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RF reference path. +Rebaseline Phase 1 before any more implementation: use the revised `docs/MITC4_FORMULATION.md` as the MITC4 gate, decide how `references/quad_02.inp` should be handled despite its `Part/Assembly/Instance` structure, and draft new sprint contracts before reworking the solver. ## Required Reading For New Agents 1. `AGENTS.md` @@ -37,13 +37,14 @@ Resolve the remaining Phase 1 reference-artifact blockers after the P1-01 throug - Execute with: `python scripts/execute.py 1-linear-static-mitc4` - Step numbering is zero-based for the harness: `step0.md` corresponds to milestone P1-01, and `step15.md` corresponds to milestone P1-16. - Every step file contains a sprint contract with objective, scope, allowed files, tests-first plan, reference-artifact policy, evaluator checklist, acceptance command, and handoff requirements. -- Current phase status: blocked at P1-15 because no Phase 1-compatible Abaqus `TYPE=S4` reference input and matching displacement CSV are available yet. +- Current phase status: rebaseline required. Do not resume the old P1-15/P1-16 path until new contracts are drafted against the revised MITC4 formulation and the `quad_02` compatibility decision. ## Phase 1 Readiness Tasks | ID | Status | Owner | Task | Source | |---|---|---|---|---| | R-010 | pending | user + verification agent | Add or define reaction-force reference artifacts, preferably `*_reactions.csv`, or decide that Phase 1 `RF` is verified by equilibrium tests until Abaqus RF CSV is available. | `docs/VERIFICATION_PLAN.md`, `docs/RESULTS_SCHEMA.md` | -| R-011 | pending | user + Abaqus compatibility agent | Add at least one Phase 1-compatible Abaqus `TYPE=S4` input case; keep `quad_01.inp` as stored S4R/NLGEOM reference provenance until S4R/nonlinear support is intentionally added. | `docs/ABAQUS_INPUT_SUBSET.md`, `docs/VERIFICATION_PLAN.md` | +| R-011 | partial | user + Abaqus compatibility agent | Review `references/quad_02.inp` and `references/quad_02_displacements.csv`: `TYPE=S4` is now available, but the input includes `Part/Assembly/Instance`, so decide whether to normalize the file into the current flat keyword subset or add a dedicated parser sprint. | `docs/ABAQUS_INPUT_SUBSET.md`, `docs/VERIFICATION_PLAN.md`, `docs/MITC4_FORMULATION.md` | +| R-012 | pending | planner + evaluator | Draft a Phase 1 redo plan and step contracts that replace or supersede the old P1-01 through P1-16 implementation path where it conflicts with the revised MITC4 formulation. | `docs/HARNESS_ENGINEERING.md`, `docs/MITC4_FORMULATION.md` | ## Phase 1 Definition Of Done Phase 1 is complete only when FESA can run a documented linear static MITC4 workflow from input to verified results without requiring Abaqus execution. @@ -74,8 +75,8 @@ Each gate should be satisfied before moving to the next implementation band unle | G1 - Build and validation | satisfied | Build system, test framework, and `scripts/validate_workspace.py` run real checks. | Validation command output | | G2 - Parser and domain | satisfied | Phase 1 `.inp` subset parses into `Domain`; unsupported features fail clearly. | Parser tests and diagnostics tests | | G3 - DOF/math/results infrastructure | satisfied | Core types, DofManager, sparse math adapters, minimal results writer, and CSV comparator are tested. | Unit tests and schema/comparator tests | -| G4 - MITC4 element readiness | satisfied | MITC4 formulation open decisions are closed and element-level tests pass. | Updated formulation doc and element tests | -| G5 - End-to-end solver | partial | Linear static path writes `U`/`RF`; stored Phase 1-compatible Abaqus reference regression is blocked by missing artifacts. | Integration/reference regression output | +| G4 - MITC4 element readiness | reopened | MITC4 formulation was rewritten from local papers; the existing element tests must be reassessed and expanded before implementation is trusted. | Revised `docs/MITC4_FORMULATION.md`, future element tests | +| G5 - End-to-end solver | reopened | Linear static path must be revalidated after the MITC4 rebuild and after the `quad_02` compatibility path is decided. | Future integration/reference regression output | ## Phase 1 Implementation Milestones All milestones are intended to become one or more self-contained sprint contracts or `phases/{phase}/stepN.md` files. Each sprint must follow `docs/HARNESS_ENGINEERING.md` and be evaluated independently. @@ -113,15 +114,19 @@ Verification should grow with the implementation bands: Current reference state: - `references/quad_01.inp` and `references/quad_01_displacements.csv` are accepted stored artifacts. - `quad_01.inp` contains `S4R`, `Part/Assembly/Instance`, `*Density`, and `NLGEOM=YES`; it is not a Phase 1 parser acceptance case as-is. +- `references/quad_02.inp` and `references/quad_02_displacements.csv` have been added by the user as an S4 reference pair. +- `quad_02.inp` uses `TYPE=S4`, but also includes `Part/Assembly/Instance`; this is a compatibility decision point, not automatic parser scope expansion. Required reference additions or decisions: -- Add at least one Phase 1-compatible `TYPE=S4` linear static `.inp` case. +- Decide whether `quad_02.inp` is normalized into the current Phase 1 keyword subset or used to justify a dedicated Abaqus/CAE `Part/Assembly/Instance` parser sprint. - Add `*_reactions.csv` or explicitly use internal equilibrium tests for Phase 1 `RF` until Abaqus RF output is available. - Add more small cases until Phase 1 can pass one single-element case, one simple multi-element plate/shell case, and one curved shell benchmark. ## Phase 1 Risk Controls - Do not implement MITC4 element stiffness until the formulation gate in `docs/MITC4_FORMULATION.md` is closed. +- Do not treat the previous P1-01 through P1-14 implementation as authoritative after the MITC4 formulation reset. - Do not use `quad_01.inp` to justify `S4R`, `Part/Assembly/Instance`, or `NLGEOM=YES` support. +- Do not use `quad_02.inp` to silently justify `Part/Assembly/Instance` support without a parser contract. - Do not compute reactions from reduced vectors only. - Do not expose MKL, TBB, or HDF5 APIs directly in solver core. - Do not narrow ids, equation numbers, or sparse indices below int64. diff --git a/PROGRESS.md b/PROGRESS.md index 0d78ac3..5ffc7a5 100644 --- a/PROGRESS.md +++ b/PROGRESS.md @@ -13,10 +13,42 @@ Every new agent session must read this file together with `PLAN.md` before plann - Do not remove history unless the user explicitly asks for archival cleanup. ## Current Status -P1-01 through P1-14 have an initial C++17 implementation with CMake/CTest validation. P1-15 and P1-16 remain blocked because the repository does not yet contain a Phase 1-compatible Abaqus `TYPE=S4` input with matching `*_displacements.csv`; the first stored pair remains `quad_01.inp` and `quad_01_displacements.csv`, with `quad_01.inp` documented as S4R/NLGEOM provenance rather than Phase 1 parser input. +Phase 1 is being rebaselined before more implementation. P1-01 through P1-14 have an initial C++17 implementation, but it is no longer authoritative until reviewed against the revised paper-based `docs/MITC4_FORMULATION.md`. The user added `quad_02.inp` and `quad_02_displacements.csv`; `quad_02` uses `TYPE=S4`, but still includes `Part/Assembly/Instance` and `*Density`, so the next planner must decide whether to normalize it or add a dedicated parser compatibility sprint. ## Completed Work +### 2026-05-04 - MITC4 formulation reset from local papers +Author: Codex + +Changed files: +- `docs/Paper/` +- `docs/MITC4_FORMULATION.md` +- `docs/ADR.md` +- `docs/NUMERICAL_CONVENTIONS.md` +- `docs/README.md` +- `docs/ABAQUS_INPUT_SUBSET.md` +- `docs/VERIFICATION_PLAN.md` +- `references/quad_02.inp` +- `references/quad_02_displacements.csv` +- `PLAN.md` +- `PROGRESS.md` + +Summary: +- Read the local MITC4 paper set under `docs/Paper/` and rewrote `docs/MITC4_FORMULATION.md` as the new Phase 1 formulation contract. +- Replaced the earlier simplified baseline with degenerated-continuum geometry, convected covariant strain components, FESA/Abaqus S4 tying-point convention, MITC transverse shear interpolation, local/global six-DOF rotation transformation, `2 x 2 x 2` Gauss integration, and thesis-backed drilling stabilization using `drilling_stiffness_scale = 1.0e-3`. +- Updated ADR-018, numerical conventions, and documentation readiness notes so the old `1.0e-6` averaged-edge baseline is no longer the active rule. +- Recorded `quad_02` as the new stored S4 reference pair while keeping its `Part/Assembly/Instance` and `*Density` features outside automatic Phase 1 parser acceptance. +- Updated `PLAN.md` so the next work is a Phase 1 redo/rebaseline, not continuation of the old P1-15/P1-16 path. + +Verification: +- `python scripts/validate_workspace.py` configured CMake, built `fesa_core` and `fesa_tests`, and ran CTest successfully. +- CTest result: 1 test executable passed. + +Follow-up: +- Draft new Phase 1 redo sprint contracts against the revised MITC4 formulation before modifying solver code. +- Decide whether `quad_02.inp` should be normalized into the current parser subset or used to justify a dedicated Abaqus/CAE `Part/Assembly/Instance` parser sprint. +- Decide whether Phase 1 `RF` gets Abaqus `*_reactions.csv` artifacts or remains verified by full-vector equilibrium tests. + ### 2026-05-01 - P1-01 through P1-14 implementation pass Author: Codex diff --git a/docs/ABAQUS_INPUT_SUBSET.md b/docs/ABAQUS_INPUT_SUBSET.md index f842bf2..366e73a 100644 --- a/docs/ABAQUS_INPUT_SUBSET.md +++ b/docs/ABAQUS_INPUT_SUBSET.md @@ -54,10 +54,12 @@ Rules: - Parser implementation must still reject unsupported features until this document and ADRs explicitly add support. - Test harnesses may use normalized or reduced derivative inputs for Phase 1 parser tests, but must keep the original Abaqus reference artifact unchanged. -Current initial reference note: +Current stored reference notes: - `references/quad_01.inp` was generated by Abaqus/CAE Learning Edition 2024. - It uses `TYPE=S4R`, `Part`, `Assembly`, `Instance`, `*Density`, and `NLGEOM=YES`, all of which are outside the current Phase 1 parser/solver subset. - Its paired `references/quad_01_displacements.csv` is still valid as a stored displacement reference artifact for future compatibility and regression work. +- `references/quad_02.inp` uses `TYPE=S4`, so it targets the Phase 1 MITC4 element formulation, and its paired `references/quad_02_displacements.csv` has the accepted displacement CSV shape. +- `quad_02.inp` still uses Abaqus/CAE `Part`, `Assembly`, `Instance`, and `*Density`; it is therefore a stored S4 reference artifact and compatibility decision point, not automatic parser acceptance as-is. ## Labels and Names Rules: diff --git a/docs/ADR.md b/docs/ADR.md index 7f36ec5..21be822 100644 --- a/docs/ADR.md +++ b/docs/ADR.md @@ -158,9 +158,9 @@ --- -### ADR-018: Phase 1 MITC4 Baseline Closure -**결정**: Phase 1 MITC4 baseline은 `docs/MITC4_FORMULATION.md`의 closed baseline decisions를 따른다. 핵심 결정은 midside transverse shear tying interpolation, averaged-edge local shell basis, 2x2 Gauss integration, `drilling_stiffness_scale = 1.0e-6`, 그리고 mandatory result output을 `U`와 `RF`로 제한하는 것이다. +### ADR-018: Phase 1 MITC4 Formulation Reset +**결정**: Phase 1 MITC4 baseline은 `docs/MITC4_FORMULATION.md`의 논문 기반 formulation contract를 따른다. 핵심 결정은 degenerated-continuum kinematics, convected covariant strain components, FESA/Abaqus S4 node-order에 맞춘 MITC transverse shear tying interpolation, `2 x 2 x 2` Gauss integration, six-DOF local/global rotation transformation, `drilling_stiffness_scale = 1.0e-3` applied to the minimum positive physical local stiffness diagonal, 그리고 mandatory result output을 `U`와 `RF`로 제한하는 것이다. -**이유**: MITC4 element implementation은 문서화된 baseline 없이 시작하면 reference 검증 전 수치 drift가 생기기 쉽다. Phase 1은 Abaqus S4 reference artifact가 아직 부족하므로, 작고 검증 가능한 baseline을 먼저 고정하고 element-level tests와 internal equilibrium tests로 보호한다. +**이유**: 이전 Phase 1 baseline은 첫 구현을 빠르게 안정화하기 위한 단순화였지만, 사용자가 추가한 MITC4 논문들과 S4 reference (`quad_02`)를 기준으로 보면 formulation detail이 부족하다. MITC4 element implementation은 convected tensor shear tying, director/local-axis policy, drilling stabilization, and integration rules가 명확해야 reference 검증 전 수치 drift를 줄일 수 있다. -**트레이드오프**: 이 baseline은 첫 구현을 안정화하기 위한 보수적 선택이며 Abaqus S4 reference 통과 후 drilling scale, warped-quadrilateral basis, stress/strain recovery, and shell resultant outputs may need refinement. `S4R`, reduced integration, and hourglass control remain out of scope. +**트레이드오프**: 기존 P1-01 through P1-14 구현은 새 formulation contract를 기준으로 재평가 또는 재작성되어야 한다. `quad_02.inp`는 `TYPE=S4`이지만 `Part/Assembly/Instance` 구조를 포함하므로, element formulation 재구현과 parser compatibility 확장은 별도 sprint로 분리해야 한다. `S4R`, reduced integration, and hourglass control remain out of scope. diff --git a/docs/MITC4_FORMULATION.md b/docs/MITC4_FORMULATION.md index d96a5ea..b8f7c26 100644 --- a/docs/MITC4_FORMULATION.md +++ b/docs/MITC4_FORMULATION.md @@ -1,257 +1,441 @@ # MITC4 Formulation ## Purpose -This document defines the baseline MITC4 formulation target for FESA Phase 1. +This document is the formulation contract for FESA's Phase 1 four-node MITC4 shell element. -It is intentionally a formulation contract, not implementation code. Exact formulas should be added and reviewed before coding the MITC4 element. +It supersedes the earlier simplified Phase 1 baseline that used an averaged-edge shell basis, analytic thickness integration, and `drilling_stiffness_scale = 1.0e-6`. The next Phase 1 implementation pass must treat this document as the MITC4 gate before coding or evaluating shell stiffness. -## Source Basis -- Dvorkin and Bathe's four-node shell element paper presents a continuum-mechanics-based, non-flat, general quadrilateral shell element for thin and thick shells and nonlinear analysis: https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf -- The paper identifies transverse shear locking as a key problem in simple 4-node shell interpolation and motivates modified transverse shear treatment: https://web.mit.edu/kjb/www/Publications_Prior_to_1998/A_Continuum_Mechanics_Based_Four-Node_Shell_Element_for_General_Nonlinear_Analysis.pdf -- OpenSees describes `ShellMITC4` as a bilinear isoparametric shell element with modified shear interpolation, four counter-clockwise nodes, and six DOFs per node: https://opensees.berkeley.edu/wiki/index.php/Shell_Element -- The MITC benchmark paper states that the MITC method is used to remedy shell locking and that the standard MITC4 employs MITC treatment for transverse shear strains; it also notes that Abaqus S4 uses Dvorkin-Bathe transverse shear interpolation: https://web.mit.edu/kjb/www/Principal_Publications/Performance_of_the_MITC3%2B_and_MITC4%2B_shell_elements_in_widely_used_benchmark_problems.pdf -- Abaqus finite-strain shell theory documentation provides useful comparison context for S4/S4R geometry, interpolation, orientation update, and transverse shear treatment, but FESA Phase 1 is linear static: https://abaqus-docs.mit.edu/2017/English/SIMACAETHERefMap/simathe-c-finitestrainshells.htm +The goal is not to reproduce Abaqus internals. The goal is to implement a documented, testable Dvorkin-Bathe-style MITC4 element that can be compared against stored Abaqus `S4` reference artifacts without running Abaqus locally. -## Phase 1 Target -Phase 1 implements a clear MITC4 baseline formulation and passes reference benchmarks before performance