System: 1: Positions Between the Wheels and Under a Wheel
Layer Young’s Poisson’s Shear Spring
Number Thickness (m) Modulus (Pa) Ratio Compliance (m3/N)
1 0.300 5.000E+09 0.35 0.000E+00
2 0.200 1.000E+09 0.35 0.000E+00
3 0.150 8.000E+08 0.35 0.000E+00
4 2.000E+08 0.35
Load Normal Shear Radius of Load Position Load Position Shear
Number Stress (Pa) Stress (Pa) Loaded Area (m) X (m) Y (m) Direction (°)
1 5.774E+05 0.000E+00 1.050E-01 0.000E+00 -1.575E-01 0.000E+00
2 5.774E+05 0.000E+00 1.050E-01 0.000E+00 1.575E-01 0.000E+00
BISAR 3.0 – Detailed Report Example Project
System: 1: Positions Between the Wheels and Under a Wheel
Position Number: 1 Layer Number: 1 X Coord (m): 0.000E+00 Y Coord (m): 0.000E+00 Z Coord (m): 1.500E-01
LoadDistance to Displacements (m) Stresses (Pa) Strains
No. Load Axis (m) Theta (°) Radial Tangential Vertical Radial Tangential Vertical Rad./Tang. Rad./Vert. Tang./Vert. Radial Tangential Vertical Rad./Tang. Rad./Vert.Tang./Vert.
1 1.575E-01 9.001E+01 9.993E-07 0.000E+00 4.950E-05 -4.723E+04 -7.377E+03 -6.449E+04 0.000E+00 -7.877E+04 0.000E+00 -4.415E-06 6.345E-06 -9.075E-06 0.000E+00 -2.127E-05 0.000E+00 2 1.575E-01 -9.001E+01 9.993E-07 0.000E+00 4.950E-05 -4.723E+04 -7.377E+03 -6.449E+04 0.000E+00 -7.877E+04 0.000E+00 -4.415E-06 6.345E-06 -9.075E-06 0.000E+00 -2.127E-05 0.000E+00
Total Stresses (Pa) XX: -1.475E+04 YY: -9.446E+04 ZZ: -1.290E+05 YZ: 0.000E+00 XZ: 0.000E+00 XY: 0.000E+00 Total Strains XX: 1.269E-05 YY: -8.831E-06 ZZ: -1.815E-05 YZ: 0.000E+00 XZ: 0.000E+00 XY: 0.000E+00 Total Displacements (m) UX: 0.000E+00 UY: 0.000E+00 UZ: 9.900E-05
Principal Values and Directions of Total Stresses and Strains
Normal Normal Shear Shear X Y Z
Stress (Pa) Strain Stress (Pa) Strain Comp. Comp. Comp.
Maximum: -1.475E+04 1.269E-05 1.0000 0.0000 0.0000
Minimax: -9.446E+04 -8.831E-06 0.0000 1.0000 0.0000
Minimum: -1.290E+05 -1.815E-05 0.0000 0.0000 1.0000
Maximum: 5.711E+04 1.542E-05 0.7071 0.0000 -0.7071
-7.186E+04 0.7071 0.0000 0.7071
MiniMax: 3.985E+04 1.076E-05 0.7071 -0.7071 0.0000
-5.461E+04 0.7071 0.7071 0.0000
Minimum: 1.726E+04 4.660E-06 0.0000 0.7071 -0.7071
-1.117E+05 0.0000 0.7071 0.7071
Strain Energy (J): 1.494E+00 Strain Energy of Distortion (J): 9.266E-01
Calculated: 02-Dec-1997 13:18:52 Print Date: 02-Dec-1997 Page: 2
12. Error Messages
The following error messages apply:
Entry of Loads in Stress and Load Mode
Field Actual Error Message
Vertical Stress The Vertical Stress Value must be greater than 0 and less than 10000 Vertical Load The Vertical Load Value must be greater than 0 and less than 10000 X Coordinate The X Coordinate Value should be between -99.9999 and 999.9999 Y Coordinate The Y Coordinate Value should be between -99.9999 and 999.9999 Horizontal Stress The Horizontal Stress Value should be between 0 and 9999.999 Shear Direction The Shear Direction Value should be between 0 and 999.9
Entry of Loads in Load and Radius Mode
Field Actual Error Message
Vertical Load The Vertical Load Value must be greater than 0 and less than 10000 Radius The Radius Value must be greater than 0 and less than 1000 X Coordinate The X Coordinate Value should be between -99.9999 and 999.9999 Y Coordinate The Y Coordinate Value should be between -99.9999 and 999.9999 Horizontal Load The Horizontal Load Value should be between 0 and 9999.999 Shear Direction The Shear Direction Value should be between 0 and 999.9
Entry of Loads in Stress and Radius Mode
Field Actual Error Message
Vertical Stress The Vertical Stress Value must be greater than 0 and less than 10000 Radius The Radius Value must be greater than 0 and less than 1000
X Coordinate The X Coordinate Value should be between -99.9999 and 999.9999 Y Coordinate The Y Coordinate Value should be between -99.9999 and 999.9999 Horizontal Stress The Horizontal Stress Value should be between 0 and 9999.999 Shear Direction The Shear Direction Value should be between 0 and 999.9
Validation of Layers
Field Actual Error Message
