Differential Equation - Why do beams break?

Question

I've never studied physics before and I'm a Maths major. But I do have a physics-related differential equation question which I need help on.

Given the beam deflection equation: $$F(x) = EI \frac{d^4u}{dx^4}$$ where $E$ is the Young's Modulus, $I$ is the moment of inertia.

I found that $u(x)$ implies the displacement, $u'(x)$ implies change in displacement, $u''(x)$ implies the bending moment and $u'''(x)$ is the sheer force of the beam.

And relating this formula to a 'breaking a spaghetti' situation- where a spaghetti eventually breaks when we bend it at both ends. Why does that happen? Does the spaghetti break into two because of the maximum bending moment of the spaghetti is exceeded? or the sheer force is exceeded? Or is there any other reasons?

I apologize if any of the terms here are inaccurate. Thank you!

Answer 1

Given the beam deflection equation: $$F(x) = EI \frac{d^4u}{dx^4}$$ where $E$ is the Young's Modulus, $I$ is the moment of inertia.

That equation is only approximately valid, and then only in the case where several simplifying assumptions apply. Those simplifying assumptions include

• The load $F(x)$ is a continuous function of position.
• The resulting deflections are small compared to beam length.
• The response is linear.
• The response is elastic.

The above is not a complete list.There are several other simplifying assumptions that go into making the equation you wrote be approximately valid.

Each of those four simplifying assumptions does not pertain in the case of applying sufficient large bending moments at the ends of a piece of spaghetti. Breaking is an obvious example of a nonlinear inelastic response. The equation simply does not apply.