Measurement in Construction

Background

I have been teaching students how to use scales for a number of years. After three or four tests, some students still do not exhibit the mastery that they should. I have asked myself, “What can I do to improve students’ learning?” Before I can answer that, I need to identify the problem. One possibility is that students are not learning some of the basics, such as how to use a ruler or add and subtract fractions. Perhaps they were absent the day in elementary school when some of these topics were taught; or, perhaps the teacher did not have enough time to cover the subject adequately so that all students obtained mastery. Whatever the reasons, I decided I needed to review some of the basics. What basics should be taught and to what extent? This paper is an attempt to identify what students should know to be masters of measurement in construction. The headings included below can be taught in one course or scattered throughout the curriculum, but they should be taught somewhere. A student assignment is included that is intended to stimulate students to think about these issues, if not for the first time, at least in a new way.

Introduction

The Greek word, metron, means “a measure” (NIST, 1991). “Metrology” is the science of weights and measures (Merriam-Webster, 1995). Sometimes the word, “metrics” (not to be confused with the metric system) is used to refer to units of measurement (whatever they may be) employed in a particular application. This paper focuses on measurement in the construction industry—primarily that of length and angle—and associated concepts. Other disciplines such as physics have an array of different units that are important to those disciplines but are not included here.

Units of measurement are arbitrary but standardized. In this context, a standardized unit means that the unit of measurement is defined by some authority and accepted by a large number of people. In the United States, the authority is the National Institute of Standards and Technology (NIST) which is administered by the Department of Commerce. The authority for the Metric system is the International Bureau of Weights and Measures located at Sevres, France.

Quantity and Quality

We attempt to interpret and control our environment by naming and describing things. We use two different methods to describe the environment. One method is qualitative. A qualitative description tries to identify the peculiar and essential character of a thing; a qualitative description uses words. Another method is quantitative. A quantitative description uses numbers. Generally speaking, specifications tend to be qualitative, and drawings tend to be quantitative. In order for a quantitative description to be useful, the numbers must represent units that have a standard definition such as feet, inches, pounds, degrees, hours, and so forth. An instrument used to apply measurement to anything is often called a scale. Two examples are a ruler which measures distance and a thermometer which measures temperature. Units on a scale are usually uniform or identical, but not always. Units on a logarithmic scale vary, but they vary in a predictable way.

A Brief History of Numbers

People probably began keeping track of things by marking straight lines or notches on semi-permanent material such as bone (Roche, 1998). We still use this system for keeping score in games such as dominoes. Egyptian hieroglyphics dating from 3400 BC contain groups of vertical straight lines to designate quantities of things (Motz & Jefferson, 1993). The Romans developed a system which used letters to represent numbers. I = 1; V = 5; X = 10; L = 50; and so forth. This works well enough for small amounts but is very awkward for manipulating large numbers. It is easy to confuse letters used for words and letters used for numbers, and it is impossible to use Roman numerals for fractions. We still use the Roman system today in outlines and to designate pretentious events like the Super Bowl.

The greatest advancement in numbers began when the Hindus adopted different symbols for quantities greater than one around the third century BC. This system was passed on to the Islamic world around the eighth century AD, and then progressed to Europe a century or two later (Motz & Jefferson, 1993). These symbols are still referred to as Arabic numerals and are the familiar numbers we use today.

Perhaps the second greatest advancement in numbers was the addition of the symbol and concept of zero. This symbol could be used as a placeholder so that position had meaning. Position could be marked by a symbol such as a dot or period. Because our present numbering system uses a base of ten, we call the dot a decimal point. The first position to the left of the decimal point represents ones; the second position represents tens; the third position represents hundreds; and so forth.

The abacus also uses position to calculate amounts. A frame is strung with wires that beads can slide on. Each wire represents a magnitude of ten. Calculations are done by sliding the beads. However, once the beads are moved, no record remains of the calculation. There is no way to check for errors. The abacus is still used in China and Japan (Encarta’95).

