.03 was used to control hourglassing in the under-integrated elements. Type 4 hourglass control is stiffness-based formulation and the hourglass forces are proportional to the displacements contributing to the hourglass modes. These forces thus counteract the accumulated hourglass deformation. This hourglass formulation is recommended in (LS-DYNA Aerospace Working Group, 2013) for impacts with low to moderate velocity. Since there is an inherent stiffening effect of a stiffness-based hourglass control, the reduced hourglass coefficient (default value is 0.1) was employed to minimise this effect.