optimization. +## Local Source Basis +The formulation below is based on the local papers in `docs/Paper/`: -Scope: -- 4-node quadrilateral shell. -- Linear static analysis. -- Linear isotropic elastic material. -- Homogeneous shell section. -- 6 DOFs per node. -- Small-strain formulation for Phase 1. -- Transverse shear interpolation based on MITC4 assumptions. -- Abaqus-compatible result signs. +- `docs/Paper/AContinuumMechanicsBasedFourNodeShellElementforGeneralNonlinearAnalysis/` + - Dvorkin and Bathe's original four-node non-flat shell element. + - Defines the degenerated-continuum geometry, displacement interpolation, convected transverse shear interpolation, constitutive transformation, and benchmark expectations. +- `docs/Paper/FourNodeQuadrilateralShellElementMITC4/` + - Restates the MITC4 geometry and displacement interpolation, gives explicit midside shear tying relations, and reports patch-test and Scordelis-Lo verification behavior. +- `docs/Paper/유한요소해석법을이용한쉘구조물의동적좌굴해석/` + - Provides the Korean thesis derivation for covariant strain, plane-stress material matrix with shear correction, 6-DOF shell transformation, drilling stabilization, and nonlinear extension notes. +- `docs/Paper/2007쉘구조물의유한요소해석에대하여/` + - Provides shell-model background, locking behavior, asymptotic behavior, and benchmark/test interpretation. +- `docs/Paper/mitc공부/` + - Treated as supporting study notes only. Use the papers above for binding formulas. -Non-scope: -- S4R reduced-integration behavior. -- Hourglass control. -- Composite sections. -- Material nonlinearity. -- Geometric nonlinearity. -- Pressure loads. -- Thermal-stress coupling. -- Mesh quality diagnostics. +## Phase 1 Scope +Phase 1 implements: -## Phase 1 Closed Baseline Decisions -The following decisions close the Phase 1 implementation gate for the first baseline C++ implementation. They are intentionally conservative and may be revised by ADR after Abaqus S4 reference cases are added. +- Abaqus `TYPE=S4` mapped to FESA `MITC4`. +- Four-node quadrilateral shell in Abaqus S4 node order. +- Linear static, small-strain, linear isotropic elastic analysis. +- Homogeneous shell section with one thickness value per element for the initial implementation. +- Six global DOFs per node: `UX`, `UY`, `UZ`, `RX`, `RY`, `RZ`. +- MITC transverse shear interpolation in a convected coordinate system. +- Constrained DOF elimination, full-vector reaction recovery, and minimum output fields `U` and `RF`. -Decisions: -- Mandatory Phase 1 result outputs are limited to nodal `U` and `RF`. Element `S`, `E`, and `SF` remain future outputs. -- In-plane membrane and bending terms use standard bilinear Reissner-Mindlin interpolation with 2x2 Gauss integration. -- Transverse shear uses MITC4 mid-side tying interpolation: +Phase 1 does not implement: + +- Abaqus `S4R`, reduced integration, or hourglass control. +- Geometric nonlinearity, material nonlinearity, dynamics, pressure loads, thermal-stress coupling, composite sections, or mesh quality diagnostics. +- Abaqus `*Orientation`, user-defined nodal normals, or shell offset behavior unless added by a later ADR. + +## Natural Coordinates and Node Order +FESA uses the standard bilinear quadrilateral natural coordinates: ```text -gamma_xz_mitc(r, s) = - 0.5 * (1 - s) * gamma_xz(0, -1) - + 0.5 * (1 + s) * gamma_xz(0, +1) - -gamma_yz_mitc(r, s) = - 0.5 * (1 - r) * gamma_yz(-1, 0) - + 0.5 * (1 + r) * gamma_yz(+1, 0) +node 1: (xi, eta) = (-1, -1) +node 2: (xi, eta) = (+1, -1) +node 3: (xi, eta) = (+1, +1) +node 4: (xi, eta) = (-1, +1) +zeta in [-1, +1] through the thickness ``` -- The four tying points are the midside natural-coordinate points `(0,-1)`, `(0,+1)`, `(-1,0)`, and `(+1,0)`. -- Local basis construction uses averaged opposite edges: +Shape functions: ```text -v1 = 0.5 * ((x2 - x1) + (x3 - x4)) -v2 = 0.5 * ((x4 - x1) + (x3 - x2)) -e1 = normalize(v1) -e2_raw = v2 - dot(v2, e1) * e1 -e2_pre = normalize(e2_raw) -e3 = normalize(cross(e1, e2_pre)) -e2 = normalize(cross(e3, e1)) +N1 = 0.25 * (1 - xi) * (1 - eta) +N2 = 0.25 * (1 + xi) * (1 - eta) +N3 = 0.25 * (1 + xi) * (1 + eta) +N4 = 0.25 * (1 - xi) * (1 + eta) ``` -- For mildly warped quadrilaterals this is an averaged midsurface basis. Severe warpage is not diagnosed as mesh quality in Phase 1, but near-zero vectors or Jacobians are invalid/singular element diagnostics. -- Integration point order for stiffness tests is `(-g,-g)`, `(g,-g)`, `(g,g)`, `(-g,g)` conceptually, with `g = 1 / sqrt(3)`. Implementation may loop over the same set in row-major order as long as assembled stiffness is invariant. -- The default drilling stiffness scale is: +Midside MITC tying points in the FESA convention: ```text -drilling_stiffness_scale = 1.0e-6 +A = ( 0, -1) edge 1-2 +B = (-1, 0) edge 1-4 +C = ( 0, +1) edge 4-3 +D = (+1, 0) edge 2-3 ``` -The current baseline applies this scale to a representative `E * thickness` drilling stabilization term. The value is reported through the element options path and must be revisited after Abaqus S4 displacement references are available. +Dvorkin-Bathe papers may use a natural-coordinate orientation where the signs on the `A/C` interpolation are reversed relative to the FESA convention above. FESA implementation must preserve the edge definitions above and use the FESA interpolation formulas in the MITC shear section. -## Nodal DOFs -Each node has: +## Degenerated-Continuum Geometry +The shell is represented as a 3D continuum degenerated through the thickness: ```text -UX, UY, UZ, RX, RY, RZ +X(xi, eta, zeta) = + sum_k N_k(xi, eta) X_k + + zeta / 2 * sum_k t_k N_k(xi, eta) Vn_k ``` -Rules: -- Translational DOFs are global translations. -- Rotational DOFs are rotations about global or transformed axes following Abaqus component convention. -- `RZ` is retained as a drilling DOF. -- Drilling stiffness is artificial in Phase 1 and must be parameterized. +where: -## Element Input Contract -`MITC4Element` requires: -- four node ids in Abaqus S4 order. -- four node coordinates. -- shell thickness. -- linear elastic material constants `E` and `nu`. -- drilling stiffness parameter. -- element id and property id for diagnostics and output. +- `X_k` is the midsurface coordinate of node `k`. +- `t_k` is the nodal thickness. Phase 1 uses `t_k = t` for all four nodes. +- `Vn_k` is the nodal director. The papers allow the director to be non-normal to the midsurface; Phase 1 derives it from geometry because Abaqus nodal normals are not in the parser subset. -Node ordering: -- Input node order follows Abaqus S4 convention. -- Positive normal follows the right-hand rule around the nodes. -- FESA maps Abaqus `TYPE=S4` to `MITC4`. -- Abaqus `TYPE=S4R` is not supported in Phase 1. +Phase 1 director policy: -## Coordinate Frames -The Phase 1 local basis construction is defined by the averaged-edge algorithm in `Phase 1 Closed Baseline Decisions`. - -Minimum requirements: -- Define a local shell normal from the quadrilateral geometry. -- Define local in-plane axes `e1` and `e2` so that `e1`, `e2`, and normal form a right-handed basis. -- Preserve Abaqus-compatible output signs. -- Document behavior for non-planar quadrilaterals. -- Use the same convention consistently for stiffness, stress/strain recovery, and result output. - -Phase 1 convention: -- Use the element midsurface geometry to compute an average basis from opposite edges. -- Use the positive shell normal implied by the input node ordering. -- Use a right-handed local basis consistently for stiffness, reaction recovery, and future result output. - -## Shape Functions -Baseline quadrilateral bilinear interpolation: +- If no parsed nodal directors exist, use the element-center midsurface normal for all four nodal directors: ```text -N1 = 0.25 * (1 - r) * (1 - s) -N2 = 