Kirchhoff plate theory
Theory and Physics
What is Kirchhoff Plate Theory?
Professor, is Kirchhoff plate theory the 2D version of Euler-Bernoulli beam theory?
Exactly. Just as the Euler-Bernoulli beam assumes "cross-sections remain perpendicular to the neutral axis," Kirchhoff plate theory assumes "straight lines through the thickness remain straight and perpendicular to the mid-surface after deformation."
Fundamental Assumptions
Kirchhoff's Assumptions:
1. Straight Normal Assumption — Straight lines perpendicular to the mid-surface before deformation remain perpendicular to the mid-surface after deformation.
2. Non-Extending Normal Assumption — Strain in the thickness direction $\varepsilon_{zz} = 0$.
3. Zero Transverse Shear Strain — $\gamma_{xz} = \gamma_{yz} = 0$.
Assumption 3 is the same as in Euler-Bernoulli beam theory. Shear deformation is neglected.
Correct. This assumption means the rotation angles are determined by the derivatives of the out-of-plane deflection:
The only degree of freedom is the deflection $w(x,y)$; the rotation angles are not independent variables.
Governing Equation
The biharmonic equation for plate deflection $w(x,y)$:
Here, $D = Et^3/(12(1-\nu^2))$ is the plate bending rigidity, and $q$ is the out-of-plane distributed load.
$\nabla^4$ is a fourth-order differential operator. It's the 2D version of the beam equation $EI w'''' = q$, right?
Exactly. Since $\nabla^4 = \nabla^2(\nabla^2)$:
Bending Moments and Shear Forces
Internal force components:
Moments are determined by the second derivatives of $w$. Bending moment is obtained by differentiating deflection twice. It has the same structure as beams.
Yes. Bending stresses in the plate are:
Maximum stress occurs at the plate surfaces ($z = \pm t/2$). It has the same structure as the beam formula $\sigma = My/I$.
Applicability Range
Up to what plate thickness can Kirchhoff plate theory be used?
Thin plates are a prerequisite. A rule of thumb is $b/t > 20$ ($b$: shorter side of the plate, $t$: thickness).
Similar to Timoshenko beam theory:
- $b/t > 20$: Kirchhoff plate theory is sufficient.
- $10 < b/t < 20$: Consider Mindlin plate theory.
- $b/t < 10$: Use Mindlin plate theory or solid elements.
Is Mindlin plate theory the 2D version of Timoshenko beam theory?
Exactly. Kirchhoff plate = 2D version of EB beam, Mindlin plate = 2D version of Timoshenko beam. The difference is whether shear deformation is considered.
Kirchhoff Plate Elements in FEM
Is implementing Kirchhoff plates in FEM difficult?
Actually, it's very difficult. Because Kirchhoff plate theory involves fourth-order derivatives of $w$, FEM implementation requires $C^1$ continuity (both displacement and rotation angles must be continuous across elements). Standard FEM ($C^0$ continuity) cannot satisfy this.
Is $C^1$ continuity that difficult?
Creating polynomial elements that achieve $C^1$ continuity in 2D is challenging. Historically, the Argyris triangle (21 DOF) and Bell triangle (18 DOF) were developed, but they have many degrees of freedom and are not practical. This is why Mindlin plate theory (which only requires $C^0$) became mainstream in FEM.
Summary
Let me summarize Kirchhoff plate theory.
Key points:
- Bending theory for thin plates neglecting shear deformation — 2D version of EB beam.
- $D\nabla^4 w = q$ — Biharmonic equation.
- Applicable for $b/t > 20$ — Limited to thin plates.
- Requires $C^1$ continuity in FEM — Difficult to implement.
- Mindlin plates (only $C^0$ needed) are mainstream in practice — Kirchhoff plate elements are rare.
So, the theory is elegant, but it yielded the spotlight to Mindlin plates because FEM implementation is difficult.
Yes. However, Kirchhoff plate theory is the foundation for analytical solutions and is indispensable as a reference solution for verifying FEM results. The Navier solution (double Fourier series solution for rectangular plates) is a classical solution based on Kirchhoff theory.
Origins of Kirchhoff Plate Theory
Gustav Kirchhoff established plate bending theory in his 1850 paper "Über das Gleichgewicht und die Bewegung einer elastischen Scheibe." His assumptions (thin plate, preservation of straight normals, neglect of mid-surface strain) remain valid today as "Classical Plate Theory (CPT)." Numerical experiments have confirmed that the error of this theory is less than 1% for forming analysis of rolled aluminum thin sheets (t<2mm).
Physical Meaning of Each Term
- Inertia Term (Mass Term): $\rho \ddot{u}$, i.e., "mass × acceleration." Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried away" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind." In static analysis, this term is set to zero, assuming "forces are applied slowly enough that acceleration is negligible." It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness Term (Elastic Restoring Force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. Now a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously the rubber band. This "resistance to stretching" is the Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong." Stiffness is "resistance to deformation," strength is "resistance to failure"—they are different concepts.
- External Force Term (Load Term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but applying "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping Term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue shaking forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum Assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small Deformation Assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic Material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-Static Assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-Applicable Cases: Large deformation/large rotation problems require geometric nonlinearity. Nonlinear material behavior like plasticity or creep requires constitutive law extensions.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify loads and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic Modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | Unify to N in mm system, N in m system. |
Numerical Methods and Implementation
Implementation of Kirchhoff Plates in FEM
How was the $C^1$ continuity problem solved?
Historically, there are three approaches.
1. High-Order Conforming Elements
2. DKT/DKQ Elements (Discrete Kirchhoff)
DKT (Discrete Kirchhoff Triangle) uses the Mindlin plate theory framework for discretization and "discretely" enforces Kirchhoff constraints (shear strain = 0) at Gauss integration points.
Applying Kirchhoff conditions after using Mindlin plate elements... clever.
DKT was proposed by Batoz, Bathe, and Ho in 1980. It has only 3 nodes and 9 DOF ($w, \theta_x, \theta_y$ per node) and is highly accurate. Nastran's CTRIA3 (bending) internally uses a DKT-type formulation.
DKQ (Discrete Kirchhoff Quadrilateral) is the quadrilateral version with 4 nodes. Similarly practical.
3. Using Mindlin Plate Elements as Thin Plates
The most practical approach. Use Mindlin plate elements (including shear deformation); if the plate is thin, shear deformation automatically becomes small. For thin plates, results identical to Kirchhoff plate theory are obtained.
So, in modern FEM, using "dedicated elements" for Kirchhoff plates is not mainstream; instead, Mindlin plate elements are used as substitutes?
Exactly. General-purpose shell elements in Abaqus, Ansys, and Nastran are all based on Mindlin (Reissner-Mindlin) theory. For thin plates, they converge to Kirchhoff's theoretical solutions.
Utilizing Analytical Solutions
How are Kirchhoff plate analytical solutions used for FEM verification?
The most famous analytical solution is the Navier solution. Deflection for a simply supported rectangular plate under uniform load $q$:
A double Fourier series...
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