Coursework – Resit Assessment Brief
UBGMW9-15-3 Computational Civil Engineering
Preamble
All assessments on this module are individual work. The work you submit must be your own work. Submitting work that is copied in part or whole from another student with or without their permission is an assessment offence.
You must fully attribute/reference all sources of information used during the completion of your submission, failure to do so constitutes plagiarism, which is an assessment offence.
If you are not familiar with the definitions of plagiarism and collusion, more information can be found here: http://www1.uwe.ac.uk/students/academicadvice/assessments/ assessmentoffences.aspx
Please ensure you are familiar with assessment procedures and policies, which can be found here: http://www1.uwe.ac.uk/students/academicadvice/assessments/ assessmentsguide.aspx
Structure of assessments
This module is assessed by two components, A and B:
• Component A is a one hour written exam and is weighted as 25 % of the final mark.
• Component B is a coursework portfolio and is weighted as 75 % of the final mark. The coursework portfolio described here asks you to consider two problems entitled:
1. Structural analysis under variable loads 2. Geotechnical slope stability
will
be
taken
as
The h
ighest
mark
obtained f
rom
your
solutions
submitted
to
Problems
1and2
your
fi
nal
coursework
mark
1
The final report on your coursework portfolio must include code routines developed for both elements in a text selectable form (no images or screenshots will be accepted).
Online blackboard submission due on 3rd August 2020.
The following two sections describe the problems you are to develop computer programs to solve. In each section, specific details of the tasks and outputs to feed in to your report are described. An overall summary of the assessment criteria is provided at the end of this docu- ment.
Structural analysis under variable loads
When dealing with variable loads the internal forces or reactions that a structure generates will vary according to a probability distribution. Then, the design of a structure is based on an output value of this distribution which has a small probability, on an absolute basis, of being exceeded. A workflow of this process is shown in Fig. 1.
1 – Generate samples for input variable UDL
2 – Compute output reactions/internal forces
3 – Plot outputs histograms
and estimate the 5% threshold
p ∼ N(μp,σp) +
MV
–
V
500 450 400 350 300 250 200 150 100 50 0
800 700 600 500 400 300 200 100
output value
+
M
-40 -35
-30 -25 V [kN]
0 -500 0
Figure1: Diagramofcomputationalanalysisforasimplysupportedbeamsubjected
to a variable uniformly distributed load (UDL).
Consider the isostatic structures shown in Figs. 2, 3, 4, 5, and the output reactions/internal forces presented in Table 1. You are asked to assess the variability of one these structures’ outputs when subjected to the shown loads. Each of the loads is assumed to follow a nor- mal distribution, e.g. for a uniform distributed load assume p ∼ N (μp , σp ) with mean μp and standard deviation σp.
Using MATLAB or other programming language generate 10000 data points for each load, according to its distribution parameters, and compute the corresponding output reactions/ internal forces.
500 1000 1500 M [kNm]
Dr Andre Jesus & Dr Richard Sandford 2 University of the West of England
p
A
1 2
34 P2
5
C
ab
M
P1
c
B
d
2
Figure2: Structure1
aaaa
P1 P2 P3 6
345
1
cp
Figure3: Structure2
b
p
P
f
Your report should include
• A description of the equations and histograms for each output reaction/internal force.
• Anestimateofthe5%thresholdoutputvalue,whichisdefinedhereasthevaluewhich is exceeded, on an absolute basis, by only 5 % of the load combination realizations.
1
46578
23
abcde
Figure4: Structure3
Dr Andre Jesus & Dr Richard Sandford 3 University of the West of England
p
2
abc
P
d e
f
1
35 4
Figure5: Structure4
Structure
Outputs
1
Bending moment at section C, bending mo- ment at section 5 and axial force at section 2
2
Axial force at bar 1-2 and shear force along sec- tion 2-3
3
Horizontal reaction at 2 and bending moment at 7 towards 5
4
Vertical reaction at 1 and bending moment at 4 towards 3
Table 1: Output reactions and internal forces
• A pseudocode or flowchart of the algorithm that underlies your analysis.