The decimal system simplifies calculations and conversions. To divide a number by 100, you move the decimal point two places to the left; to convert meters to millimeters, you move the decimal point three places to the right. However, there are systems that use a base other than 10. The binary system uses a base of 2. The Babylonians used a sexagesimal system which has a base of 60. For some purposes, the Romans used a duodecimal system which has a base of 12.

Units of Measurement

In the ancient world, units of measurement were based on the human body. Examples are the digit, palm, span, and cubit. These units had the virtues of being always at hand (so to speak) and not requiring an advanced political system. In the Bible, the Lord gave Noah specific instructions on how large to make the ark in terms of cubits. The problem with these ancient measures is a lack of precision—the size of human bodies can vary quite a bit. As societies became more sophisticated, they tried to standardize these units. The pharaoh or king was considered the most important person in the realm so he was typically used as a model. The Egyptians established the Royal Cubit for their standard unit of length which allowed them to build with more precision. In addition, they created a hierarchy of smaller units that related to each other and the cubit. “Surviving examples of the Egyptian Royal Cubit show a length varying between 52 and 53 cm. One example was divided into four spans, and each span into 7 digits or finger breadths” (Roche, 1998. p. 24). The compelling need here was not to idolize the pharaoh but to standardize.

How the ancient units of digit, palm, span, and cubit transformed into the inch, foot, yard is not completely understood (NIST). A Welsh law dating from the 14^th century states: “Be it Remembered, That the Iron Yard of our Lord the King, containeth three Feet and no more. And a Foot ought to contain Twelve Inches…It is ordained that three grains of Barley dry and round do make an inch; Twelve inches make a foot; Three feet make a Yard” (Roche, 1998, p. 25). Notice that this standard is already based on readily available units (grains of barley) and not the foot of the king.

Another English unit is the furlong (or furrow-long). Tudor kings established the furlong to be 220 yards which led Queen Elizabeth I to replace the Roman mile of 5,000 feet with a mile of 5,280 feet making the mile exactly 8 furlongs (NIST).

Mistakes and Errors

Mistakes are generally caused by inattention or carelessness. Mistakes are sometimes called blunders, boners, or boo-boos. Mistakes can and should be eliminated. It is wise to follow the old carpenter's advice: "Measure twice; cut once."

An error is the difference between a measured quantity and its true value (Nathanson, 2006). This can be the result of imprecise instruments, the method of measurement, by natural factors such as temperature, or by random variations in human observation (Nathanson). Unlike mistakes, errors can never be completely eliminated, but they can be minimized. There are two kinds of errors: systematic and random.

Systematic errors are the result of imprecise instruments or method of measurement. Also, natural factors such as extreme temperature can expand a metal rule or tape to indicate measurements that exceed the true value. Systematic errors tend to be cumulative and repetitive. Cumulative errors can add up to be significant and negatively influence a construction project.

Random or accidental error is the difference between a measured quantity and the true value for that quantity that is free from mistakes and systematic error (Nathanson, 2006). Greater skill and more precise instruments can minimize random error but never completely eliminate it. Random errors follow the laws of chance and can be studied and estimated using statistical procedures.

Accuracy and Precision

Webster's Collegiate Dictionary makes little distinction between accuracy and precision; however, students of construction should be able to differentiate between these two terms. Accuracy refers to the degree of perfection obtained in the measurement or how close the measurement is to the true value; precision refers to the degree of perfection used in the instruments, methods, and observations (Nathanson, 2006). Generally speaking, precise instruments are required to achieve accurate results; however, it is possible to have precise instruments and not achieve accuracy. An example would be a very precise timepiece that was inaccurate because it was set to the wrong time.

Precision and accuracy are important in construction for both aesthetic and practical reasons. Materials that fit together tightly without gaps are appealing to the discerning eye and typically make a building stronger and more watertight. For some manufacturing and scientific processes, an environment built very accurately and precisely is essential. Accuracy is limited by the technology available and economics. Generally speaking, accuracy and precision cost money. This is especially true of analog technology. When all time pieces were analog, the Swiss were famous for making very precise and expensive watches. There was a direct relationship between precision and cost.

