When a structural engineer has to figure out what size a beam/joist/lintel has to be to support a load without buckling under the strain, a great deal of information is required. He/she would probably refer to a Table of Sections, an extensive list which describes the properties of beams of various types, which are shown in cross- section. Some of these properties are explained below.
When a beam is subjected to a load, it will be in compression on one side, and under tension on the other. The division between the two is referred to as the neutral axis, since it is not subjected to tension nor compression. In beams of symmetrical section, this axis passes through the centre, or centroid of that section.
The geometric property of the section's shape and area is known as the Second Moment of Area, I, and represents the way in which the area of the section is distributed about the neutral axis. It may often be called the Moment of Inertia.
For instance, a rectangular beam of depth D and breadth B has a Second Moment of Area, I = BD³/12 (see Figure 1). Depending on the units used in the equation, I takes those units to the fourth power (e.g. mm4 or inches4 ).
Timber beams are usually rectangular. However, since there is no stress in the centre, and maximum stress at the upper and lower face, a rectangular beam is wasteful in material. For construction with steel, it is more economical to use an I-beam.This puts the material where it is most needed, at the top and bottom,with a web to link them.
The second moment of area for this type of section is 1/12(BD³ - bd³) (units4). Again, if it is symmetrical, the neutral axis will pass through the centroid.
What is most significant is that the second moment, though proportional to the width, is also proportional to the cube of the depth. If a square 10 mm beam is made 90mm wider, its second moment will be 10 times bigger. If the beam is made 90mm deeper, its second moment will be 1000 times bigger!
Another piece of information vital to the engineer refers to the material itself. The modulus of elasticity, or Young's modulus (E), can be considered as a measure of stiffness. Typical values are:
Mild Steel E = 200 kN/mm²
Aluminium E = 70 kN/mm²
Timber E = 10 kN/mm²
It is no surprise that steel is much stiffer than timber, but due to variations in the structure of different materials (especially timber), these values only provide a rough guide. Different grades of steel, for example, each have a specified E value.
So, given a beam of certain dimension and shape, and formed from a given material, a very useful value can be derived.The product EI is called the flexural rigidity, and neatly sums up the properties covered above. The beauty of it lies in the fact that so much can be learned about a beam's behaviour, without even knowing its length or the load it is to bear.
Application
If we are to discover anything about stress in the neck of a steel strung instrument, it would be ideal to consider it first as a simple beam supporting a load, that load being the tension of the strings. Though this is very different to a beam supporting, say, a brick wall, the effect on the beam is pretty much the same, in that the load causes a deflection, or bending. It is also the case that, like the engineer, the object is to minimise the deflection given a limiting set of parameters, such as dimensions and choice of material.
To start, the cross-section of the neck must be considered, to find the second moment of area. Unfortunately, such a shape is not dealt with in the engineer's Table of Sections. It is almost an ellipse cut in half, but not quite. It could be more pointed, relating more to a triangle, or more rectangular, but with softened corners. Since the neck is tapered, the width is variable, as may be the depth. The back curvature will also be continuously redefined between the headstock and the body. The fingerboard may also vary in depth, and may be cambered, and that camber may increase along the neck, as with conical fingerboards. With every small change in dimension, there is also a change in area, hence neutral axis, hence second moment, and hence rigidity.
To embark on this veritable nightmare, the most simple of cases will be shown as an example.
The neck sawn in two here has a total depth of 20mm, a width of 40mm, and is semi-circular with a flat fingerboard. The presence of fret slots will be assumed to have no effect on the outcome!
It is shown as a graph, so that any point can be referred to easily as (x, y) co-ordinates. It can be said, therefore, that the neck section is contained within an area between x = -20 and +20, and y = 0 and +20. Using p r² / 2 would find the area, but since the intention is to move on to more complex shapes, it is divided into horizontal strips, from which the neutral axis (shown on the graph as an arbitrary line) can be calculated.
To find the Second Moment of Area about the neutral axis requires the deduction of two unknown quantities. The method to be used here avoids some pretty grisly calculus. Known as the parallel axes theorem, it states that "the moment of inertia about any axis is equal to the moment of inertia about a parallel axis through the centroid plus the area times the square of the distance between the axes." In other words, while the neutral axis is unknown, it can be found by calculating the Second Moment of Area about a convenient axis which is parallel to it. In this instance, the x-axis ( y = 0 ) is convenient.
The area of the section on the graph is found by calculating the area of each strip, and adding them all together. Only the approximate area can be found for each, so the thinner the strips, the more there will be and the more accurate the area figure. For this example, 20 strips will be calculated, according to the table below.
The first column lists y values from 0 to 20. The second column shows corresponding x values. Note that there are two x values, positive and negative, for each y value. The area of each strip ( dA ) is found by roughly deducing its average length.
As can be seen from the figure, the strip bounded by +/-x1 and +/- x2 will have an average length (shown dotted) of x1 + x2. Therefore the third column shows the sums of consecutive x values, to be taken as strip areas, since the strips are one y-unit (1mm) wide.
Now these can be added together to give the total area of the section ( A = SdA ). The fourth column multiplies each strip area by its distance from the x-axis, which can be taken to be the corresponding y value. Again, these results are added ( SydA ). The last column is the strip area times the square of the distance ( y²dA ). This column is added to give the second moment of area about the x-axis ( Sy²dA = Ix ).
The centroid (h) will lie on the y-axis ( because the section is symmetrical ), at a height h = SydA / A = 8.45 mm.
The second moment of area about the centroid ( I ) = Ix - Ah² or 61951.11 - ( 625.69 x 8.452 ) = 17275 mm4.
The accuracy can be checked against the true value for area, which is p r² / 2 or p x 200 = 628.32 mm².
The error will be 628.32 - 625.69 / 628.32 x 100 % = 0.42 %
The theoretical height of the centroid ( 4r / 3p ), is 8.488mm, an error of 0.48 %.
The true I value, given by ( 9p² - 64 ) r4 / 72p, will be 17561 mm4, an error of 1.62 %.
This is quite impressive, given that only 20 strips were used. ( Formulae taken from Table of Sections for semi-circle.)
The arithmetic required for this "simple" example is pretty extensive, and one has yet to include the fingerboard, truss rod channels and reinforcement material, let alone the variations in dimension which will be experienced.