The structure and numerical values that each student has to consider are made avail- able on Blackboard Learning Materials > Coursework > Coursework values html file, or by following the URL https://blackboard.uwe.ac.uk/bbcswebdav/ pid-7216458-dt-content-rid-16362959_2/courses/UBGMW9-15-3_19jan_ 1/my_values.html
Furthermore, the results of validation tests against which you can test if your program is func- tioning correctly can be found here:
https://blackboard.uwe.ac.uk/bbcswebdav/pid-7318412-dt-content-rid-17299225_ 2/courses/UBGMW9-15-3_19jan_1/my_ex1_results%281%29.html
Dr Andre Jesus & Dr Richard Sandford 4 University of the West of England
Geotechnical slope stability
An important task in geotechnical engineering is to assess the propensity for a slope to col- lapse. It is common to analyse the stability of cohesive soil slopes by considering limiting plastic equilibrium. To carry out a limiting plastic equilibrium analysis, it is first necessary to define the failure mechanism, which is specified by the geometry of the failure surface. The mass of soil bounded by this failure surface is assumed to move over this surface as a free body in equilibrium. The forces and moments acting to induce failure are then compared with the resistance to slip that is mobilised along the assumed failure surface.
A variety of different failure surfaces can be considered, but a common choice is a circular seg- ment in two-dimensions. An important analysis case is that relevant to short-term conditions, immediately after a cutting is made or an embankment is built. In the short-term, there is in- sufficient time for excess pore water pressures to dissipate; such conditions are referred to as undrained. The shear strength, τ, along a failure surface in undrained conditions is constant and denoted as cu. The difficulty in carrying out a limiting equilibrium analysis is the choice of failure surface. The key task is therefore to find the critical failure surface, that is the failure surface along which failure is most likely to occur and, hence, gives the lowest factor of safety.
Figure 6: Example of the slope stability problem
Figure 6 is an example of the class of problem that you are to address. The figure shows a two-dimensional slope of constant inclination. The soil consists of a cohesive homogeneous soil of undrained strength, cu, and unit weight, γ. The slope overlies a stiff strata. The ge-
Dr Andre Jesus & Dr Richard Sandford 5 University of the West of England
ometry and material parameters shown in Figure 6 are an example for illustration – you have been assigned an individual problem, with a set of geometric and material properties that are individual to you and can be downloaded from: Blackboard Learning Materials > Course- work > Coursework values html file, or by following the URL https://blackboard. uwe.ac.uk/bbcswebdav/pid-7216458-dt-content-rid-16362959_2/courses/ UBGMW9-15-3_19jan_1/my_values.html.
Your task is to determine the safety factor against collapse for the slope geometry and ma- terials to which you have been assigned. The material properties (γ and cu) relevant to your individual problem are given on the diagram together with your slope geometry (which can be read-off from the scale). You are to consider only rotational failure along circular slip surfaces, but are to vary the radius and centre coordinates of the failure surface in order to find the min- imum safety factor against collapse. A bounding box, termed the ’search area’, is provided to limit the bounds on the search of your circle centre coordinates. The approach to minimising the safety factor by varying the location of the slip circle centre and its radius is your choice, although recommendations and possibilities are detailed in the supporting materials accom- panying the lectures.
For a particular choice of circular slip surface, the safety factor, SF is calculated as: SF = resisting moment
disturbing moment where the disturbing moment is given as:
disturbing moment = W d
and the resisting moment due to shear along the slip plane is given as:
resisting moment = cuR2θ
(0.1)
(0.2)
(0.3)
In these equations, W is the weight of the soil bounded within the failure surface (NB: which is NOT the same as the unit weight, γ), d is the horizontal distance from the slip circle centre to the centre of gravity of the soil mass bounded within the failure surface, R is the slip-circle radius and θ is the angle subtended by the slip surface (see Figure 7). Note that W and d are typically found by dividing the soil bounded with the failure surface into slices or rectangular segments and then taking area-moments about a convenient point. Substitution of Equations 0.2 and 0.3 into Equation 0.1 gives:
SF = resisting moment = cuR2θ (0.4) disturbing moment W d
To aid the validation of the computer program you will develop, a particular slope geometry is shown in Figure 8. For the particular circular slip line shown (i.e. the given circle centre
Dr Andre Jesus & Dr Richard Sandford 6 University of the West of England
position and radius), and for γ=18.5kN/m3 and cu=40kPa, the safety factor against collapse is 1.44 (correct to 2 decimal places). Demonstrating that your computer program can correctly calculate this safety factor is a valuable task and one you should document in your report. [You might find it valuable to note that for this problem: θ=84.06◦, R=17.43m and d=6.54m]. Please note that the solution of 1.44 is for this particular slope and material parameters to serve the purpose of validating your code – it is NOT the solution to your individual problem.
