Situations provided by: Prof. Don Coduto, Civil Engineering Department, Cal Poly Pomona
Edited by: Prof.
Phil Rosenkrantz, Industrial
and Manufacturing Engineering Department, Cal
Poly Pomona
The following situations are examples from the field of
civil engineering where variation occurs and statistical method either are
or could be applied. In some instances the current practice may not be very
sound based on statistical theory. These applications are useful for analyzing
and discussing the use of statistics in the practice of civil engineering.
Traffic/Transportation
Engineering
Given: Distribution of hourly
traffic volume on a proposed highway
Find: The hourly volume to
be used for design
Commentary: The design volume
dictates the required number of lanes, etc. For most highways, the design
volume is that which corresponds to the 30th busiest hour in
the year.
Discussion: What are the ramification
of using this "rule of thumb" for estimating the design volume?
How can the "distribution" affect how the rule works (ie. distribution
of interarrival times: normal vs. exponential),
For Further Study: Queuing
Theory
Given: A series of measurements
from the field, each of which is made using an instrument that has a certain
precision.
Commentary: Many measurements
in surveying are made using a series of instrument setups. For example,
consider a piece of property whose boundary consists of 8 line segments.
The orientation of one of these segments with respect to true north is known
to a precision of 20" of arc. A surveyor sets up a "total station"
(an instrument that measures angles and distances) at each of the 8 corners,
and measures the angles to a precision of 20" and the distances to
a precision of 0.01 ft. These measurements are then "adjusted"
so that satisfy the rules of geometry.
Find: The precision of the
computed orientations of the other property lines. It is not 20"!
For Further Study: "Precision"
and "accuracy" are two very important aspects of measurement.
Also see "Repeatability and Reproducibility (R&R) Studies".
Given: The Uniform Building
Code dictates certain design values for the live load on building elements
(live load is that load induced by furniture, inventory, occupants, moveable
objects, etc., as contrasted to dead load (the structure itself), earthquake
load, wind load, etc.). For example, the code-specified design load for
classrooms is 40 lb/ft^2.
Find: The probability that
this design load will be exceeded.
Commentary: I've seen some
statistical data on live loads based on actual measurements in buildings.
I couldn't find it in my files, but I think the COV is about 0.10, and the
probability of exceeding is about 0.05.
A design method called "load and resistance factor
design" (LRFD), which our students learn in a junior-level course,
uses "load factors" and "resistance factors" that have
been developed from extensive reliability analyses. Although design engineer
simply uses the specified factors in a typical design, it is important to
understand how the building codes developed them.
Earthquake
Engineering/Seismology
Given: A proposed dam to be
built at a certain site, the probability of earthquakes of various sizes
occurring on faults at various locations, the predicted peak horizontal
ground acceleration at the dam site from each of these earthquakes.
Find: The design peak ground
acceleration, which is the one that corresponds to a certain probability
of being exceeded during the design life of the dam.
Given: Stream flow records
for the Santa Ana River near Corona.
Find: The stream flow (ft^3/s)
that corresponds to a particular recurrence interval. This flow can then
be used to design dams, levees, etc.
Commentary: After a devastating
flood in 1938, the Army Corps of Engineers built Prado Dam to protect Orange
County. This dam and its spillway can be seen from the 91 freeway (the spillway
has 1776-1976 painted on it). The reservoir behind this dam is normally
empty, and is intended to capture the excess flood waters in the event of
another major flood, thus keeping the stream flow below the dam at manageable
levels. This dam was designed based on the best hydrologic data and analyses
then available. However, subsequent analyses performed in the 1970s and
1980s found the flood that corresponds to a certain recurrence interval
(200 years?) is much greater than had previously been considered. Such a
flood could overtop the dam, causing it to fail, thus producing massive
flooding in Orange County. Because of this, the Army Corps of Engineers
is currently building a massive flood control project along the Santa Ana
River, which includes a new dam upstream of Prado (Seven Oaks Dam), new
levees, and the raising of Prado Dam. This is a massive construction project,
all of which is based on a statistical analysis of hydrologic data.
Given: A series of groundwater
samples obtained at different locations and depths in an aquifer, and the
concentration of a certain chemical in each sample.
Find: The probability that
the concentration at any point in the aquifer exceeds some specified value.
Given: Four soil samples obtained
from a certain stratum of soil and the results of laboratory consolidation
tests on each sample. These test results are used to compute the settlement
that will occur if a certain load is placed on this strata.
Find: Considering only variations
due to the sample locations (i.e., assuming the sampling method, testing,
and analysis introduce no uncertainty), compute the probability that the
settlement will exceed some specified value.
Commentary: I've seen data
that indicate COV values of 0.26 to 0.52 from multiple samples obtained
from "homogeneous" strata. Thus, analyses based on mean values
could produce results that are seriously in error.
Related analysis: Given the
cost of sampling and testing, how many samples should be taken
Traffic/Transportation
Engineering
Given: An access road for a
new recreation area is to have a toll booth. The distribution of vehicles
per minute, the time required to process each vehicle, and the length of
roadway required for each waiting vehicle are given.
Find: The required "storage
area", which is the minimum required length of roadway between the
toll booth and the adjacent highway such that the probability of waiting
vehicles extending onto the highway does not exceed a certain value.
Commentary: The acceptable
probability depends on the consequences of being exceeded. For example,
if the highway is a busy one-lane road, a very low design value would be
used, whereas if it is a low-volume four-lane road, perhaps a higher probability
would be acceptable.
A similar analysis also could be done based on the probability
of exceeding some maximum acceptable waiting time.
Based on the results of such analyses, we might decided
to have more than one toll booth. If so, how often would both booths need
to be in service?
For Further Study: Queuing
Theory (study of waiting lines)
Given: A series of samples
of wastewater ("water" from a sewer) and results of biological
oxygen demand (BOD) tests
Find: The design BOD, which
is the value that corresponds to a certain probability of not being exceeded.
This value would be used to design the wastewater treatment plant.
Commentary: A deterministic
approach would use the mean BOD and apply some factor of safety.
Traffic/Transportation
Engineering
Given: An intersection between
a residential street and a major artery in a suburban community. The residential
street has a stop sign, but the artery does not.
Find: The probability that
a vehicle stopped at the stop sign will have to wait more than a specified
period before making a left turn onto the artery.
Commentary: If this probability
is too high a traffic signal may be warranted. Traffic engineers perform
this kind of analysis to determine where to place traffic signals.
For Further Study: Poisson
processes. Interarrival times.
Given: Anticipated distribution
of cargo ship arrivals at a port in the year 2010, and the mean time required
for a ship to occupy a berth.
Find: The number of berths
required so the probability of a ship having to wait more than a certain
number of hours to enter a berth is no more than x.
Commentary: A large amount
of construction is currently under way at the ports in Long Beach and Los
Angeles. Portions of this construction would be based on these kinds of
analysis.
For Further Study: Queuing Theory, Simulation