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The reason the channel wants to twist and the I beam does not, is because the I beam is symmetrical about a vertical line and the channel is not.

Fig. 4.41

More generally, the effect of the cross-sectional shape of a structural element interacts with the type of loading to produce different types of structural behaviour. This structural behaviour can become extremely complicated for general shaped elements. The important point is to appreciate what causes simple or complex behaviour.

For axially loaded elements, if all parts are to be equally stressed, that is, uniform stress distribution (see Fig. 3.29), then the load has to be applied to the element at a particular point. This point is confusingly called the centre of gravity of the cross-section, a better term is centre of area. This point is the same point that balances a uniform stress distri-bution. To illustrate this concept imagine a tee shaped platform carrying people of equal weight equally spaced on the platform.

Fig. 4.42

These people are supposed to represent a uniform stress distribution. But where is the bal-ance point? Unlike the see-saw the balbal-ance point has to be found in two dimensions. Two pictures can be drawn, one along the tee and one across it.

Advanced concepts of stress 123

Fig. 4.43

In the AB direction the balance point will be nearer B than A, but for the CD direction the loading is symmetrical so the balance point is midway between C and D. The balance points are really lines and where these lines intersect is the centre of area.

Fig. 4.44

So if the axial loads are applied through the centre of area a uniform stress distribution will be caused. If another shaped cross-section (or platform) is used then the centre of area will move. The balance lines will always be along any axes of symmetry, so for shapes with two axes of symmetry the centre of area will be at the intersection of the axes.

Fig. 4.45

Notice that for the box section the centre of area is not actually within the cross-section.

Where there is only one axis of symmetry, the unsymmetrical balance line will vary with the geometrical dimensions of the cross-section.

124 Building Structures: From Concepts to Design

Fig. 4.46

And where there is no axis of symmetry, the centre of area can only be found by calculation.

Fig. 4.47

For each cross-sectional shape there is only one position for the centre of area and the posi-tion depends on the shape and, if the secposi-tion is not doubly symmetric, the dimensions of the cross-section. In other words the centre of area is a section property. In actual structures it is not easy to ensure that axial loads are always applied through the centre of area.

As well as affecting the behaviour of elements when axially loaded the cross-sectional shape also affects the way they behave when they bend as beams. When a beam bends, part of the cross-section is in compression and part is in tension (see Fig. 3.38). Where the stress changes from compression to tension the stress is zero. The question is where is this point?

And the answer is where it needs to be to satisfy horizontal equilibrium (see Fig. 3.46).

This is not a helpful answer but if no simplifying assumptions are made the position of zero stress is not easy to find. If the Engineer’s theory (see page 65) is used the position can be found. With this theory, plane sections are expected to remain plane and the structural material to be linear elastic. This results in the points of zero stress due to bending being in a straight line. Because this is a line of zero stress it is often called the neutral axis. For a rectangular beam loaded laterally there has been a tacit assumption (see Section 3.4) that the neutral axis is across the beam at mid-depth.

Advanced concepts of stress 125

Fig. 4.48

Because the horizontal (push/pull—see Fig. 3.45) forces must balance, then, for cross-sec-tions that are symmetric about a horizontal axis that are being loaded vertically, the neutral axis will be at mid-depth.

Fig. 4.49

Because of this symmetry, the stress due to bending at the top of the section will be equal to the stress at the bottom of the section.

Fig. 4.50

For sections that are not symmetrical about a horizontal axis, the neutral axis will not be at mid-depth, but where will it be? Surprisingly and very fortunately it passes through the centre of area.

Fig. 4.51

126 Building Structures: From Concepts to Design

As the Engineer’s theory is being used, the bending stresses vary linearly with the depth of the beam. Where the neutral axis is not at mid-depth this linear variation means that the stresses at the top and the bottom will no longer be equal.

Fig. 4.52

That the neutral axis passes through the centre of area may seem fortuitous but this is not so. To see why consider a cross-section in two parts, one double the area of the other. The centre of area is the balance point for constant stress over the whole area of the cross-sec-tion. In this case it means the stress f is constant over the two parts of the seccross-sec-tion.

Fig. 4.53

The force in each part of the cross-section is the area multiplied by the constant stress f.

The moments about the balance point (see Fig. 1.41) will be equal if the position is as shown in Fig. 4.53. For bending, plane sections remain plane and due to linear elasticity the stress is directly proportional to the movement. For this section, if it is rotated about the balance point (centre of area—see Fig. 3.37) the movement of one part will be twice that of the other. This means where the movement is double, the stress will be double.

Advanced concepts of stress 127

Fig. 4.54

This diagram shows that for the part of the section with area A, the stress is 2f giving a force of 2fA in one direction. For the part of the section with an area of 2A the stress is f giving a force of 2Af in the opposite direction. Therefore the push/pull forces are equal and opposite as required for horizontal equilibrium (see Fig. 3.45). This in principle is why the neutral axis goes through the centre of area and this principle applies to a cross-section of any shape or any number of ‘parts’.