Note that to consider a variety of different combinations of the circle centre positions and circle radii in a time-efficient manner, it is necessary to implement a test as to whether a particular slip circle intersects the inclined or horizontal portions of the slope surface. Supplementary information is provided in the Appendix to help you to find the intersection points.
Figure 7: Parameters involved in the calculation of the safety factor Your report should include:
1. A description of the mathematical equations needed to find the safety factor against col- lapse.
2. The results of a validation case to demonstrate that your code can calculate the safety factor correctly for a particular choice of circle centre coordinates, slip circle radius and parameters that specify the geometry and strength of the slope.
3. Justificationofyourapproachtofindthecriticalslipcircleradiusandcentrecoordinates.
4. Pseudocodeoraflowchartshowingyourapproachto(i)findthesafetyfactorforagiven combination of slip-circle centre coordinates and slip-circle radius, and (ii) optimise the slip circle centre coordinates and slip-circle radius to find the critical safety factor.
Dr Andre Jesus & Dr Richard Sandford 7 University of the West of England
Figure 8: Validation problem geometry
5. A graphical presentation of the dependence of the safety factor on the slip circle centre coordinates.
6. Your calculation of the critical safety factor (as well as the circle centre coordinates and slip-circle radius that generated the critical safety factor).
Assessment criteria
Your report should contain the following and you will be assessed according to the criteria described in Table 2.
• Problem description: A summary of the problem you are attempting to solve, to include the assumptions needed to obtain a solution and any mathematical elaboration of the equations that are used within your computer program. (15%)
• Program development: The pseudocode or flowchart used to solve the problem, to- gether with an explanation and justification for your chosen numerical approach to solve the problem. Note that you are also required to submit, as part of your report, the code used to generate your results. (25%)
• Presentation of the results: To include plots showing the outputs from your work and accompanying text to describe their meaning. This section should include the outcomes of any validation exercises you undertake to demonstrate the correct functioning of the programs you develop. (50%)
• Concluding comments: To explain how your computer program could be extended or generalised for increased functionality. (10%)
Dr Andre Jesus & Dr Richard Sandford 8 University of the West of England
%
Descriptor
Problem description (15%)
Program de- velopment (25%)
Presentation of results (50%)
Concluding comments (10%)
80-100
Outstanding
Problem descriptions stated with outstanding clarity, with complete mathemati- cal treatment
Outstanding program de- velopment, with com- plete and thorough justification for chosen approach
Outstanding clarity of presentation with fully annotated plots, com- plete and fully cor- rect results and vali- dation test outcomes
Outstanding clarity of comments on the generalisa- tion of the computer program
70-79
Excellent
Problem descriptions stated with excellent clarity, with comprehen- sive math- ematical treatment
Excellent program de- velopment, with clear justification for chosen approach
Excellent clarity of presentation with well annotated plots, com- plete and fully cor- rect results and vali- dation test outcomes
Excellent clarity of comments on the generalisa- tion of the computer program
60-69
Very good: 65-69 Good: 60-64
Problem descrip- tions stated with clarity, with mostly complete mathemati- cal treatment
Program de- velopment presented that ad- dresses the main aims of the task with clear justification
Very good/good clarity of presentation with well annotated plots, mostly complete and correct results and some vali- dation test outcomes
Very good/good clarity of comments on the generalisa- tion of the computer program
