Consolidation CSM8 User Manual - Routledgecw.routledge.com/textbooks/.../Consolidation_CSM8... · Consolidation_CSM8: User’s Manual 3 Below the Basic data 1 section is the FDM node

Consolidation_CSM8

A spreadsheet tool for determining ground settlement with time of layered soils undergoing one-dimensional consolidation by the Finite Difference Method (FDM).

USER’S MANUAL

J. A. Knappett (2012)

This user’s manual and its associated spreadsheet (‘Consolidation_CSM8.xls’) accompanies Craig’s Soil Mechnics, 8th Edition (J.A. Knappett & R.F. Craig).

Consolidation_CSM8: User’s Manual

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1. INTRODUCTION This manual will explain how to use the spreadsheet analysis tool ‘Consolidation_CSM8.xls’ to determine the ultimate settlement and settlement-time relationship for layered soils undergoing one-dimensional consolidation, using the finite difference method. The spreadsheet has the following features:

• Ability to model up to 3 distinct consolidating soil layers with different properties, combined with any number of free-draining or impermeable layers/boundaries;

• Modelling soil profiles of any depth with a resolution of 500 datapoints over this depth (i.e. within a 100 m deep soil profile, excess pore water pressures can be calculated every 0.2 m);

• Ability to correct for a non-instantaneous construction period. • Automatic produce an A4 output sheet with the settlement-time curve shown graphically.

This manual is structured as follows:

Section 2 The basic structure of both the workbook (Consolidation_CSM8.xls) and the worksheet used to perform the analyses will be described and the principle of operation will be highlighted.

Section 3 This section will describe how to use this simple tool to analyse a range of different

ground conditions by considering worked examples from the main text.

Section 4 The final section describes the library of different FDM nodes implemented within the spreadsheet tool and provides further detail of the governing equations.

2. PROGRAMME DESCRIPTION The spreadsheet analysis tool consists of two worksheets. The first, Main sheet, is the worksheet which is used to interact with the spreadsheet. Physical data about the consolidating geomaterials are input and the finite difference mesh is also created in this worksheet. The second worksheet, Summary report, produces an A4 output sheet summarising the initial excess pore pressure distributions within the different materials, the settlement-time curve for the ground surface and the ultimate settlement of this point. The structure of the workbook is shown in Figure 1. Main sheet: The Basic data 1 section contains cells for user input data, including the vertical spacing between Finite Difference (FD) nodes (∆z), the properties of the consolidating materials, and parameters which control the numerical solution (the factor β). The parameter b controls the time difference between each step of the solution (∆t) – the smaller the value of β, the smaller ∆t will be and the more precisely the curve will be defined. However, because of the size limitation within MS Excel spreadsheets, only N = 236 time steps may be analysed, limiting the total time that can be modelled to t = N(∆t). If any of the consolidating materials are thick with low permeability, it may be necessary to increase the value of β to calculate more of the settlement-time curve; however, β must always be ≤ 0.5 to retain numerical stability. It should be noted that the ultimate settlement will always be calculated irrespective of how much of the settlement-time curve is calculated.


