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I came across the following relations in the plate theory. I, however, don’t understand the reason for the negative signs in Mx and My. There is, however, no negative sign in Mxy. I am guessing, this is because of some sign convention, but that doesn’t make much sense to me. Please help me out.

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Thanks in advance.

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  • $\begingroup$ The sign of Mx, My is assigned by sign convention, that in turn is relied upon to identify which face of an element is in tension. I think Mxy/Myx does have +/- sign, evident by the combination of moments using Wood-Armer Method - Mx* = Mx + |Mxy| and My* = My + |Mxy|. $\endgroup$
    – r13
    Commented Jul 29, 2022 at 0:51

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In 2D, you can choose a convention, but in 3D, it is much more convenient to see the moments as vectors with 3 components, each twisting around an axis with the right hand rule (strictly speaking, there is a convention, because Cartesian coordinate system in 3D can be right-handed (standard) or left-handed).

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From the figure, $M_y$ vector is pointing towards you in 2D. If you have a stress distribution in $x$ direction on the black edge $\sigma_x(z)$ (positive in $x$ direction), then the moment must be: $$M_y = -\int\limits_{-h/2}^{h/2} z\cdot \sigma_x(z) \mathrm{d}z$$

With bending moments, when you use vectors, you are always in 3D, even if the bending goes on in 2D, because the bending vector will be perpendicular to your 2D plane.

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The negative sign in the stress integrals for the bending moments around x and y axes is a matter of sign convention. Sign convention is a matter of choice. It just means 'whatever sign you decide to apply to moments in one direction and moments in the other direction'. An easy way to visualise it is to look out along the x and y axes from the origin and define say +ve moments to be clockwise around each axis (sometimes called a 'RH rule'), or if you prefer, anticlockwise around each axis ('LH rule'). Right/Left hand rule can be determined by pointing your thumb along the axis from the origin, and the direction of your curled fingers indicates the direction of positive moment.

In terms of the twisting moments, on the diagram they have shown 'Mxy' for twisting on both axes. This is fairly common practice in these kinds of diagrams, but can be confusing. The 'correct' way to show it would be 'Mxy' on one axis and 'Myx' on the other. Showing both as 'Mxy' is a simplification based on the fact that Mxy = - Myx (whereas 'My' does not usually = 'Mx'). They have therefore just given the formula for the one twisting moment.

Note that this simplification could be problematic in some circumstances, for example if you were analysing the capacity of a reinforced concrete slab subject to biaxial bending and twisting moments. In this case correct calculation of the sign of the twisting moment for each axis is important.

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  • $\begingroup$ Thank you for your response. I have a follow-up question. What sign convention do your refer to here? I believe, this moment would cause a positive bending, thus having a positive sign, and not a negative. Or is there another convention? $\endgroup$ Commented Jul 29, 2022 at 1:55
  • $\begingroup$ The sign convention is a matter of choice. It just means 'whatever sign you decide to apply to moments in one direction and moments in the other direction'. An easy way to visualise it is to look out along the x and y axes from the origin and define say +ve moments to be clockwise around each axis, or if you prefer, anticlockwise around each axis. (I've updated my answer to include this) $\endgroup$
    – PM-14
    Commented Jul 29, 2022 at 10:13

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