Analysis
What are Dip and Dip direction
30/11/08 19:53 Filed in: Geomechanics
One of the keys to understand rock mechanics is to
understand the joint patterns within the rock.
Critically important in understand the joint
pattern is to understand the orientation of the
joints. To do this we use Dip, Dip Direction and
the related measurements.
The
first concept to understand is the dip of a
joint. We shall initially consider a single
planar joint within the rock such as a shear
or fault. To define the direction of the joint
in three dimensional space we typically use
two values. By doing this we can use the
conventional 360 degrees that is commonly used
to define angles and directions. The two
angles typically used are called Dip and Dip
Direction.
To consider Dip and Dip Direction we need to consider a line on the plane of the joint. Imagine a grid of lines on this plane, with one set of lines running across the plane at the same height, effectively forming contours not rising or falling across the plane. The second set of lines on this grid will run perpendicular to the contours. This second set of lines would be the steepest line that could be formed on the plane. This line is the key line from which we measure Dip and Dip Direction.
The Dip Direction can be determined by viewing the plane and the line we have drawn on plan. The orientation of the line can be specified using the 360 degrees of a compass bearing. At this point it is worth noting that this value is not absolute, but that the baseline needs to be specified. Normally the baseline orientation is North but this could be true north, grid north or magnetic north.
Having specified the direction of the line, we need to specify how steep the line is. We do this using the Dip of the joint. To obtain the dip we have to cut the plane of the joint along the Dip Direction of the joint. Having cut the joint plane the Dip of the joint can be measured by measuring the angle from the horizontal of the plan of the joint. This means that a steeper dip gives a steeper joint up to a maximum of 90 degrees. Typically dip is only measured as a positive angle between 0 and 90 degrees.
Dip
and Dip Direction are the typical values that
are quoted to give the joint orientation.
There are however a few other dimensions that
are used to specify the orientation of
features. Strike is a value used instead of
Dip Direction. The Strike is a line at 90
degrees to the Dip Direction. Referring back
to our original diagram the Strike follows the
contour lines running parallel to the slope.
Strike and Dip Direction are therefore
interrelated. It is important to note that
Strike may not be defined using the
conventional 360 degree compass, but may be
defined as being an angle East or West of
North such as N72E, 72 degrees East of North.
Strike can be represented on a plan by a long
line following the direction of the strike and
a short bar at 90 degrees showing the
direction of the dip.
Another method of defining Dip and Dip Direction is Trend and Plunge. Trend and plunge tend to be used with linear features such as tunnels, shafts or roads. Trend tends to follow the centreline of these features such as the axis of a tunnel and is defined in the same way as Dip Direction. The plunge is then defined in the same way as the dip so the steeper the plunge the steeper the slope of the tunnel.
It is important to understand Dip and Dip Direction when designing rock structures. When using these measurements of the orientation of joints you must clarify how it has been defined by the geologist to ensure that you are applying the measurements in the same manner as they have been defined.
The
first concept to understand is the dip of a
joint. We shall initially consider a single
planar joint within the rock such as a shear
or fault. To define the direction of the joint
in three dimensional space we typically use
two values. By doing this we can use the
conventional 360 degrees that is commonly used
to define angles and directions. The two
angles typically used are called Dip and Dip
Direction.
To consider Dip and Dip Direction we need to consider a line on the plane of the joint. Imagine a grid of lines on this plane, with one set of lines running across the plane at the same height, effectively forming contours not rising or falling across the plane. The second set of lines on this grid will run perpendicular to the contours. This second set of lines would be the steepest line that could be formed on the plane. This line is the key line from which we measure Dip and Dip Direction.
The Dip Direction can be determined by viewing the plane and the line we have drawn on plan. The orientation of the line can be specified using the 360 degrees of a compass bearing. At this point it is worth noting that this value is not absolute, but that the baseline needs to be specified. Normally the baseline orientation is North but this could be true north, grid north or magnetic north.
Having specified the direction of the line, we need to specify how steep the line is. We do this using the Dip of the joint. To obtain the dip we have to cut the plane of the joint along the Dip Direction of the joint. Having cut the joint plane the Dip of the joint can be measured by measuring the angle from the horizontal of the plan of the joint. This means that a steeper dip gives a steeper joint up to a maximum of 90 degrees. Typically dip is only measured as a positive angle between 0 and 90 degrees.
