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When beams do not have simple rectangular cross-sections, as is often the case, the actual behaviour can be quite complex. Fig. 3.52 shows a number of common non-rectangular shapes used for beams and Fig. 3.55 shows the distribution of the bending stresses for an I beam. Fig. 3.64 illustrates the idea of shear area used in the simple design of non-rectan-gular cross-sectional shapes.

The idea of the new shear stress shown in Fig. 4.5 is crucial to the understanding of the complex behaviour of beams with non-rectangular cross-sections. Although this shear stress is required for the equilibrium of a small cube cut from a beam, what does it contribute to the behaviour of a beam with a rectangular cross-section? Suppose there are two beams of equal depth, one on top of the other and separated by a perfectly slippery surface.

Fig. 4.27

Advanced concepts of stress 117 The bent shape of the two beams will be the same, with the top of each beam shortening and the bottom lengthening (see Fig. 2.13). This means ABCDE of the top beam gets longer and ABCDE of the bottom beam gets shorter. Note that point C is at midspan.

Fig. 4.28

Along the slippery surface ABCDE there will be a relative movement between the top and bottom beams. At midspan the movement is zero and the relative movement increases to become a maximum at the ends A and E. If the two beams are to act as one beam there would no relative movement along ABCDE. If this movement is to be zero there must be a force to stop the movement and the force will be proportional to the relative movement.

So the force will be zero at midspan and a maximum at the ends.

Fig. 4.29

It is the existence of these forces that causes the new shear stress. Because it is acting along the beam it is often called horizontal shear (or in timber design rolling shear). It is this horizontal shear stress that alters the bending stress distribution from a’two beam’ to a ‘single beam’ one.

Fig. 4.30

118 Building Structures: From Concepts to Design

Not only does the size of the horizontal shear stress vary along the beam but it also varies within the depth of the beam. The size of the stress at any point within the depth of the beam is related to the size of the horizontal force being transferred. This force is due to the change in size of the bending stress across a slice. The difference in the size of the bending stresses has an out of balance horizontal force on any horizontal cut and this force is bal-anced by the horizontal shear stresses at the cut.

Fig. 4.31

By taking a series of horizontal cuts across the slice, the size of the horizontal shear stresses can be found. At the top and bottom faces of the beam this stress is zero whilst at mid-depth it is at a maximum. The actual shape of the distribution is parabolic, the same shape as the distribution of the vertical shear stress (see Fig. 3.62).

Fig. 4.32

If an I section is used for a beam the horizontal shear stress still exists but the distribution is rather different. Because of the changed distribution of bending stresses, the largest part of the bending stress is in the top and bottom flanges (see Fig. 3.55). The maximum horizontal shear stress is still at mid-depth but because the section is ‘made up’ of flanges and a web the horizontal shear force has to be transmitted from one part of the beam to another. If the section is ‘exploded’ it can be seen how the horizontal shear forces ‘join’ it together.

Advanced concepts of stress 119

Fig. 4.33

What is happening is that the change in push force in each half of the top flange is being transmitted to the top of the web by a horizontal shear force.

Fig. 4.34

The total change in flange force is then transmitted by a horizontal shear force to the web underneath the flange.

Fig. 4.35

120 Building Structures: From Concepts to Design

At the mid-depth of the beam, the total change in web and flange force is being transmitted by a horizontal shear force.

Fig. 4.36

Similarly for the change in bending tensile forces. Note that the word force has been used rather than stress because the stress will depend on the relative thickness of the flanges and the web. The horizontal shear stress in the web is balanced by the vertical shear stresses, but the horizontal shear stresses in the flange are acting in the plane of the flange and have to be balanced by shear stresses acting across the flange.

Fig. 4.37

The cross flange shear stress is zero at the outer edges and increases linearly towards the web. Because the web shear under the flange is the sum of the change of both the left and right-hand push forces, there is an increase at this point. The stress in the web then varies parabolically with a maximum at mid-depth. These shear stresses can be plotted on the cross-section. These shear stresses in built-up sections are often called the shear flow.

Strictly the shear flow is a force, as it is the shear stress times the thickness.

Advanced concepts of stress 121

Fig. 4.38

The horizontal forces in each flange are in horizontal equilibrium and the vertical shear stresses are in vertical equilibrium with the shear force. However, if a channel section is used for a beam, the shear flow can be obtained from the diagram for the I beam.

Fig. 4.39

Here, although the horizontal forces in each web are in overall equilibrium, these forces are some distance apart which means there is a moment.

Fig. 4.40

This moment is trying to twist the channel rather than bend it. This example of the channel has been introduced to show how structural actions can become complicated even for a

‘simple’ beam and a ‘simple’ section like a channel.

122 Building Structures: From Concepts to Design