If I have a stiff metal plate, and several forces acting on it at specific points, how would I calculate the resulting shape?
What information, besides the forces acting on it, would I need?
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$\begingroup$ may be useful en.wikipedia.org/wiki/Shear_modulus $\endgroup$– Adrian HowardCommented Nov 10, 2019 at 19:38
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$\begingroup$ It is necessary to set the boundary conditions at the edges of the plate. $\endgroup$– Alex TrounevCommented Nov 10, 2019 at 23:54
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$\begingroup$ Thank you, everyone for your comments and answers - it gives me plenty to investigate $\endgroup$– rdmsglCommented Nov 11, 2019 at 14:27
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3 Answers
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You need to know the shape and exact thickness of the plate, the material it is made from, its processing history (heat treatment, cold work, etc.), the points of application of the forces, and their magnitudes.
In the mechanical engineering field, calculations like this are routine and are typically handled by modeling programs that run inside any of the popular computer-aided design and drafting (CAD) packages that run on PC's. They will furnish detailed stress and deflection maps of the part in question as a function of loading.
answered Nov 10, 2019 at 20:13
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$\begingroup$ Heat treatment does not affect the shape. Only the Modulus of Elasticity and Poisson's ratio affect the shape as far as mechanical properties. Heat treatment affects the strength, and it is used as a check against calculated stress values to predict yielding or breaking. $\endgroup$ Commented Nov 11, 2019 at 1:42
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$\begingroup$ heat treatment affects the strength, as in the 7000 series of precipitation-hardening aluminum alloys. You must specify the tempering treatment to know its yield strength. $\endgroup$ Commented Nov 11, 2019 at 6:10
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$\begingroup$ Yes, but the op is asking about deflection. Heat treatment does not affect deflection. The deflected shape is a function of $E$, $\nu$ and geometry. $\endgroup$ Commented Nov 11, 2019 at 12:37
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$\begingroup$ heat treatment affects strength which affects deflection under load. $\endgroup$ Commented Nov 11, 2019 at 18:38
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$\begingroup$ Strength is just a limit of stress. Like in a beam, deflection does not depend on strength so long as the part does not yield. $\delta = \frac{F \ell^3}{3 E I}$ is the standard equation for cantilever beam, and strength is not included. Strength is included in the safety factor calculation ${\rm SF} = \frac{S_y}{\sigma_{\rm VM}}$ $\endgroup$ Commented Nov 11, 2019 at 19:16
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Apart from geometric measures and material characteristics, then together with the forces applied, you need to specify how and where the plate is fixed.
Then, mainly according to the ratio of thickness wrt width / length, various approximated approaches can be taken, and we are speaking of plates and shells.
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The math behind bending even simple cases with rectangular or circular plates is very complex. You can look through the classic book of Roark Formulas for Stress and Strain Chapter 11, page 451 to see for your self.
In page 502, the simplest case of a rectangular plate is shown which is simply supported on all edges and a uniform load is applied perpendicular to the plate.
The maximum deflection $y_{\rm max}$ is given as a function of geometry and material properties, but also with coefficient $\alpha$ which non-linearly depends on the aspect ratio $a/b$ of the plate.
As a result, engineers often rely on FEA Software that can handle far more complex shapes that analytically possible and give accurate but approximate results for deformations, stresses and reaction forces.
For example, a rectangular plate with a hole in it under uniform load is solved below using Autodesk Fusion 360 (free for personal use) and deflection results shown in color contours.
answered Nov 11, 2019 at 1:54
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