Beams
 

 

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When designing a beam, i.e. when calculations are made in order to choose a beam which is strong enough to carry its loads, and which at the same time makes economical use of the material, at least four items must be considered:

  1. The beam must have a bending strength adequate to resist the bending moments.
  2. There must be no danger of failure due to shear forces.
  3. The amount the beam bends, i.e. the deflection of the beam, must not be excessive.
  4. There must be no danger of lateral (sideways) buckling.

Any one of these items may be the deciding factor in the design of the beam.

Consider the behaviour of a beam of elastic material such as timber or steel.  Imagine the beam to consist of layers of longitudinal fibres similar to a leaf spring on an old motor vehicle but with all the layers being securely cemented together.

If a load is placed on the top of the beam, which is supported at each end, the longitudinal fibres near the top of the beam will become shorter as a result of the bending and are therefore stressed in compression.  The fibres near the bottom of the beam will become longer and are thus stressed in tension.

One of the layers, along the centre or centroid of the beam section, will remain unstressed and this is called the Neutral Layer.

The compressive stress will not be uniform over the portion of the beam above the neutral layer.  It will be greater in the longitudinal fibre at the top of the beam and the stress in the remaining fibres will get progressively less towards the neutral layer.

Below the neutral layer, the stress, which is now tensile, increases towards a maximum at the extreme bottom longitudinal fibre.  The neutral layer, where there is no stress, can be proved by mathematics to pass through the centre of gravity (centroid) of the cross-section.  Thus in a symmetrical beam, such as a rectangular one, the neutral axis is at mid-depth.

Since the fibres near the top and bottom of the beam are more highly stressed than those near the neutral axis, it is an advantage to have as much material as is practical as far as possible from the neutral avis.  Hence the shape of commonly used steel beam sections.  Most of the steel is concentrated in the flanges where it is most effective in resisting bending.  The web on the other hand must have sufficient steel to resist the shear forces.

Failure of the beam will occur due to the crushing of the extreme fibres in compression or tearing of the extreme fibres which are in tension.

The material of the beam is obviously important.  A steel beam is much stronger than a timber beam of identical dimensions.  The shape of the cross section of the beam is important too; the depth of the beam is more important than width for resisting bending.  It can be proved that the bending strength increases proportionally to the square of the depth but only in direct proportion to the breadth.

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Deflection

A beam may be strong enough to resist safely the bending moments and shear forces and yet be unsuitable because its deflection under the calculated safe load is excessive.  Apart from being unsightly and giving an impression of insecurity, excessive deflection can cause cracking of plaster ceilings and partitions.  For steel beams, the deflection should not exceed 1/360 of its span and for timber it is .003 times its span.

The amount a beam deflects depends on the way it is supported, the amount and position of the load, the span of the beam, the size and shape of its cross-section and the nature of the material.  All other factors being equal, a beam with its ends fixed deflects less than a beam with its ends simply or freely supported.  The effect of span on deflection is very important.

Assuming that two beams are identical in size, with the same magnitude of load but one is twice as long as the other then the longer of the two beams will deflect 8 times greater than the shorter beam.

For a given amount of material, the deepest beam is the best for limiting deflection as well as being the most economical for resisting the bending moments due to the loads on the beam.  For large spans, beams may have to be of larger sizes than are necessary for resisting bending moments in order that the deflections may be kept to reasonable limits.

Another important factor which has to be taken into account when calculating deflection is the nature of the material.  It is known that force cannot be applied to a material without altering its dimensions.  Hooke's Law states, that for an elastic material, stress is proportional to strain and that stress divided by strain is called modulus of elasticity, E.  The greater the value of E, the stiffer is the beam, i.e. the greater is its resistance to being bent.  It is possible to obtain aluminium alloys which have strengths approaching those of mild steel but their E values are only about one third of that for steel.  It therefore follows that, for identical conditions, aluminium beams would have to be bigger than steel beams in order to limit deflections.  They would of course be much lighter in weight than steel and are used in the construction of aircraft.

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Sideways Buckling

To resist a given bending moment and also to keep deflections small, it has been stated that the deepest beam is the most suitable.  If, however, the depth is made too great in proportion to the breadth, the beam may buckle sideways due to a column effect as the result of compressive stresses in the top fibres.

When a beam is embedded in a reinforced concrete floor, there is no need to consider the buckling tendency since the floor slab provides sufficient lateral restraint but when the beam is not restrained laterally or is only partially restrained laterally by cross-beams, the permissible bending strength for a slender beam must be reduced in accordance with the requirements of the relevant British Standard Code of Practice.

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Structural Analysis

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Last Edited :  22 January 2011 14:26:17