Mathematics for Construction and the Built Environment
The purpose of this assignment is to:
Outcome 1: Be able to apply analytical methods to construction problems.
Outcome 2: Be able to apply analytical methods to surveying and setting out procedures. Outcome 3: Be able to apply statistics to construction problems.
Outcome 4: Be able to apply analytical methods to engineering problems.
For all the following tasks, please show all the steps clearly in the solution and justify the use of relevant theories, formulae, techniques, etc.
Task 1
Scenario:
In a construction project, the equipment and materials are some of the main resources. The equipment may include loaders, dozers, etc. and the material may include oil, tires, spare parts, etc. A construction project leader should determine the resource required, i.e. equipment and materials required. In the following cases, the calculations to determine resource (e.g. material) requirements are faced.
Task 1.1
A bulldozer is estimated to cost $86,000 new and to have a useful life of 5 years with a salvage value of $14,000. The company believes that a realistic MARR would be 10%. Taxes, insurance, and storage should amount to an additional 8%, which results in an overall cost of money of 10 + 8, or 18%. To recover ownership cost, what is the appropriate amount per hour that must be charged for the equipment usage if the expected use rate of the equipment is 1,200 hours per year?
For D3, you should assess whether the hourly charge for the bulldozer is appropriate, and provide an evaluation of how change in MARR, taxes, insurance, and storage cost, and yearly usage of the bulldozer can affect the ownership cost. You need to provide a detail evaluation for each factor.
Scenario:
In project planning, assuming that construction of a power station will take three years. The cost of construction is $1.2 million and it is equally spread over three years ($400,000 in year 1 and $400,000 in the following year 2 and year 3). We also assume that the power station has an operating life of 18 years. The entire project, therefore, spans 21 years. To simplify matters we assume that there is no inflation during the entire life of the project, and that all benefits and costs are certain as assumed. We also presume that during the 18 years of the power station operation the structure must be maintained at some costs. These operating costs are $50,000 per year. As for benefits, the power station project will produce 5,000 MWH of electricity each year (after construction) at a cost of $0.05 KWH. This cost represents a savings of $0.02 per KWH over the next best method of electricity generation. These savings in electricity cost constitute a benefit. The power station will also produce a reservoir for recreation. We assume that there will be 50,000 person days per year of recreation benefits and that the value of these recreation benefits is $1.00 per person per day.
Task 1.2
Initially, the annual total benefits and total costs for the power station project should be calculated. You are also requested to perform calculations for cost analysis on Net Present Value (NPV) for the project whole life span of 21 years for annual interest rate is
a) 8%
b) Assuming the interest rate is 6% per annum, how much the cost of saving per KWH can the power company adjust such that she still maintain a profit on the NPV at the end of the 21 years span?
The formula used to calculate the NPV is:
NPV = ∑t=1T(benefitt - costt)/(1+r)t
where B represents benefits in the project C represents costs in the project
r means the annual interest rate
t refers to the number of year in the project life span
For M1, you should identify and apply appropriate theories of cost analysis.
Use the benefit-cost-ration (BCR) to determine what annual interest rate is justifiable to demonstrate the construction of the power station is a worthwhile project. It is assumed that the annual interest rate remains the same throughout the project life span. The formulae for the BCR is given as follow:
BCR = (Σri=1Bt/(1+r)t)/(Σri=1Bt/(1+r)t)
where B represents benefits in the project C represents costs in the project
r means the annual interest rate
t refers to the number of year in the project life span
BCR is larger than 1 implying the project is a worthwhile project. Because the benefits of the project exceed the project cost when BCR > 1 at the minimum acceptable rate of return (MARR).
For D2, Plan, manage and organize calculation, diagrams and substantial activities of organization a building of a power plant. The calculations and planning should include the depreciation, change in interest rate, change in operation costs, labor costs throughout the power plant life span, the net present value of all concerned items of the power plants, a thorough optimization plan on the plant evaluation and the developed statistical data supporting the sustainability of the plant operation.
Interest
Task 2.1
(a) As an engineer, you are requested to solve this surveying problem. Calculate the area and side-widths of the cross-section in the following figure.
Scenario:
In the geometric design of roads, drains, etc., the setting out of curves is an important aspect of the engineer's work. An engineer may use trigonometry to support setting out as follows.
Task 2.2
You need to perform calculations to support the setting out of a horizontal curve. Two straights intersecting at a point B have the following bearings, BA 270°, BC 110°. They are to be joined by a circular curve which must pass through a point D which is 150m from B and the bearing of BD is 260°. Determine (a) the required radius and (b) the tangent lengths.
Task 3
Scenario:
As for environmental protection measure to measure the efficiency of a gas combustion, thirty eight laboratory measurements are made of the work developed by combustion of gases within an enclosed container.