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Below the Basic data 1 section is the FDM node library. As the formulation of the basic equation governing the excess pore water pressure at any node depends on the cells around it in the previous time step and on the material properties of the layer, there are three different internal cell formulations (one for each consolidating material). For nodes on the boundary of an impermeable layer some of the adjacent cells will be inactive (e.g. ‘within’ the impermeable boundary). For nodes along the interface of two different consolidating materials the calculation of excess pore water pressure depends on the relative material properties of the soil above and below the interface. As a result, special versions of the basic node formulae are required to correctly model the boundary conditions within the model. The formulae employed are described in more detail in Section 4 of this manual. The FDM node library section contains one example of each formula required, which may be copied into appropriate cells in the Drawing area to build up a complete FD mesh. The mesh is defined as a column of cells and not a 2-D grid as the consolidation is one-dimensional. The first column of the drawing area (cell T18 downwards) represents the initial conditions (at t = 0) and is used to enter the initial values of the excess pore water pressure at each depth which are to dissipate during consolidation. The cells from the FDM node library are copied as appropriate into the second column (cell U18 downwards). This column will determine the excess pore water pressures at t = 0 + ∆t. Once they have been entered into this column to define the layering, the cells from U18 downwards may be selected and copied and pasted into all of the columns to the right, each of which represents a further time step in the analysis. The calculations at these subsequent time steps are automatically calculated on copying the FD mesh across. A worked example of this procedure is shown in Section 3. In the ultimate settlement section the material at each node must be defined from the dropdown list. The calculations to the right of these cells then calculate the total settlement of the piece of material immediately above each node using Equation 4.8 from the main text. These incremental settlements are then summed to give the total integrated one-dimensional settlement over the full profile of soil which has been modelled. The result is the ultimate settlement Σsoed (cell O13). In the Basic data 2 section, the total depth of the soil profile modelled (including any free draining layers) is entered – the value may be obtained from the last entry in the depth scale (column R). Construction period may also be entered here – the spreadsheet always calculates the solution for instantaneous rise in excess pore water pressure, and subsequently corrects for a gradual increase in this using the method outlined in Section 4.9 of the main text. In the Settlement-time section, the excess pore water pressure distribution with depth at each time is integrated in row 5; the relative amount of consolidation U is then calculated using these values in row 6, and these are multiplied by Σsoed to determine the settlement at each time for instantaneous application of the initial excess pore water pressures (denoted by sc, in row 7). These are then corrected for the construction period entered in cell T3 to determine corrected time (tcorr) and settlement (scorr) points using the method outlined in Section 4.9 of the main text in rows 9 and 10 respectively. It is this data which is plotted in the Summary report worksheet. Summary report: In the Project ID section, basic information relating to the project or example being analysed is inputted, along with details of user who prepared the calculations for auditing purposes. This information is not required for analysis, but appears in the header boxes when the output sheet is


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printed, and should be included as a matter of course. If the workbook is use to analyse data from a borehole log or CPT sounding (see Chapter 6 in the main text) then the identifier (ID) of the hole/sounding or grid coordinates may be input for reference. The Input data summary section summarises the material properties of the consolidating materials used in the analysis and shows the initial conditions (excess pore water pressure distribution, ui). These are plotted using differently coloured lines for the three different materials, so this plot also shows the soil layering. Any free draining layers are shown with zero excess pore water pressure. In the Output data section, the settlement-time curve is plotted for the total time analysed and the ultimate settlement is also shown. The axes have no units directly marked – the units for time and settlement are noted in the lower right corner of the input data summary section, which are defined by the units entered by the user in cells D2 and D4 of Main sheet. The Data query section allows for direct numerical output of the ground surface settlement at any time specified by the user, for use in subsequent analyses. Both worksheets are protected so that only data input cells can be edited by the user. This is to prevent accidental over-typing of formulae which may affect the functionality of the spreadsheet. However, the protection is not password protected and so may be removed (Tools > Protection > Unprotect Sheet… in Microsoft Excel) if users wish to investigate the detailed programming of the spreadsheet.

Figure 1a: Workbook structure (Main sheet shown)


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Figure 1b: Workbook structure, contd. (Summary report shown) 3. APPLICATION TO WORKED EXAMPLES IN MAIN TEXT To illustrate how the spreadsheet may be used to model different soil profiles, loading type/duration and hydrostatic conditions, this section will consider worked examples (4.2, 4.4, 4.5, and 4.6) presented in Chapter 4 of the main text. This will also serve to validate the FDM spreadsheet tool. Output sheets from Consolidation_CSM8 for all of these examples are provided in the Appendix. Example 4.2 – Two layers of consolidating material The problem geometry is shown in Figure 2. This problem demonstrates:

• how to model consolidation induced by stress change; • how to model ground with multiple soil layers; • how to model a layer of highly permeable soil using a free-draining conditions.


