By its very definition a beam is a structural element that resists bending forces. These bending forces or bending moment will introduce a mechanical stress to the beam. This mechanical stress is defined in terms of force per unit area and may be represented in US customary units as lbs/ft2, lbs/in2, etc. or in metric units as kN/m2, kN/mm2, etc.
The magnitude of the bending stress in a beam varies linearly with respect to the neutral axis (for a symmetric section, the neutral axis is located at the centroid about which the beam is bending) of the section and can be tensile or compressive in nature. For a simply supported beam that resists gravity loads which causes a bending force, the beam will undergo compressive stresses on its top face (the face for which the loads are acting in to) and tensile stresses on its bottom face (the face for which the loads are acting away from). An example of this type of beam could be a roof beam that resists dead loads acting on its top face like roof sheathing and building material and also resists live loads such as snow loads and roof top mechanical systems. To further illustrate the concept of tensile and compressive stresses, take a plastic ruler and hold it out in front of you and place both hands (palms down) on it and bend it such that it bends downward in a shape like a smile. The top surface of the ruler experiences compressive stress and the bottom surface or underside experiences tensile stress.
The bending stress of any beam at any location along its depth can be determined by:
Ïƒ = My / I
Where Ïƒ is the bending stress, M is the bending moment or bending force applied to the beam, y is the distance from the neutral axis to the location you are finding the stress at and I is the moment of inertia.
Often the bending stress is calculated at the top and bottom face of the beam to determine the maximum stresses (tensile and compressive) acting on the beam for purposes of design and analysis.
As an example let's calculate the bending stress at the top and bottom faces for an unsymmetric steel beam. The beam is a standard wide flange section with a steel plate welded to its bottom surface and is simply supported. The beam is 20 ft long and loaded with an applied uniform load of 300 lbs/ft. The moment of inertia for the beam with respect to the x-axis is Ix = 305 in4. The neutral axis is located 16.5 in. from the bottom surface of the beam and the top surface of the beam is located 18.5 in. from the neutral axis.
At the top surface of the beam, also termed in this case as the extreme compression fiber the bending stress is calculated as follows.
Ïƒc = Myt / Ix
For M we will use a previously determined solid mechanics calculation in that the maximum bending moment, M = wl2 / 8.
M = wl2 / 8 = (300 lbs/ft)(20 ft)2 / 8 = 15,000 lb â€“ ft = 15 k â€“ ft (k = 1,000 lbs)
yt = 18.5 in.
Ix = 305 in4
Ïƒc = (15 k â€“ ft)(12 in./ft)(18.5 in.) / (305 in4)
Ïƒc = 10.92 k/in.2 = 10.92 ksi
This is the maximum compressive stress on the beam acting at the top surface (extreme compression fiber) of the beam.
At the bottom surface of the beam, which is called the extreme tension fiber, we will calculate the bending stress in the same way.
Ïƒt = Myb / Ix
M = 15 k â€“ ft
yb = 16.5 in.
Ix = 305 in4
Ïƒc = (15 k â€“ ft)(12 in./ft)(16.5 in.) / (305 in4)
Ïƒc = 9.74 k/in.2 = 9.74 ksi
This is the maximum tensile stress on the beam acting on the bottom surface of the beam.
Gere, James M. (2004). Mechanics of Materials (6th ed.). USA: Brooks/Cole â€“ Thomson Learning.