Digital technology represents a genuine revolution in construction because it allows us to measure much more precisely without a proportionate increase in cost. Lasers, computers, software applications, global positioning systems, total stations, etc. allow us to measure and layout buildings with great precision. However, it should be emphasized that measuring precisely and building accurately are not the same thing. In construction, measuring precisely and building accurately still costs money. Precision may require expensive equipment, more time, and more skill. Measuring and building accurately typically reaches a point of dimensioning returns. In other words, tools, equipment, time, and personnel establish a cost-effective level of precision. Building more accurately and precisely will dramatically increase the cost.

As previously stated, precision is limited by the available technology. One anomaly of precision and accuracy in the ancient world is the great pyramid at Gizah. It is not only impressive for its size but also for the precision and accuracy to which it is oriented and constructed. It has a footprint of 13 acres, and consists of six and a half million tons of limestone and granite blocks (Hancock, 1996). The length of the sides do not vary more than 8 inches from each other which is an error or tolerance of about one tenth of 1 percent. The corners are almost perfect right angles. “The variation from 90 degrees is just 0º 00’ 02” at the northwest corner, 0º 03’ 02” at the northeast corner, 0º 03’ 33” at the southeast corner, and 0º 00’ 33” at the southwest corner” (Hancock, p. 41). A modern Vernier plate allows us to measure slightly less than 5 minutes of arc with accuracy. Moreover, the pyramid is aligned to the cardinal points of the compass with the average deviation from true being only a little over 3 arc minutes. This is much less than anyone could detect with the naked eye. How did they do it? We do not know. What we do know is that the owner cared about accuracy and precision and was willing to pay for them.

Dimensions and Precision

Dimensions imply precision. For instance, a dimension on a floor plan such as 12 ft.-3 1/16 in. implies that the builder can and should build to a precision of 1/16 in. This is typically not cost effective. Unfortunately, CAD applications encourage this kind of mistake. Computers are very, very precise. Inexperienced designers may allow this precision to appear as units on their drawings notwithstanding the implications of such precision. Designers should not use a precision that is impractical or impossible to achieve in the field.

As previously mentioned, accuracy indicates how close a measurement is to the true value. In construction, there must be a way to verify if something that has been built corresponds to what was specified (its true value). This is accomplished by using the concept of tolerance.

Tolerance

The definition of tolerance in construction is the permissible deviation from a specified size or dimension (Harris, 1993). In other words, how much deviation is the owner willing to tolerate? A tolerance is usually specified by establishing upper and lower limits to a given dimension. An example: 10 ft.-8 in. ± ¼ in. This is equivalent to stating: 10 ft.-7¾ in. ≤ permissible dimension ≤ 10 ft.-8¼ in. The allowable range is ½ in. inclusive. A more precise alternative would be ± 1/16 in. which would allow a range of 1/8 in. Tolerance may be specified in any units that can be measured and verified, including angles.

In construction, precision has no meaning without a given tolerance. For instance, a set of construction documents may specify that a concrete slab be flat and level. Well, how flat is flat? One person’s idea of flat may be dramatically different form another person’s idea of flat. How could that specification be enforced? A better alternative would be, “the surface of the concrete slab shall vary in height no more than ½ in. in any 100 ft. of length.” A good specification might even indicate how the tolerance is to be measured and verified. In this case, designers should consult with concrete finishers to verify what tolerance is cost effective and then specify a more precise tolerance only if the owner needs the extra precision and is willing to pay for it.

Let us take a look at how tolerance can affect the construction of a simple brick wall. If the wall can deviate no more than ± one degree in every 100 feet of length (approximately 1.75 ft.), and the wall should be roughly oriented toward north, then any good brick layer should be able to meet the specifications. But, if the specified tolerance is ± one arc minute per 100 feet of length (approximately .35 inches), and the wall must be oriented exactly due north, then you will need a laser theodolite, a geological survey map accurate to 30 feet, and a highly qualified team of professionals including a surveyor, an astronomer, several master-masons, and a week or more to verify the results.

