Mohr-Coulomb con ángulo de fricción interna variable con la presión
Dr. Alejo O. Sfriso
Mohr-Coulomb con ángulo de fricción interna variable
Universidad de Buenos Aires SRK Consulting (Argentina) AOSA
2
materias.fi.uba.ar/6408 latam.srk.com www.aosa.com.ar
[email protected] [email protected] [email protected]
La curva de resistencia intrínsica de los geomateriales
(Bishop 1966)
1
Mohr-Coulomb con ángulo de fricción interna variable Mohr-Coulomb con ángulo de fricción interna variable
3
El criterio de Mohr-Coulomb adaptado a CRI realistas (planteo) Los suelos y rocas tienen CRI curvas en el diagrama 𝜏 − 𝜎 En geotecnia se emplea Mohr-Coulomb (una recta) • Razones históricas (Coulomb 1776) • Interpretación de ensayos (primera mitad s. XX) • Soluciones analíticas para problemas prácticos – Empujes: Rankine, Coulomb, Terzaghi – Fundaciones: Terzaghi, Meyerhof, Brinch-Hansen – Taludes: Bishop, Spencer, Morgestern-Price • (Tentadoramente) fácil de calibrar (con 𝑐 y 𝜙 tangentes) Con la geomecánica computacional no necesitamos soluciones analíticas (ni estamos en el siglo XX)
El criterio de Mohr-Coulomb adaptado a CRI realistas (solución) Solución: emplear 𝝓 𝒑 (respetando el vértice 𝒄𝒑) 𝑓, = 𝜎. − 𝜎/ + 𝜎/ + 𝜎. 𝑠𝑖𝑛 𝝓 𝒑 − 2𝑐 𝑐𝑜𝑠 𝝓 𝒑 = 0 reescrito como 𝑓, = 𝜎. − 𝜎/ + 𝜎/ + 𝜎. − 2𝒄𝒑 𝑠𝑖𝑛 𝝓 𝒑 = 0 𝜏
𝒄𝒑 se calibra para reproducir 𝜎) 𝒄𝒑 =
𝜎) 1 −1 2 𝑠𝑖𝑛 𝜙 𝜎) ⁄3
𝑐*
𝝓𝒑
𝜎
4
2
Mohr-Coulomb con ángulo de fricción interna variable
¿Porqué Mohr-Coulomb con 𝜙 𝑝 ?
• Reproduce comportamiento en 𝜏 − 𝜎 • Respeta definiciones históricas (𝑐 y 𝜙) • Produce superficies de curvatura simple 𝜃=
𝜋 6
recto
Mohr-Coulomb con 𝜙 𝜎. en extensión superficie curva si 𝜙 𝜎.
curvo
𝜋 6 superficie recta si 𝜙 𝑝 (igual a M-C en compresión) 𝜃=−
Mohr-Coulomb con ángulo de fricción interna variable
5
¿Porqué no usar parámetros tangentes? El empleo de parámetros tangentes puede conducir a resultados erróneos en cálculos analíticos
Dirección de falla no realista: cinemática errónea
Dirección de falla realista: cinemática confiable
τ Cohesión alta: sobre-estima resistencia de “círculos” poco profundos Cohesión conservativa pero razonable
6
𝜎
3
Mohr-Coulomb con ángulo de fricción interna variable
Leps (1970): enrocados
• Basado en resultados de ensayos triaxiales • Tiene (cualitativamente) en cuenta el efecto de la densidad relativa • Es función de 𝜎? • Extrapola presiones bajas 𝝓 = 𝜙@ − Δ𝜙 · 𝑙𝑜𝑔/@ 𝝈𝒏 3 · 𝑐𝑜𝑠 G 𝜙 𝝈𝒏 = 𝒑 3 − 𝑠𝑖𝑛 𝜙
7
(Leps 1970)
Mohr-Coulomb con ángulo de fricción interna variable
Bolton (1986): arenas
• Basado en resultados de ensayos triaxiales • Separa fricción mineral (𝜙) ) de dilatancia (𝜓) • Es función de 𝑝
(Bolton 1986)
𝝓 − 𝜙) = 3º 𝐷L 𝑄 − 𝑙𝑛 8
100 𝒑 𝑝NOP
− 3º
4
Geotechnical Issues – Strength, Stability and Seepage
degrees higher (e.g. about 2° to 4°) when plane tests are compared with triaxial tests on the same material. There is noticeably less crushing of particles: hence the two empirical curves in Figure 15.
Shear Strength of Rockfill, Interfaces and Rock Joints, and their Points of Contact in Rock Dump Design
Mohr-Coulomb con ángulo de fricción interna variable
4
Estimating the shear strength triaxialof rockfill
N.R. Barton
def. plana
Barton As(2008): emphasised in all reports of rockfill shear strength, including Barton and Kjærnsli (1981), the degree of compaction and porosity achieved when building a dam or when preparing relevant laboratory samples is all enrocados important. The particle roughness and smoothness is also fundamental. Figure 14 illustrates an empirical •
scheme developed by the writer, for estimating the likely R-value for rockfills, whether for rounded gravels or for rough quarried rock. The high (relatively uncompacted) porosities in mining rock dumps clearly places such dumps in the middle-to right-hand areas of this diagram, and even sharp angular particles (relevant for Basadowaste en rock, but perhaps not always for tailings) are unlikely to generate ‘R-values’ above 5 to 7, as also suggested in Figure 9. d50 (mm)
ensayos triaxiales • Considera Figure tamaño 15 Particle size strongly effects the strength of contacts points in rockfill. Triaxial or plane shear also influences behaviour. Empirical S/UCS reduction factors for estimating S when de partículas, evaluating equation 3. porosidad y resistencia5deInterface shear strength Interface shear strength, as between a (too smooth) rock foundation and a rockfill dam, seems to be governed los granos by the ‘weakest link’ rule. If the roughness JRC of the interface, registered by amplitude/length profiling, is too low in relation to particle size (d50), the interface strength is controlled by JRC, and sliding occurs along the interface, as along the bottom face of a rock joint. If on the other hand, the interface roughness is sufficient to give good interlock to the rockfill particles, sliding will occur preferentially within the rockfill, 𝒏particle smoothness or roughness dependent manner, with influence also of the porosity. in an ‘R-controlled’ /@illustration of the interface problem, and (probable) relevant controlling parameters is shown in A schematic Figure 16.
𝝓 − 𝜙Q = −𝑅 · 𝑙𝑜𝑔 3 · 𝑐𝑜𝑠 G 𝜙 𝝈𝒏 = 𝒑 3 − 𝑠𝑖𝑛 𝜙
𝝈 𝑆
9
Rock Dumps 2008, Perth, Australia
(Barton 2008)
13
Figure 14 An empirical method for estimating the equivalent roughness R of rockfill as a function of porosity and particle origin, roundedness and smoothness. Barton and Kjærnsli (1981) As a result of the literature survey of numerous rockfill test data, Barton, 1980 and Barton and Kjærnsli, 1981 developed a simple strength factoring scheme for estimating S as a function of UCS (or σc), when particle size (d50) varied over a wide range. The points A and B in Figure 15 were used to illustrate S-value estimation for a rock with UCS = 150 MPa, when d50 was 23 mm (S ≈ 0.3x150 = 50 MPa) and when d50 was 240 mm (S ≈ 0.2x150 = 30 MPa), in the case of interpreting triaxial strength data. Note the higher factors apparently needed when planar (and large-scale) shear is involved. Friction angles are typically several
12
Rock Dumps 2008, Perth, Australia
5