Dr Andre Jesus & Dr Richard Sandford 9 University of the West of England
50-59
Competent: 55-59 Adequate: 50-54
Problem descriptions stated with adequate clarity, with basic math- ematical treatment
Program de- velopment presented that ad- dresses some as- pects of the task with partial justification
Competent/ adequate clarity of presenta- tion with plots, some complete and correct results and limited val- idation test outcomes
Competent/ adequate clarity of comments on the generalisa- tion of the computer program
40-49
Weak
Problem descriptions lacking clar- ity, with minimal or only par- tially correct mathemati- cal treatment
Program de- velopment presented that ad- dresses limited as- pects of the task with limited justification
Limited clarity of presenta- tion with few plots, incomplete results and limited val- idation test outcomes
Limited clarity of comments on the generalisa- tion of the computer program
30-39
Poor (FAIL)
Problem descriptions unclear, with incomplete or incorrect mathemati- cal treatment
Program develop- ment that is incom- plete with very limited justification
Poor clarity of presen- tation with very few plots, in- complete and incor- rect results and limited validation test out- comes
Poor clarity of com- ments on the generalisa- tion of the computer program
Dr Andre Jesus & Dr Richard Sandford 10 University of the West of England
<30
Very poor (FAIL)
Problem descriptions very un- clear, with no math- ematical treatment
Program de- velopment
that
to
the
and justification
fails address brief lacks
Very poor
clarity presenta- tion lacking plots, in- complete and incor- rect results and very limited val- idation test outcomes
of
Very poor
clarity comments on the generalisa- tion of the computer program
of
Table 2: Assessment Criteria
Dr Andre Jesus & Dr Richard Sandford 11 University of the West of England
Appendix - A guide to solving the Slope Stability Problem
The slope stability problem can be addressed by developing the following five functions: slope_safety_factor, calculate_intersection, choose_intersection, calculate_disturbing, calcu- late_restoring. This appendix details the content of those five functions to aid your coding developments. For each of the five functions, the purpose of the function is summarised, and suggested input and output variables are described, with reference to Table 3. Some comments are also added on suggested techniques to undertake the task involved in writing each func- tion. In carrying out your work, it is important to set up a coordinate system (e.g. place the origin at the toe of the slope) and consistently adhere to that coordinate system.
Variable
safety factor
undrained shear strength
bulk unit weight
slope width
slope height
slip circle radius
coordinates of the slip circle centre
coordinates at the toe of the slope*
coordinates at the crest of the slope*
coordinates of the first intersection point between the slip circle and the slope profile
coordinates of the second intersection point between the slip circle and the slope profile
lower x component of the intersection point forming between the slip circle and the inclined portion of the slope*
upper x component of the intersection point forming between the slip circle and the inclined portion of the slope*
Symbol unit -
kPa kN/m3 m
m
m
m
m
m
m
m m m
SF cu
γ
w
h
R (xc, yc)
(xtoe, ytoe) (xcrest, ycrest) (x1, y1)
(x2, y2) xslope,L xslope,U
Table 3: Nomenclature used in the Appendix for the slope stability problem
* NB: all coordinates need to be taken relative to a common origin - such as the toe slope
Dr Andre Jesus & Dr Richard Sandford 12 University of the West of England
Figure 9: Slope stability problem nomenclature
Dr Andre Jesus & Dr Richard Sandford 13 University of the West of England
1. FUNCTION 1: slope_safety_factor
• Function purpose: This function carries out the overall calculation of the safety fac- tor (using Equation 0.1) and controls the calls to the other functions. Since Function 1 is the controlling function (i.e. it calls a series of other functions) it is suggested you start your developments by looking at Function 2.
