Publicada originalmente en J. Keenan y J. Standard Atmosphere Supplements. Oficina de Impresiones del Gobierno de los EE. Empleadas con permiso del doctor Edward E. Obert, Universidad de Wisconsin. Fuente : John R. Howell y Richard O. Buckius, Fundamentals of Engineering Thermodynamics , version S. Nueva York: McGraw-Hill, , p. R Fuente: Gordon J.
Magnesio 0. Psia P 1. P sat 0. Keenan, Frederick G. Empleadas con permiso de los autores. Fuente: Kenneth Wark, Thermodynamics, 4a. Originalmente publicadas en J. Oficina de Impresiones del Gobierno de EE.
Maximum deflection will be at the center; however, the contact stress over the footing will be uniform q per unit area. A rigid foundation resting on the same clay will show a uniform settlement Figure 3. The contact stress distribution will take a form such as that shown in Figure 3. Foundations on sand For a flexible foundation resting on a cohesionless soil, the distribution of contact pressure will be uniform Figure 3. However, the edges of the foundation will undergo a larger settlement than the center.
This occurs because the soil located at the edge of the foundation lacks lateral-confining pressure and hence possesses less strength. The lower strength of the soil at the edge of the foundation will result in larger settlement.
A rigid foundation resting on a sand layer will settle uniformly. The contact pressure on the foundation will increase from zero at the edge to a maximum at the center, as shown in Figure 3. The latter, of course, requires elaborate field instrumentation. However, from the results available at present, fairly good agreement is shown between theoretical considerations and field conditions, especially in the case of vertical stress. Stresses and displacements in a soil mass Figure P3.
The uniformly distributed vertical loads on the area are also shown. Determine the vertical stress increase at A and B due to the loaded area. A and B are located at a depth of 3 m below the ground surface.
Determine the vertical stresses due to a loaded area at A, B, C, and D. All points are located at a depth of 1. References Ahlvin, R. Borowicka, H. Bridge Struct. Boussinesq, J. Burmister, D. Fox, L. Giroud, J. SM1, pp. Harr, M. Holl, D. Jones, A. Kezdi, A. Rethati, Handbook of Soil Mechanics, vol. Melan, E. Mech, vol. Newmark, N. Osterberg, J. Stresses and displacements in a soil mass Peattie, K. Poulos, H. R63, University of Sydney, Australia, Chapter 4 Pore water pressure due to undrained loading 4.
A knowledge of the increase of pore water pressure in soils due to various loading conditions without drainage is important in both theoretical and applied soil mechanics. If a load is applied very slowly on a soil such that sufficient time is allowed for pore water to drain out, there will be practically no increase of pore water pressure. However, when a soil is subjected to rapid loading and if the coefficient of permeability is small e. Figure 4. In this chapter, mathematical formulations for the excess pore water pressure for various types of undrained loading will be developed.
If drainage from the soil is not allowed, the pore water pressure will increase by u. The increase of pore water pressure will cause a change in volume of the pore fluid by an amount Vp. It should be noted that compression, i.
Since the change in volume of the pore fluid, Vp , is equal to the change in the volume of the soil skeleton, V , we obtain from Eqs. A summary of the soil types and their parameters and the B values at saturation that were considered by Black and Lee is given in Table 4. It is obvious from this figure that, for stiffer soils, the B value rapidly decreases with the degree of saturation.
This is consistent with the experimental values for several soils shown in Figure 4. Table 4. Pore water pressure due to undrained loading Figure 4. As noted in Table 4. An example of such behavior in saturated varved Fort William clay as reported by Eigenbrod and Burak is shown in Figure 4. Let the increase of pore water pressure be equal to u. However, in reality, this is not the case, i. The magnitude of A for a given soil is not a constant and depends on the stress level.
For highly overconsolidated clay soils, the magnitude of A at failure i. Catherines Till? Parallel orientation of clay particles could cause the strength of clay and thus Af to vary with direction. Kurukulasuriya et al. Pore water pressure due to undrained loading shows the directional variation of Af with overconsolidation ratio.
