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enter image description here

The above beam is loaded by a distributed load per unit length of the referential scale defined by the vector field q = q(x) and a distributed moment load vector per unit length m = m(x). As a consequence of the external loads, the beam is deformed into the so-called current state where the external loads are balanced by an internal section force vector F = F(x) and an internal section moment vector M = M(x)

The internal force F has normal component $N$ and shear components $Q_y$ and $Q_z$ and the internal moment has the respective components $M_x$, $M_y$, and $M_z$. The equations of equilibrium are as follows:

enter image description here

What I don't understand is why is the cross product of the unit vector $i$ with $dF$ equal to zero? Thank you.

Source of the tutorial

user134613
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2 Answers2

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Because unit vector $i$ is parallel to the $dF$.

In the diagram, they show $F$ in an angle, but it's not correct. In a beam bending stress and the resultant force are always axial, parallel with the centroid of the beam.

kamran
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  • So both $dF$ and $F$ cancel right? So the equation is wrong? – user134613 Feb 01 '22 at 20:08
  • What you said is not right. Because if that is the case, then i * F should also be equal to zero. – Rameez Ul Haq Feb 01 '22 at 20:08
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    not necessarily. But the cross product of unit vector i is parallel to dF, both align with the x-axis. and we know the cross-product of parallel vectors is zero. – kamran Feb 01 '22 at 20:12
  • @RameezUlHaq, I never said F or dF is zero! They are parallel to the x-axis so is the unit vector i, therefore their cross product is zero! – kamran Feb 01 '22 at 20:17
  • Thank you Kamran! – user134613 Feb 01 '22 at 21:00
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    @user134613, thank you for seeing my point. – kamran Feb 01 '22 at 21:06
  • @kamran, i still think you are wrong. Because F and dF both must lie on the same axis, since dF means infinitesimal small change in F. So if the cross product of i and dF is zero if both are lying on the same axis, then cross product of i and F must also be zero because then F also lies on the same axis as i. I still don't think this answer what you have given is correct. – Rameez Ul Haq Feb 02 '22 at 06:31
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Maybe they are assuming dF is extremely small, for a length change of dx, as compare to dM (which is relatively much larger than dF). So we can put dF to be equal to zero for simplifications.

Rameez Ul Haq
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