Mechanics of Materials Bending Stres Problem

[Delta-One]

Mechanics of Materials Bending Stress Problem

Hi,

I have a homework problem for mechanics of materials involving bending stress. Here is the exact wording:

"Show that the maxiumum bending stress for a beam of rectangular cross-section is Omax = Mc/I [(2n + 1) / (3n)] if instead of Hooke's law, the stress-strain relationship is O^n = Ee, where n is a number dependent on the material."

--NOTE: Omax is the maxiumum bending stress

Essentially the diffence is not using Hooke's law: O = Ee (e is the strain). Using Hooke's law yields the flexure formula: Omax = Mc/I.

So far I have got O^n = Omax^n * (y/c)
but when I insert this into the equation dM = ydF or dM = yOdA I am uncertain how to obtain I (the moment of Inertia)

Any help would be greatly appreciated.

 

[mathmike]

the moment of inertia is calculated by the eq. [1/12*b*h^3]

 

[thepatientmental]

First of all, great job on getting started with the problem! Let's go through the steps to solve this problem together.

Step 1: Understanding the given information
The problem states that instead of using Hooke's law, we are using a stress-strain relationship of O^n = Ee, where n is a material-dependent constant. This means that the maximum bending stress, Omax, can no longer be calculated using the flexure formula Omax = Mc/I.

Step 2: Deriving the bending stress equation
To find the maximum bending stress, we need to start with the basic equation for bending stress: O = My/I. In this case, we are using a non-linear stress-strain relationship, so we need to substitute O^n = Ee for O. This gives us (O^n) = My/I. Now, we can rearrange this equation to solve for Omax: Omax = (My/I)^(1/n).

Step 3: Finding the moment of inertia
Next, we need to find the moment of inertia, I. This can be done by using the formula for the moment of inertia of a rectangular cross-section: I = (bh^3)/12. However, in this case, we are dealing with a beam of rectangular cross-section, so we need to consider the height, h, as the distance from the neutral axis to the top or bottom edge of the beam, which we will call c. This gives us I = (bc^3)/12.

Step 4: Substituting and simplifying
Now, we can substitute this moment of inertia into our equation for Omax: Omax = (My)/[(bc^3)/12]^(1/n). Simplifying this further, we get Omax = My/(b^(1/n)c^(3/n)).

Step 5: Solving for Omax
To find the maximum bending stress, we need to solve for Omax by finding the maximum moment, M. To do this, we can use the equation dM = ydF, where dM is the differential moment, y is the distance from the neutral axis to the differential force dF, and dF is the differential force acting on a small area dA. We can also use the equation dM = yOdA, where dM is the differential moment, y is the distance from the neutral axis to the differential force dF, and dA