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Quantitative concepts and skills

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The weight of rocks is a force on underlying surfaces. ... A block of rock sits on an inclined plane, and there is nothing holding it back ... – PowerPoint PPT presentation

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Title: Quantitative concepts and skills


1
Module 1-2B
Density of Rocks
B. Some applications
Quantitative concepts and skills Unit
conversions Forward calculations using trial and
error Logic functions
Calculating stresses originating from the weight
of overlying rocks
2
PREVIEW
The weight of rocks is a force on underlying
surfaces. If the surface is horizontal
(perpendicular to gravity), the load creates a
normal stress, which is the weight (force)
divided by the area of contact. If the surface
is inclined, the load creates both a normal
stress and a shear stress on the contact surface.
The magnitude of the stresses depends on the
angle. The stresses can be calculated from the
load, the area of surface contact, and a
trigonometric ratio. This module considers two
types of problems. Both types are discussed in
your textbook. The first (covered in Chapter 2)
involves the stress, strength, and factor of
safety for a rock roof resting on one or more
columns in an underground room (or cave or
tunnel). The second (Chapter 4) involves the
normal and shear stresses, cohesion force, and
inclination angle for a slab of rock resting on
an inclined surface. The structure is the same
for all of the problems. The spreadsheets do a
forward calculation. You assume specific values
for the independent variables, and the
spreadsheet calculates the factor of safety. A
logic function looks at the factor of safety and
states whether the design is acceptable. Then,
you modify the assumed input values until the
logic function returns the desired result.
Derringh, E., Computational Engineering Geology,
Prentice Hall, 1998
3
  • Will rock columns
  • hold up the roof?

A block of rock sits on a column of rock. This
situation can arise if a room or tunnel is
excavated underground and some of the rock is
left to support the roof. If the stress on the
column due to the weight of the rock held up by
the column exceeds the compressive strength of
the column, the column should fail, and the roof
should fall. The weight of the block, and
therefore the stress on the column, depend on the
density of the rock in the roof.
Weight of block held up by column
Stress on column
Cross-sectional area of column
B
Strength of column
Factor of safety (FS)
Stress on column
50 cm
If FS
4
Column holding a roof, 2
7 m
Does this column hold this section of the roof?
4 m
12 m
0.5 m
0.75 m
Recreate this spreadsheet and figure out the cell
equations for the empty cells. The next slide
gives the results.
Units of stress and strength are Pascals (Pa).
MPa is megaPascals.
Formula in Cell E28 is a logic function IF(E15/E2
4E26,Yes,NO)
5
Column holding a roof, 3
6
Column holding a roof, 4
Exercises with this spreadsheet 1. Trial and
error Find the greatest thickness (i.e., depth
below ground) that the column will support,
keeping the factor of safety at 1.5. 2. What is
the minimum area for a column that would support
this 7-m roof (FS 1)? 3. Continuing the
preceding question Starting with a column that
can just barely support the roof (FS 1) --
double the thickness of the roof rock, and find
how many columns it would take to support the new
roof?
7
A variation -- Columns holding the roof of a
tunnel.
A 7.5-m-wide tunnel is to be dug 550 m into a
hillside of granite, 120 m below the top of the
ridge. The columns are to have an area of .375 m
by 1 m and consist of a material with an
unconfined compressive strength of 79 MPa. How
may columns will you need, not allowing for a
factor of safety (i.e., FS 1)?
Use the spreadsheet of the preceding problem and
trial and error.
120 m
7.5 m
550 m
0.375 m
1 m
8
Columns holding the roof of a tunnel, 2
So 400 columns arent enough. Keep going until
the No becomes a Yes.
Did you get tired of the trial and error and
calculate the answer directly using algebra?
Does that answer agree with the spreadsheet
result?
9
  • Will a rock slab slide
  • down an incline?

A block of rock sits on an inclined plane, and
there is nothing holding it back except its
weight against the underlying material and the
friction between the block and the underlying
material. This can happen when stratified
sedimentary rocks dip out of a hillside or out
into a road cut.
A component of the blocks weight acts along the
incline. This is the driving force tending to
make the block move down the slope.
The resisting force is the force of static
friction plus a force of cohesion between the
block and the underlying material. The force of
static friction is a coefficient times the
component of the blocks weight acting
perpendicular to the incline (forcing the block
against the underlying material). The force of
cohesion is another coefficient times the contact
area between the block and the underlying
material.
The question is, What is the ratio of total
resisting force to the driving force?
10
Rocks on inclined plane, 2
21 m
25 m
Arkose (? 2.03 g/cm3)
13 m
Is this slab stable on this slope?
17º
µ 0.25 c 15 kPa
Adapted from Derringh, E., Computational
Engineering Geology, Prentice Hall, New Jersey,
1998, fig. 4.5.
Recreate this spreadsheet and figure out the cell
equations for the empty cells. The next slide
gives the results.
Force of static friction m W cos(a) where m
coefficient of static friction. Force of
cohesion cA where c coefficient
(cohesion stress) and A contact area.
Either ok or BEWARE
11
Rocks on inclined plane, 3
12
Rocks on inclined plane, 4
  • Exercises with this spreadsheet
  • Trial and error. Find the maximum angle for
    which the slope is stable for this slab of rock.
  • Calculate the arctangent of the coefficient of
    static friction. Compare this to the result you
    get if you let the cohesion stress be zero and
    again find the maximum slope that is stable. How
    do you explain this result?
  • 3. Suppose there was an earthquake, and the
    additional downward force per unit mass is 0.2 g
    during the downstroke of the ground shaking.
    Would the slope be stable?

13
End of Module Assignments
Assignment due 30 Sept 2003
  • Do the three exercises in Slide 6 and turn in the
    spreadsheet with the answer for each of them.
  • Answer the question asked in Slide 7 and turn in
    the spreadsheet supporting your answer.

Assignment due 2 Oct 2003
  • Do the three exercises in Slide12 and turn in the
    spreadsheet with the answer for each of them.
  • This module illustrates problem solving by
    forward calculation using trial and error. Write
    a paragraph on this method of problem solving.
    What are the pros and cons of this method?
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