0.25 * (1 + r) * (1 - s) -N3 = 0.25 * (1 + r) * (1 + s) -N4 = 0.25 * (1 - r) * (1 + s) +G1_c = dX_mid / dxi at (0, 0) +G2_c = dX_mid / deta at (0, 0) +Vn_k = normalize(G1_c x G2_c), k = 1..4 ``` -where `r, s` are natural coordinates in `[-1, 1]`. +- If `G1_c x G2_c` is near zero, reject the element with an invalid/singular element diagnostic. +- This flat-or-mildly-warped policy is sufficient for the Phase 1 S4 baseline. A true non-flat nodal-director policy requires a later ADR and benchmark update. -Implementation requirements: -- Compute shape function derivatives with respect to natural coordinates. -- Build the surface Jacobian and local derivatives. -- Detect invalid or near-zero Jacobian as a singular/invalid element diagnostic, not as a mesh quality metric. +The nodal local director axes are: -## Strain Treatment -The baseline element separates: -- membrane strain terms. -- bending curvature terms. -- transverse shear strain terms. -- artificial drilling stabilization. +```text +V1_k = normalize(EY x Vn_k) +V2_k = Vn_k x V1_k +``` -MITC4 requirement: -- Use standard displacement interpolation for membrane and bending terms in Phase 1. -- Use MITC transverse shear interpolation to alleviate shear locking. -- Do not replace MITC4 with plain full-integration Reissner-Mindlin Q4. +If `EY x Vn_k` is near zero, use a deterministic fallback axis before normalization: -The Phase 1 transverse shear tying formula is the midside interpolation stated in `Phase 1 Closed Baseline Decisions`. +```text +V1_k = normalize(EZ x Vn_k) +if still near zero, V1_k = normalize(EX x Vn_k) +V2_k = Vn_k x V1_k +``` -## Numerical Integration -Phase 1 baseline: -- In-plane integration: 2x2 Gauss for membrane, bending, MITC shear, and drilling stabilization. -- Thickness integration: homogeneous linear elastic shell section is integrated analytically using `t`, `t^3 / 12`, and shear correction `5 / 6`. -- Benchmark literature commonly reports 2x2 in-plane Gauss integration for S4/MITC4-style 4-node elements and 2-point thickness integration in comparative shell studies. +The implementation must keep `V1_k`, `V2_k`, and `Vn_k` orthonormal and right-handed. -Rules: -- Do not introduce reduced integration or hourglass control for S4R behavior in Phase 1. -- Do not optimize integration before reference benchmarks pass. -- Integration point ordering for output must be documented before stress/strain reference comparisons. +## Displacement and Rotation Interpolation +The local five-DOF MITC4 displacement field is: + +```text +u(xi, eta, zeta) = + sum_k N_k u_k + + zeta / 2 * sum_k t_k N_k q_k + +q_k = -V2_k * alpha_k + V1_k * beta_k +``` + +where: + +- `u_k` is the nodal translation vector. +- `alpha_k` is the rotation about `V1_k`. +- `beta_k` is the rotation about `V2_k`. +- Drilling rotation `gamma_k` about `Vn_k` does not enter the continuum strain field. + +FESA stores six global DOFs per node. Global nodal rotations are mapped to local director rotations by: + +```text +theta_k = [RX_k, RY_k, RZ_k]^T + +[alpha_k, beta_k, gamma_k]^T = L_k theta_k + +L_k = + [ EX dot V1_k EY dot V1_k EZ dot V1_k + EX dot V2_k EY dot V2_k EZ dot V2_k + EX dot Vn_k EY dot Vn_k EZ dot Vn_k ] +``` + +The element-level local-to-global transformation is block diagonal: + +```text +u_local_e = T_e u_global_e +K_global_e = T_e^T K_local_e T_e +f_global_e = T_e^T f_local_e +``` + +Each node block maps `[UX, UY, UZ, RX, RY, RZ]` to `[u_x, u_y, u_z, alpha, beta, gamma]`. + +## Covariant Strain Definition +Use convected covariant components for the MITC4 strain construction: + +```text +g_i = dX / dr_i, where r_i = xi, eta, zeta + +eps_ij = + 0.5 * (du/dr_i dot g_j + du/dr_j dot g_i) +``` + +For the Phase 1 small-strain implementation: + +- Membrane and bending terms are derived directly from the degenerated-continuum displacement field. +- `eps_33 = 0` is enforced by the shell assumption. +- The transverse shear rows `eps_13` and `eps_23` are not taken directly at the Gauss point. They are replaced by the MITC assumed transverse shear interpolation below. + +Use the internal strain vector order: + +```text +[ eps_11, eps_22, eps_33, gamma_23, gamma_13, gamma_12 ]^T +``` + +where engineering shear strains are: + +```text +gamma_ij = 2 * eps_ij +``` + +If an implementation uses another row order internally, it must include an explicit permutation test against this documented order. + +## MITC Transverse Shear Interpolation +The key MITC4 assumption is that the transverse shear tensor components are interpolated from midside tying points, instead of being evaluated directly at every Gauss point from the displacement field. + +The original paper form is: + +```text +eps_13 = 0.5 * (1 + r2) * eps_13^A + 0.5 * (1 - r2) * eps_13^C +eps_23 = 0.5 * (1 + r1) * eps_23^D + 0.5 * (1 - r1) * eps_23^B +``` + +With FESA's Abaqus-style natural coordinate convention and tying-point labels, use: + +```text +eps_13_MITC(xi, eta) = + 0.5 * (1 - eta) * eps_13^A + + 0.5 * (1 + eta) * eps_13^C + +eps_23_MITC(xi, eta) = + 0.5 * (1 - xi) * eps_23^B + + 0.5 * (1 + xi) * eps_23^D +``` + +The tying values are the direct covariant shear strains evaluated at midside points: + +```text +eps_13^A = eps_13_DIRECT( 0, -1) +eps_13^C = eps_13_DIRECT( 0, +1) +eps_23^B = eps_23_DIRECT(-1, 0) +eps_23^D = eps_23_DIRECT(+1, 0) +``` + +Implementation rule: + +- Build the direct covariant strain-displacement rows from the geometry and displacement interpolation. +- Evaluate only the needed direct transverse shear rows at `A`, `B`, `C`, and `D`. +- Interpolate those rows to the integration point using the formulas above. +- Do not compute MITC shear from local Cartesian `gamma_xz`/`gamma_yz` first; the tying is on convected tensor components. + +For sign and regression checks, define: + +```text +q_k = -V2_k * alpha_k + V1_k * beta_k +``` + +The paper edge formulas can be used as a diagnostic check for the direct tying rows: + +```text +eps_13^A ~ 1/8 * [ + (u1 - u2) dot 0.5 * (t1 Vn1 + t2 Vn2) + + (X1 - X2) dot 0.5 * (t1 q1 + t2 q2) +] + +eps_13^C ~ 1/8 * [ + (u4 - u3) dot 0.5 * (t4 Vn4 + t3 Vn3) + + (X4 - X3) dot 0.5 * (t4 q4 + t3 q3) +] + +eps_23^B ~ 1/8 * [ + (u1 - u4) dot 0.5 * (t1 Vn1 + t4 Vn4) + + (X1 - X4) dot 0.5 * (t1 q1 + t4 q4) +] + +eps_23^D ~ 1/8 * [ + (u2 - u3) dot 0.5 * (t2 Vn2 + t3 Vn3) + + (X2 - X3) dot 0.5 * (t2 q2 + t3 q3) +] +``` + +These compact formulas are not a license to skip the general direct-row evaluation. They are a sign/orientation cross-check derived from the same midside strain definitions. + +## Local Cartesian Frame and Material Law +The constitutive law is defined in a local orthonormal Cartesian frame at each integration point with `sigma_33 = 0`. + +Build the integration-point local basis from the covariant basis: + +```text +e3_hat = normalize(g_3) +e1_hat = normalize(g_2 x e3_hat) +e2_hat = e3_hat x e1_hat +``` + +If `g_2 x e3_hat` is near zero, use a deterministic fallback that still creates a right-handed orthonormal basis and emits a diagnostic if no valid basis exists. + +For isotropic linear elasticity, use the local plane-stress shell material matrix with transverse shear correction `kappa = 5/6`: + +```text +D_hat = E / (1 - nu^2) * + [ 1 nu 0 0 0 0 + nu 1 0 0 0 0 + 0 0 0 0 0 0 + 0 0 0 kappa*(1-nu)/2 0 0 + 0 0 0 0 kappa*(1-nu)/2 0 + 0 0 0 0 0 (1-nu)/2 ] +``` + +This matrix uses the documented strain order: + +```text +[ eps_11, eps_22, eps_33, gamma_23, gamma_13, gamma_12 ]^T +``` + +For a general non-flat element, transform the local Cartesian material relation to the convected coordinate strain vector: + +```text +E_hat = T_xi_to_x E_convected +D_convected = T_xi_to_x^T D_hat T_xi_to_x +``` + +The transformation matrix must be constructed from direction cosines between the local orthonormal