Dip
and Dip Direction are the typical values that
are quoted to give the joint orientation.
There are however a few other dimensions that
are used to specify the orientation of
features. Strike is a value used instead of
Dip Direction. The Strike is a line at 90
degrees to the Dip Direction. Referring back
to our original diagram the Strike follows the
contour lines running parallel to the slope.
Strike and Dip Direction are therefore
interrelated. It is important to note that
Strike may not be defined using the
conventional 360 degree compass, but may be
defined as being an angle East or West of
North such as N72E, 72 degrees East of North.
Strike can be represented on a plan by a long
line following the direction of the strike and
a short bar at 90 degrees showing the
direction of the dip.
Another method of defining Dip and Dip Direction is Trend and Plunge. Trend and plunge tend to be used with linear features such as tunnels, shafts or roads. Trend tends to follow the centreline of these features such as the axis of a tunnel and is defined in the same way as Dip Direction. The plunge is then defined in the same way as the dip so the steeper the plunge the steeper the slope of the tunnel.
It is important to understand Dip and Dip Direction when designing rock structures. When using these measurements of the orientation of joints you must clarify how it has been defined by the geologist to ensure that you are applying the measurements in the same manner as they have been defined.
Monte Carlo analysis in Excel
30/11/08 19:48 Filed in: Analysis
Monte-Carlo analysis is one of those tools that I
have always found useful to keep in my pocket as an
engineer. I like to find an elegant solution to
problems limiting the number of variables and
amount of analysis that has to be done. Sometimes a
brute force approach can not however be avoided and
Monte Carlo analysis is a very effective way of
solving a complex risk or probability problem.
Because I don't use it much specialist tools such as @risk are not really an option for me. The basic version of Excel is however more than capable of performing Monte-Carlo analysis with relative simplicity.
The first stage is to build your model with the usual input and output parameters using the same format as you would normally do. Once the is done create a new worksheet for the input parameters that you want to use in he Monte-Carlo analysis. To generate the random numbers to put into the Monte-Carlo analysis you will need to use a Rand() function for each input parameter you want to consider. You can then use Excel's built in distribution functions to generate your output value or you can create your own - I will consider some of the different distribution functions in later posts.
If you have done this correctly, every time you amend your spreadsheet now the values should change on this worksheet. Now link the input values on your model worksheet to the values on the inputs worksheet. The entire model should now change each time values are adjusted on the spreadsheet. At this point it is worth taking time to check how your model is performing. Just pressing the delete key on an empty cell should run one Monte-Carlo step. It is worth doing this a few times because Monte-Carlo analyses often test the limit of a spreadsheet with combinations of very high and very low values.
It is possible to manually run a Monte-Carlo analysis manually with the spreadsheet in this form. The power of this solution is however when it is automated, and to do that we can make use of a Macro.
First create a new worksheet to hold the outputs from the model. Now create a new Macro. The typical code that should be used for the Macro is something like:
Sub MonteCarlo()
With Application
.Calculation = xlManual
End With
For Counter = 1 To 500
Sheets("Model").Select
Calculate
Output1 = Range("L3").Value
Output2 = Range("H17").Value
Set curCell = Worksheets("Results").Cells(Counter, 2)
curCell.Value = Counter
Set curCell = Worksheets("Results").Cells(Counter, 3)
curCell.Value = Output1
Set curCell = Worksheets("Results").Cells(Counter, 4)
curCell.Value = Output2
Next Counter
With Application
.Calculation = xlAutomatic
End With
End Sub
This Macro runs the Monte-Carlo analysis 500 times. The results from the analysis are taken from cells L3 and H17 from the worksheet Model. They are then put into columns 3 and 4 of the worksheet Results.
Post processing of the output data should be done in a separate workbook. If this is not done then re-running the Monte-Carlo analysis will take much longer as the post processing will be done for each step of the Monte-Carlo analysis.
Because I don't use it much specialist tools such as @risk are not really an option for me. The basic version of Excel is however more than capable of performing Monte-Carlo analysis with relative simplicity.
The first stage is to build your model with the usual input and output parameters using the same format as you would normally do. Once the is done create a new worksheet for the input parameters that you want to use in he Monte-Carlo analysis. To generate the random numbers to put into the Monte-Carlo analysis you will need to use a Rand() function for each input parameter you want to consider. You can then use Excel's built in distribution functions to generate your output value or you can create your own - I will consider some of the different distribution functions in later posts.
