This section is from the book "Safe Building", by Louis De Coppet Berg. Also available from Amazon: Code Check: An Illustrated Guide to Building a Safe House.
(German, Neutrale Achse; French, Axe neutre.) The neutral axis of a body, or figure, is an imagin- Neutral Axis. ary line upon which the body, or figure, will always balance, provided the body, or figure, is acted on by no other force than gravity. The neutral axis always passes through the centre of gravity, and may run in any direction. In calculating transverse strains, the neutral axis designates an imaginary line of the body, or of the cross-section of the body, at which the forces of compression and tension meet. The strain on the fibres at the neutral axis is always naught. Extreme Fibres. On the upper side of the neutral axis the fibres are compressed, while those on the lower side are elongated. The amount of compress-ion or elongation of the fibres increases directly as their distance from the neutral axis; the greatest strain, therefore, being in the fibres along the upper and lower edges, these being farthest from the neutral axis, and therefore called the extreme fibres. It is necessary to calculate only the ultimate resistance of these extreme fibres, as, if they will stand the strain, certainly all the other fibres will, they all being nearer the neutral axis, and consequently less strained. Where the ultimate resistances to compression and tension of a material vary greatly, it is necessary to so design the cross-section of the body, that the "extreme fibres" (farthest edge) on the side offering the weakest resistance, shall be nearer to the neutral axis than the "extreme fibres" (farthest edge), on the side offering the greatest resistance, the distance of the "extreme fibres" from the neutral axis being on each side in direct proportion to their respective capacities for resistance. Thus, in cast-iron the resistance of the fibres to compression is about six time9 greater than their resistance to tension; we must therefore so design the cross-section, that the distance of the neutral axis from the top-edge will be six-sevenths of the total depth, and its distance from the lower edge one-seventh of the total depth.
To find the neutral axis of any plane-figure, some writers recommend cutting, in stiff card-board, a duplicate of the figure (of which the neutral axis is sought), then to experiment until it balances on the edge of a knife, the line on which it balances being, of course, the neutral axis. This is an awkward and unscientific method of procedure, though there may be some cases where it will recommend itself as saving time and trouble.
The following general formula, however, covers every case: To find the neutral axis M-N in any desired direction, draw a line X-Y at random, but parallel to the desired direction. Divide the figure into any number of simple figures, of which the areas and centres of gravity can be readily found, then the distance of the neutral axis M-N from the line X-Ywill be equal to the sum of the products l of each of the small areas, multiplied by the distance of the centre of gravity of each area from X-Y, the whole to be divided by the entire area of the whole figure. An Example will make this more clear.
How to find Neutral Axis.
Find the horizontal neutral axis of the cross-section of a deck-beam, standing vertically on its bottom-flange.
Draw a line (X-Y) horizontally (Fig. 1), then let d1, d11, d111, represent the respective distances from X-Y of the centres of gravity of the small subdivided simple areas a1, + a11, + a111, then let a stand for the whole area of section, that is: a1, + a11, + a111, =a, then the required distance (d) of the neutral axis M-N from X-Y, will be d = (a1d1+a11d11 + a111d111 )/a To find the centre of gravity of the figure, we might find another neutral axis, but in a different direction, the point of intersection of the two being the required centre of gravity. But as the figure is uniform, we readily see that the centre of gravity of the whole figure must be half-way between points A and B.
The centre of gravity of a circle is always its centre. The centre of gravity of a parallelogram is always the point of intersection of its two diagonals. The centre of gravity of a triangle is found by bi-secting two sides, and connecting these points each with its respective opposite apex of the triangle, the point of intersection of the two lines being the required centre of gravity, and which is always at a distance from each base equal to one-third of the respective height of the triangle. Any line drawn through either centre of gravity is a neutral axis.