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 


σ_{max }= maximum stress 
σ_{t max} = 6
M_{max} 
σ_{c}_{
max} = _ 6 M_{max} 
H = height 
B H^{2} 
B H^{2} 
B = width 


M_{max}
= 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 
Y_{max} = _ P L^{3} 
Y_{max} = _ 5 W L^{4} 

48 E I 
384 E I 
. 
Beam fixed at both ends 
Y_{max} = _ P L^{3} 
Y_{max} = _ W L^{4} 

192 E I 
384 E I 
where Y_{max} = 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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Knowhere 

Page 95 
Building stability  Beam stress and deflection 

