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.

The law of the lever is well known. The distance of a force from its fulcrum or point where it takes effect is called its leverage. The effect of the force at such point is equal to the amount of the force multiplied by its leverage.

The effect of a force (or load) at any point of a beam is called the moment of the force (or load) at said point, and is equal to the amount of the force (or load) multiplied by the distance of the force (or load) from said point, the distance measured at right angles to the line of the force. If therefore we find the moments - for all of the forces acting on a beam - at any single point of the beam we know the total moment at said point, and this is called the bending-moment at said point. Of course, forces acting in opposite directions will give opposite moments, and will counteract each other; to find the bending-moment, therefore, for any single point of a beam take the difference between the sums of the opposing moments of all forces acting at that point of the beam.

Now on any loaded beam we have two kinds of forces, the loads which are pressing downwards, and the supports which are resisting upwards (theoretically forcing upwards). Again, if we imagine that the beam will break at any certain point, and imagine one side of the beam to be rigid, while the other side is ten ling to break away from the rigid side, it is evident that the effect at the point of rupture will be from one side only; therefore we must take the forces on one side of the point only. It will be found in practice that no matter for what point of a beam the bending moment is sought, the bending moment will be found to be the same, whether we take the forces to the right side or left side of the point. This gives an excellent check on all calculations, as we can calculate the bending moment from the forces on each side, and the results of course should be the same.

Now to find the actual strain on the fibres of any cross-section of the beam, we must find the bending moment at the point where the cross-section is taken, and divide it by the moment of resistance of the fibre, or, m/r = s

Where m = the bending moment in lbs. inch.

Where r = the moment of resistance of the fibre in inches.

Where s = the strain.

The stress, of course, will be equal to the resistance to cross-breaking the fibres are capable of. In the case of beams which are of uniform cross-section above and below the neutral axis, this resistance is called the Modulus of Rupture (k). It is found by experiments and tests for each material, and will be found in Tables IV and V. We have, then, for uniform cross-sections: - v =k

Where v = the ultimate stress per square inch.

Where k = the modulus of rupture per square inch. Inserting this and the above in the fundamental formula (1), viz.: v = s.f, we have: k = m/r.f, or

Bendlng moment.

Transverse strength uniform cross-section.

m/(k/f) = r (18)

Where m = the bending moment in lbs. inch at a given point of beam.

Where r = the moment of resistance in inches of the fibres at said point.

Where (k/f) =the safe modulus of rupture of the material, per square inch.

If the cross-section is not uniform above and below the neutral axis, we must make two distinct calculations, one for the fibres above the neutral axis, the other for the fibres below; in the former case the fibres would be under compression, in the latter under tension. Therefore, for the fibres above the neutral axis, the ultimate stress would be equal to the ultimate resistance of the fibres to compression, or v = c.

Transverse strength section not uniform.

Inserting this in the fundamental formula (1), we have: - c=m/r.f, or m/(c/f).= r (19)

Where m = the bending moment in lbs. inch, at a given point of beam. Where r = the moment of resistance in inches of the fibres at said point.

Where (c/f. ) =the safe resistance to crushing of the material, per square inch.

For the fibres below the neutral axis, the ultimate stress would be equal to the ultimate resistance of the fibres to tension, or, v=t.

Inserting this in the fundamental formula (1) we have: t = m/r. f, or m/(t/f) = r (20)

Where m = the bending moment in lbs. inch at a given point of beam.

Upper fibres.

Lower fibres.

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