• Inputs: This function takes as its input:
(a) the strength parameters cu and γ
(b) any two parameters needed to specify the slope geometry e.g. w and h
(c) the three parameters needed to specify the geometry of the slip circle, xc, yc, R
• Outputs: This function takes as its output: (a) the safety factor: SF
• Techniques needed: The main technique needed is the ability to make a calls func- tions. The relevant MATLAB syntax that is needed is:
[output_1, output_2] = my_function(input_1, input_2)
where:
my_function
is the name of the function being called and:
output_1, output_2
are the output arguments (those coming back from my_function) and:
input_1, input_2
are the input arguments (those being passed to my_function)
• Other notes: The suggested structure of slope_safety_factor is as follows. A call should first be made to choose_intersection (which itself will contain calls to calcu- late_intersection). Calls should be then to calculate_intersection to obtain the final intersection points. Then calls should be made to disturbing_moment and restor- ing_moment. The final task of slope_safety_factor is to take the ratio of the restoring and disturbing moment, according to Equation 0.1, to calculate the safety factor.
Dr Andre Jesus & Dr Richard Sandford 14 University of the West of England
2. FUNCTION 2: calculate_intersection
• Function purpose: This function returns the point(s) of intersection between a
generic straight line and a circle.
• Inputs: This function takes as its input:
(a) the three parameters needed to specify the geometry of the slip circle, xc, yc, R
(b) the set of parameters needed to define the straight line - if using LINECIRC, this will be the slope and y-intercept of the straight line (defined in the same
coordinate system as used to define the geometric properties of the slip circle)
• Outputs: This function returns as its inputs:
(a) the x-coordinates of the intercept point or points
(b) the y-coordinates of the intercept point or points
• Techniques needed: The code for LINECIRC, as posted on Blackboard, can be used for this directly. You need to copy and paste this code into a MATLAB function, validating that it works correctly by comparing with some hand-calculations. Alter- natively, there is an earlier short video on Blackboard showing you how to develop this function from first principles.
• Other notes: In general, a line may pass through a circle (generating two intersec- tion points), OR it may be a tangent to a circle (generating one intersection point) OR it may not pass through the circle as all (generating no intersection points) - see Figure 10. In the last case (no intersection points), LINECIRC will return NaN (Not a Number) values to indicate that there are no intersection points. []Note that, if you wish to test if a NaN is encountered in Matlab, there is an in-built function, "isnan" available].
Figure 10: Intersections of a straight line and a circle: LINE 1 has two intersection points (shown in blue), LINE 2 has one (shown in orange) and LINE 3 has none
Dr Andre Jesus & Dr Richard Sandford 15 University of the West of England
3. FUNCTION 3: choose_intersection
• Function purpose: This function will enable you to work out the intersection points between the slip circle and your slope profile. To do this, choose_intersection will need to make use of calculate_intersection, so you must have developed Function 2 before progressing to develop this function.
The slope profile can be considered to be made up of three straight lines: two that are horizontal and one that is inclined. The horizontal line at the top of the slope will be called the crest line and the horizontal line at the bottom of the slope will be called the toe line. As shown in Figure 11, there are four possible ways in which the slip circle can cut the slope:
– Case 1: both intersection points are on the inclined portion of the slope
– Case 2: one intersection point is on the crest line, the other is on the inclined
portion
– Case 3: one intersection point is on the inclined portion, the other is on the toe
line
– Case 4: one intersection point on the crest line, the other is on the toe line.
[Please note that your problems have been constrained so that it is not possible for both intersection points to be on the crest line or both to be on the toe line.]
To be able to determine the intersection points for a given slip circle and your slope profile, you need to be able to determine which of the four cases listed above applies to a particular slope profile and choice of slip circle. Once you know which of the four cases applies, you can then determine the intersection points straightforwardly by calling the function calculate_intersection.
• Inputs: This function takes as its inputs:
(a) the three parameters needed to specify the geometry of the slip circle: xc, yc, R
(b) the width, w, the height, h of the slope and the coordinates of the toe of the
slope, xtoe and ytoe, so that the slope and y-intercept of the line defining the inclined portion of the slope can be readily calculated.
