Trac model formulation

Trac load data was obtained from Tyre Stress Seal, a software package developed by the Council for Scientic and Industrial Research (CSIR) of South Africa. Tyre Stress Seal was compiled from research conducted on dierent wheel loads with the stress in motion (SIM) system. The inter-active software allows user dened functions such as:

a) Selection of desired wheel size.

b) Specifying the vertical load and tyre ination pressure (TiP) of selected wheel. c) Dening the output display format (block mode, contours or interpolated data). d) Exporting the 3D stress distributions to an output le.

Five dierent wheel sizes at various vertical loads and TiPs (Table 3.8) were exported from Tyre Stress Seal and formed the bases of the trac formulation for the FE seal models. Two of the ve wheel sizes were heavy vehicle (HV) wheel loads and the

Table 3.8: Wheel size and load combination as derived from Tyre Stress Seal.

Size Class Vertical load [kN] TiP [kPa]

12R22.5 HV 15 20 35 40 50 520 620 720 800 950 1000 315-80R22.5 HV 20 35 40 50 75 100 520 620 720 800 950 1000 265-65R17 LV 5.4 190 245-70R16C LV 5.4 190 195R15C LV 6.0 360 84

remainder were passenger vehicles, also referred to as light vehicle (LV) wheel loads. The trac model therefore makes provision to address the discrepancies in damage which arise in applying dierent wheel loads to the seal structure. Tyre Stress Seal provides only one vertical load and one corresponding TiP for the three LVs. Each vertical load and TiP combination as presented in Table 3.8, was related from its 3D macro-scale output to a 2D meso-scale load function. The procedure is illustrated in Figure 3.16 and summarised as follows:

a) Export Tyre Stress Seal data in block format (raw data format).

b) Select the two centre blocks (Pin 11 & Pin 12) to represent the contact stress at centre of the tyre and select the two edge blocks (Pin 7 & Pin 16) to represent the contact stress underneath the side-wall of the tyre.

c) Adjust the measured pin stress data to satisfy pneumatic tyre theory and convert the stress signals into 2D force functions.

Figure 3.16: Typical SIM output and 2D force functions adjustment procedure.

The latter process resulted in losing the eect of the third dimension, therefore, em- phasis was placed on translating the 3D stress situation to a 2D force system for the longitudinal direction i.e. the direction of travel. Forces which are present in the longitudinal direction are the vertical force and the longitudinal forces. From the onset of the trac model formulation it was decided to limit the longitudinal inu- encing factors to: free rolling, acceleration and braking wheels, axle speed variation and vertical inclines. No other factors such as wheel turn, wheel spin, wheel lock or any damping eects were considered. Although cognisant of the eects which these inuencing factors introduce, especially the contribution of the tread rubber prop- erties to the longitudinal shear force, the objective was to obtain general structural seal responses with various classes of trac in the 2D domain.

Transforming the complex pin stress signals to simplied functions, while adhering to pneumatic tyre theory in the longitudinal direction, encompassed the adjustment

3. METHODOLOGY

process. During this process the pin stress signals were converted to 2D force func- tions as dened in Equation 3.32.

Fz,y,x = σz,y,x× aggregate nominal size × unit width (3.32)

where:

Fz,y,x =force functions

σz,y,x =pin stress

unit width = 1 mm

The shape of longitudinal force function Fx was replaced with parabola segments.

The number of segments varied between two and three and depended on the original pin stress function. The roots of each segment were inherited from the original Fx

and the integral of the newly segmented Fx resulted in the rolling resistance Fr. In

some cases the parabola peak values were slightly adjusted and the vertical force Fz

was skewed with an iterative process to satisfy Equation 3.33. Although the trans- verse force Fy had no eect in a vertical-longitudinal 2D simulation, it followed a

similar transformation procedure than Fz.

Fr =

Fz(∆x)

Rs (3.33)

where:

Rs =undeected tyre radius

∆x = horizontal oset of the vertical resultant for a free rolling wheel

In general the longitudinal force is of little importance on thicker surface layers such as asphalt or concrete, but in the case of a thin seal surfaces it has a signi- cant eect on the response of the layer. Longitudinal forces were therefore further adjusted for driven (acceleration) and brake wheels. Based on the work of Moore (1975), an assumption was made that the brake force coecient and relative traction increased linear towards peak values of 1.0 at a slip ratio of 0.2. According to Miller et al. (2001) general wheel slip ranges between 0% and 3%, therefore a slip value of 3%was selected to account for the braking force Fb or traction force Fdas presented

in Equation 3.34. These forces were superimposed on the existing longitudinal force, adjusting the overall Fx.

Fb,d

Fz

= msb,d (3.34)

where:

m =linear increase rate sb,d =brake or drive slip ratio

86

In the case of an incline (φ) the horizontal component of the translated vertical force (Fzsin(φ)) was included to account for the total relative traction or brake force

coecient as dened in Equation 3.35. Ftotal Fz = Fb,d Fz + Fzsin(φ) Fzcos(φ) (3.35)

It is acknowledged that the tyre road contact stress is a complex scenario and the approach followed in this study is a simple relation thereof. However, the quality of the SIM data provided satisfactory results during the model verication process. A summary of the peak vertical and longitudinal forces per wheel size and load combination is presented in Appendix B.