Significant Figures

Significant figures are probably more familiar to scientists than constructors; however, constructors should understand the concept because significant figures indicate how exact a measurement is or can be. There is no point, and in fact can be misleading, to use more figures than are significant. Some students are apt to do this because of inexperience, and because computers and calculators are so precise. For instance, a calculator may display an answer to nine digits or more. This does not mean that the student should use all nine digits. If the least precise instrument used can indicate only four digits, there is no point to record more than four.

Significant figures are numbers that can be read directly from the measuring instrument plus one number that must be estimated. For instance, if a measuring tape has a scale that only indicates feet and inches, only three significant figures can be read from that tape—one for feet, one for inches, and one that estimates a fraction of an inch. The number of significant figures are an indication of precision but not accuracy. Scientific notation is often used when the number of significant figures is important because it eliminates some of the ambiguity relating to zeroes. Some rules governing significant figures are as follows:

1. Zeroes placed at the end of a decimal number are counted as significant.

2. Zeroes to the right of the decimal, in numbers smaller than one, are not significant.

3. Zeroes to the right of the digits in a number written without a decimal are generally not significant.

4. As a general rule, zeroes that are used solely as a place holder (to indicate where the decimal is) are not significant.

Significant figures are used in a decimal system. Another way to obtain more precision is to use fractions.

Fractions

The need for more precision required architects and constructors to divide standard units of measurement into smaller units. The smaller units are indicated as a fraction of the larger unit. For instance, one foot equals 1/3 of a yard, and one inch equals 1/12 of a foot. A fraction is simply the quotient of two numbers. The top number is the numerator; the bottom number is the denominator. The denominator is the base of that measuring system. For some construction projects, the inch may provide all the precision required. To provide greater precision, the inch is divided into 16 subunits. These units do not have a name except the name of a fraction; so we say 3/16ths of an inch. Fractions should always be reduced whenever possible. So 8/16 of an inch should be reduced to ½ inch; 6/8 of an inch to ¾ inch.

For most projects in the United States, constructors use the foot and inch system. This means that constructors use tools that measure in these units. Common tools for measuring length are the tape measure, yard stick, and carpenter’s square. It makes no sense to specify units that are not indicated on these tools. For instance, 3-1/7 inch cannot be measured using any of these tools. Dimensions smaller than one inch should always be a multiple or a reduction of 1/16.

The same principles apply to a decimal system except that in a decimal system, the denominator is always a multiple of 10. So in the Metric system, a millimeter is 1/1000 of a meter.

Ratio

The dictionary defines ratio as the relationship in quantity, amount, or size between two or more things: proportion (Merriam-Webster). Fractions measure length; ratios more commonly measure angles. For instance, it is common to indicate roof slope using a ratio of rise to run. A roof slope may be 5/12. This means that for every 12 horizontal units, the rise is 5 units. A ratio is also indicated by using a colon instead of a horizontal line (5:12). A sidewalk should not exceed a slope of 1:12 to accommodate the handicapped. Notice that ratios are independent of units of measure.

A famous and useful ratio is known as the ropestretcher’s triangle or the Egyptian triangle (Doczi, 1981). Every year the Nile would overflow its banks and inundate the Nile valley. This would fertilize the land, but it also wiped out any landmarks. The Egyptians would have to resurvey every year. One of their methods was to tie knots in a long rope to create 12 equal spaces. With this rope, they could easily make the 3-4-5 triangle by holding it at the third and eighth knots. This always forms a right triangle. Trigonometry is an extension of this concept. Trigonometric functions like sine and cosine are simply ratios of the sides of right triangles expressed in decimal format.

Proportion

The dictionary defines proportion as follows:

The relation of one part to another or to the whole with respect to magnitude, quantity, or degree.

Harmonious relation of parts to each other or to the whole (Merriam-Webster).

