## Quantities, symbols, units and formulas

The metric system, and especially that part of it called the SI (le SystÒme International d'UnitÕs or, in plain English, the International System of Units), is by far the simplest and most rational system of units devised.

One of the main reasons for this is the simple compatibility of the metric system with our world-wide numerals and arithmetic based on the 10 digits and their position relative to a decimal point. This results from the practical system of attaching to unit names (symbols) standard prefixes that stand for some of the powers of 10 such as 0.001 (milli), 1000 (kilo). For instance, since the prefix kilo (k) stands for 1000, 1 kilometre (km) equals 1000 metres (m), and any change from metres to kilometres or vice versa simply involves a decimal point or zeros as shown below.

The Common Prefixes and Units

 Prefix & Symbol Meaning Value Factor micro (μ) one millionth 0.000 001 10-6 milli (m) one thousandth 0.001 10-3 centi (c)* one hundredth 0.01 10-2 deci (d)* one tenth 0.1 10-1 kilo (k) a thousand 1000 103 mega (M) a million 1,000,000 106 giga (G) a thousand million 1,000,000,000 109

* The prefixes `centi' and `deci' are only used with the metre. Centimetre is a recognised unit of length but centigram is not a recognised unit of mass.

Tables of measures for mass, length, area and volume are set out below

MASS SI Base unit: kilogram (kg)

 1000 micrograms (μg) = 1 milligram (mg) 1000 milligram (mg) = 1 gram 1000 grams (g) = 1 kilogram (kg) 1000 kilograms (kg) = 1 megagram (Mg) = 1 tonne (t)

LENGTH SI Base unit: metre (m)

 1000 micrometres (μm) = 1 millimetre (mm) 10 millimetres (mm) = 1 centimetre (cm) 10 centimetres (cm) = 1 decimetre (dm) 100 centimetres (cm) = 1 metre (m) 1000 millimetres (mm) = 1 metre (m) 1000 metres (m) = 1 kilometre (km)

AREA SI unit: square metre (m²)

 100 square millimetres (mm²) = 1 square centimetre (cm²) 10 000 square centimetres (cm²) = 1 square metre (m²) 1000 000 square millimetres (mm²) = 1 square metre (m²) 10 000 square metres (m²) = 1 hectare (ha) 100 hectares (ha) = 1 square kilometre (km²)

VOLUME SI unit: cubic metre (m³)

 1000 cubic centimetres (cm³) = 1 cubic decimetre (dm³) 1 cubic decimetre (dm³) = 1 litre (L) 1000 cubic decimetres (dm³) = 1 cubic metre (m³) = 1 kilolitre (kL) Or alternatively, for use with liquids and gases: 1 cubic centimetre (cm³) = 1 millilitre (mL) 1000 millilitres (mL) = 1 litre (L) 1000 litres (L) = 1 kilolitre (kL) = 1 cubic metre (m³) 1000 kilolitres (kL) = 1 megalitre (ML)

Symbols and base units we use in Structures

You should try to memorise the symbols and units listed below. These symbols and units will be used in all the structural subjects. (Many symbols are unfortunately Greek letters)

 The sum of (Greek letter sigma; capital letter S) Mass m kg Mass density kg/m³ (Greek letter rho) Second moment of area I mm4 (also called moment of inertia) Section modulus Z (or W) mm³ Modulus of elasticity E GPa (kN/mm²) (also called Young's modulus) Linear strain dimensionless (Greek letter epsilon) Original length lo mm (units must be compatible) Difference in length l mm (Greek letter delta; capital letter D) Normal stress (or f) MPa (N/mm²) (Greek letter sigma small letter s) Shear stress MPa (N/mm²) (Greek letter tau) Moment of a force M newton metre (N m) Deflection mm (Greek letter delta; small letter d) Slenderness ratio Sr dimensionless Radius of gyration r mm Velocity v m/s (meter per second) Acceleration a m/s² (meter per second per second) Gravitational acceleration g m/s² (meter per second per second) Force or weight F N (newton) also [kgm/s²]

Students need to familiarise themselves with these quantities, symbols and units. If you learn them by heart it'll be easier to grasp and take in the subject matter of Structures 1.

Please note:

If a student in a calculation has a correct numerical answers but without displaying the unit the answer will be discarded and receives no marks.

The following will emphasize the importance of the units:
If someone borrowed 10 dollars from you and the borrower settles his debt with 10 cents you wouldn't be happy about it, although the number is identical. If someone borrows 10 dollars from you, you would insist of the same unit, wouldn't you? This simple example points out that the unit is more important then the correct answer.

MPa has already been mentioned in some subjects and by now you'll be familiar with the unit MPa (Strength of material). This unit is of great significance in structures. The following relationship must be learned by heart:

 1 Pa = 1 N/m2 1 kPa = 1 kN/m2 1 MPa = 1 MN/m2 = 1 N/mm2

The meaning of MPa is very important because you'll do a lot of stress calculations in Structures 1. Look for prefixes like k and M in the table above.

Formulae

Students can find the formulae in the appropriate Section topics. However, the most common formulae are listed below:

 Force = mass × acceleration (F = m × a) Weight = mass × gravitational acceleration    (W = m × g) Stresses (tension and compression) = force / area ( = F / A) Bending stress is more complex (tension          and compression occurs simultanuously)         Bending stress = Bending moment / section modulus = BM / Z) Modulus of elasticity       E = Stress / strain (E = / ) Moment of inertia        I = Breath × depthý/12 (I = b × dý/ 12    This is valit for a rectangular cross-sectional    area

Density for major structural material

 Material Mass density (kg/m3) Weight density (kN/m3) Concrete (reinforced) 2500 kg/m3 25 kN/m3 Concrete (unreinforced) 2300 kg/m3 23 kN/m3 Brickwork 1900 kg/m3 19 kN/m3 Timber (Softwood) 600 to 800 kg/m3 6 to 8 kN/m3 Timber (Hardwood) 800 to 1100 kg/m3 8 to 11 kN/m3 Steel 7850 kg/m3 78.5 kN/m3

By closely looking at the units we can easily work out the correct answer of a propblem by substituting the units into the formula.
Consider the follwing example to work out the weight of a structural component or member:
To calculate the weight of a component or member we use the formula:

Weight (W) = Density × Volume

Remember unit for density is kg/m3 and the unit for volume is m3 but the unit for weight is measured in newton.

We need to convert the mass into a weight figure.

 Weight = mass × gravitational acceleration W = m × g
(g = 9.81 m/s2 but we use 10 m/s2)

Having converted the mass unit into a weight figure we can now calculate the weight of any structural component or member in newtons by using:

W = kN/m3 × m3

Examlpe 1:
Calculate the dead load (DL) for a concrete slab, size 4.0 m × 3.5 m of 172 mm thickness . Density of concrete is 2500 kg/m3

Solution:
First convert mass density in weight density. 2,500 kg/m3 = 25,000 N/m3 = 25 kN/m3

Now we can calculate the weight of the slab:

 W = 4.0 × 3.5 × 0.172 × 25 = 60 kN

Example 2:
Calculate the live load (LL) for a room of a residential building, size 5.5 m × 3.8 m. The LL according to AS 1170 Part 1 (Dead and live load) is 1.5 kPa).
Remember 1 kPa = 1 kN/m2

Solution:
Using the formula LL = m (length) × m (width) × kN/m2

 LL = 5.5 × 3.8 × 1.5 = 31.35 kN