In order to design the most efficient beam to carry a given load while spanning a given

distance, it is critical to know the maximum compressive and tensile stresses it will be

subjected to.  Once these stresses have been determined a beam material can be selected

whose compressive and tensile strengths are at least equal to them.  The maximum tensile

stress occurs at the midpoint of the bottom edge of a beam. And the maximum compressive

stress occurs at the midpoint of the top edge.  The equations describing these stresses for a

uniformly loaded, simply supported beam are as follows:

.

 Maximum tensile stress Max. compressive stress where σmax = maximum stress σt max =  6 Mmax σc max = _  6 Mmax H = height B H2 B H2 B = width Mmax = max. bend. mom.

.

The equations are identical except for the sign. Notice that the maximum stress is indirectly

related to the height and width of the beam and thus its moment of inertia.  In particular

the maximum stress is decreased by the square of the height.  Thus a beam with greater

height can be made from material that has much lower tensile and compressive strength

than a beam with a lesser height.  As is the case with a column, if the maximum stresses

induced in a beam by the load are greater than the tensile or compressive strength of its

material, it will fail either by being permanently deformed or by breaking.

It is also important to know what the maximum amount of bending, or deflection will be for

the beam for a given load.  Anyone who has crossed a sagging plank over a stream can

attest to that!  The equations describing this deflection are:

.

Loaded in middle

Uniformly loaded

.

Simply supported beam

Ymax = _  P L3

Ymax = _  5 W L4

48 E I

384 E I

.

Beam fixed at both ends

Ymax = _  P L3

Ymax = _  W L4

192 E I

384 E I

 where   Ymax = maximum deflection Ι = moment of inertia P = load (Newtons or lbs.) E = modulus of elasticity

.

Note that the deflection of a simply supported beam is four to five times that of a similarly

loaded beam whose ends are fixed to its supports.  Recall that fixing both ends of a column

also reduces its tendency to buckle compared to a free standing column.

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 Page 95 - Building stability - Beam stress and deflection
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