To take the intermediate principal stress into consideration Figure 4. Pore water pressure due to undrained loading A comparison of Eqs. This can be illustrated by deriving an expression for the excess pore water pressure developed in a saturated soil undrained condition below the centerline of a flexible strip loading of uniform intensity, q Figure 4.
Example 4. Estimate the excess pore water pressure that will be developed due to the loading at A and B. This is a plane strain case. Pore water pressure due to undrained loading From Eq. In that case, the soil specimen was allowed to undergo axial and lateral strains.
However, in oedometer tests the soil spec- imens are confined laterally, thereby allowing only one directional strain, i. For such a case, referring to Figure 4. Lambe and Whitman reported the following C values: Vicksburg buckshot clay slurry 0. The results of those tests for Monterey no. From Table 4. An increase in the initial relative density of compaction as well as an increase in the effective confining pressure does increase the soil stiffness.
Pore water pressure due to undrained loading Table 4. Calculate the increase of pore water pressure at M immediately after application of the load for the cases given below.
Figure P4. Estimate the height of water h1 that a piezometer would show immediately after the application of the surcharge. SMI, pp. Eigenbrod, K. Henkel, D. Kenney, T. SM3, pp. Kurukulasuriya, L. Oda, and H. Simons, N. Reiner, ed. Terzaghi, K. Veyera, G. Charlie, D. Doehring, and M. Chapter 5 Permeability and seepage 5. The continuous void spaces in a soil permit water to flow from a point of high energy to a point of low energy.
Permeability is defined as the property of a soil that allows the seepage of fluids through its interconnected void spaces. This chapter is devoted to the study of the basic parameters involved in the flow of water through soils. The cross-sectional area of the soil is equal to A and the rate of seepage is q. Some typical values of the coefficient of permeability are given in Table 5. At any other temperature T , the coefficient of permeability can be obtained from Eq.
A criterion for investigating the range can be furnished by the Reynolds number. Hansbo reported the test results of four undisturbed natural clays. On the basis of his results Figure 5. The value of n for the four Swedish clays was about 1. There are several studies, however, that refute the preceding conclusion. Figure 5. Constant-head test The constant-head test is suitable for more permeable granular materials. The basic laboratory test arrangement is shown in Figure 5. The outflow water is collected in a measuring cylinder, and the duration of the collection period is noted.
Falling-head test The falling-head permeability test is more suitable for fine-grained soils. Water from the standpipe flows through the specimen. Permeability from consolidation test The coefficient of permeability of clay soils is often determined by the consolidation test, the procedures of which are explained in Sec.
Equation 5. A small quantity of silts and clays, when present in a sandy soil, may substantially change the coefficient of permeability. A theoretical solution for the coefficient of permeability also exists in the literature. This is generally referred to as the Kozeny—Carman equation, which is derived below.
It was pointed out earlier in this chapter that the flow through soils finer than coarse gravel is laminar. The interconnected voids in a given soil mass can be visualized as a number of capillary tubes through which water can flow Figure 5. Let SV be equal to the surface area per unit volume of soil bulk. The factor S in Eq. Referring to Figure 5. This relation is the Kozeny—Carman equation Kozeny, ; Carman, Comparing Eqs.
However, serious discrepan- cies are observed when the Kozeny—Carman equation is applied to clayey soils. From the plot, it appears that all three relations are equally good.
More recently, Chapuis proposed an empirical relationship for k in conjunction with Eq. This can be extended to natural, silty sands without plasticity. It is not valid for crushed materials or silty soils with some plasticity. Mention was made in Sec. However, under a low hydraulic gradient, laminar flow conditions usually exist. Kenney et al. The uniformity coefficients of these specimens, Cu , ranged from 1. Modification of Kozeny—Carman equation for practical application For practical use, Carrier modified Eq.
Substituting these values into Eq. Carrier further suggested a slight modification of Eq. Sieve No. Given: the void ratio of the sand is 0. The dis- crepancies between the theoretical and experimental values are shown in Figures 5. These results are based on consolidation—permeability tests Olsen, , Olsen developed a model to account for the variation of perme- ability due to unequal pore sizes. Several other empirical relations were proposed from laboratory and field permeability tests on clayey soil.