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Figure 2: Example 4.2

There are two consolidating layers in this example (upper and lower clays); the sand is comparatively very permeable and will be modelled as a free-draining layer (i.e. only the two clay layers give time dependent settlement). The values of mv and k are given in the main text, but the values of cv must be found for entering into the spreadsheet. Using Equation 4.16 from the main text, for the upper clay:

yearmsmm

kcwv

v228

3

1092.01091.2

81.91035.0101

=×=⋅×

×== −

−

−

γ

For the lower clay:

yearmsmm

kcwv

v229

3

1125.01084.7

81.91013.0101

=×=⋅×

×== −

−

−

γ

Given the low permeability of both of the clay layers, time will be measured in years with depths and settlements in m. The coefficients of consolidation are given above and the permeability of the clay is 3.15 × 10-3 m/year for the upper clay. This upper layer has material of lower permeability above it (the sand) but material of lower permeability below it (the lower clay). It can therefore only drain upwards (1-way drainage). The permeability of the lower clay is 3.15 × 10-4 m/year. This layer has material of lower permeability above it (the upper clay) and below it (more sand is assumed) so this layer can drain to both the top and bottom (2-way drainage). The soil properties for the two consolidating layers can then be entered as shown in Figure 3. The parameter β has been set as 0.4 so that the time step ∆t = 0.1 years. The total depth is set to 15 m and the construction period to 0 years, as the load application is instantaneous in the example.


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Figure 3: Data input, Example 4.2 The initial conditions, layering and FD mesh then need to be set-up in the drawing area. ∆z has been set at 0.5 m (see Figure 3) so that there will be nodes at all of the layer boundaries. The node at z = 9 m depth is the upper boundary between the sand and the upper clay. This is a free-draining upper boundary as all of the material above is free-draining (Horizontal drain in the FDM node library). The boundary between the two clay layers is at 12 m depth – as the upper clay is clay 1 and the lower clay is clay 2, this must have the special boundary node for transition between materials (1) and (2). The bottom boundary of the lower clay layer is again free-draining, so this is a free-draining lower boundary for material 2. All of these cells are copied and pasted from the FDM node library into the second column of the drawing area as shown in Figure 4.

Figure 4: Specification of FD mesh and initial conditions, Example 4.2


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The initial excess pore water pressures are equal to the induced total stresses from the embankment construction. These are estimated as being constant in each of the layers at 82 kPa in the upper clay layer (node at 9 m to node at 11.5 m inclusive) and 77 kPa in the lower clay layer (node at 12 m to node at 15 m inclusive) – the same estimate was made in the hand calculations in the main text. These are entered into the first column of the drawing area (column T in the spreadsheet) as shown in Figure 4. The internal cells within the two consolidating layers are then entered into the FD mesh by copy and paste as shown in Figure 5.

Figure 4: Specification of FD mesh and initial conditions, Example 4.2 (contd.) The column with the FD mesh (column U in the worksheet) is then copied across to all columns to the right. The calculated ultimate settlement of the ground surface (in cell O13) is 115 mm, which compares favourably with the hand calculations from the main text (116 mm). The advantage of having used the spreadsheet is that the full settlement-time curve is also available. Example 4.4 – Variable distribution of excess pore pressure within a consolidating layer This example consists of determining the excess pore water pressure distribution within a half-closed layer 10 m thick after consolidation has been in progress for 1 year. This problem additionally demonstrates:

• how to model a variable distribution of excess pore pressure within a consolidating layer; • how to extract the predicted excess pore pressure distribution with depth at a given instant in

time. The value of cv is 7.9 m2/year. The initial distribution of excess pore water pressure from the example is given in Table 1. In this example, there is no k or mv given. The spreadsheet is still able to determine the excess pore pressures using the finite difference method as these calculations only


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depend on cv. However, with no value of k or mv, neither the ultimate settlement nor the settlement-time curve can be determined (as Σsoed cannot be found). The soil properties of the single consolidating layer are entered as shown in Figure 6.

