Soil Mechanics 2015/2016
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1 Soil Mechanics 015/016 EXERCISES - CHAPTER The purpose of this exercise is to study in a simplified form the movements of the Tower of Pisa (Italy) foundation due to the compressibility of the clay layers on which it is built. Figure 1 shows the geological profile including the parameters known from laboratory tests. Weight of the tower: T Circular foundation, 19.6 m in diameter Centre of gravity G Length GO =.6 m Sand NC Clay Stress distribution Sand NC Clay Dense Sand 1. Consider first the vertical movement of the tower (without rotation). Assuming that the sand layers are non-deformable, the settlement is produce by consolidation of the two existing clay layers. Use Terzaghi s one-dimensional consolidation theory, assuming that the stresses that the towers transmits to the ground are distributed with depth as if they were applied to increasing surfaces limited by lines forming 30º with the vertical (see figure).
2 With these conditions: a) Find the final settlement of point O due to consolidation of the two clay layers. b) Find the time necessary, from the end of the tower s construction, for the settlement of point O being 95% of the final settlement. c) At present, the average settlement of the tower is.75 m. Compare this result with the one obtained in a). What do you think is the reason for the difference? d) The tower was finished in 185. In 1991 was still moving. Is this fact coherent with the results in a) and b)? What do you think was happening? NOTE: You can use the midpoint of each layer as a representative point for the calculations.. Consider now the effect of the foundation s rotation. For this, use the Theory of Elasticity with long-term parameters ν' = 0.3 and E' to be determined comparing the result in 1.a) with the expression: Elastic semi-space Rigid circular footing s Settlement at O: σ med a = ( ν ʹ ) Eʹ O 1 On the other hand, the moment-rotation relation for a rigid circular footing is: α (radians) = ( ) 3M 1 ν ʹ 4Eʹa 3 where M is the moment acting on the footing that can be calculated as M = W(e 0 + e) with W: weight of the tower; e: eccentricity due to the foundation s rotation and e 0 : additional eccentricity due to lack of symmetry of the problem, estimated as e 0 = 30 cm. a) Find the total final settlement of point B b) What would happen to the tower after an instantaneous rotation α 0 of the foundation (because an earthquake, ground heterogeneity, etc.)?
3 6. A clay layer with C v = 10-3 cm /s has a settlement of 1 cm after one month of being loaded at the surface. The clay layer has a thickness of 0 m and sits on a very hard impermeable layer. What will be the surface settlement three months after (i.e., four months after being initially loaded)? 8 1 exp > T T 0. Recall: 4 U ( T ) T T During a preload test on a plastic clay layer with thickness e = 4 m (double drained) and applied load σ, the settlement after three months of consolidation is 15 cm. Obtain the settlement of a 10 m-thick layer of the same soil (simply drained) after five months of consolidations, under a load 50% larger than σ. Assume U(T) (4T/) ½ 6.4 How are related, approximately, the times necessary to reach a degree of consolidation U = 50% in the two cases shown in the figure? Explain the differences in the consolidation process between the two cases. NOTE: K hor = 10K vert ; the bottom layer is made of gravel.
4 Mecànica de sòls 015/016 EXERCICIS TEMA Se desea estudiar de forma simplificada el movimiento de la cimentación de la Torre de Pisa (Italia) debido a la compresibilidad de los estratos de arcilla sobre los que está construida. La figura 1 representa el perfil del terreno y en ella se incluyen los parámetros conocidos a partir de resultados de laboratorio. 1. Considérese en primer lugar el movimiento vertical de la torre (sin giro). Si se supone que los estratos de arenas son indeformables, el asiento se produce por consolidación de las dos capas de arcilla existentes. Se utilizará la teoría unidimensional de la consolidación de Terzaghi, suponiendo que la tensión que la torre transmite al terreno se reparte en profundidad como si estuviera aplicada en superficies crecientes (delimitadas por líneas que forman 30º con la vertical).
5 En estas condiciones: a) Calcular el asiento final del punto O debido a la consolidación de los dos estratos de arcilla. b) Hallar el tiempo necesario que debe transcurrir desde la finalización de la torre para que el asiento del punto O sea el 95% de su asiento final c) El asiento medio de la torre es en la actualidad de.75 m. Compare este resultado con el obtenido en a). A qué cree que es debida la diferencia? d) La torre fue finalizada en 185. En el año 1991 seguía moviéndose. Es coherente esta circunstancia con los resultados obtenidos en a) y en b)? Qué cree que estaba ocurriendo? NOTA: Puede usarse el punto medio de cada estrato como punto de cálculo característico.. Se considera en segundo lugar el efecto del giro de la cimentación. Para ello se usará la teoría de la elasticidad, con parámetros a largo plazo ν' = 0.3 y E' a determinar comparando el resultado de 1.a) con la expresión: s Asiento en O: σ med a = ( ν ʹ ) Eʹ O 1 Por otra parte, la relación momento-giro en el caso de zapata rígida circular es: ( ν ʹ ) 3M 1 α (radianes) = 4Ea ʹ 3 donde M es el momento sobre la cimentación que puede escribirse en la forma M = W(e 0 + e) con W: peso de la torre; e: excentricidad debido al giro de la cimentación y e 0 : excentricidad adicional debida a una falta de simetría inicial en el problema, evaluable en e 0 = 30 cm. a) Obtener el asiento final total del punto B b) Qué le pasaría a la torre si se produjera un giro instantáneo α 0 en la cimentación (por un terremoto, heterogeneidad del terreno, etc.)?
6 6. Un estrat d argila amb C v = 10-3 cm /s te un assentament de 1 cm després d un mes d haver-se carregat a la superfície. La potència de l estrat és de 0 m i està sobre un estrat impermeable molt dur. Quin serà l assentament a la superfície tres mesos després (i.e., quatre mesos després d haver-se carregat)? 8 1 exp > T T 0. Recordar: 4 U ( T ) T T Durant un assaig de precàrrega sobre un estrat d argila plàstica de potència e = 4 m (drenat pels dos costats) amb una càrrega aplicada σ, l assentament després de tres mesos de consolidació és de 15 cm. Obtenir l assentament d un estrat de 10 m de potència del mateix sòl, drenat per un costat, després de cinc mesos de consolidació, amb una càrrega un 50% superior a σ. Es pot suposar que U(T) (4T/) ½ 6.4 Com estan relacionats, aproximadament, els temps necessaris per a produir un grau de consolidació U = 50% en els dos casos de la figura? Explicar les diferències en el procés de consolidació entre ambdós casos. NOTA: K hor = 10K vert ; l estrat inferior està format per graves.
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