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Euler beam equation is defined as follows:

$$M=-EI{\frac {d^{2}w}{dx^{2}}}$$

There is a negative sign, so if the second derivative of the deflection is negative, the moment is positive. When the beam curves down, the second derivative is negative. So the moment on the beam should be positive when the beam bends down.

But normally moment is defined the other way around:

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Here the moment is positive when the beam curves up. So is there a contradiction or am I misunderstanding something?

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  • $\begingroup$ Who inserts your minus sign? I do not think there is a standard convention here. It depends on your coordinate system. Perhaps the author's $w$ axis points downward? $\endgroup$ – mike stone Apr 15 at 14:07
  • $\begingroup$ I copied the formula from Wikipedia, en.m.wikipedia.org/wiki/Euler%E2%80%93Bernoulli_beam_theory $\endgroup$ – S. Rotos Apr 15 at 15:51
  • $\begingroup$ The wiki article makes it clear that their $M(x)$ is the moment exerted on the right side of the beam cut at $x$ by the part to the left of $x$. Thus, when the beam bends up the torque exerted by the LHS is trying to bend it down. Hence the sign. Other accounts may use different definitions. $\endgroup$ – mike stone Apr 15 at 16:00
  • $\begingroup$ @S.Rotos My mechanics of materials references do not have a negative sign on the right side of the Euler Beam Equation $\endgroup$ – Bob D Apr 15 at 20:03
  • $\begingroup$ @BobD Interesting, my textbook has the negative sign. It must the just that the direction of positive moment has been defined differently.. $\endgroup$ – S. Rotos Apr 16 at 8:54

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