Topic 5: Beams II - Bending Stress

• 5.1 Beams - Bending Stress
  • 5.1a Centroids and the Moment of Inertia
  • 5.1b Bending Stress - Example 1
• 5.2 Beams - Bending Stress (cont)
  • 5.2a Bending Stress - Example 2
  • 5.2b Bending Stress - Example 3
• 5.3 Beams - Horizontal Shear Stress
  • 5.3a Horizontal Shear Stress - Example 1
  • 5.3b Horizontal Shear Stress - Example 2
  • Additional Bending & Shear Stress Examples: 
  • #1, #2, #3, #4, #5, #6 [Previous test problems]
• 5.4 Beams - Beam Selection
  • 5.4a Beam Selection - Example 1
  • Additional Beam Selection Examples: #2, #3, #4, #5, #6, #7 [Previous test problems]
• 5.5 Calculus Review (Brief)
• 5.6a Beams - Problem Assignment 1 - Inertia, Moment, Stress [required]
• 5.6b Beams - Problem Assignment 2 - Bending Stress [supplementary]
• 5.6c Beams - Problem Assignment 3 - Horizontal Shear Stress [supplementary]
• 5.6d Beams - Problem Assignment 4 - Bending & Shear Stress [required, except #5]
• 5.7a Beams - Problem Assignment 5 - Beam Selection [required, except #5]
• 5.7b Beams - Problem Assignment 6 - Beam Deflection [supplementary]
• 5.8   Beams -Topic Examination

BEAMS: BENDING STRESS
In the proceeding topic (Beams - Shear Forces and Bending Moments) we examined how to determine the Shear Forces and Bending Moments in a loaded beam. This is particularly valuable since the Axial Stress (tension and compression) and the Shear Stress (vertical and horizontal) which develop in a loaded beam depend on the values of the Bending Moments and the Shear Forces in the beam. Determining the axial stress - which is often known as the Bending Stress in a beam; and determining the shear stress - often called the Horizontal Shear Stress (for reasons we will discuss) is important in two ways. First, it will enable us to determine if a particular loaded beam is safe under the applied loading. Second, it will enable us to select the best beam (from a table of beams) for a particular loading. Both of these are very important processes for the safety and efficiency of a beam.

Before we continue with determining the Bending Stresses and the Horizontal Shear Stresses in a loaded beam we must examine two related topics which are used in the determination of the bending and shear stresses - Centroids and The Moment of Inertia.
Please select Centroids and The Moment of Inertia , review the material, and return to this page.

The Bending Stress: We will first develop a relationship for the bending stress which develops in a loaded beam. This relationship is known as the Flexure Formula. In Diagram 1 we have shown a simply supported beam loaded at the center. It deflects (or bends) under the load.

In Diagram 2, we have shown the left end section of the beam. As discussed previously, when examining bending moments, horizontal forces act on the cross sectional face of the beam section. We have shown only the horizontal forces along the top and bottom in Diagram 2a, but the forces act across the whole cross section as shown in the side view in Diagram 2b. The horizontal forces decrease from maximum at the outer edges to zero at the neutral axis (an axis running through the centroid of beam cross section).

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 F_x 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 y_max 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 theterm from the summation sign (which we may since it is a constant), we find:
; 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 y_max / 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 and is known as the Flexure Formula.

Flexure Formula: M y / I

We now will look at a simple example to see the application of the flexure formula.
Select: Topic 5.1b:Beams - Bending Stress Example 1

When finished with Bending Stress - Example 1,
Continue to: Topic 5.2:Beams - Bending Stress (cont)
or Select:
Topic 5: Beams - Table of Contents
Strength of Materials Home Page