They are summarized in Table 5. Example 5. Use the equation proposed by Samarsinghe et al. In most cases, the anisotropy is more predom- inant in clayey soils compared to granular soils. In anisotropic soils, the directions of the maximum and minimum permeabilities are generally at right angles to each other, maximum permeability being in the horizontal direction. The flow line is a line along which a water particle at O will move from left to right.
For the definition of an equipotential line, refer to Sec. Note that in anisotropic soil the flow line and equipotential line are not orthogonal. The coefficients of permeability in the x and z directions are kh and kv , respectively. In Figure 5. Direction n is perpendicular to the equipotential line at O, and so it is the direction of the resultant hydraulic gradient.
Substitution of these into Eq. It can be seen that, for given values of kh and kv , Eqs. All tests were conducted after full saturation of the compacted soil specimens. The results show that kh and kv are functions of molding mois- ture content and confining pressure. If the stratification is con- tinuous, the effective coefficients of permeability for flow in the horizontal and vertical directions can be readily calculated.
Flow in the horizontal direction Figure 5. Owing to fabric anisotropy, the coefficient of permeability of each soil layer may vary depending on the direction of flow. Considering the unit length of the soil layers as shown in Figure 5. Note that for flow in the horizontal direction which is the direction of stratification of the soil layers , the hydraulic gradient is the same for all layers.
Permeability and seepage Figure 5. Several techniques are presently available for determination of the coefficient of permeability in the field, such as pumping from wells and borehole tests, and some of these methods will be treated briefly in this section. Pumping from wells Gravity wells Figure 5. The coefficient of permeability of the top permeable layer can be determined by pumping from a well at a constant rate and observing the steady-state water table in nearby observation wells.
The steady-state is established when the water levels in the test well and the observation wells become constant.
Substituting these into the above equation for rate of discharge gives Figure 5. According to Kozeny , the maximum radius of influence, R Figure 5. This is shown in Figure 5. The relation for the coefficient of permeability given by Eq. If the well partially penetrates the permeable layer as shown in Figure 5. Artesian wells The coefficient of permeability for a confined aquifier can also be deter- mined from well pumping tests. Pumping is continued until a steady state is reached.
In this method, an auger hole is made in the ground that should extend to a depth of 10 times the diameter of the hole or to an impermeable layer, whichever is less. Water is pumped out of the hole, after which the rate of the rise of water with time is observed in several increments.
There are several other methods of determining the field coefficient of permeability. For a more detailed description, the readers are directed to the U. Bureau of Reclamation and the U. Department of the Navy The coefficient of permeability of the layer is 0. Shape and size of the soil particles. Void ratio. Permeability increases with increase in void ratio. Degree of saturation.
Permeability increases with increase in degree of saturation. Composition of soil particles. For sands and silts this is not important; however, for soils with clay minerals this is one of the most important factors. Permeability depends on the thickness of water held to the soil particles, which is a function of the cation exchange capacity, valence of the cations, and so forth.
Other factors remaining the same, the coefficient of permeability decreases with increasing thickness of the diffuse double layer. Soil structure. Fine-grained soils with a flocculated structure have a higher coefficient of permeability than those with a dispersed structure.
Viscosity of the permeant. Density and concentration of the permeant. The principle can be explained with the help of Figure 5. When dc electricity is applied to the soil, the cations start to migrate to the cathode, which consists of a perforated metallic pipe. Since water is adsorbed on the cations, it is also dragged along. When the cations reach the cathode, they release the water, and the subsequent build up of pressure causes the water to drain out. This process is called electroosmosis and was first used by L.
Casagrande in for soil stabilization in Germany. Rate of drainage by electroosmosis Figure 5. The surface of the clay particles have negative charges, and the cations are concentrated in a layer of liquid. In contrast to the Helmholtz—Smoluchowski theory [Eq. Electroosmosis is costly and is not generally used unless drainage by conventional means cannot be achieved. Gray and Mitchell have studied the factors that affect the amount of water transferred per unit charge passed, such as water content, cation exchange capacity, and free electrolyte content of the soil.
