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I'd like to calculate the bending moment of a cantilever, fixed at its base, and submitted to a certain stress on a specific spot, but I can't find the proper definition of this bending moment (first time I'm encountering this name, and google didn't not help me so far).

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See the Treasure of Civil Engineering Resources. – Keep these mind Feb 10 '13 at 18:09
up vote 1 down vote accepted

Let us assume that the cantilever is fixed on a wall at the RHS end, and use that end as reference for measuring the distance, $x$, along the cantilever away from the wall. Let us also assume that the force $F$ is applied at a point $a$ units away from the wall (obviously the $a\le L$, where $L$ is the length of the canrilever). The bending moment at some point P which is distance $x$ from the wall, the bending moment is the algebraic sum of the moments of all loads lying to the right of the point P, and taken with respect to the point P itself. So we can write the forllowing:

for $x\le a$ the bending moment $M(x) =-F((L-x))$

for $x\gt a$ the bending moment $M(x)=0$ as there are no loads beyond the point $x=a.$

When you combine this with the equation for the deflection, v, you get

$EI{\frac {d^2v}{dx^2}}=M(x)$,

where $E$ is the Young modulus of the material, and $I$ is the second moment of area of the cantilever (also known as moment of inertia, calculated in a similar way). You need to solve this equation using youe boundary conditions, to get the diflection at various values of $x$. Hope this helps.

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Thanks for your explanation, it raises a new question, let's look at the deflection passed the point of application of the force (a), it obeys the following equation: $\frac{d^2v}{dx^2} =0$ and has the general solution: $v(x>a) = v(a) + \alpha (x-a)$. Supposing that the end of my cantilever is not fixed, on the right side, what conditions do I have to constrain the value of $\alpha$? (conservation of length or something along this line?) – Learning is a mess Feb 11 '13 at 10:39
Ok, I think that I've got the answer to my question, we also need to impose continuity of the derivative of the deflection at $x=a$, and so we have: $v(x>a) = v(a) + v'(a)(x-a)$? – Learning is a mess Feb 11 '13 at 10:45
@Learningisamess Sorry, I have just read your comment. Yes, that is right, you do need to impose boundary conditions and contimuity requirement. Note that the cantilever remains straight (is not bent) beyond the point of application of the load, unless the cantilever itself has some continuous weight distribution w [N/m]. The analysis must be different in this case. – JKL Feb 11 '13 at 11:10
Thank you for clearing this out, I am still a bit puzzled by this definition $M(x)=−F((L−x))$ in the sense that it means that the bending moment is non local quantity on the cantilever, I'll have to play a bit with it ) – Learning is a mess Feb 11 '13 at 11:15

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