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Algebra for Athletes.com

3.3 Bending Stress

Another type of stress you find in the weight room is bending stress.  When a bar is fixed on one end and has a load placed on the other end, it will experience a moment and bend.  The figure below shows some exaggerated bending.

Notice how the bending causes the top of the bar to be in tension and the bottom part to be in compression.  The stress will be the greatest at the top and bottom edges of the bar.  To determine the best size for a bar having a moment like this, engineers use formulas for bending stress.  The variable thatís usually used for bending stress is sigma, , the Greek letter for "s".  The formula for the bending stress of a bar with a circular cross-section is given by the equation: 

where            = the bending stress on the bar                      (3-5)

                        M = the moment on the bar

                        r = the radius of the bar 

The formula for the bending stress of a bar with a rectangular cross-section is given by the equation: 

where             = the bending stress on the bar                                     (3-6)

            M = the moment on the bar

             h =  the height of the bar

                        b = the base width of the bar 

Example 3.7:  The force is put on a rectangular bar that gives it a moment 36,000 in. lb.  (Remember that M = distance x force.)  The height, h, of the bar is 2 in. and its base width, b, is 1 in.  How much bending stress will the bar have with this moment?

Solution: The bending stress is determined directly from Equation 3-6.

The bar will experience 54,000 psi of bending stress.

3.3.1 Properties of Exponents 

As you can see, the equations for bending stress get pretty complicated.  To work with these kinds of equations, you need to understand the properties of exponents.  The equations below show the laws for exponents:

  (3-7)                    (3-11)                     (3-14) 

           (3-8)                          (3-12)    (3-15)

  (3-9)                  (3-13)            (3-16)

         (3-10) 

The following examples show how the exponent equations may be used to solve some bending stress problems. 

Example 3.8:  Figure 3.17 shows the cross-section of a square bar.  A manufacturer is designing a bar which can bend in either direction-x or direction-y as shown in the figure.  To resist bending in both directions, the bar has been given a square cross-section.  Using Eq. 3-6, find the formula that will allow you to calculate the required width, a, of a bar with a square cross-section for any values of M and s.  In other words, solve Eq. 3-6 for a if a = b = h. 

Solution:  If a = b = h, Eq. 3-6 could be written as:

To solve the equation fora, we can first combine the a's. Using Eq. 3.7: 

Therefore:

We then try to get the a³ term on one side.  We do this by dividing both sides of the equation by s and multiplying both sides by a³. 

Solving for a means getting its exponent to 1.  You can do this by taking the third root of each side of the equation. 

From Eq. 3-16:                                   

From Eq. 3-12:                                   

Therefore:                            

Using Eq. 3-14, we can get the equation into a more standard from:

The value of a is determined by the equation:

Example 3.9: The equation for bending stress of a bar with a rectangular cross-section, , was derived from a general equation for bending stress, , and two equations which apply specifically to rectangular cross-sections.  The two equations are: 

                      and                  

Using the three equations and the laws of exponents, derive Eq. 3-6. 

Solution:                               

We first substitute the equations for c and I:

The h's are all moved to the denominator by using Eq. 3-11: 

Eq. 3-13 is then used to reduce the h's:    

Therefore: