Stresses.docx

Stresses: We will now go through a relatively brief derivation to arrive at the flexure formula. We first write an expression for the bending moment produced by the horizontal forces with respect to the neutral axis (which is a line passing through the centroid of the beam cross sectional area - shown in Diagram 2a). The expression for the bending moment is simply the sum of the forces times the perpendicular distance to the neutral axis, or: We now note that we can express the force Fx as ; that is, the force acting on any small horizontal strip of area (dA) is the product of axial stress at that point and the amount of area (dA). (This simply comes from the definition of axial stress = Force/Area). We can now rewrite the expression for the bending moment as: We now will rewrite this expression one more time by noting that the horizontal forces and accompanying stresses increase linearly from zero at the neutral axis to a maximum value at an outer edge. We can then write: , where y is the distance from the neutral axis to area dA, and ymax is the distance from the neutral axis to the outer edge of the beam cross-section, as shown in Diagram 2a. We can then write the stress at an arbitrary y as: . We now substitute this expression into our relationship for the bending moment and obtain ; then rewriting slightly and factoring out the we find:

term from the summation sign (which we may since it is a constant),

; and finally we recognize that the summation term remaining is simply the Moment of Inertia (I) about the centroid of the beam cross section.. We now rewrite one last time, arranging terms and isolating the stress term by itself, we finally obtain: M ymax / I That is, the maximum "Bending Stress" at some location along the beam is equal to the bending moment, M, at that location "times" the distance, y, from the neutral axis to the outer edge of the beam "divided" by the moment of inertia, I, of the beam cross sectional area. If this seems somewhat confusing, it will become clearer as we work through several examples. While the formula above was derived for the maximum stress, it actually holds for the stress at any point in the beam cross section