basis and contravariant/covariant convected bases. For Phase 1, a flat-element implementation may compute the B-matrix directly in the local Cartesian frame only if tests prove it is equivalent to the convected formulation for the accepted reference cases. + +## Element Stiffness and Numerical Integration +The baseline stiffness is: + +```text +K_e = integral_V B^T D B dV +``` + +Phase 1 baseline integration: + +- Use `2 x 2 x 2` Gauss integration in `(xi, eta, zeta)` for the first rebuilt MITC4 implementation. +- This follows the thesis derivation's `2 x 2 x 2` natural-coordinate integration and the Dvorkin-Bathe elastic recommendation of `2 x 2` in the midsurface with two through-thickness points. +- Do not introduce reduced integration or hourglass control. +- Analytic through-thickness integration may be introduced later only after equivalence tests preserve the MITC shear, drilling, and reference displacement behavior. + +At every integration point: + +1. Compute `g_i`, the metric/Jacobian, local Cartesian basis, and material transform. +2. Build direct membrane/bending strain rows. +3. Replace transverse shear rows with MITC interpolated rows. +4. Accumulate `B^T D B |J| w_xi w_eta w_zeta`. + +Detect near-zero Jacobian, invalid basis, missing property/material, or non-positive thickness as invalid/singular element diagnostics. Do not report these as mesh quality diagnostics in Phase 1. ## Drilling DOF Stabilization -Decision: -- Phase 1 uses small artificial drilling stiffness. -- Default scale: `drilling_stiffness_scale = 1.0e-6`. +The continuum MITC4 formulation has five physical DOFs per node. FESA keeps six global DOFs, so the local drilling rotation `gamma` has no physical strain contribution and must be weakly stabilized. -Requirements: -- Expose a parameter such as `drilling_stiffness_scale`. -- Provide a deterministic default. -- Make the default small enough not to dominate physical response. -- Include benchmark sensitivity checks if reference results are sensitive to the value. -- Report the value in result metadata. - -Baseline rule: +Baseline decision: ```text -k_drill = drilling_stiffness_scale * representative_element_stiffness -representative_element_stiffness = E * thickness +drilling_stiffness_scale = 1.0e-3 +k_ref = min_positive_diagonal(K_local_without_drilling) +k_drill = drilling_stiffness_scale * k_ref ``` -The representative stiffness may be refined after the first Abaqus S4 reference cases are available. +Rules: -## Element Outputs -Phase 1 minimum: -- element stiffness matrix. -- element equivalent nodal internal force for full-vector residual/reaction recovery. -- optional stress/strain output after displacement benchmarks are stable. +- `K_local_without_drilling` is the element local stiffness after MITC integration and before artificial drilling stabilization. +- Use the minimum positive diagonal entry over the physical local DOFs. Ignore zero, negative, NaN, or non-finite entries. +- Add `k_drill` to each nodal local `gamma` diagonal before transforming to global coordinates. +- If no positive reference diagonal exists, emit a singular/invalid element diagnostic. +- The scale must be configurable and written to result metadata or analysis diagnostics. +- Reference comparisons must include a drilling sensitivity check if displacement results change materially with the scale. -Future output: -- local shell stresses. -- local shell strains. -- section forces and moments. -- integration point and section point data. +This replaces the earlier arbitrary `1.0e-6 * E * thickness` rule. The `1.0e-3` scale follows the local thesis derivation's six-DOF stabilization note and is still an artificial numerical parameter, not a physical stiffness. -## Required Element-Level Tests -Before integration with the global solver: -- shape functions sum to one. -- derivatives satisfy expected bilinear identities. -- element stiffness dimensions are `24 x 24`. -- stiffness is symmetric for linear elastic Phase 1. -- rigid body translations produce near-zero internal strain energy. -- rigid body rotations do not create physical membrane/bending stiffness beyond documented drilling effects. -- constant membrane patch behavior. -- bending-dominated sanity case. -- drilling stiffness sensitivity check. +## Internal Force and Reaction Recovery +For Phase 1 linear static analysis: -## Pre-Implementation Gate -Phase 1 baseline implementation may proceed because these items are now recorded above: -- transverse shear tying point equations. -- local shell basis algorithm for flat and mildly warped quadrilaterals. -- default drilling stiffness scale and parameter name. -- integration point set and ordering convention. -- Phase 1 mandatory output scope: `U` and `RF` only. +```text +f_int_e = K_e u_e +R_full = K_full U_full - F_full +``` -Stress/strain recovery locations for `S`, `E`, and `SF` remain future decisions and must not be implemented as mandatory Phase 1 outputs. +Rules: -## Reference Benchmarks -MITC4 baseline acceptance should include: -- single-element membrane test. -- single-element bending test. -- cantilever shell strip. -- simply supported square plate. -- Scordelis-Lo roof. -- pinched cylinder. -- twisted beam. +- Reactions must be recovered from the full global vector, not only from the reduced free-DOF system. +- `RF` output follows the Abaqus-style component convention documented in `docs/NUMERICAL_CONVENTIONS.md`. +- Element stress/strain/resultant outputs are not mandatory until displacement and reaction reference behavior is stable. -Distorted mesh tests should be added after the baseline passes, but Phase 1 does not implement general mesh quality diagnostics. +## Required Tests Before Reimplementation +The rebuilt MITC4 element must be protected by tests before implementation code is accepted. + +Element-level tests: + +- Shape functions sum to one and have correct derivatives at center, corners, and tying points. +- Center normal, local director axes, and integration local Cartesian axes are orthonormal and right-handed. +- The local-to-global rotation transform maps global rotations to `alpha`, `beta`, `gamma` as documented. +- Direct tying rows at `A`, `B`, `C`, `D` agree with finite-difference strain perturbations. +- MITC shear rows at Gauss points are interpolated from tying rows, not evaluated directly at the Gauss points. +- Stiffness is symmetric for linear elastic Phase 1. +- Rigid-body translations and rotations produce near-zero physical strain energy, with only documented drilling stabilization effects. +- The element has no spurious zero-energy modes beyond expected rigid-body behavior after drilling stabilization is accounted for. + +Patch and benchmark tests: + +- Constant membrane states. +- Pure bending. +- Pure shear. +- Pure twist. +- Thin cantilever or plate strip to expose shear locking. +- Scordelis-Lo roof after the basic flat tests pass. +- Pinched cylinder or twisted beam as later Phase 1/Phase 2 benchmark expansion. + +Reference regression tests: + +- Compare FESA `U` against stored Abaqus displacement CSV files using absolute and relative tolerances. +- Keep `RF` verified by full-vector equilibrium until matching Abaqus reaction CSV files are provided or explicitly deferred. + +## Current Reference Artifacts +Stored artifacts currently known: -Current stored reference artifact: - `references/quad_01.inp` - `references/quad_01_displacements.csv` +- `references/quad_02.inp` +- `references/quad_02_displacements.csv` -Compatibility note: -- `quad_01.inp` is an Abaqus-generated S4R/NLGEOM reference input and is not a Phase 1 MITC4 parser acceptance case. -- It should be used as an external reference artifact and future compatibility target unless a normalized Phase 1 `S4` input is