If you have done this correctly, every time you amend your spreadsheet now the values should change on this worksheet. Now link the input values on your model worksheet to the values on the inputs worksheet. The entire model should now change each time values are adjusted on the spreadsheet. At this point it is worth taking time to check how your model is performing. Just pressing the delete key on an empty cell should run one Monte-Carlo step. It is worth doing this a few times because Monte-Carlo analyses often test the limit of a spreadsheet with combinations of very high and very low values.
It is possible to manually run a Monte-Carlo analysis manually with the spreadsheet in this form. The power of this solution is however when it is automated, and to do that we can make use of a Macro.
First create a new worksheet to hold the outputs from the model. Now create a new Macro. The typical code that should be used for the Macro is something like:
Sub MonteCarlo()
With Application
.Calculation = xlManual
End With
For Counter = 1 To 500
Sheets("Model").Select
Calculate
Output1 = Range("L3").Value
Output2 = Range("H17").Value
Set curCell = Worksheets("Results").Cells(Counter, 2)
curCell.Value = Counter
Set curCell = Worksheets("Results").Cells(Counter, 3)
curCell.Value = Output1
Set curCell = Worksheets("Results").Cells(Counter, 4)
curCell.Value = Output2
Next Counter
With Application
.Calculation = xlAutomatic
End With
End Sub
This Macro runs the Monte-Carlo analysis 500 times. The results from the analysis are taken from cells L3 and H17 from the worksheet Model. They are then put into columns 3 and 4 of the worksheet Results.
Post processing of the output data should be done in a separate workbook. If this is not done then re-running the Monte-Carlo analysis will take much longer as the post processing will be done for each step of the Monte-Carlo analysis.
Shear banding in geomechanics
29/11/08 08:25 Filed in: Geomechanics
Shear bands a bands of high shear strains within a
material. They are common features of certain
geotechnical problems such as slope stability
failures. Simple numerical models, including FLAC
analyses can simulate the development of shear
bands when modeled correctly, and the development
of shear bands within the model can have a
significant influence on the results of the model.
In order to consider the formation of shear bands within numerical models, we will consider a simple model of a shear box test on a Mohr-Coulomb material.
When the two sides of the box slide apart, the shearing occurs across a very thin band as the soil grains slide across each other. In this case the shear band that is formed is very thin. We can simulate the shear box test relatively simply in most general numerical analysis packages, such as FLAC. In order to do this we need to discretise the problem, i.e. divide the problem into small elements. Typically there is a preference for a coarse discretisation as it gives the most efficient and simplest models.
In this model the shear box has been divided into a 4 x 4 grid giving 16 elements/zones. The base of the box has been fixed with roller boundary conditions on the three sides, as has the top of the box. A constant force is applied to the side of the box to model the shear force applied during the test.
We need to consider how this model will behave when the force applied to cause the shearing starts. The top section of the shear box has started to elastically deform under the shearing force.
The block of soil in the top section of the ground will behave roughly as a cantilever beam giving a linear variation in the stress shear stress through the model. Based on this stress distribution, we can obtain a basic stress distribution for elements within the model. This stress profile assumes that the elements behave as constant stress elements, a common assumption among general numerical analysis programs.
The stresses in elements 1 and 2 is therefore roughly an average of the theoretical stress and can only be correct at a few locations. The stress profile calculated by the model is therefore only an approximation of the actual theoretical stress profile. In order to consider the implications of this averaging we can increase the shearing in the theoretical model further.
The load, and hence the stress, in the section has been increased to the point where theoretical plastic shear failure will occur as indicated by the blue line. It can also be seen that the average stress in element 2 is not high enough to cause plastic failure, therefore the model will not exhibit plasticity at this load. Assuming a linear stress profile and the stress in the elements is the average theoretical stress across the element, the stress in element 2 will only be 75% of the limiting shear stress. In order to cause shear failure in element 2 to occur, the load on the model needs to be further increased.
In this case the theoretical stress has been increased to 33% above the shear stress limit so that the stress in element 2 will achieve the shear stress limit. We can now plot the distorted form of the model under additional load with the elements undergoing plastic deformation highlighted in blue.
In this case the numerical model is overestimating the strength of the ground by 33%. This is obviously not a conservative solution and indicates the problems that shear bands and other similar plasticity based problems can cause when carrying out numerical analyses.