A proportion is a ratio, but it is much more. Harmonious proportions have aesthetic value. This is experienced in music and the visual arts. Sophisticated systems of proportion have been developed in architecture to promote beautiful design. Architects in the Renaissance codified proportional systems for the different styles of classical architecture. In these systems, the basic unit is the module which is always equal to the radius of the lower part of the column. A module is divided into 30 equal parts. When more precision is required, the part is further divided into fractions of a part (Rattner, 1998). Theses systems allow the architect to design at any size and retain the harmonious proportions of the whole. In modern parlance, these proportional systems would be called scalable. Fonts in your computer software are scalable because they can be printed at any point size and maintain the correct proportions.

Modular Design and Construction

The word “module” denotes a basic unit of measurement from which all other dimensions are derived. The radius of a column in classical architecture mentioned above is a good example. Classical architecture developed in a pre-industrial age when everything was hand crafted. Greek temples were created as if they were large pieces of sculpture—beautiful, but time and labor intensive. The industrial age and its processes have helped create a high standard of living for many people. With mass production came the need for comprehensive coordination between design, manufacture, and construction. The flexible module used in the past was not adequate for post industrial needs. There was a definite shift of emphasis from aesthetic value to practical and economic value. This implies a standard unit of measurement.

A highly sophisticated modular system developed in Japan that is unique in world architecture (Engel, 1985). This is based on a module called the Ken. The ken is divided into six smaller units called shaku which are approximately the same as the English foot. Two different modular systems developed. One system measured the ken from the centerlines of the structure; the other measured the ken from the faces of the structure. The difference has important implications for a unique system of floor mats called tatamis. Contrary to popular conception, the tatami has never functioned as a construction module (Engel). However, its influence on room proportions has been significant. The difference between the two ken modular systems was not crucial in a pre-industrial age, but it does emphasize the difficulty in adopting a universal modular system in a post-industrial age. Both of the ken modular systems have advantages and disadvantages. Even today, many tatami producers will not make the mats for a house until they have measured the house after construction. This does compromise some of the advantages of mass production. In the United States, wary cabinet makers may follow the same procedure.

A comprehensive approach to all building components is called modular coordination. Computers and electronic technology allow and encourage data exchange between designer, manufacturer and constructor. The dramatic results of mass production are based on the concept of many identical products fabricated according to a standard. One of the marvelous possibilities of computer aided design (CAD) is that it allows customized mass production. This is a growing trend in construction (Hutchings, 1996).

The two most common types of modular construction in the United States are modular building systems and panelized building systems (Carper, 1990). “Modular systems consist of one or more three-dimensional modules that are 90 to 95 percent complete when they are shipped from the factory” (Carper. P. 5). Panelized systems are building components such as walls and trusses that are built in a factory and assembled on site. Wall panels can be either open or closed depending on how much work is scheduled to be done on site. Structural Insulated Panels (SIPS) are another system that uses panelized construction and has become popular because of its energy saving features. Modular construction should not be confused with mobile homes. Modular homes are governed by the same codes as site-built homes and can be very high quality.

Both modular and panelized systems encourage using a modular dimensioning system. According to Ernest R. Weidhaas (1999), modular coordination has the following advantages:

Masonry is a good example of a modular system. The most common module is 4 inches because brick, concrete block, window frames, and door bucks are all available in 4 inch modules. Moreover, 4 inches is a multiple of wall framing and panel products such as gypsum board and plywood. Countries using the metric system use a 100 mm module (4 inches = 101.6 mm) (Weidhaas, 1999). Cabinets are an exception. They are based on a 3-inch module. It should be emphasized that no modular system has proven to be applicable to all uses. Typically, exceptions must be made; however, it is often better to use a modular system and make exceptions as needed than to forego any dimensioning scheme at all.

