Math 2.0

Mechanics of deformable solids Definition

Biharmonic plate equation

Builds on Kirchhoff–Love plate theoryBiharmonic equationLaplacian

Definition

Let Ω⊂ℝ2\Omega \subset \R^2 be the mid-plane of a thin isotropic plate of thickness hh, with Young's modulus EE and Poisson's ratio ν\nu, and let D=Eh3/(12(1−ν2))D = Eh^3/(12(1 - \nu^2)). Let qq be the transverse load per unit area and ww the deflection, in the same direction. The biharmonic plate equation is D Δ2w=qin Ω,Δ2w=∂4w∂x4+2∂4w∂x2∂y2+∂4w∂y4,D \, \Delta^2 w = q \quad \text{in } \Omega, \qquad \Delta^2 w = \frac{\partial^4 w}{\partial x^4} + 2 \frac{\partial^4 w}{\partial x^2 \partial y^2} + \frac{\partial^4 w}{\partial y^4}, where Δ\Delta is the Laplacian. It is completed by two boundary conditions on each part of ∂Ω\partial \Omega. These are w=∂nw=0w = \partial_n w = 0 on a clamped edge, and w=0w = 0 with vanishing normal bending moment on a simply supported edge.

Remarks

Example: a simply supported rectangle 0<x<a0 < x < a, 0<y<b0 < y < b under the load q=q0sin⁡(πx/a)sin⁡(πy/b)q = q_0 \sin(\pi x/a) \sin(\pi y/b) has w=q0Dπ4(1/a2+1/b2)2sin⁡πxasin⁡πyb.w = \frac{q_0}{D \pi^4 (1/a^2 + 1/b^2)^2} \sin\frac{\pi x}{a} \sin\frac{\pi y}{b}. Check: Δ2\Delta^2 multiplies this product of sines by π4(1/a2+1/b2)2\pi^4 (1/a^2 + 1/b^2)^2, and w=Δw=0w = \Delta w = 0 on the edges, which on a straight edge is equivalent to w=0w = 0 with zero normal moment.

Non-example: the membrane equation −TΔw=q-T \Delta w = q, for a stretched membrane with tension TT, is a second-order equation and describes a body with no bending stiffness.

The equation follows from Kirchhoff–Love plate theory and is a case of the biharmonic equation with a right-hand side. Expanding qq in double sine series gives Navier's solution for any load on a simply supported rectangle.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

Show the path as a graph
Biharmonic equation, concept, no evidence levelBiharmonic equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptKirchhoff–Love plate theory, concept, no evidence levelKirchhoff–Love plate …conceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptBiharmonic plate equation, definition, no evidence levelBiharmonic plate equa…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Biharmonic equation concept
  2. Euclidean space object
  3. Function concept
  4. Kirchhoff–Love plate theory concept
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Vector field concept
  11. Gradient concept
  12. Divergence concept
  13. Laplacian concept
  14. Biharmonic plate equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Kirchhoff–Love_plate_theory
About this page
TagEMRAQ3V9BD
URIm2://EMRAQ3V9BD
Kinddefinition
WritingW0 stub
MSC74K20, 35J30, 74G05
DefinesBiharmonic plate equation
UsesKirchhoff–Love plate theory, Biharmonic equation, Laplacian
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.