Distance Calculation Techniques

The skill of estimating relies on the estimator’s ability to accurately calculate lengths, heights, areas, volumes, and weights. In actual practice, however, accuracy may not be enough, especially in fast-phase construction projects. Speed is essential, so the need to know quick calculation techniques is inevitable. Here are some techniques for calculating distances.

Direct Measurement
The most direct way to measure distances like length and height is by the use of measuring instruments like measuring tape, laser, pedometer etc...  However, in situations when it is impossible to actually measure the distance, there are alternative methods that can be used.

Triangle Method
Using the Pythagorean Theorem, many hard to measure distances can be calculated. Here are some possible situations where the triangle method can be applied. 

Unknown height

H= √(D^2-L^2 )

Unknown length

L= √(D^2-H^2 )

Unknown diagonal distance

D= √(L^2+H^2 )

Pacing
Pacing is an approximate method of calculating surface distances. By knowing your pace factor (meter per pace), you can calculate the approximate distance by multiplying it by the number of steps. Do not use pacing in estimating items that need precision especially those involving expensive materials.

Getting your pace factor
Find a flat road where you can measure at least 20 meters straight. Mark the beginning and the end of 20 meters. Walk straight through the 20-meter line while counting your steps. Remember to walk on a normal pace the way you walk on a daily basis. Record the number of steps that you take to cover the distance. Repeat this step three times. Divide 20m by the number of steps and get the average result of the three trials. The number you will get is your pace factor. If it takes you 26 steps in average to walk the 20 m distance, then your pace factor is 20/26 = 0.769meter per pace.

Using your pace factor
Set eyes on the distance that you want to measure by pacing. Put your ankle at the starting point and start walking at a normal pace and count your steps as you go until the end of the distance you are measuring. Multiply the number of steps you counted by your pace factor. The resulting number is the approximate distance measurement. 100 steps, for example, is 100 x 0.769 = 76.9 meters.

Using GPS
A modern way to do approximate horizontal distance measurement is by the use of GPS or global positioning system. There are a number of apps available that allow users to measure distances using GPS. Examples of such apps are the apps used by runners to track their performance.

Using leveling hose
Leveling hose with water inside is an old yet efficient tool to measure vertical distances or level differences. Just find a flat vertical surface that you can mark on. Match the water level with the level of the higher line or surface that you are measuring and mark on the flat surface the water level at the other end. The height difference can be measured using a measuring tape.

Excavation and Backfill Calculation

Calculating excavation and backfill is not as easy as getting the volume of soil. Angle of repose and compaction need to be considered.

A. Angle of repose

Angle of repose is the angle loose materials form from the horizontal when it is placed freely on a flat surface. So when excavations are made, it is not only the volume of the soil right under the specified area that needs to be calculated but also the surrounding area around the perimeter that forms ramps because of angle of repose of the soil that became loose after excavation.

Angle of repose depends on the materials’ coefficient of internal friction. Here are the approximate average values of angle of repose for specific types of materials.

Table 1 –  Average Angle of Repose of Different Materials

Type of Material	Angle of repose (degrees)
Dry Sand	27.5
Moist Sand	37.5
Wet Sand	30
Ordinary Dry Earth	32.5
Ordinary Moist Earth	35
Ordinary Wet Earth	27.5
Gravel	39
Gravel, Sand and Clay	25

You can actually measure the angle of repose of a material. Get a sample of the loose material. Pour it on a flat surface and measure the angle it makes from the horizontal.

By applying the law of tangents, we can calculate the horizontal distance X that will be added to the volume calculation.

tanθ=  DEPTH/x

x=  DEPTH/(tanθ)

Example:

Calculate the total volume of soil to be excavated for 10m x 20m pit with the depth of 2m. Assume the soil to be ordinary moist.

Given:

            Depth = 2m

            θ = 35 degrees for ordinary moist earth

            Area = 10m x 20m

Solution:

            x = DEPTH / tan θ

               = 2m / tan35

            x = 2.86 m

Base dimension: 10m by 20m

Top dimension: (10m+2x) by (20m+2x)

Base area = A1 = 10 x 20 = 200 m²

Top area = A2 = (10+2*2.86)(20+2*2.86) = 404.3 m²

V = 2/3(200+404.3+sqrt(200+404.3)

V =  592.44 m³

B. Compaction

Backfills are usually compacted up to 25% of the fill volume. On the other hand, excavated materials are expected to expand up to 25% in volume when loaded for transport. By considering compaction and loosening up, we could get the idea.

Volumes of fills need to be multiplied by 1.25 to allow compaction.

V _fill = V _actual x 1.25

Example:

Find the required volume of fill for 10m x 10m x 3m pit. Assume the pit to have sheet piles on the edges to prevent soil from slipping down.

Given:

            V _actual = 10 x 10 x 3 = 300 m³

Solution

            V _fill = V _actual x 1.25 = 300 x 1.25 = 375m³

Bigger is NOT Always Better - Pipe Clogging and Duct Choking

Clogging and choking have always been a consistent problem with piping and duct systems whether it be hydraulic or pneumatic. The common layman's solution to this is to use a bigger pipe or duct. This intuitive solution, however, is not always effective. In contrary, using larger diameter might actually be the source of pipe clogging and duct choking problems.

The Math

There are two major factors that influence the pipe/duct size selection process - flowrate or discharge and pressure. Flowrate and pressure are included in two independently different formulas which may be hard for some to correlate.

Q = AV ... (Equation 1)

Where:
Q = Flow Rate
A = Cross-Sectional Area
V = Velocity

P = F / A  ... (Equation 2)

Where:
P = Pressure
F = Force
A = Area

By analyzing these two equations, the science behind the relationship between clogging and pipe size can be understood. As you can see, Cross-Sectional Area is common to both Equations 1 & 2. By substitution, we can say that:

A = Q/ V
A = F / P

But,

A = A

Then,
Q / V = F / P

or

P = V F/Q  ... (Equation 3)

In practice, how is clogging dealt with? Pressure is applied to force the obstruction out. To prevent clogging, sufficient pressure should be specified at the design stage. The conclusions about Pressure that can be drawn out of the equations above are as follows:

1. Pressure is inversely proportional to area (from Equation 2). This means that as the cross-sectional area increases, the pressure decreases.

2. Pressure is directly proportional to design velocity (from Equation 3). This means that as the velocity increases, the pressure increases.

3. Velocity is inversely proportional to cross-sectional area (from Equation 1). This means that as the cross-sectional area increases, velocity increases.

One can see right away that by considering pressure’s major role in declogging, using bigger pipes or duct is not the solution because using larger diameter increases the cross-sectional area and increasing cross-sectional area decreases pressure and velocity. At the design stage, the designer must specify a sufficient design velocity and pressure that will not allow clogging. It can be done by identifying the mass, density and mass flow of the possible clogging materials and calculating the necessary force and pressure to keep such materials moving across the system. The viscosity of the fluid and the roughness coefficient of the pipe or duct must always be considered to arrive at an efficient design.

Self-cleaning action in waste pipes

Waste pipes that use gravity to allow water to move are the most prone to clogging because these merely rely on the slope of the pipe. The ideal slope to allow self-cleaning action is 2% (2 units vertical and 100 units horizontal). If the slope is too steep, the water would flow faster than the solid waste so it will be left out and accumulate at the bottom of the pipe and would eventually clog the pipe. If the slope is too small, the velocity will not be enough to carry the solid waste which will result in sedimentation, accumulation, and clogging. For gravity drainage systems, the design slope which dictates the design velocity is critical.