• Outputs: This function returns as its output:
(a) the coordinates of the two points of intersection forming between the slip circle
and the slope profile: (x1, y1), (x2, y2)
• Techniques needed: This function will rely on you making extensive use of conditional statements. To be able to make a decision as to which of the four cases applies, you will need to be able to make use of if-elseif-else construct in MATLAB. When you call calculate_intersection with the parameters specifying the inclined line of the slope profile, you will be able to distinguish between the four cases shown in Figure 11. Specifically, one set of suitable tests would be:
– Case 1 if: xslope,L ≥ xcrest AND xslope,U ≤ xtoe – Case 2 if: xslope,L < xcrest AND xslope,U ≤ xtoe – Case 3: if: xslope,L ≥ xcrest AND xslope,U > xtoe
Dr Andre Jesus & Dr Richard Sandford 16 University of the West of England
– Case 4: if xslope,L < xcrest AND xslope,U > xtoe
where xslope,L is the lower x-component of the intersection point between the slip circle and the inclined portion of the slope and xslope,U is the upper x-component of the intersection point between the slip circle and the inclined portion of the slope. As an example, see Figure 12, which shows the position of the points of intersection between the sloping portion of the slope profile and the slip circle, with the positions of xslope,L and xslope,U labelled. This example is for Case 2, and it is readily apparent from this figure that xslope,L is less than xcrest and xslope,U is less than xtoe – consistent with the tests stated above.
You need to implement tests such as those listed above as conditional statements to be able to distinguish between Cases 1-4.
• Other notes: Once you know which of the four cases applies, it is then straight- forward to be able to determine the intersection points themselves. This is done by calling calculate_intersection with the slip circle geometry and the parameters specifying the appropriate lines, depending on which of the four cases applies. For example, if Case 2 applies, to determine the intersection point on the crest line, a call to calculate_intersection is needed, passing the slip circle parameters and the parameters specifying the crest line. Of course, this will return, in general, two in- tersection points and you will need to use another if-else construct to deduce which of the two is needed (for this example, it will be the intersection point with the lower x value). It is a good idea is to plot your slope, the slip circle and the intersection points to check that what you have done is correct.
Developing this function is, perhaps, the biggest challenge to undertaking the coursework – think logically and work methodically (it boils down to a sequence of conditional statement tests).
Dr Andre Jesus & Dr Richard Sandford 17 University of the West of England
(a) both intersection points are on the inclined portion of the slope
(b) one intersection point is on the crest line, the other is on the inclined portion
(c) one intersection point is on the inclined portion, the other is on the toe line
(d) one intersection point is on the crest line, the other is on the toe line
Figure 11: Possible cases for the intersection of a circular slip surface with the slope
Dr Andre Jesus & Dr Richard Sandford 18 University of the West of England
Figure 12: Figure to show the variables, xslope,L and xslope,H
Dr Andre Jesus & Dr Richard Sandford 19 University of the West of England
4. FUNCTION 4: calculate_disturbing
• Function purpose: This function will enable you to calculate the disturbing mo- ment, once you know the intersection points. You therefore need to have completed the developments of calculate_intersection and choose_intersection before develop- ing this function.
• Inputs: This function takes as its inputs:
(a) the bulk unit weight of the soil, γ
(b) the coordinates of the two intersection points, (x1, y1) and (x2, y2)
(c) the width, w, the height, h of the slope and the coordinates of the toe of the
slope, xtoe and ytoe, so that the slope and y-intercept of the line defining the inclined portion of the slope can be readily calculated.
• Outputs: This function returns as its output: (a) the disturbing moment, MD
• Techniques needed: The most straightforward way to determine the disturbing moment is to divide the soil bounded between the two intersection points (as deter- mined using choose_intersection) into a series of small filaments. A loop will then be needed to step through each filament and to determine its moment contribution. It is suggested you make use of the for-loop construct in Matlab to carry out this task.
