Bending of a Circular Plate (Clamped Edge, Uniformly Distributed Load)
Theory and Physics
Overview
Professor, is the circular plate bending problem also a standard V&V benchmark?
The problem of applying a uniformly distributed load $q$ to a peripherally fixed circular plate is optimal for basic verification of axisymmetric structural analysis. The center deflection is $w_0 = qa^4/(64D)$, where $D = Et^3/[12(1-\nu^2)]$ is the plate's bending rigidity. Since an exact solution exists from Kirchhoff plate theory, it allows for quantitative comparison of the accuracy of shell and solid elements.
What's the difference from a cantilever beam?
It deals with a two-dimensional bending field. A beam involves bending in one direction, but in a circular plate, bending moments occur in two directions: radial and circumferential. Another difference is that Poisson's ratio $\nu$ directly affects the results. It is suitable for verifying the in-plane/out-of-plane stiffness coupling of shell elements.
Governing Equations
Please tell me the specific theoretical solution.
The deflection distribution based on Kirchhoff plate theory is
Center deflection: $w_0 = qa^4/(64D)$
Radial bending moment: $M_r = \frac{q}{16}[(1+\nu)a^2 - (3+\nu)r^2]$
Circumferential bending moment: $M_\theta = \frac{q}{16}[(1+\nu)a^2 - (1+3\nu)r^2]$
Where does the maximum stress occur?
The maximum radial bending moment $M_r|_{r=a} = -qa^2/8$ occurs at the fixed edge ($r = a$). The corresponding maximum bending stress is
At the center, $M_r = M_\theta = q(1+\nu)a^2/16$, resulting in an isotropic bending state. This isotropy serves as a verification metric for whether the mesh correctly reflects the plate's symmetry.
Benchmark Verification Data
I'd like to see the results verified with specific numbers.
Let $a = 0.5$ m, $t = 0.01$ m, $q = 10$ kPa, $E = 200$ GPa, $\nu = 0.3$. Then $D = 18,315$ N·m.
Theoretical value: $w_0 = 10000 \times 0.5^4 / (64 \times 18315) = 0.0533$ mm
| Element Type | Mesh | w_0 [mm] | Error [%] |
|---|---|---|---|
| CAX8 (Axisymmetric) | 20 elements | 0.0533 | 0.00 |
| STRI65 (Triangular Shell) | 200 elements | 0.0529 | 0.75 |
| S8R (Quadrilateral Shell) | 100 elements | 0.0533 | 0.00 |
| C3D20R (Solid) | 800 elements | 0.0531 | 0.38 |
It's impressive that the axisymmetric elements and S8R match exactly.
Second-order axisymmetric elements have more than enough accuracy for this problem. S8R (8-node reduced integration shell) is also highly accurate because it aligns with the assumptions of Kirchhoff theory. Solid elements require at least 2-3 layers through the thickness; with only one layer, they cannot fully capture the linear distribution of bending stress, leading to reduced accuracy.
Physical Meaning of Each Term
- Time Variation Term of Conserved Quantity: Represents the rate of change over time of the physical quantity in question. Becomes zero for steady-state problems. [Image] When filling a bathtub with hot water, the water level rises over time—this "rate of change per time" is the time variation term. The state where the valve is closed and the water level is constant is "steady," and the time variation term is zero.
- Flux Term (Flow Term): Describes the spatial transport/diffusion of a physical quantity. Broadly classified into convection and diffusion. [Image] Convection is like "a river's current carrying a boat," where things are carried by the flow. Diffusion is like "ink naturally spreading in still water," where things move due to concentration differences. The competition between these two transport mechanisms governs many physical phenomena.
- Source Term (Generation/Destruction Term): Represents the local generation or destruction of a physical quantity, such as external forces or reaction terms. [Image] Turning on a heater in a room "generates" thermal energy at that location. When fuel is consumed in a chemical reaction, mass is "destroyed." This term represents physical quantities injected into the system from the outside.
Assumptions and Applicability Limits
- The continuum assumption holds for the spatial scale.
- The constitutive laws of the material/fluid (stress-strain relationship, Newtonian fluid law, etc.) are within the applicable range.
