Appendix - Summary of Sinkage and Shearing Approaches

Bekker for penetration forces:(1)
p=( k c b + k φ ) D n C D ˙ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiabg2 da9maabmaabaWaaSaaaeaacaWGRbWaaSbaaSqaaiaadogaaeqaaaGc baGaamOyaaaacqGHRaWkcaWGRbWaaSbaaSqaaiabeA8aQbqabaaaki aawIcacaGLPaaacaWGebWaaWbaaSqabeaacaWGUbaaaOGaeyOeI0Ia am4qaiqadseagaGaaaaa@44BF@
Where:
  • D = the sinkage of the track link in the direction perpendicular to the link surface
  • p = pressure
  • b = track width
  • C = damping per unit area coefficient
  • kc, kΦ, n = empirically determined constants
Janosi shear force:(2)
τ=( c+ptanφ )( 1 e j k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaey ypa0ZaaeWaaeaacaWGJbGaey4kaSIaamiCaiGacshacaGGHbGaaiOB aiabeA8aQbGaayjkaiaawMcaamaabmaabaGaaGymaiabgkHiTiaadw gadaahaaWcbeqaaiabgkHiTmaalaaabaGaamOAaaqaaiaadUgaaaaa aaGccaGLOaGaayzkaaaaaa@48C6@
Where:
  • τ = shear stress
  • j = shear displacement
  • c = cohesion
  • Φ= angle of internal friction
  • k = empirically determined constant
According to the Janosi approach, the shear stress increases while increasing the shear displacement. The maximum shear stress is:(3)
τ max = c + p tan φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccqGH9aqpcaWGJbGaey4k aSIaamiCaiGacshacaGGHbGaaiOBaiabeA8aQbaa@4319@

However, if a value is above a certain value of the shear displacement, then the shear stress is decreased. This is due to the soil failure changing the soil parameters (c and Φ). It's recommended to use a simple approach, where the shear displacement affects the maximum shear stress and not the soil parameter. The term for the proposed approach is a modified Janosi approach.

(4)
τ max ={ c+ptanφ j< j max ( c+ptanφ ) e j max j k 1 j max j j u ( c+ptanφ )r j> j u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiGac2gacaGGHbGaaiiEaaqabaGccqGH9aqpdaGabaqaauaa baqadmaaaeaacaWGJbGaey4kaSIaamiCaiGacshacaGGHbGaaiOBai abeA8aQbqaaiaaywW7caaMf8UaaGzbVdqaaiaadQgacqGH8aapcaWG QbWaaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaakeaadaqadaqaai aadogacqGHRaWkcaWGWbGaciiDaiaacggacaGGUbGaeqOXdOgacaGL OaGaayzkaaGaamyzamaaCaaaleqabaWaaSaaaeaacaWGQbWaaSbaaW qaaiGac2gacaGGHbGaaiiEaaqabaWccqGHsislcaWGQbaabaGaam4A amaaBaaameaacaaIXaaabeaaaaaaaaGcbaaabaGaamOAamaaBaaale aaciGGTbGaaiyyaiaacIhaaeqaaOGaeyizImQaamOAaiabgsMiJkaa dQgadaWgaaWcbaGaamyDaaqabaaakeaadaqadaqaaiaadogacqGHRa WkcaWGWbGaciiDaiaacggacaGGUbGaeqOXdOgacaGLOaGaayzkaaGa aGjbVlaadkhaaeaaaeaacaWGQbGaeyOpa4JaamOAamaaBaaaleaaca WG1baabeaaaaaakiaawUhaaaaa@7A76@
where:
  • jmax = the maximum shear displacement which affects increasing the shear stress
  • ju = the ultimate shear displacement
  • there is no effect on the shear stress while increasing the shear displacement above this point, ju>jmax
  • k1 = constant; r = the maximum shear ratio, 1>r>0

For τmax to be a continuous function, the following relationship should be maintained:

(5)
r= e j max j u k 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCaiabg2 da9iaadwgadaahaaWcbeqaamaalaaabaGaamOAamaaBaaameaaciGG TbGaaiyyaiaacIhaaeqaaSGaeyOeI0IaamOAamaaBaaameaacaWG1b aabeaaaSqaaiaadUgadaWgaaadbaGaaGymaaqabaaaaaaaaaa@41FC@

Thus:

(6)
k 1 = j max j u l n r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaBa aaleaacaaIXaaabeaakiabg2da9maalaaabaGaamOAamaaBaaaleaa ciGGTbGaaiyyaiaacIhaaeqaaOGaeyOeI0IaamOAamaaBaaaleaaca WG1baabeaaaOqaaiaadYgacaWGHbGaamOBaiaaykW7caWGYbaaaaaa @4541@

A plot of non-dimensional shear stress τ* versus non-dimensional shear displacement j* is provided in the following figure.

Where:

(7)
τ*= τ c+ptanφ ;j*= j k ; j max * = j max k =10; j u * = j u k =17;r=0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdqNaai Okaiabg2da9maalaaabaGaeqiXdqhabaGaam4yaiabgUcaRiaadcha ciGG0bGaaiyyaiaac6gacqaHgpGAaaGaaGzbVlaacUdacaaMf8Uaam OAaiaacQcacqGH9aqpdaWcaaqaaiaadQgaaeaacaWGRbaaaiaaywW7 caGG7aGaaGzbVlaadQgadaqhaaWcbaGaciyBaiaacggacaGG4baaba GaaiOkaaaakiabg2da9maalaaabaGaamOAamaaBaaaleaaciGGTbGa aiyyaiaacIhaaeqaaaGcbaGaam4AaaaacqGH9aqpcaaIXaGaaGimai aaywW7caGG7aGaaGzbVlaadQgadaqhaaWcbaGaamyDaaqaaiaacQca aaGccqGH9aqpdaWcaaqaaiaadQgadaWgaaWcbaGaamyDaaqabaaake aacaWGRbaaaiabg2da9iaaigdacaaI3aGaaGzbVlaacUdacaaMf8Ua amOCaiabg2da9iaaicdacaGGUaGaaGynaaaa@714F@


Figure 1. Non-dimensional shear stress versus non-dimensional shear displacement