Depth (m) 0 2 4 6 8 10 Pressure (kPa) 60 54 41 29 19 15

Table 1: Initial excess pore water pressure distribution

Figure 6: Data input, Example 4.4 The value of β is set to 0.198 so that there will be a set of calculations conducted at 1 year (because of rounding, the closest set of calculations is at t = 1.003 years). The value of β was found by trial-and-error, examining the cumulative time in the tcorr row (row 9). The layer has 1-way drainage as it is half closed (only able to drain to the surface). The total depth is set to 10 m and the construction period to 0 years, as the example involves instantaneous application of the excess pore water pressures in Table 1. The initial excess pore water pressures and FD mesh are shown in Figure 7. ∆z is set at 1 m to provide a greater level of detail compared to the hand calculations in the main text, so the values of excess pore water pressure at odd depths were determined by linear interpolation between the values given in Table 1. In the FD mesh, the uppermost cell is of the ‘upper free-draining’ type and the lowermost cell is of the ‘lower impermeable’ type.



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Once the FD mesh cells have been copied to the right, the excess pore pressures can be extracted from column BH when the corrected time tcorr = 1.003 years. The corresponding depths can be similarly extracted from column R. These values are plotted in Figure 8 and compared to the values found from the hand calculations reported in the main text (Table 4.4).

Figure 8: Comparison of excess pore pressures after 1 year of consolidation Example 4.5 – Trapped layer under artesian pressure The problem geometry is shown in Figure 9. This problem demonstrates:

• how to model pumping (water table change) and artesian conditions; • how to model a process (in this case pumping) that takes a certain period of time to complete.

Figure 9: Example 4.5


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The total depth is set to 12 m and the construction period to tc = 2 years which is the duration of pumping. Because data is required in the longer-term (5 years after the start of pumping) a large value of β is used so that the ∆t is as large, without violating the stability criterion (β ≤ 0.5). The upper and lower sand will be modelled as free-draining materials so the only consolidation data requiring input is for the clay layer. Because there are high permeability materials above and below the trapped clay, drainage will be two-way. The data entry is shown in Figure 10. Creation of the FD mesh and initial pore water pressure conditions are shown in Figure 11.

Figure 10: Data input, Example 4.5



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The upper 4 m thick layer of sand is modelled as free-draining and the cell on the interface between the sand and top of the clay (i.e. at 4 m exactly) is also of this type. The lowermost cell of the clay layer (at z = 12 m) is also free draining due to the high permeability sand beneath. The initial excess pore water pressure distribution is shown in Figure 9 – the value at the base of the layer was calculated and then the values at the intermediate depths were found by linear interpolation (the pore pressure distribution is linear in Figure 9). Once the FD mesh cells have been copied to the right, the instantaneous settlement (sc) versus time data can be extracted from rows 7 and 14 respectively; the settlement versus time data after correcting for the 2 year pumping period can be extracted from rows 10 (scorr) and 9 (tcorr) respectively. These show excellent agreement with the values computed by hand in the main text (Table 4.3), as shown in Figure 12.

Figure 12: Comparison of settlement-time curves (Example 4.5) Example 4.6 – Consolidating material with trapped sand seam The problem geometry is shown in Figure 13. This problem demonstrates:

• how to model a change in total stress (surface loading) taking a certain period of time to complete;

• how to model thin, highly permeable sand seams (or engineered horizontal drains) of small (negligible) thickness.

• how to use the data query to determine settlement at any instance in time.


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Figure 13: Example 4.6 Example 4.6 is split into two parts. In part (a), surface filling over a period of 1 year loads a single layer of clay beneath sand with one-way drainage. In part (b), the same loading is applied over the same time period, but a thin sand seam of negligible thickness exists within the clay layer which modifies the drainage conditions (see Figure 13). Part (a) – no sand seam: The basic data required is shown in Figure 14. The total depth was set to 14 m and the construction period to tc = 1 year.

Figure 14: Data input, Example 4.6(a)


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Details of the FD mesh are shown in Figure 15. It should be noted that the lower-most node of the clay layer is of the ‘lower impermeable’ type. The fill applies an additional vertical total stress to the entire soil profile of magnitude 3 m × 20 kN/m3 = 60 kPa; this change in total stress must be initially carried everywhere by the pore water, giving the initial distribution of excess pore water pressure shown in Figure 15.