For these problems, calculation of flow is generally made by use of graphs referred to as flow nets. To derive the equation of continuity of flow, consider an elementary soil prism at point A Figure 5.
This is an equipoten- tial line. The flow is in one direction only, i. The lengths of the two soil layers LA and LB and their coefficients of permeability in the direction of the x axis kA and kB are known. The total heads at sections 1 and 3 are known. For one-dimensional flow, Eq.
Integration of Eq. For flow through soil A the boundary conditions are 1. From the first boundary condition and Eq. Permeability and seepage From Eq. As discussed in Sec. An equipotential line is a line joining the points that show the same piezometric elevation i.
The permeable layer is isotropic with respect to the coefficient of permeability, i. Note that the solid lines in Figure 5. In drawing a flow net, the boundary conditions must be kept in mind. For example, in Figure 5. AB is an equipotential line 2. EF is an equipotential line 3. BCDE i. GH is a flow line The flow lines and the equipotential lines are drawn by trial and error.
It must be remembered that the flow lines intersect the equipotential lines at right angles. The flow and equipotential lines are usually drawn in such a way that the flow elements are approximately squares. Drawing a flow net is time consuming and tedious because of the trial-and-error process involved. Once a satisfactory flow net has been drawn, it can be traced out. Some other examples of flow nets are shown in Figures 5.
Calculation of seepage from a flow net under a hydraulic structure A flow channel is the strip located between two adjacent flow lines.
To calculate the seepage under a hydraulic structure, consider a flow channel as shown in Figure 5. The equipotential lines crossing the flow channel are also shown, along with their corresponding hydraulic heads. Let q be the flow through the flow channel per unit length of the hydraulic structure i. Combining Eqs. We could construct flow nets with all the flow elements drawn as rectangles. In that case the width-to-length ratio of the flow nets must be a constant, i.
Part a i : To reach A, water must go through three potential drops. Part a iii : Points A and C are located on the same equipotential line. So water in a piezometer at C will rise to the same elevation as at A, i. From Figure 5. The procedure can best be explained through a numerical exam- ple. Consider the dam section shown in Figure 5. To find the pressure head at point D Figure 5.
Between points F and G, the variation of pressure heads will be approximately linear. Let us now consider the case of constructing flow nets for seepage through soils that show anisotropy with respect to permeability. For two-dimensional flow problems, we refer to Eq.
The steps for construction of a flow net in an anisotropic medium are as follows: 1. To plot the section of the hydraulic structure, adopt a vertical scale. With the scales adopted in steps 1 and 3, plot the cross-section of the structure. Draw the flow net for the transformed section plotted in step 4 in the same manner as is done for seepage through isotropic soils. Draw a flow net for the transformed section. Replot this flow net in the natural scale also.
One important fact to be noticed from this is that when the soil is anisotropic with respect to permeability, the flow and equipotential lines are not nec- essarily orthogonal. Permeability and seepage 5. Rarely in nature do such ideal conditions occur; in most cases, we encounter stratified soil deposits such as those shown in Figure 5.
When a flow net is constructed across the boundary of two soils with different permeabilities, the flow net deflects at the boundary. This is called a transfer condition. Soil layers 1 and 2 have permeabilities of k1 and k2 , respectively. The dashed lines drawn across the flow channel are the equipotential lines. Let h be the loss of hydraulic head between two consecutive equipoten- tial lines. It is useful to keep the following points in mind while constructing the flow nets: 1.
So the flow elements in layer 2 will be rectangles. An example of the construction of a flow net for a dam section rest- ing on a two-layered soil deposit is given in Figure 5. For the rate of flow from point 1 to point 0 through the channel shown in Figure 5. However, for the case of flow across the boundary of one homogeneous soil layer to another, Eq.
Consider the problem of determining the hydraulic heads at various points below the dam shown in Figure 5. Since the flow net below the dam will be symmetrical, we will consider only the left half. The steps or determining the values of h at various points in the permeable soil layers are as follows: 1.
Roughly sketch out a flow net. Based on the rough flow net step 1 , assign some values for the hydraulic heads at various grid points. These are shown in Figure 5. Note that the values of h assigned here are in percent. Consider the heads for row 1 i. The flow condition for these grid points is similar to that shown in Figure 5. If these values are substituted into Eq. This is called the relaxation process. The residual R is calculated by substituting values into Eq.
