Given the schematic illustrations (Figure 5.1) of the simple cubic (loosest) packing and tetrahedral (closest) packing arrangements of idealized, non-deformable solid particles of dry sand: (A) Verify that the maximum void ratio and porosity of the idealized sandy soil, in its loosest state, are indeed 0.91 and 0.476 (47.6%), respectively. A single solid particle with radius R would fit a cubical unit volume with dimensions 2R x 2R x 2R (8R¹). (Show all pertinent volume-weight relationships and calculations.) (B) Verify that the minimum void ratio and porosity of the idealized sandy soil, in its closest state, are indeed 0.34 and 0.26 (26.0%), respectively. In this case, the cubical unit volume is 5.657R³, versus 8R³ in simple cubic packing. (Show all pertinent volume-weight relationships and calculations.) (C) If the specific gravity of the soil solids is 2.65, and water is carefully added so as not to disturb the condition of the sand, show that the saturated water content of the idealized sandy soil, in its loosest state, is 0.343 (34.3%). (D) If the specific gravity of the soil solids is 2.65, and water is carefully added so as not to disturb the condition of the sand, show that the saturated water content of the idealized sandy soil, in its closest state, is 0.128 (12.8%). (a) Simple cubic: radius R Coordination number =6) Layer spacing= 2R Unit volume = 8R³ Void ratio 0.91 Porosity = 47.6% 80 (b) Tetrahedral: radius R Coordination number = 12) Layer spacing= 2R(2/3)0.5 Unit volume = 4(2R³)0.5 Void ratio= 0.34 Porosity 26.0% Figure 5.1 Uniform spheres in (a) simple cubic packing order and (b) tetrahedral close packing order.
Given the schematic illustrations (Figure 5.1) of the simple cubic (loosest) packing and tetrahedral (closest) packing arrangements of idealized, non-deformable solid particles of dry sand: (A) Verify that the maximum void ratio and porosity of the idealized sandy soil, in its loosest state, are indeed 0.91 and 0.476 (47.6%), respectively. A single solid particle with radius R would fit a cubical unit volume with dimensions 2R x 2R x 2R (8R¹). (Show all pertinent volume-weight relationships and calculations.) (B) Verify that the minimum void ratio and porosity of the idealized sandy soil, in its closest state, are indeed 0.34 and 0.26 (26.0%), respectively. In this case, the cubical unit volume is 5.657R³, versus 8R³ in simple cubic packing. (Show all pertinent volume-weight relationships and calculations.) (C) If the specific gravity of the soil solids is 2.65, and water is carefully added so as not to disturb the condition of the sand, show that the saturated water content of the idealized sandy soil, in its loosest state, is 0.343 (34.3%). (D) If the specific gravity of the soil solids is 2.65, and water is carefully added so as not to disturb the condition of the sand, show that the saturated water content of the idealized sandy soil, in its closest state, is 0.128 (12.8%). (a) Simple cubic: radius R Coordination number =6) Layer spacing= 2R Unit volume = 8R³ Void ratio 0.91 Porosity = 47.6% 80 (b) Tetrahedral: radius R Coordination number = 12) Layer spacing= 2R(2/3)0.5 Unit volume = 4(2R³)0.5 Void ratio= 0.34 Porosity 26.0% Figure 5.1 Uniform spheres in (a) simple cubic packing order and (b) tetrahedral close packing order.
Chapter2: Loads On Structures
Section: Chapter Questions
Problem 1P
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Subject is soil Mechanics
Transcribed Image Text:Given the schematic illustrations (Figure 5.1) of the simple cubic (loosest) packing and
tetrahedral (closest) packing arrangements of idealized, non-deformable solid particles of
dry sand:
(A) Verify that the maximum void ratio and porosity of the idealized sandy soil, in its
loosest state, are indeed 0.91 and 0.476 (47.6%), respectively. A single solid particle
with radius R would fit a cubical unit volume with dimensions 2R x 2R x 2R (8R¹).
(Show all pertinent volume-weight relationships and calculations.)
(B) Verify that the minimum void ratio and porosity of the idealized sandy soil, in its
closest state, are indeed 0.34 and 0.26 (26.0%), respectively. In this case, the cubical
unit volume is 5.657R³, versus 8R³ in simple cubic packing. (Show all pertinent
volume-weight relationships and calculations.)
(C) If the specific gravity of the soil solids is 2.65, and water is carefully added so as not
to disturb the condition of the sand, show that the saturated water content of the
idealized sandy soil, in its loosest state, is 0.343 (34.3%).
(D) If the specific gravity of the soil solids is 2.65, and water is carefully added so as not
to disturb the condition of the sand, show that the saturated water content of the
idealized sandy soil, in its closest state, is 0.128 (12.8%).
(a)
8
(b)
Simple cubic: radius R
Coordination number = 6
Layer spacing= 2R
Unit volume = 8R³
Void ratio 0.91
Porosity = 47.6%
Tetrahedral: radius R
Coordination number = 12)
Layer spacing= 2R(2/3)0.5
Unit volume = 4(2R³)0.5
Void ratio = 0.34
Porosity = 26.0%
Figure 5.1 Uniform spheres in (a) simple cubic packing order and (b) tetrahedral
close packing order.
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