Figure 15: Specification of FD mesh and initial conditions, Example 4.6(a)

Once the FD mesh has been copied to the right, the total settlement and settlement after 3 years can be found from the Summary report sheet. The final settlement is 183 mm (c.f. 182 mm calculated by hand in the main text) and 66 mm of settlement has occurred after 3 years using the data query (c.f. 61 mm in the main text). Part (b) – including sand seam: To model the sand seam, a number of changes must be made to the model described previously. Firstly, as shown in Figure 13, the presence of the seam divides the clay into an upper region with two-way drainage and a lower region with one-way drainage. These two regions are also different thicknesses (changing H). As the drainage paths are now different, the clay must be modelled with two distinct consolidating materials (though having the same cv and k). These changes are shown in Figure 16, where clay1 represents the upper region and clay2 the lower region.


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Figure 16: Data input, Example 4.6(b)

Changes must also be made to the FD mesh. The ‘internal cell’ node at z = 12.5 m is replaced with the ‘horizontal drain’ node type. The cells beneath this are replaced with the ‘internal cell’ nodes for material clay2 and as before, the lower-most node must be of the ‘lower impermeable’ type, but now for material clay2. These changes are shown in Figure 17. Note that the initial excess pore water pressure distribution is unchanged

Figure 17: Specification of FD mesh and initial conditions, Example 4.6(b)


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Once the FD mesh has been copied to the right, the total settlement and settlement after 3 years can be found from the Summary report sheet. The final settlement is still 183 mm as the thickness of clay that will eventually consolidate has not been changed by the seam, which has negligible thickness. However, 157 mm of settlement can now be seen to occur after 3 years using the data query (c.f. 157 mm in the main text). By shortening the drainage path lengths within the clay layer, the sand seam has sped-up the consolidation process significantly. 4. FDM NODE LIBRARY This section describes the different nodal formulae which are available within Seepage_CSM8.xls and provides the theoretical formulation of each. These are split into three separate tables over pages 16 – 17 inclusive:

• p.16: Basic nodes for modelling impermeable boundaries and general soil nodes • p.17: Nodes for modelling layer boundaries between different materials • p.17: Nodes for modelling free-draining layers/boundaries and thin horizontal drains

Node type Governing equation Representation in node library

Internal cell Material (1) ( )

( )jijijiv

jiji uuuz

tcuu ,,1,12

1,1, 2−+

∆

∆+= +−+

Upper impermeable

boundary Material (1)

( )( )jiji

vjiji uu

z

tcuu ,,12

1,1, 22 −

∆

∆+= ++

Lower impermeable


( )( )jiji

vjiji uu

z

tcuu ,,12

1,1, 22 −

∆

∆+= −+


( )jijijiv

jiji uuuz

tcuu ,,1,12

2,1, 2−+

∆

∆+= +−+

Lower impermeable


( )( )jiji

vjiji uu

z

tcuu ,,12

2,1, 22 −

∆

∆+= −+


( )jijijiv

jiji uuuz

tcuu ,,1,12

3,1, 2−+

∆

∆+= +−+

Lower impermeable


( )( )jiji

vjiji uu

z

tcuu ,,12

3,1, 22 −

∆

∆+= −+


17


Layer boundary Materials (1) – (2)

( ) ( )

( )( )jiji

v

jivv

ji

uuz

tc

uH

tc

H

tcu

,112,121

,22

2122

1

1121, 5.05.0

+−

+

+∆

∆+

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ∆−

∆−+=

α

αα

Layer boundary Materials (2) – (3)

( ) ( )

( )( )jiji

v

jivv

ji

uuz

tc

uH

tc

H

tcu

,123,122

,23

3232

2

2231, 5.05.0

+−

+

+∆

∆+

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ∆−

∆−+=

α

αα

In the table above the parameter α is given by:

2

1

1

212 H

Hkk

⋅=α 3

2

2

323 H

Hkk

⋅=α


Horizontal drain (or internal cell,

free-draining layer) 01, =+jiu

Upper free-draining boundary

01, =+jiu

Lower free-draining boundary

01, =+jiu

N.B. A layer which is perfectly free to drain (infinite permeability) cannot support any excess pore pressure so u = 0 at all times and at all points within the free-draining material.

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Consolidation CSM8 User Manual - Routledgecw.routledge.com/textbooks/.../Consolidation_CSM8... · Consolidation_CSM8: User’s Manual 3 Below the Basic data 1 section is the FDM node