The corrected heads are shown in Figure 5. According to Eq. Permeability and seepage 6. Find the corrected head in a manner similar to that in step 4.
These values are shown in Figure 5. These values are given in Figure 5. With the new heads, repeat steps 3—7. This iteration must be carried out several times until the residuals are negligible.
With these values of h, the equipotential lines can now easily be drawn. To evaluate the seepage force per unit volume of soil, consider a soil mass bounded by two flow lines ab and cd and two equipotential lines ef and gh, as shown in Figure 5. The soil mass has unit thick- ness at right angles to the section shown. The self-weight of the soil mass Figure 5. Harza investigated the safety of hydraulic structures against pip- ing. The maximum exit gradient can be determined from the flow net.
A factor of safety of 3—4 is considered adequate for the safe performance of the structure. Harza also presented charts for the maximum exit gradient of dams constructed over deep homogeneous deposits see Figure 5.
Using the notations shown in Figure 5. Once the weighted creep length has been calculated, the weighted creep ratio can be determined as Figure 5.
Table 5. For structures other than a single row of sheet piles, such as that shown in Figure 5. How- ever, Harr , p. The approximate hydraulic heads at the bottom of the prism can be evaluated by using the flow net. The subsoil is fine sand. Since the calculated weighted creep ratio is 3.
Over a period of time, this process may clog the void spaces in the coarser material. Such a situation can be prevented by the use of a filter or protective filter between the two soils.
For example, consider the earth dam section shown in Figure 5. If rockfills were only used at the toe of the dam, the seepage water would wash the fine soil grains into the toe and undermine the structure. Hence, for the safety of the structure, a filter should be placed between the fine soil and the rock toe Figure 5. For the proper selection of the filter material, two conditions should be kept in mind. The size of the voids in the filter material should be small enough to hold the larger particles of the protected material in place.
The filter material should have a high permeability to prevent buildup of large seepage forces and hydrostatic pressures in the filters. The proper use of Eqs. Consider the soil used for the construction of the earth dam shown in Figure 5.
Let the grain-size distribution of this soil be given by curve a in Figure 5. The acceptable grain-size distribution of the filter material will have to lie in the shaded zone. Based on laboratory experimental results, several other filter design cri- teria have been suggested in the past. These are summarized in Table 5.
In this section, some of these solutions will be considered. Dupuit assumed that the hydraulic gradient i is equal to the slope of the free surface and is constant with depth, i. However, in the derivation of the equation, no attention has been paid to the entrance or exit conditions.
Schaffernak proposed that the phreatic surface will be like line ab in Figure 5. The seepage per unit length of the dam can now be determined by considering the triangle bcd in Figure 5. Schaffernak suggested a graphical procedure to determine the value of l. This procedure can be explained with the aid of Figure 5. Extend the downstream slope line bc upward. Draw a vertical line ae through the point a.
This will intersect the projection of line bc step 1 at point f. With fc as diameter, draw a semicircle fhc. Draw a horizontal line ag.
With c as the center and cg as the radius, draw an arc of a circle, gh. With f as the center and fh as the radius, draw an arc of a circle, hb.
Casagrande showed experimentally that the parabola ab shown in Figure 5. So, with this modification, the value of d for use in Eq. Casagrande suggested that this relation is an approximation to the actual condition. Fuente: Chemical and Process Thermodynamics 3a. Adaptado con permiso de Pearson Education, Inc. Acetileno C2H2 Usada con permiso. Vapor Temp. Klein y F. Data, 16, , con modificaciones para ajustarla a la Escala Internacional de Temperaturas de Data, 22, , Vapor Pres.
Gallagher y George S. Tillner-Roth y H. Chem, Ref. Data, vol. Obert, Universidad de Wisconsin. Standard Atmosphere Supplements. Oficina de Impresiones del Gobierno de Estados Unidos, Publicada originalmente en J. Keenan y J. C6H6 C4H10 C10H22 C2H6O
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