A possible solution to this problem is to re-mesh the problem.
Now consider the case where the grid density has been increased by a factor of 2 in each direction. This results in a revised stress profile.
In this case the overstress required to cause plasticity is reduced to 14% and so the revised grid has significantly improved the performance of the model. We can plot how the accuracy of a model varies with increasing mesh density.Based on the mesh regenerating approach used above a density of 11 in each direction zones is required to get results with less than 5% overstress. Increasing the density of the zones does however reduce the speed of the solution of a typical model by an amount between the increase in density of the zones, or the square of the increase in the density of the zones. In this case then the models solution speed would decrease by between 7 times and 57 times. These could be large reductions in solution speeds for large models.
This is a very simplified way of looking at shear banding but the principals and logic will help understand some of the more complex problems. Lack of shear banding within certain types of geomechanical models is a critical reason why many models fail to produce credible results. Critically models without the ability to model shear banding will often overestimate soil strength. A modeller therefore needs to understand how shear banding forms and how to model it to create an effective and credible model.
In order to consider the formation of shear bands within numerical models, we will consider a simple model of a shear box test on a Mohr-Coulomb material.
When the two sides of the box slide apart, the shearing occurs across a very thin band as the soil grains slide across each other. In this case the shear band that is formed is very thin. We can simulate the shear box test relatively simply in most general numerical analysis packages, such as FLAC. In order to do this we need to discretise the problem, i.e. divide the problem into small elements. Typically there is a preference for a coarse discretisation as it gives the most efficient and simplest models.
In this model the shear box has been divided into a 4 x 4 grid giving 16 elements/zones. The base of the box has been fixed with roller boundary conditions on the three sides, as has the top of the box. A constant force is applied to the side of the box to model the shear force applied during the test.
We need to consider how this model will behave when the force applied to cause the shearing starts. The top section of the shear box has started to elastically deform under the shearing force.
The block of soil in the top section of the ground will behave roughly as a cantilever beam giving a linear variation in the stress shear stress through the model. Based on this stress distribution, we can obtain a basic stress distribution for elements within the model. This stress profile assumes that the elements behave as constant stress elements, a common assumption among general numerical analysis programs.
The stresses in elements 1 and 2 is therefore roughly an average of the theoretical stress and can only be correct at a few locations. The stress profile calculated by the model is therefore only an approximation of the actual theoretical stress profile. In order to consider the implications of this averaging we can increase the shearing in the theoretical model further.
The load, and hence the stress, in the section has been increased to the point where theoretical plastic shear failure will occur as indicated by the blue line. It can also be seen that the average stress in element 2 is not high enough to cause plastic failure, therefore the model will not exhibit plasticity at this load. Assuming a linear stress profile and the stress in the elements is the average theoretical stress across the element, the stress in element 2 will only be 75% of the limiting shear stress. In order to cause shear failure in element 2 to occur, the load on the model needs to be further increased.
In this case the theoretical stress has been increased to 33% above the shear stress limit so that the stress in element 2 will achieve the shear stress limit. We can now plot the distorted form of the model under additional load with the elements undergoing plastic deformation highlighted in blue.
In this case the numerical model is overestimating the strength of the ground by 33%. This is obviously not a conservative solution and indicates the problems that shear bands and other similar plasticity based problems can cause when carrying out numerical analyses.
A possible solution to this problem is to re-mesh the problem.
Now consider the case where the grid density has been increased by a factor of 2 in each direction. This results in a revised stress profile.
In this case the overstress required to cause plasticity is reduced to 14% and so the revised grid has significantly improved the performance of the model. We can plot how the accuracy of a model varies with increasing mesh density.Based on the mesh regenerating approach used above a density of 11 in each direction zones is required to get results with less than 5% overstress. Increasing the density of the zones does however reduce the speed of the solution of a typical model by an amount between the increase in density of the zones, or the square of the increase in the density of the zones. In this case then the models solution speed would decrease by between 7 times and 57 times. These could be large reductions in solution speeds for large models.
This is a very simplified way of looking at shear banding but the principals and logic will help understand some of the more complex problems. Lack of shear banding within certain types of geomechanical models is a critical reason why many models fail to produce credible results. Critically models without the ability to model shear banding will often overestimate soil strength. A modeller therefore needs to understand how shear banding forms and how to model it to create an effective and credible model.