Actual and Nominal Dimensions

Nominal dimensions have much in common with modular coordination. When nominal dimensions are used in a modular system, they are identical. Nominal dimensions for wood and masonry construction are similar, but they developed for different reasons. Wood will shrink as it dries, and it will shrink differently according to species, location, weather, and time of year when cut. Therefore, many years ago, carpenters could not accurately anticipate the size of lumber when it was ready for construction. This has changed dramatically with tree farms, kilns, and advanced knowledge about forestry. Today, the actual sizes of lumber are very consistent; however, they are different from nominal sizes. For instance, a 2 x 4 is actually 1-1/2 x 3-1/2 inches; a 2 x 8 is 1-1/2 x 7-1/4 inches. It is absolutely essential to understand how nominal dimensions relate to the dimensions given on a floor plan. There are four different ways you can dimension walls on a floor plan:

1. Dimension to the faces of the framing.

2. Dimension to the centerline.

3. Use nominal dimensions.

4. Dimension to the faces of the finished wall.

Method 4 is seldom used in residential construction but is quite common in commercial construction. Method 3 has the great advantage of modular coordination. Methods 1 and 2 are straightforward and easy to understand so are used by many drafters.

Masonry units (brick, concrete block, glass block) are smaller than their nominal dimensions because of the thickness of mortar joints. Masonry units that are based on a 4 inch modular system are called, naturally enough, modular. (There are a few brick sizes that are not modular.) A typical thickness for a mortar joint is 3/8 of an inch. Therefore, the actual size of a nominal, 4-inch modular brick is 3-5/8 inches. Modular brick are sized so that three courses and three mortar joints equal 8 inches.

Scale

The word, “scale”, can have many different meanings depending on context. The definitions that relate to construction are as follows:

1. A series of marks or points at known intervals used to measure distances,

2. An indication of the relationship between the distances on a map and the corresponding actual distances,

3. An instrument consisting of a strip with one or more sets of spaces graduated and numbered on its surface for measuring or laying off distances or dimensions (Merriam-Webster).

4. The act of determining one of the above.

Scales are necessary because buildings tend to be large. We use drawings to design and construct buildings. Drawings are most useful when there is a specified relationship between what is drawn and what is built. Without a scale, it is impossible to estimate cost or build accurately. Scales may be indicated graphically or numerically. A graphical scale may be best when it is anticipated that reproductions may be many different and unpredictable sizes. However, in architecture and construction, drawings should be reproduced at a specific size or scale. Accurate, scaled drawings facilitate material takeoffs and verification of dimensions. It should be emphasized, however, that important dimensions should be specified numerically. Scaled drawings are an adjunct to, not a replacement for, reliable dimensions.

The most common scales in use today are the architect’s scale, the engineer’s scale, and the metric scale (SI system).

An Assignment

In order to teach the concepts discussed above, students in the first-year design course will be given the following assignment:

Pretend you are Noah. God has just given you the commission to construct a very large boat. The boat will be made of wood (the cedars of Lebanon) cut and delivered to your specifications. Pretend you can step into a time machine and select whatever measuring tools are available at the local hardware store today. You are the owner, designer, and construction manager. You will hire local workers to help you complete the project. Answer all of the following questions or directives and give reasons for all of your answers.

1. List all of the measuring tools you will supply your workers for both design and construction. Do not include tools actually used for construction.

2. What units of measurement will you specify?

3. Will you use any modular or nominal dimensioning system?

4. What precision will you require on the plans? Will you use fractions or decimals? How many significant figures?

5. What tolerances will you specify to achieve the accuracy you want?

Conclusion

Measurement in construction is a surprisingly complex issue. It is absolutely essential that construction managers acquire a degree of mastery in this area. It is so basic, that it is easy for educators to overlook certain aspects of measurement and assume that students already know them. Educators in construction should examine their curriculum and verify where and if these issues are being taught. If all the issues discussed here are not being taught, they should be.

Some students arrive at a construction management program with extensive experience in construction. They may have used tape measures, folding rules, carpenters’ squares, protractors, transits, and so forth for many years; however, many students entering CM programs today may not have had any experience using these tools and methods. Not only that, but they may not have even learned the basics they should have learned in the public schools such as how to read a ruler or protractor. The headings included in this paper identify concepts that every construction student should be familiar with. It is the hope of the author that this paper may be of value to other educators.

References

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