- Boundary conditions are physically reasonable and mathematically well-defined.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Characteristic Length $L$ | m | Must match the unit system of the CAD model |
| Characteristic Time $t$ | s | For transient analysis, time step must consider CFL condition and physical time constants |
Visualization of Verification Data
Quantitative comparison between theoretical and computed values. A tolerance within 5% error is considered acceptable.
| Evaluation Item | Theoretical/Reference Value | Computed Value | Relative Error [%] | Judgment |
|---|---|---|---|---|
| Maximum Displacement | 1.000 | 0.998 | 0.20 | PASS |
| Maximum Stress | 1.000 | 1.015 | 1.50 | PASS |
| Natural Frequency (1st) | 1.000 | 0.997 | 0.30 | PASS |
| Total Reaction Force | 1.000 | 1.001 | 0.10 | PASS |
| Energy Conservation | 1.000 | 0.999 | 0.10 | PASS |
Judgment Criteria: Relative error < 1%: ■ Excellent, 1–5%: ■ Acceptable, > 5%: ■ Needs Review
Numerical Methods and Implementation
Shell Element vs Solid Element
Should I use shell elements or solid elements?
Decide based on the thickness/radius ratio $t/a$. If $t/a < 0.1$, Kirchhoff plate theory holds, so shell elements are efficient. For thick plates with $t/a > 0.1$, shear deformation (Reissner-Mindlin) cannot be ignored, so use solid elements or thick shell elements.
Abaqus's S8R (thick plate compatible) can be used stably from thin to thick plates, but for extremely thin plates ($t/a < 0.01$), be cautious of membrane locking. S8R5 (thin plate specific) is suitable for this case.
What are the precautions when solving a plate with solid elements?
A minimum of 2 layers of second-order elements (C3D20R) are required through the thickness. With one layer, the linear bending stress distribution can only be evaluated at a single integration point, leading to insufficient accuracy. Linear elements (C3D8) require 4 or more layers through the thickness, causing computational costs to skyrocket. C3D8I with incompatible modes is an effective compromise.
Mesh Design Points
What is the best way to mesh a circular plate?
A radial mapped mesh from the center is ideal, but there is the problem of element degeneration at the center point. There are two countermeasures.
1. Place triangular/wedge elements at the center: Converge to a single point at the center and expand with quadrilaterals around it (spider web).
2. Offset the center: Convert the center to a square using an O-grid type mesh. This avoids the singular degeneration at the center.
Increase mesh density for 2-3 elements near the fixed edge because the bending moment gradient is large there.
Is unstructured mesh (triangular shell) also acceptable?
STRI65 (6-node triangle) or S6 (second-order triangle) can achieve practical accuracy. However, convergence is slower compared to quadrilateral shells, requiring 1.5–2 times the number of elements for equivalent accuracy. It's a trade-off between the convenience of automatic meshing and computational cost.
Implementation by Solver
Please tell me how to set it up in each solver.
Abaqus: S8R is standard. Define thickness with *SHELL SECTION, uniformly distributed load with *DSLOAD, and fixed edge with *BOUNDARY, ENCASTRE.
Nastran: CQUAD8 shell + PSHELL to define thickness and material. PLOAD2 for surface pressure load. SPC1 for fixed constraint.
Ansys: SHELL281 (8-node shell). Define section with SECTYPE,1,SHELL, surface pressure load with SFE.
CalculiX: *SHELL SECTION (equivalent to S8). Abaqus-compatible input. Load with *DLOAD for surface pressure.
How is the switch between Mindlin theory and Kirchhoff theory controlled?
Abaqus's S8R defaults to Reissner-Mindlin (considering shear deformation). It asymptotically approaches Kirchhoff theory in the thin plate limit. Nastran's CQUAD8 is similar. The solver does not automatically switch; rather, the element formulation itself is based on thick plate theory, and for thin plates, it naturally exhibits Kirchhoff-like behavior.
Low-Order Elements
Low computational cost and easy to implement, but accuracy is limited. Coarse meshes may lead to significant errors.
Higher-Order Elements
Achieve higher accuracy with the same mesh. Computational cost increases, but often the required number of elements is lower.
Newton-Raphson Method
Standard method for nonlinear problems. Quadratic convergence within the convergence radius. Convergence is judged by $||R|| < \epsilon$.
Time Integration
Explicit Method: Conditionally stable (CFL Condition). Implicit Method: Unconditionally stable but requires solving simultaneous equations at each step.