added. -- The displacement CSV can still exercise `U` field comparison infrastructure once node ids and component mapping are supported. +Compatibility notes: + +- `quad_01.inp` contains `S4R`, `Part/Assembly/Instance`, `*Density`, and `NLGEOM=YES`; it remains future compatibility provenance, not a Phase 1 parser acceptance case. +- `quad_02.inp` contains `TYPE=S4`, which is the correct element target for Phase 1, but it also uses `Part/Assembly/Instance`. The next Phase 1 planning pass must either normalize this input into the Phase 1 flat keyword subset or explicitly add a parser sprint for this Abaqus/CAE structure. Do not silently expand parser support while implementing the element. + +## Implementation Checklist +The next MITC4 implementation pass should proceed in this order: + +1. Lock the shape-function, natural-coordinate, node-order, and tying-point tests. +2. Implement geometry/director/local-axis construction with diagnostics. +3. Implement the five-DOF degenerated-continuum displacement interpolation. +4. Implement global six-DOF to local `[alpha, beta, gamma]` transformation. +5. Implement direct covariant strain rows and finite-difference tests. +6. Replace transverse shear rows with MITC tying interpolation. +7. Implement material transform and `2 x 2 x 2` stiffness integration. +8. Add drilling stabilization in local coordinates. +9. Transform element stiffness and internal force to global coordinates. +10. Run element patch tests before assembly/reference regression work. ## Future Extensions Geometric nonlinearity: -- Add updated geometry, current frame handling, tangent stiffness, and Newton-Raphson integration. -- Preserve `AnalysisState` element/internal state extension points. + +- Update current geometry and nodal directors in `AnalysisState`. +- Add finite-rotation updates for `V1`, `V2`, and `Vn`. +- Add tangent stiffness, geometric stiffness, and Newton-Raphson convergence checks. Thermal-stress coupling: -- Add temperature field state. -- Add thermal strain contribution. -- Add material expansion data. -- Add result fields for temperature and thermal strain/stress. + +- Add temperature field state and thermal strain contribution. +- Add material expansion data and thermal output fields. S4R: -- Add only after a separate ADR and formulation document update. -- Requires reduced integration and hourglass control decisions. -## Open Decisions Before Coding -- Exact stress/strain recovery locations. -- Whether local coordinate transforms from Abaqus input are deferred or rejected. +- Add only after a separate ADR and formulation update. +- Requires reduced integration, hourglass control, and separate reference artifacts. + +Stress/resultant output: + +- Define section point locations, component labels, local coordinate convention, and Abaqus CSV comparison format before making `S`, `E`, or `SF` mandatory. + +## Remaining Open Decisions +The MITC4 stiffness formulation is now defined for the Phase 1 rebuild. Remaining decisions are outside the first stiffness pass: + +- Exact stress/strain/resultant recovery locations and component labels. +- Whether to parse Abaqus `*Orientation` or nodal shell normals. +- Whether `quad_02.inp` should be normalized into the existing Phase 1 parser subset or used to justify a dedicated `Part/Assembly/Instance` parser sprint. +- Reference tolerances for `quad_02` after the input compatibility path is chosen. diff --git a/docs/NUMERICAL_CONVENTIONS.md b/docs/NUMERICAL_CONVENTIONS.md index ecf96ac..3550364 100644 --- a/docs/NUMERICAL_CONVENTIONS.md +++ b/docs/NUMERICAL_CONVENTIONS.md @@ -158,7 +158,7 @@ Phase 1 retains `RZ` at each shell node and assigns a small artificial drilling Rules: - The value must be parameterized, not hard-coded deep inside the MITC4 kernel. -- The default value must be documented in `docs/MITC4_FORMULATION.md` before implementation. +- The Phase 1 default follows `docs/MITC4_FORMULATION.md`: `drilling_stiffness_scale = 1.0e-3` applied to the minimum positive physical local stiffness diagonal before the local-to-global six-DOF transform. - The artificial stiffness should be small relative to representative membrane or bending stiffness, but large enough to avoid a singular stiffness matrix for unconstrained drilling modes. - Reference comparisons should include sensitivity checks if the drilling stiffness materially changes displacement results. @@ -208,7 +208,7 @@ Before implementing solver modules that depend on numerical conventions: - Ensure singular diagnostics can reference node id, DOF component, element id, property id, and set name. ## Open Decisions -- The exact default drilling stiffness scale. -- The final MITC4 local basis algorithm for warped quadrilateral elements. -- Which shell stress/strain/resultant outputs are mandatory in Phase 1. +- Stress/strain/resultant recovery locations and component labels after `U` and `RF` are stable. +- Whether to parse Abaqus `*Orientation` or nodal shell normals. +- The future nodal-director policy for truly non-flat or strongly warped quadrilateral elements. - Optional non-displacement CSV formats, such as reaction force or stress/resultant exports. diff --git a/docs/Paper/2007쉘구조물의유한요소해석에대하여/2007쉘구조물의유한요소해석에대하여.md b/docs/Paper/2007쉘구조물의유한요소해석에대하여/2007쉘구조물의유한요소해석에대하여.md new file mode 100644 index 0000000..f54e988 --- /dev/null +++ b/docs/Paper/2007쉘구조물의유한요소해석에대하여/2007쉘구조물의유한요소해석에대하여.md @@ -0,0 +1,558 @@ +# 쉘 구조물의 유한요소해석에 대하여 + +On the Finite Element Analysis of Shell Structures + +이필승\*·노혁천\*\* + +Lee, Phill-Seung·Noh, Hyuk-Chun ····························································································································································································································· 1. 서 론 + +Based on recent research works, important concepts on the finite element analysis of shell structures and the relations among them are presented in this paper. We review the basic shell mathematical model, which is the underlying mathematical model of the continuum mechanics based shell finite elements. The asymptotic theory of shell structures then is reviewed and we present how to evaluate the asymptotic behavior in finite element solutions. S-norm is introduced as an error measure of finite element solutions and we show "locking" in the convergence curves of shell finite element solutions. We discuss the concept of Keywords : shell structures, finite elements, asymptotic behavior, uniform optimal convergence, benchmark tests ····························································································································································································································· 요 지 본 논문에서는 최근 주요 연구들을 토대로 쉘 구조물의 유한요소해석에 대하여 중요한 개념들과 그 연관관계를 고찰한다. 감절점 쉘 유한요소의 수학모델인 기본쉘수학모델을 살펴본다. 쉘 구조물의 두께가 얇아짐에 따라 일어나는 쉘 구조문제의 세가지 극한거동들(휨지배거동, 막지배거동, 혼합지배거동)에 대한 쉘의 점근거동 이론을 소개하고 점근거동을 유한요소해석 을 통해 찾아내는 방법을 알아본다. 유한요소해의 오차를 s-norm으로 평가하는 방법을 소개하고 이를 이용하여 쉘 유한요소 의 잠김현상이 유한요소해의 수렴곡선에 어떻게 나타나는지 살펴본다. 쉘 구조물의 유한요소해석에서 균일최적수렴의 개념을 논의한다. 마지막으로 이상적인 쉘 유한요소의 조건을 알아보고 쉘 유한요소의 성능평가를 위한 방법론을 제시한다. + +핵심용어 : 쉘 구조물, 유한요소해석, 점근거동, 균일최적수렴, 성능시험 + +ments and propose how to perform benchmark tests of shell finite elements. 계란의 외피가 얼마나 큰 외력에 견딜 수 있는가 하는 것 + +은 일반 대중들에게도 잘 알려져 있는 사실이다. 그 밖에도 우리는 자연 속에서 갑각류의 표피나 조개 껍질과 같은 여 러 종류의 쉘 구조물들(shell structures)을 접할 수 있다. 이 런 자연의 예들은 쉘 구조가 매우 이상적이고 효과적인 구 조물의 한 형태라는 것을 간접적으로 증명하고 있다. 인간이 만들어낸 쉘 구조물들도 셀 수 없이 다양하고 많으며 우리 는 쉘 구조물들 속에서 살고 있다고 해도 과언이 아니다. 유한요소법(finite element method)은 쉘 구조물의 선형 및 비선형 해석에 가장 널리 쓰이는 방법으로 지난 수십 년 동안 쉘 구조물의 유한요소해석에 대한 연구가 활발하게 이 루어져 오고 있다(Bathe, 1996; 최창근, 2002; Chapelle and Bathe, 2003; Noh, 2006). 