• Other notes: To determine the mass of each filament, you will need to work out the area of the filament. The area of the ith element is calculated by multiplying its width, ∆xi and its height, hi. To calculate the height, hi, you will need to work out the y-coordinates of the top and bottom of the filament. The y-coordinate at the top is straightforward (as the slope profile is made of straight lines). You will need to use conditional statements (if-elseif-else) again to be able to decide which straight line segment portion applies to each filament. The y-coordinate at the bottom of the slip circle is determined from the equation defining a circle. Once the area of the ith filament is known, its mass is obtained by multiplying by the unit weight, γ. Once the mass is known, its moment contribution is obtained by multiplying by the moment arm, xi (horizontal distance from the slip circle centre to the filament centre). The total disturbing moment is calculated by summing the contributions from each of the individual filaments. Summarising the above into one equation, we have:
N
MD = γ(hi∆xi)xi
1
[Please note that there are different approaches to carry out the task of calculating the disturbing moment – for example, you could find the distance to the centre of gravity, d, analytically from the polygon defining the perimeter of the soil region undergoing failure. However, the approach above is perhaps the most intuitive and easiest to imple- ment].
where N is the number of filaments.
Dr Andre Jesus & Dr Richard Sandford 20 University of the West of England
Figure 13: Variables used in the calculation of the disturbing moment
Dr Andre Jesus & Dr Richard Sandford 21 University of the West of England
5. FUNCTION 5: calculate_restoring
• Function purpose: This function will enable you to calculate the restoring moment, once you know the intersection points. You therefore need to have completed the developments of calculate_intersection and choose_intersection before developing this function.
• Inputs: This function takes as its inputs:
(a) the undrained shear strength of the soil, cu
(b) the coordinates of the two intersection points, (x1, y1) and (x2, y2)
(c) the width, w, the height, h of the slope and the coordinates of the toe of the
slope, xtoe and ytoe, so that the slope and y-intercept of the line defining the inclined portion of the slope can be readily calculated.
• Outputs: This function returns as its output: (a) the restoring moment, MR
• Techniques needed: The restoring moment is readily calculated once the length of the arc enclosing the body of soil undergoing failure is known. This can be calcu- lated with recourse to geometry. Specifically, with reference to Figures 14 and 15, the arc length is calculated as:
where: where:
The restoring moment, MR, is then calculated by multiplying the arc length by the undrained shear strength and the slip circle radius:
MR = LarccuR
This function should just be just two or three lines long! Matlab has inbuilt trigono-
metric functions – make sure you don’t get degrees and radians mixed up!
Larc = Rθ θ=2arcsin( s )
2R s=(x1 −x2)2 +(y1 −y2)2
Dr Andre Jesus & Dr Richard Sandford 22 University of the West of England
Figure 14: Figure showing the parameter, s
Figure 15: Figure showing the parameter, s
Dr Andre Jesus & Dr Richard Sandford 23 University of the West of England
6. NEXT STEPS
(a) Validation: The problem brief gives a particular geometry of slope, a particular slip circle radius and centre coordinates and particular strength parameters – see Figure 8. Using these values, your code developed as a result of completing the aforemen- tioned function development should return the safety factor value of 1.44. PLEASE note that this safety factor value is applicable to the validation problem geometry and strength parameters (i.e. those listed in this pdf) and NOT your individual problem. You are not given the solution to your individual problem – it is for you to find the critical safety factor value.
(b) Optimisation: The final step, needed to achieve a high mark, is to optimise to find the lowest (i.e. the critical) values of the safety factor. This is readily done using a grid-search approach. You are encouraged to develop a separate function, which will make use of a series of for-loops to attempt many different values for the possi- ble centre coordinates of the slip circle in your defined ’search area’. For each (xc, yc) coordinate pairing that you consider, you will need to consider a range of different R values, such that you consider the range of slip circles that extends from the circle that just intersects the slope profile to one that touches the stiff strata. You will then need to record the value of R that gives the minimum safety factor – this will be the critical safety factor for the current choice of xc and yc. Finally, you will need to select the global minimum – the lowest safety factor recorded for all pairings of xc and yc that you considered. This is a relatively simple development, once you have the five functions detailed in the appendix working correctly.
(c) Plotting: A contour plot, showing the minimum safety factor obtained for each of the (xc,yc) values considered in the search area is a valuable way to display the results. This is readily achieved in Matlab using the command:
where
contour(x_Values, y_values, SF_results)
x_Values, y values
are vectors containing all the pairings of xc, yc and SF_results
contains the corresponding safety factor results.
Dr Andre Jesus & Dr Richard Sandford 24 University of the West of England