Visualization of Verification Data
Quantitative comparison between theoretical and computed values. A tolerance within 5% error is considered acceptable.
Evaluation Item Theoretical/Reference Value Computed Value Relative Error [%] Judgment Maximum Displacement 1.000 0.998 0.20 PASS Maximum Stress 1.000 1.015 1.50 PASS Natural Frequency (1st) 1.000 0.997 Related Topics
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Should I use shell elements or solid elements?
Decide based on the thickness/radius ratio $t/a$. If $t/a < 0.1$, Kirchhoff plate theory holds, so shell elements are efficient. For thick plates with $t/a > 0.1$, shear deformation (Reissner-Mindlin) cannot be ignored, so use solid elements or thick shell elements.
Abaqus's S8R (thick plate compatible) can be used stably from thin to thick plates, but for extremely thin plates ($t/a < 0.01$), be cautious of membrane locking. S8R5 (thin plate specific) is suitable for this case.
What are the precautions when solving a plate with solid elements?
A minimum of 2 layers of second-order elements (C3D20R) are required through the thickness. With one layer, the linear bending stress distribution can only be evaluated at a single integration point, leading to insufficient accuracy. Linear elements (C3D8) require 4 or more layers through the thickness, causing computational costs to skyrocket. C3D8I with incompatible modes is an effective compromise.
What is the best way to mesh a circular plate?
A radial mapped mesh from the center is ideal, but there is the problem of element degeneration at the center point. There are two countermeasures.
1. Place triangular/wedge elements at the center: Converge to a single point at the center and expand with quadrilaterals around it (spider web).
2. Offset the center: Convert the center to a square using an O-grid type mesh. This avoids the singular degeneration at the center.
Increase mesh density for 2-3 elements near the fixed edge because the bending moment gradient is large there.
Is unstructured mesh (triangular shell) also acceptable?
STRI65 (6-node triangle) or S6 (second-order triangle) can achieve practical accuracy. However, convergence is slower compared to quadrilateral shells, requiring 1.5–2 times the number of elements for equivalent accuracy. It's a trade-off between the convenience of automatic meshing and computational cost.
Please tell me how to set it up in each solver.
Abaqus: S8R is standard. Define thickness with *SHELL SECTION, uniformly distributed load with *DSLOAD, and fixed edge with *BOUNDARY, ENCASTRE.
Nastran: CQUAD8 shell + PSHELL to define thickness and material. PLOAD2 for surface pressure load. SPC1 for fixed constraint.
Ansys: SHELL281 (8-node shell). Define section with SECTYPE,1,SHELL, surface pressure load with SFE.
CalculiX: *SHELL SECTION (equivalent to S8). Abaqus-compatible input. Load with *DLOAD for surface pressure.
How is the switch between Mindlin theory and Kirchhoff theory controlled?
Abaqus's S8R defaults to Reissner-Mindlin (considering shear deformation). It asymptotically approaches Kirchhoff theory in the thin plate limit. Nastran's CQUAD8 is similar. The solver does not automatically switch; rather, the element formulation itself is based on thick plate theory, and for thin plates, it naturally exhibits Kirchhoff-like behavior.
Low-Order Elements
Low computational cost and easy to implement, but accuracy is limited. Coarse meshes may lead to significant errors.
Higher-Order Elements
Achieve higher accuracy with the same mesh. Computational cost increases, but often the required number of elements is lower.
Newton-Raphson Method
Standard method for nonlinear problems. Quadratic convergence within the convergence radius. Convergence is judged by $||R|| < \epsilon$.
Time Integration
Explicit Method: Conditionally stable (CFL Condition). Implicit Method: Unconditionally stable but requires solving simultaneous equations at each step.
Visualization of Verification Data
Quantitative comparison between theoretical and computed values. A tolerance within 5% error is considered acceptable.
| Evaluation Item | Theoretical/Reference Value | Computed Value | Relative Error [%] | Judgment |
|---|---|---|---|---|
| Maximum Displacement | 1.000 | 0.998 | 0.20 | PASS |
| Maximum Stress | 1.000 | 1.015 | 1.50 | PASS |
| Natural Frequency (1st) | 1.000 | 0.997 | Related Topicsこの記事の評価 ご回答ありがとうございます！ 参考に なった もっと 詳しく 誤りを 報告 |