Plate - bending moment equation

In summary, the conversation discusses solving an exemplary case of a cantilever plate subjected to pressure using different methods. One method involves calculating stress from maximum bending moment and deflection from a differential equation, while the other involves using the Timoshenko's Plates & Shells formula. The results from the first method are incorrect, possibly due to the asymmetrical boundary conditions, but a solution is found through the second method.
  • #1
FEAnalyst
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TL;DR Summary
How to calculate the stress and deflection for a cantilever plate using bending moment equation ?
Hi,

I'm trying to solve an exemplary case of a cantilever plate (one long edge fixed, all other edges free) subjected to pressure. I've already calculated this using approximation to beam of unit width and the results are good but I would like to use another method too. In a polish book I've found two examples - one plate with shorter edges simply supported and one with shorter edges clamped. I tried solving my example in a similar manner but the results are meaningless. The method is analogous to the approach used for beams - calculate stress from maximum bending moment and deflection from differential equation.

Here's a scheme of the plate:
plate.png

I was advised to use the following formula for bending moment: $$ \displaystyle{ M_{x}=\frac{qa^{2}}{2}-qax+ \frac{q x^{2}}{2}} $$ The bending moment equation for a plate is: $$\displaystyle{ D w^{\prime \prime}=-M_{x}}$$ Thus: $$\displaystyle{ D w^{\prime \prime}=- \frac{qa^{2}}{2}+qax- \frac{q x^{2}}{2}}$$ $$\displaystyle{ D w^{\prime}=- \frac{qa^{2}x}{2}+ \frac{q a x^{2}}{2} - \frac{q x^{3}}{6}+C_{1}}$$ $$\displaystyle{ D w=- \frac{qa^{2}x^{2}}{4}+ \frac{q a x^{3}}{6} - \frac{q x^{4}}{24}+C_{1}x+D_{1}}$$ Introducing boundary conditions to find constants of integration: $$\displaystyle{ x=0 \Longrightarrow w=0}$$ $$\displaystyle{ x=0 \Longrightarrow w^{\prime}=0}$$ $$\displaystyle{ C_{1}=\frac{qx \left( 3 a^{2} - 3ax + x^{2} \right) }{6}}$$ $$\displaystyle{ D_{1}=- \frac{q x^{2} \left( 6 a^{2} - 8ax + 3 x^{2} \right)}{24}}$$ $$\displaystyle{ Dw=- \frac{q a^{2} x^{2}}{4}+ \frac{q x^{2} \left( 3 a^{2} - 3ax + x^{2} \right)}{6}- \frac{qx^{2} \left( 6a^{2} - 8ax + 3x^{2} \right)}{24}+ \frac{qax^{3}}{6}- \frac{qx^{4}}{24}}$$ $$\displaystyle{ x=a \Longrightarrow w_{max}}$$ Now when I substitute ##a## to the previous equation I get ##Dw=0## which makes no sense.

When it comes to stresses, they will be highest at the fixed edge where ##x=0## so: $$M_{x \ max}=\frac{qa^{2}}{2}$$ $$\sigma_{x \ max}=\frac{M_{x \ max}}{\frac{h^{3}}{12}}$$ but when I substitute values, the stress results are completely incorrect.

What's wrong with these calculations ? Is the method itself not applicable in this case ? Examples from the book featured plates with symmetric boundary conditions but it wasn't mentioned that this approach won't work for assymetric boundary conditions. Or maybe I just made some mistake ?

Thanks in advance for help
 
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  • #2
Have you checked Timoshenko's Plates & Shells? I don't know if this is included, but that would be the most likely source I can think of.
 

Related to Plate - bending moment equation

What is the plate-bending moment equation?

The plate-bending moment equation is a mathematical expression used to calculate the bending moment of a plate subjected to external loads. It takes into account the material properties of the plate, the applied loads, and the geometry of the plate.

How is the plate-bending moment equation derived?

The plate-bending moment equation is derived from the principles of mechanics and the theory of elasticity. It involves solving differential equations to determine the internal stresses and deformations of a plate under external loads.

What are the assumptions made in the plate-bending moment equation?

The plate-bending moment equation assumes that the plate is made of a homogeneous and isotropic material, and that it is thin compared to its other dimensions. It also assumes that the plate is subjected to small deformations and that the applied loads are static and do not change over time.

How is the plate-bending moment equation used in engineering?

The plate-bending moment equation is used in engineering to design and analyze structures made of thin plates, such as beams, bridges, and shells. It helps engineers determine the maximum stresses and deformations that a plate can withstand, and to ensure that the structure will not fail under the applied loads.

What are the limitations of the plate-bending moment equation?

The plate-bending moment equation is limited to plates that are thin and have simple geometries, such as rectangular or circular shapes. It also does not take into account the effects of shear forces, which can be significant in some cases. Additionally, the equation assumes that the material properties of the plate are constant, which may not be the case in some materials.

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