13. Plate Theory

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AERSP 301Plate Theory

Jose Palacios

August 2008


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Today

Plate Theory

Analyzing energy method (Stationary Principle of Total PotentialEnergy)

Similar to process for FEM

Consider plate as a whole (not discretized)

Displacement Functions change

Final Exam: August 14 2008 @ 10:00 RCOE


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Plate Theory

Beam

Plate


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Plate Theory


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Plate Theory


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Plate Theory


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Plate Theory

Non-zero strains (in vector form):

These are the strains/deformations that we are concerned with whenconsidering analysis of a plate


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Plate Theory


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Plate Theory

Stress-Strain Relations for plates:

But for a plate (2 in-plane directions) we must consider Poissons effects


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Plate Theory

Strains in each direction are functions of stress in both directions:


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Plate Theory

For an isotropic material, there are only 2 independent constantsrelating stress to strain

For isotropic materials we need to know Youngs Modulus, Shear

Modulus, and poissons ratio. If we know 2 of these constants, we can

calculate the third.


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Plate Theory

Put strain/stress relations into matrix form:


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Plate Theory


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Plate Theory

Now that we have stress-strain relations, we use Energy methods(Stationary Principle) to finish the analysis

Start with Strain Energy

We only had to consider 1 axial stress and strain

For plates we are concerned with two axial stresses and strains as wellas shear stress and strain


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Plate Theory

For plates:


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Plate Theory

Integrate over thickness, and area of plate


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Plate Theory

(Example of non-isotropic plate)


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Plate Theory

Similar to plates and bars


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Plate Theory

At this point, for bars and beams we discretized the structure usingthe finite element method


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Plate Theory

For the plate, like the beam or bar, we need to assume a displacement function

Assumed Modes Method assume the displacement of the plate can bewritten as the sum of a number of mode shapes, mn, and correspondingcoefficients, Amn.

1 1 ),(),( m nmnmn yxAyxw


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Plate Theory

The assumed displacements must match the geometric boundaryconditions of the plate and be 2 times differentiable.

Mode shapes must match BCs

2 times differentiable since strain energy is a function of curvatures

Simply-supported plate (BCs)


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Plate Theory

Clamped plate


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Plate Theory

Once the displacement of the plate is assumed, then the Total strainenergy of the plate can be determined.

As before the work potential is a function of the external loading and

assumed displacement.

We can then write out the total potential energy and apply thestationary principle to get equilibrium equations and determine theresulting displacements, strains, and stresses of the plate


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Plate Theory example

Example


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We have D already, need to determine the curvatures!


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Look at the integrand kTDk in the expression for U

Since the our assumed displacement has only one term (one mode)the product, kTDk, is a scalar


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Perform the integrals for each term:

Using these expressions, the strain energy can be evaluated as


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What about W, work potential?


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Know our final results depends on the loading(work potential)


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Plate Theory

To get accurate results

Use multiple mode shapes

Displacement due to loading should match the mode shapes(examples)

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13. Plate Theory