Thickness The Thickness Value should be greater than 0 and less than 100 Modulus of Elasticity The Modulus of Elasticity Value should be greater than 0 and less than
1E20
Poisson’s Ratio The Poisson’s Ratio Value should be greater than 0 and less than 1 Spring Compliance The Spring Compliance Value should be between 0 and 1E+10
Validation of Positions
Field Actual Error Message
Appendix 1
BISAR Calculations with Slip between Layers (Shear Spring Compliance Concept)
A1.1 Theoretical Background
One of the possibilities of BISAR is the capability to account for (full or partial) slip. This type of calculation is made with aid of the shear spring compliance, a parameter which should not be confused with the well-known friction coefficient.
Within BISAR, it is not possible to use the ‘classic’ friction coefficient, because its value differs for static and dynamic conditions. Use of this parameters would require BISAR to be able to cope with discontinuities (step functions). The mathematics behind the BISAR model, however, assumes continuous relations for all its parameters.
To solve this problem, the designers of BISAR have developed the concept of shear spring compliance. In this approach the interface between two (horizontal) pavement layers is represented by an infinite thin inter-layer of which the strength is described by means of a spring compliance. Physically it assumes that the shear stresses at the interface cause a relative horizontal displacement of the two layers, which is proportional to the stresses acting at the interface.
The physical definition of the standard shear spring compliance, AK, is given by relative horizontal displacement of layers
AK = [m3/N]
stresses acting at the interface
which relation is treated mathematically through the parameter α, defined as
in which
a = radius of the load, m
E = modulus of the layer above the interface, Pa ν = Poisson’s Ratio of that layer
α = friction parameter, with 0 ≤ α ≤1
(α= 0 means full friction, α= 1 means complete slip).
The reduced shear spring compliance,ALK expressed in m, is defined as
One of the values of AK and ALK is input for the BISAR program. The value of α, called interface friction, used in all computations is derived from the input (either AK or ALK).
The friction parameter αshould not be considered as a classic friction coefficient. The interface friction parameter depends on the diameter of the applied load and is therefore not a pure material property. Within calculations with loads of different diameters, different values for α apply for one ALK or AK value as physical characteristic for a specific layer interface. It is
AK
therefore formally not correct to express a percentage of slip as a proportion of the spring compliance for full slip.
On the other hand, it remains difficult to assign or justify a specific value for AK (ALK).
Therefore, it is recommended to always perform a series of calculations with different values for ALK as a kind of sensitivity analysis. A numerical variation in ALK from zero to, say, 100 times the radius of the loaded area covers the range from full friction to (practically) full slip (α= 0.99).
The physical meaning (see above definition of AK) of such input values should be considered in connection with the moduli of the layers in the structure and the corresponding shear spring compliance (AK) values, with aid of the relation
1 + ν AK = ALK .
E
Appendix 2
The radial direction within fixed and local co-ordinate systems
The input for BISAR is expressed in terms of a fixed Cartesian co-ordinate system (X,Y,Z). The actual BISAR calculations, however, to determine the response of a load at a certain position in terms of resulting stresses, strains and displacements are carried out in a local cylindrical co-ordinate system (r,θ,z) for each load. An outline of both systems is given in Figure 2-1. L is the centre of a load in the X-Y plane at the top of the structure and is the origin of the local cylindrical system. P corresponds to an arbitrary position in the structure, with P’ as projection of P on the surface plane.