Task 3.1
The initial condition of a volume of 2,5 in3 is measured for each test. The test results of work developed by combustion of gas are provided as follows:
76, 78, 81, 82, 84, 84, 86, 86, 87, 88, 88, 88, 89, 91, 91, 92, 92, 92, 94, 94, 98, 101, 103, 103, 103, 104, 104, 106, 108, 108, 112, 113, 114, 116, 116, 118, 118, 119
(i) Develop histograms diagram for work by gas combustion using an appropriate cell size to present the data such that an acceptable shape of distribution can be obtained.
(ii) Identify the relative frequency histogram of work by gas combustion.
For M3, you should use properly statistical language and a range of methods of presentation such as tables and diagrams.
Scenario:
As an engineer, presentation of construction data and use of statistical methods to solve problems involving estimation are essential. Use probability functions such as mean, variance, standard of deviation, and covariance to solve the problem.
Task 3.2
A sample of five tests was taken to determine the unconfined compression strength (in tons/foot2) of soil, with the test results shown in table 3.2.
Table 3.2 Soil strength of sampled soil
| Sample number |
Strength of tested soil (Tons/foot2) |
| 1 |
2.8 |
| 2 |
3.5 |
| 3 |
6.9 |
| 4 |
4.6 |
| 5 |
4.3 |
(i) Compute the variance and standard deviation of the soil strength.
(ii) Use coefficient of variation to determine the safety of the soil when it is used in a site construction.
Scenario:
As an engineer, use of statistical methods to solve problems involving prediction and quality control is essential. Use probability estimation and confidence level in sampling distribution to answer task 3.3.
Task 3.3
Analyse the probability distributions for discrete and continuous data with the following situations.
a) Concrete blocks are tested and it is found that, on average, 7% fail to meet the required specification. To pick up 9 concrete blocks from a batch of concrete blocks in a construction site, , determine the probabilities that less than three blocks out from the chosen 9 concrete blocks will fait o meet the specification.
b) A construction company wants to recruit a technical assistant to work in a new site and the offered salary is HK$11,500. Current research has indicated that the average monthly income for a technical assistant in construction industry is above 13K Hong Kong dollars. It is also known that the standard deviation of income is $9000. A random sample of 30 technical assistants' income is shown in Table 3.3. Use sample data from the table to predict whether the construction company is able to recruit the assistant? Use the level of confidence to support your claim on this recruitment matter.
i) Use z value to predict the chance of hiring a technical assistance using an offered salary of HK$11,500.
ii) Applying the z value theorem, determine how much monthly income would be able to recruit a new technical assistance, by the construction company, to work in the new site with a chance of 90% successful.
The formula for finding the z value is given as follows:
z = (X‾-u)/(σ/√N)
where μ is the population mean, σ is the population standard deviation, and N is the sample size.
X‾ is the sample mean.
The z value table of normal sampling distribution is shown in Appendix 1.
Table 3.3 Sample of monthly income of technical assistance
| 13800 |
16500 |
18500 |
| 12500 |
12000 |
11800 |
| 11200 |
9800 |
14500 |
| 14300 |
11800 |
16500 |
| 10050 |
13600 |
11500 |
| 12800 |
15500 |
10900 |
| 9600 |
13900 |
10800 |
| 8900 |
20000 |
15000 |
| 11300 |
11600 |
11800 |
| 12800 |
13500 |
12800 |
For D1 question - From the z value equation, discuss the difference of the sampling distribution if z value computed is to represent a large population while the sample size taken is 40. The degree of freedom of the sampling distribution would be a main issue. Use data in Table 3.3 for discussion. Validate your conclusion when small sample of data is going to be applied to a large population which represents the whole technical assistance employment sector in construction industry in Hong Kong.
Task 4
Scenario:
In a construction project, vector analysis is required to solve problems in engineering mechanics.
Task 4.1
Solve the following engineering problem using vector analysis. P4.1:
The floor crane and the driver have a total weight of 1135 Kg with a center of gravity at G .
(a) If the crane is required to lift a 227-Kg drum, determine the normal reaction on both the wheels at A and both the wheels at B when the boom is in the position shown.
(b) Determine the largest weight of the drum that can be lifted without causing the crane to overturn when its boom is in the position shown.
(Hint: Draw the free body diagram for the crane system first.)
Figure 4.1
M2 - Apply relevant theories and techniques, with justification, analyze the shear forces and bending moments exerted on the base of the floor crane when the drum is at the position shown, then draw the shear forces and bending moment diagram
Scenario:
In a construction project, a slope excavation is shown by change of slope shape. Use calculus to find the area enclosed by curves represented the prior and final slope shape as shown in Figure 4.2
Task 4.2
The two slope shapes were represented by two curves y=x2 and y2=8x. Use integral calculus to the area shown by the shaded lines in Figure. In this area is rotated 360° about the x-axis, determine the volume of the soil of revolution produced. The volume of revolution V, obtained by rotating area A through one revolution about the x-axis is given by the following expression:
V = a∫b Πy2dx
V = a∫b Πy2dx
