Deflection Lab
The deflection (in engineering) of a structure is the measure of how a material is bended or lowered because of a load. The first equation for deflection
is:
is:
Moment of Inertia Equation
Were‘b’ is the base and ‘h’ is the height. The base is multiplied by height (which is cubed). To get the
final product, you have to divide all of this by 12. This is the moment of
inertia for a rectangular prism or a beam.
The equation for maximum deflection is:
final product, you have to divide all of this by 12. This is the moment of
inertia for a rectangular prism or a beam.
The equation for maximum deflection is:
Maximum Deflection Equation
Were ‘F’ is force or usually weight, and ‘l’ is length. The ‘E’ is the modulus of
elasticity and “IXX” is the moment of inertia (from the earlier equation).
elasticity and “IXX” is the moment of inertia (from the earlier equation).
So, suppose we have to find the modulus of elasticity for a wooden beam. First we would need to find the dimensions to the wooden beam. The length of the beam is 80 inches, the base is 3.4 inches, and the height is 1.5 inches.
When we plug the numbers into the equation, we get:
When we plug the numbers into the equation, we get:
With the .96in4, we would plug it into the maximum deflection equation, but first we need to rewrite the equation to solve for the modulus of elasticity. Let’s say the load is 170lbs and the maximum deflection is 1.25 inches. So: Now we simply substitute: As you can see above, the load is 183.6lbf.
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Now that we have the modulus of elasticity, we could solve for a load we don’t know because the modulus of elasticity never changes with orientation. Let’s say the maximum deflection is 1.35 inches. First we need to solve for ‘F’: |
Those were the calculations when the side of the wooden beam was in “landscape” orientation. So what would happen if we made the orientation of the beam “portrait”?
Let's start with the moment of inertia equation:
Let's start with the moment of inertia equation:
(The “landscape” orientation is ‘B’ and the portrait orientation is ‘A’)
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If we switch the orientation, then base and the height will change. So instead of 3.4 inches for our base and 1.5 inches for our height, the base will now be 1.5 inches and the height will be 3.4 inches. So when we plug the numbers into the equation: We get a completely different number from before, but why?
Our second moment of inertia is greater than before; therefore it produces more resistance to deformation. So the wooden beam would bend more if it was in “landscape” because the orientation and shape give it less stiffness than a “portrait” orientation. Let’s say we have to calculate the maximum deflection with the new moment of inertia. The weight of the load is 170lbf and the modulus of elasticity is the same as before because it’s the same wooden beam. So: |
So what would happen if the beam was hollow? It’s simple; we would just have to calculate the moment of inertia a different way:
Where ‘b1’ is the size dimension (biggest dimension) and ‘b2’ is the smaller dimension, and like the base, ‘h1’ is the size dimension and ‘h2’ is the smaller dimension.
Conclusion
Deflection is important to engineering and engineers. Understanding an object’s moment of inertia and modulus of elasticity can help engineers reduce the weight and costs for building structures. Understanding the equations can help an engineer make our world a more efficient and safe one.