Solving Euler-Bernoulli Beam Equation: Second Area of Moment

[mike amory]

Using the Euter-Bernoulli beam equation , solving for the SECOND AREA OF MOMENT, my answer is a numerical value. In laymans terms what does this number signify in relation to the bending of a beam?

 

[SteamKing]

Can you be more specific? What is the second area of moment? Is it like the second moment of area? Please provide details of your calculation.

 

[mike amory]

Iy = bh^3 + Ad^2 <br /> <br /> Iy = second moment of area<br /> b = horizontal width of section<br /> h = height of section<br /> A = area of section<br /> d = vertical distance of the section from nuetral axis<br /> <br /> b = 17<br /> h = 6.6<br /> A = 97.8<br /> d = 2.6

 

[mike amory]

Sorry, the correct equation is:

Iy=Ad^2 + bh^3/12

 

[SteamKing]

The quantity EI is known as the flexural rigidity of a beam. E is the Young's modulus of the material used to construct the beam, and it also represents the ratio of stress to strain for the material, such that

stress = E * strain

The second moment of area I (also known as the moment of inertia) is a geometric property of the cross-section of the beam. In practical terms, the greater the value of I, the stiffer the beam, and for a given beam loading, a higher value of I results in a lower value of deflection.

 

[jefswat]

Also, if you tell me the loading, what the material is (E), and what the moment of inertia (I) are, I can tell you how much the member will rotate and deflect. E and I are critical to making the jump from forces to some form of deformation. Euler-Bernoulli is actually a simplification that ignores a lot of important factors that really don't matter when you have high span length/member depth ratios (99% of normal beams in reality).

Moment of inertia is a quantity that requires a reference point. Take your plate for example. The bh^3/12 term is actually the moment of inertia of the section with respect to its centroid (the centroid is the "center" in an average sense, but it can get complicated). Ad^2 is a correction term, if you want the moment of inertia with respect to something besides the centroid.

We both assumed you were referring to the structural engineering definition since you mentioned Euler-Bernoulli. Moment of inertia is also the constant relating moment on a body and its angular acceleration. In the dynamic sense, mass is for linear motion what moment of inertia is for rotational motion.

 

[mike amory]

what I am trying to find is the longitudinal strength of a wooden boat hull , using an athwartship section taken amidship , and treating the whole as a box beam .

 

[SteamKing]

This type of calculation is a little tricky for a naval architect, especially for a wooden boat. Generally, one only includes the material which extends continuously over about the middle 40% of the length of the boat (i.e., 20% forward and 20% aft of the midships location).

Also, if the boat has a lot of shape fore and aft of amidships, your moment of inertia value will be accurate only for a small portion of the length of the vessel.