그러나 쉘 구조물은 쉘의 형상, 경계조건, 하중 등에 따라 아주 다양한 거동을 보이며 + +····························································································································································································································· 기 때문에 쉘 구조물의 유한요소해석에 앞서 근본적인 쉘 이론 및 물리적 거동에 대한 이해가 필수적이다(Chapelle and Bathe, 1998; Lee and Bathe, 2002; Chapelle and + +Bathe, 2003). 이러한 쉘 구조물의 거동에 대한 이해의 핵 심은 쉘 구조물의 두께가 얇아짐에 따라 나타나는 점근거동 (asymptotic behavior)을 연구하는 것이다(Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003). 쉘 구조물의 점근 거동은 일반적으로 휨지배(bending dominated)거동, 막지배 (membrane dominated)거동, 혼합(mixed)지배거동 등으로 나 누어진다. 일반적으로 공학(engineering)에서 주어진 문제를 풀기 위 한 접근방법은 실험(experiment)과 관찰(investigation)을 통 하여 물리적 거동을 살펴본 후 그 물리문제의 중요한 특성/ 인자를 찾아내고 그에 따른 여러 가지 가정들(assumptions) 을 사용하여 문제를 단순화시켜 수학모델(mathematical model)을 만든다. 수학모델은 주어진 물리문제를 이론적으로 접근할 수 있는 방법을 제공해 준다. 수학모델의 해는 이론 + +특히 쉘의 두께가 얇을 경우 거동의 특성이 매우 민감해지 + +\*정회원 · 교신저자 · 삼성중공업(주) 건설부문 과장 (E-mail : phillseung.lee@samsung.com) + +적인 방법으로 구해질 수 있으나 풀고자 하는 문제가 매우 복잡할 때는 해를 구하기가 거의 불가능하다. 수치해석 (numerical analysis)을 이용하면 복잡한 물리문제의 근사해 (approximation)를 구할 수 있고 그 결과를 실험/관찰 결과 와 비교하면 수학모델이나 수치해석의 적정성을 평가할 수 있다. + +이와 같은 일련의 작업들, 즉, 물리적 거동, 수학모델, 수 치해석은 서로 밀접하게 연관되어 있다. 그러므로, 쉘 구조 물의 유한요소해석을 명확하게 이해하기 위해서는 쉘 구조 물의 물리적 거동에 대한 이해, 쉘의 수학모델(mathematical shell model)에 대한 이해와 쉘 유한요소에 대한 이해가 동 시에 체계적이고 심도 있게 이루어져야 한다. 세가지 모두에 대한 통합적인 이해가 없을 경우 쉘 구조물의 유한요소해석 에 있어서 중대한 오류를 범할 수 있다. 본 논문의 목적은 이 각각의 세가지 부분에 대한 이해와 이들이 서로 어떻게 유기적으로 관계를 맺고 있는지를 최근 주요 연구들(Lee and Bathe, 2002; Chapelle and Bathe, 2003; Bathe, Chapelle and Lee, 2003; Hiller and Bathe, 2003; Lee and Bathe, 2005)을 중심으로 정리하여 고찰해 보고 이상적 인 쉘 유한요소의 성질과 쉘 유한요소의 성능평가 방법을 제시하는 것이다. 특히 쉘 구조물의 설계를 위하여 유한요소 해석을 수행하는 기술자들(engineers)이나 유한요소법을 공부 /연구하는 학생/연구자들의 "쉘 구조물의 유한요소해석에 대 한 이해"를 돕고자 하는데 글의 초점을 맞추었다. + +유한요소법에 의해 쉘 구조물을 효과적으로 해석하기 위해 서는 신뢰할 만한 쉘 유한요소를 사용해야 한다는 것은 자 명하다. 일반적으로 변위법에 의해 정식화(displacement based formulation)된 쉘 유한요소는 사용된 근사함수 (interpolation function)의 차수(order)에 상관없이 휨지배 및 혼합지배거동을 하는 쉘 구조물에 대하여 과도한 강성을 나 타낸다(Bathe, 1996; Chapelle and Bathe, 2003). 이를 잠 김현상(locking phenomenon)이라 하며 효과적인 쉘 유한요 소의 개발에 있어서 극복해야 할 어려운 문제들 중의 하나 이다. 이상적인 쉘 유한요소는 여러 가지 점근거동과 다양한 형상의 쉘 구조문제에 있어서 균일최적수렴(uniform optimal convergence)을 보여야 하며 이상적인 쉘 유한요소를 개발하 는 것은 매우 어려운 일이다. + +본 논문에서는 먼저 쉘의 대표적인 수학모델을 설명하고 쉘 구조물의 점근거동에 대한 기본이론을 살펴본 후 임의의 쉘 구조물의 점근거동을 어떻게 알아낼 수 있는지를 알아본 다. 또한 쉘 유한요소해의 오차(error)를 평가하는 규준을 살 펴보고 이를 바탕으로 잠김현상 발생 시의 쉘 유한요소의 수렴과 균일최적수렴에 대해 알아본다. 마지막으로 이상적인 쉘 유한요소와 쉘 유한요소의 성능평가에 대하여 논한다. 앞 으로 논의하는 내용은 등방성재료(isotropic material)에 대한 선형탄성(linear elastic)이론에 국한된다. + +# 2. 쉘의 수학모델(mathematical shell model) + +기본쉘수학모델(basic shell mathematical model)은 쉘 구 조물의 휨(bending)거동, 면내(membrane)거동, 면외전단 (transverse shearing)거동과 그 상호연관(coupling)관계를 모 두 표현할 수 있는 가장 일반적인 쉘 수학모델로 3차원 연 속체 역학으로부터 유도된 감절점 쉘 유한요소(Ahmad, Irons and Zienkiewicz, 1970; Bathe, 1996)와 동일한 변형 률 항들을 가지고 있다(Chapelle and Bathe, 1998; Chapelle and Bathe, 2003; Lee and Bathe, 2005). 즉, 기본쉘수학 모델은 감절점 쉘 유한요소의 수학모델인 것이다. 본 장에서 는 미분기하학(differential geometry)을 이용하여 쉘의 형상 (geometry)과 변형거동(kinematics)을 살펴보고 기본쉘수학모 델의 유도를 보여준다. + +# 2.1 쉘의 형상 (shell geometry) + +쉘은 두께(thickness)가 얇은 3차원 구조물이다. 두께가 얇 다는 특징 때문에 쉘의 형상은 쉘의 중심면으로 이루어진 2 차원 영역과 두께에 의하여 정의 될 수 있다. 여기서는 기 본쉘수학모델에서 쓰이는 쉘의 형상에 관한 개념을 미분기 하학을 통하여 보여준다. + +![](_page_1_Figure_9.jpeg) + +그림 1. 쉘의 형상 + +λ + +φ + +$$\vec{a}_{\alpha} = \frac{\partial \vec{\phi}(\xi^1, \xi^2)}{\partial \xi^{\alpha}} \tag{1}$$ + +$$\vec{a}^{\alpha} \cdot \vec{a}_{\beta} = \delta^{\alpha}_{\beta} \tag{2}$$ + +미분기하학의 정의들을 유도하기 위하여 아인슈타인 합표 시규약(Einstein summation convention)을 사용한다. α, β, , µ는 1에서 2까지 변하며 i, j, k는 1에서 3까지 변하는 첨자(index)들이다. 쉘의 중심면(midsurface)은 그림 1에서 보이는 2차원 영역 ω에서 S로의 사상(mapping)을 나타내는 함수 에 의하여 정의 된다. 중심면의 공변기저벡터(covariant base vector)는 다음과 같이 나타내어진다. (1) 위의 식의 공변(covariant)기저벡터에 대응하는 반변기저벡터 (contravariant base vector)는 다음의 관계에 의하여 얻어진다. (2) 여기서 는 α와 β가 같을 때 1이고 다를 때 0인 δ β α + +$$\vec{a}_3 = \frac{\vec{a}_1 \times \vec{a}_2}{\left|\vec{a}_1 \times \vec{a}_2\right|} \tag{3}$$ + +$$\overrightarrow{\Phi}(\xi^{l}, \xi^{2}, \xi^{3}) = \overrightarrow{\phi}(\xi^{l}, \xi^{2}) + \xi^{3}\overrightarrow{a}_{3}(\xi^{l}, \xi^{2}) \tag{4}$$ + +$$\Omega = \left\{ (\xi^{1}, \xi^{2}, \xi^{3}) \in \Re^{3} | (\xi^{1}, \xi^{2}) \in \omega, \xi^{3} \in \left[ -\frac{t(\xi^{1}, \xi^{2})}{2}, \frac{t(\xi^{1}, \xi^{2})}{2} \right] \right\}$$ +(5) + +Kronecker symbol이다. 쉘의 중심면에 수직인 벡터는 중심 면의 공변(covariant)기저벡터들의 벡터곱(vector product)으로 정의 된다. (3) 쉘의 3차원 기하형상은 다음 식으로 표현된다. (4) 여기서 변수 ξ , ξ , ξ 의 영역은 다음과 같다. (5) 여기서 t는 쉘의 두께이다. + +$$a_{\alpha\beta} = \vec{a}_{\alpha} \cdot \vec{a}_{\beta} \tag{6}$$ + +$$a^{\alpha\beta} = \vec{a}^{\alpha} \cdot \vec{a}^{\beta} \tag{7}$$ + +위의 정의들을 이용하여 쉘의 중심면에서 surface 텐서들 (tensors)을 정의 할 수 있다. 첫 번째 기본텐서는 2D metric 텐서로 공변형(covariant type)은 다음 식과 같다. (6) 위 식의 반변형(contravariant type)은 다음과 같이 나타난다. (7) 두 번째 기본텐서는 곡률(curvature)텐서로서 쉘 중심면의 곡 + +$$b_{\alpha\beta} = \vec{a}_3 \cdot \vec{a}_{\alpha,\beta} \tag{8}$$ + +$$b^{\alpha}_{\beta} = a^{\alpha\lambda}b_{\lambda\beta} \tag{9}$$ + +$$c_{\alpha\beta} = b_{\alpha}^{\lambda} b_{\lambda\beta} \tag{10}$$ + +률에 관한 정보를 담고 있다. 공변형(covariant type)은 다음 과 같다. (8) 또한 위 식의 합성형(mixed)텐서는 다음 식과 같다. (9) 세 번째 기본텐서는 다음과 같이 정의 된다. (10) 쉘의 중심면에서의 벡터를 라고 하면 이 벡터의 공변미분 w + +$$w_{\alpha|\beta} = w_{\alpha,\beta} - \Gamma^{\lambda}_{\alpha\beta} w_{\lambda} \tag{11}$$ + +Γαβ λ + +$$\Gamma^{\lambda}_{\alpha\beta} = \vec{a}_{\alpha,\beta} \cdot \vec{a}^{\lambda} \tag{12}$$ + +$$\vec{g}_i = \frac{\partial \vec{\Phi}(\xi^1, \xi^2, \xi^3)}{\partial \xi^i} \tag{13}$$ + +$$\vec{g}_{\alpha} = \vec{a}_{\alpha} - \xi^3 b_{\alpha}^{\lambda} \vec{a}_{\lambda} \tag{14}$$ + +$$\vec{g}_3 = \vec{a}_3 \tag{15}$$ + +$$\vec{g} \cdot \vec{g}_j = \delta_j^i \tag{16}$$ + +# 2.2 쉘의 변형거동(shell kinematics) + +$$\overrightarrow{U}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}, \xi^{\mathsf{3}}) = \overrightarrow{u}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) + \xi^{\mathsf{3}} \theta_{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) \overrightarrow{a}^{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}})$$ + +$$\tag{17}$$ + + (11) 여기서 는 면에서의 Christoffel symbol이다. (12) 식 (4)로부터 3차원 공변(covariant)기저벡터를 얻을 수 있다. (13) 그리고 위 식으로부터 다음 식들이 얻어진다. (14) (15) 3차원 반변(contravariant)기저벡터는 다음과 같이 정의 된다. (16) 쉘의 변형거동에 있어서 기본이 되는 가정은 변형전 쉘의 중심면에 수직인 직선은 변형중에도 직선을 유지하며 늘어 나거나 줄어들지 않는다는 것이고, 이때 쉘의 변위는 다음과 같이 표현된다. (17) 여기서 는 쉘의 중심면의 미소변위(infinitesimal translation)를 나타내고 는 쉘의 중심면에 수직인 직선의 미소회전(infinitesimal rotation)을 나타낸다. 는 회전 벡터 이고 는 그 회전 벡터에 의한 변위를 u ξ 1 ,ξ 2 ( ) θ λ ξ 1 ,ξ 2 ( ) θ λa λ θ ξ 3 θ λa λ u a 1 2 a 3 a3 + +$$e_{ij} = \frac{1}{2} (\vec{g}_i \cdot \vec{U}_{,j} + \vec{g}_j \cdot \vec{U}_{,i})$$ +(18) + +$$\vec{a}^2$$ +, $\vec{a}^3$ (= $\vec{a}_3$ )에 의하여 주어진다. +선형해석을 위한 3차원 Green-Lagrange 공변변형률 +(covariant strain) 텐서의 선형부분은 다음과 같이 정의 된다. + $e_{ij} = \frac{1}{2}(\vec{g}_i \cdot \vec{U}_{,j} + \vec{g}_{j} \cdot \vec{U}_{,i})$ (18) +여기서 + $\vec{U}_{,i} = \frac{\partial \vec{U}(\xi^1, \xi^2, \xi^3)}{\partial \xi^i}$ . (19) +식 (14), (15)와 (17)을 식 (18)에 대입시키면 변형률 텐서의 공변(covariant)항들을 얻을 수 있다. + $e_{\alpha\beta} = \gamma_{\alpha\beta}(\vec{u}) + \xi^3 \chi_{\alpha\beta}(\vec{u},\vec{\theta}) - (\xi^3)^2 \kappa_{\alpha\beta}(\vec{\theta})$ (20a) + $e_{\alpha3} = \zeta_{\alpha}(\vec{u},\vec{\theta})$ (20b) + $e_{33} = 0$ (20c) +여기서 + $\gamma_{\alpha\beta}(\vec{u}) = \frac{1}{2}(u_{\alpha|\beta} + u_{\beta|\alpha}) - b_{\alpha\beta}u_3$ (21a) + $\chi_{\alpha\beta}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha|\beta} + \theta_{\beta|\alpha} - b_{\beta}^{\lambda}u_{\lambda|\alpha} - b_{\alpha}^{\lambda}u_{\lambda|\beta}) + c_{\alpha\beta}u_3$ (21b) + $\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda}\theta_{\lambda|\alpha} - b_{\alpha}^{\lambda}\theta_{\lambda|\beta})$ (21c) + $\zeta_{\alpha}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda})$ (21d) +등방성재료(isotropic material)에 대한 평면응력조건(plane + +. (19) + +$$e_{\alpha\beta} = \gamma_{\alpha\beta}(\vec{u}) + \xi^3 \chi_{\alpha\beta}(\vec{u}, \vec{\theta}) - (\xi^3)^2 \kappa_{\alpha\beta}(\vec{\theta})$$ + (20a) + +$$e_{\alpha\beta} = \zeta_{\alpha}(\vec{u}, \vec{\theta}) \tag{20b}$$ + +$$e_{33} = 0$$ + (20c) + +의 공변(covariant)항들을 얻을 수 있다. +$$e_{\alpha\beta} = \gamma_{\alpha\beta}(\vec{u}) + \xi^3 \chi_{\alpha\beta}(\vec{u},\vec{\theta}) - (\xi^3)^2 \kappa_{\alpha\beta}(\vec{\theta}) \qquad (20a)$$ + +$$e_{\alpha3} = \zeta_{\alpha}(\vec{u},\vec{\theta}) \qquad (20b)$$ + +$$e_{33} = 0 \qquad (20c)$$ +역기서 +$$\gamma_{\alpha\beta}(\vec{u}) = \frac{1}{2}(u_{\alpha|\beta} + u_{\beta|\alpha}) - b_{\alpha\beta}u_3 \qquad (21a)$$ + +$$\chi_{\alpha\beta}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha|\beta} + \theta_{\beta|\alpha} - b_{\beta}^{\lambda}u_{\lambda|\alpha} - b_{\alpha}^{\lambda}u_{\lambda|\beta}) + c_{\alpha\beta}u_3 \qquad (21b)$$ + +$$\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda}\theta_{\lambda|\alpha} - b_{\alpha}^{\lambda}\theta_{\lambda|\beta}) \qquad (21c)$$ + +$$\zeta_{\alpha}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda}) \qquad (21d)$$ +등방성재료(isotropic material)에 대한 평면응력조건(plane) + +$$\chi_{\alpha\beta}(\vec{u}, \vec{\theta}) = \frac{1}{2}(\theta_{\alpha|\beta} + \theta_{\beta|\alpha} - b_{\beta}^{\lambda} u_{\lambda|\alpha} - b_{\alpha}^{\lambda} u_{\lambda|\beta}) + c_{\alpha\beta} u_{3}$$ +(21b) +$$\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda} \theta_{\lambda|\alpha} - b_{\alpha}^{\lambda} \theta_{\lambda|\beta})$$ +(21c) +$$\zeta_{\alpha}(\vec{u}, \vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda} u_{\lambda})$$ +(21d) +동방성재료(isotropic material)에 대한 평면응력조건(plane + +$$\kappa_{\alpha\beta}(\vec{\theta}) = \frac{1}{2}(b_{\beta}^{\lambda}\theta_{\lambda|\alpha} - b_{\alpha}^{\lambda}\theta_{\lambda|\beta})$$ + (21c) +$$\zeta_{\alpha}(\vec{u},\vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda})$$ + (21d) +한방성재료(isotropic material)에 대한 평면응력조건(plane + +$$\zeta_{\alpha}(\vec{u}, \vec{\theta}) = \frac{1}{2}(\theta_{\alpha} + u_{3,\alpha} + b_{\alpha}^{\lambda}u_{\lambda})$$ + (21d) +통방성재료(isotropic material)에 대한 평면응력조건(plane + + (21d) 등방성재료(isotropic material)에 대한 평면응력조건(plane + +$$\sigma^{\alpha\beta} = C^{\alpha\beta\lambda\mu} e_{\lambda\mu} \tag{22a}$$ + +$$\sigma^{\alpha\beta} = \frac{1}{2} D^{\alpha\lambda} e_{\lambda\beta} \tag{22b}$$ + +$$C^{\alpha\beta\lambda\mu} = \frac{E}{2(1-v)} \left( g^{\alpha\lambda} g^{\beta\mu} + g^{\alpha\mu} g^{\beta\lambda} + \frac{2v}{1+v} g^{\alpha\beta} g^{\lambda\mu} \right)$$ +(23a) + +$$D^{\alpha\lambda} = \frac{2E}{1+\nu}g^{\alpha\lambda} \tag{23b}$$ + +g αβ g αβ=g α g β ( ) ⋅ + +V U + +$$\int_{\Omega} C^{\alpha\beta\lambda\mu} e_{\alpha\beta}(\overrightarrow{U}) e_{\lambda\mu}(\overrightarrow{V}) dV + \int_{\Omega} D^{\alpha\lambda} e_{\alpha\beta}(\overrightarrow{U}) e_{\lambda\beta}(\overrightarrow{V}) dV$$ + +$$= \int_{\Omega} \overrightarrow{F} \cdot \overrightarrow{V} dV \qquad (24)$$ + +F + +$$\vec{V}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}, \xi^{\mathsf{3}}) = \vec{v}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) + \xi^{\mathsf{3}} \eta_{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}}) \vec{a}^{\lambda}(\xi^{\mathsf{I}}, \xi^{\mathsf{2}})$$ +(25) + +# 3. 쉘 구조물의 점근거동 + +stress condition)을 적용하면 면에 수직인 응력은 0이며 (σ33 = 0) 이때 응력과 변형률의 상관관계는 다음과 같다. (22a) (22b) 위의 식 (22)에서 (23a) (23b) 여기서 E는 재료의 탄성계수(elastic modulus)이고 v는 포 아손비(Poisson's ratio)이며 는 식 (14)의 3차원 반변 (contravariant)기저벡터로 정의된 metric텐서이다 . 쉘 구조물에 강체운동(rigid body motion)이 일어나지 않 도록 적절한 변위경계조건이 주어지면 식 (20)에서 (23)까지 를 이용하여 기본쉘수학모델(basic shell mathematical model) 의 지배변분식(variational equation)을 얻을 수 있다. 해를 구하는 과정은 모든 임의의 시험함수(test function) 에 대하여 다음 식 (24)와 변위 경계조건을 만족시키는 미지변위 를 찾는 것으로 나타내어질 수 있다. (24) 여기서 는 쉘 구조물에 작용하는 외력(external loading) 이며 시험함수는 변위경계조건을 만족시켜야 한다. (25) 휨(bending), 막(membrane), 면외전단(transverse shearing) 작용들은 쉘 구조물이 하중을 지지하는 기본적인 원리 (mechanism)이다. 그러므로 하중 재하 시 쉘 구조물은 휨, 막, 전단 에너지를 그 내부에 저장하게 된다. 쉘의 두께가 얇아짐에 따라 쉘의 전단 에너지는 무시할 만큼 작아지므로 쉘 구조물의 에너지는 주로 휨 에너지와 막 에너지에 의해 구성된다고 할 수 있다. 쉘은 그 두께가 얇아짐에 따라 특정한 한계거동 - 휨지배 (bending dominated)거동, 막지배(membrane dominated)거동, 혼합(mixed)지배거동 - 을 보이게 되며 이를 쉘의 점근거동 (asymptotic behavior)이라 한다. 쉘의 두께가 얇아짐에 따라 쉘 구조물이 주로 휨 거동에 의해 하중을 지탱할 경우 그 + +3.1 점근거동의 분류 + +U∈Ψ + +$$\varepsilon^3 A_b(\vec{U}, \vec{V}) + \varepsilon A_m(\vec{U}, \vec{V}) = \vec{F}(\vec{V}), \forall \vec{V} \in \vec{\Psi}$$ + (26. 여기서 $\varepsilon$ 는 쉘의 두께와 전체 쉘 구조물 크기의 비 $(t/L)$ $\varepsilon^3 A_b(\cdot,\cdot)$ 는 휨 에너지, $\varepsilon$ $A_b(\cdot,\cdot)$ 은 막 및 전단 에너지에 대응하는 겹선형식들(bilinear forms)이며, $\vec{U}$ 는 변위장의 해 $\vec{V}$ 는 시험함수, $\vec{\Psi}$ 는 Sobolev 공간(space) $^1$ , $\vec{F}(\cdot)$ 는 외력에 대응하는 선형식(linear form)을 나타낸다. 일반적으로 쉴의 두께가 얇을 때 전단 에너지는 막 에너지에 비해 매우작으므로 $\varepsilon$ $A_m$ 을 막 에너지에 대응하는 항이라고 부를 수 있다 + +ερ + +$$\vec{F}(\vec{V}) = \varepsilon^{\rho} \vec{G}(\vec{V}) \tag{27}$$ + +G Ψ′ Ψ + +$$1 \le \rho \le 3 \tag{28}$$ + +$$\overrightarrow{\Psi}_0 = \left\{ \overrightarrow{V} \in \overrightarrow{\Psi} \middle| A_m(\overrightarrow{V}, \overrightarrow{V}) = 0 \right\}$$ + (29) + +얇아짐에 따라 쉘 구조물이 휨과 막거동 두 가지 모두에 의 해 외력을 지지할 경우 혼합(mixed)지배 쉘 구조물이라 한 다. 쉘 구조물의 점근거동은 쉘의 형상(geometry), 경계조건 (boundary condition), 하중(loading)에 따라 달라진다(Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003; Chapelle and Bathe, 2003). 식 (24)에서 구해진 선형 쉘 이론의 변분식(variational form)을 두께(t)에 대하여 정리하여 t의 고차항을 제거하면 다음과 같이 간략화하여 나타낼 수 있다. Find such that (26) 여기서 ε 는 쉘의 두께와 전체 쉘 구조물 크기의 비(t/L), Ab(·,·)는 휨 에너지, ε Ab(·,·)은 막 및 전단 에너지에 대응 하는 겹선형식들(bilinear forms)이며, 는 변위장의 해, 는 시험함수, 는 Sobolev 공간(space)1 , 는 외력 에 대응하는 선형식(linear form)을 나타낸다. 일반적으로 쉘 의 두께가 얇을 때 전단 에너지는 막 에너지에 비해 매우 작으므로 ε Am을 막 에너지에 대응하는 항이라고 부를 수 있다. ε이 작아짐에 따른 쉘의 점근거동을 살펴 보기 위해 (ρ = load scaling factor)가 곱해진 외력(scaled loading)을 사용한다. (27) 여기서 는 ( 의 dual space2 )에 속하며 ρ는 실수이 다. 식 (26)의 좌변의 각 항은 ε 과 ε에 비례하므로 ρ는 1보다 크거나 같고 3보다 작거나 같은 실수임을 알 수 있다. (28) 다음의 공간(space)은 쉘의 점근거동에 중요한 역할을 한다. (29) 공간은 순수 휨(pure bending)을 나타내는 변위의 공간 이며 막 및 전단 에너지를 0으로 만들 수 있는 모든 변위 의 형태들(patterns)을 포함한다. 이 공간(space)이 단지 모든 변위를 0으로 하는 변위의 형태들만을 가지고 있을 때 쉘 구조물에서 순수 휨거동이 구속되었다고 하며 그러한 쉘을 순수 휨이 구속된 쉘(inhibited shell)이라 한다. 반면에 쉘이 모든 변위가 0인 형태(pattern)가 아닌 순수 휨 모드(mode) 를 가지고 있을 경우 그 쉘 구조물을 순수 휨이 구속되지 않은 쉘(non-inhibited shell) 구조물이라 한다. 쉘의 점근거 Ψ0 + +탄성문제에서 변위경계조건을 만족시키는 Sobolev 공간은 으로 표기되며 정의는 다음과 같다. 1 + +쉘 구조물을 휨지배 쉘(bending dominated shell) 구조물이 라 부르며 막거동에 의해 하중을 지탱할 경우 막지배 쉘 (membrane dominated shell) 구조물이라 한다. 또한 두께가 여기서 이다(Bathe, 1996). H0 H0 1 ( ) Ω = V: V L2 ∈ Ω( ); ∂Vi ∂xj ------- L2 ∈ Ω( ); Vi =0 at prescribed displacement boundary ⎩ ⎭ ⎨ ⎬ ⎧ ⎫ ( ) Ω = V: Ω ∫ 3 ∑ Vi ( )2 dΩ<+∞ + +2 공간위의 모든 선형식 들의 공간을 의 dual space(쌍대공간)라고 부른며 로 표기한다. i = 1 Ψ F Ψ Ψ′ + +| 표 1. 쉘의 점근거동의 분류 | | | | | +|----------------------|--------------------------------------|-------------------------|--|--| +| | | | | | +| 경우 | 하중
순수 휨을 유발하는 외력 | 분류 | | | +| 순수 휨이 구속되지 | ∃
Ψ V∈ 0
such that
G V( )≠0 | (i) 휨지배거동 | | | +| 않은 쉘 구조물,
Ψ0≠{ }0 | 순수 휨을 유발하지 않는 외력 | (ii) 불안정한 막지배 또는 혼합지배거동 | | | +| | ,
G V( ) = 0
∀
Ψ V∈ 0 | | | | +| | Admissible membrane loading | | | | +| 순수 휨이 구속된 | G∈Ψm′ | (iii) 막지배거동 | | | +| 쉘 구조물,
Ψ0 = 0{ } | Non-admissible membrane loading | | | | +| | G∉Ψm′ | (iv) 혼합지배거동 | | | + +Ψ0≠{ }0 + +U ∈Ψ0 + +$$A_{b}(\overrightarrow{U}, \overrightarrow{V}) = \overrightarrow{G}(\overrightarrow{V}), \ \forall \overrightarrow{V} \in \overrightarrow{\Psi}_{0}$$ + +$$\tag{30}$$ + +Ψ0= 0{ } Ψm Ψ + +U ∈Ψm + +![](_page_4_Picture_9.jpeg) + +그림 2. 쉘의 변형거동 + +$$A_{m}(\overrightarrow{U}^{m}, \overrightarrow{V}) = \overrightarrow{G}(\overrightarrow{V}), \forall \overrightarrow{V} \in \overrightarrow{\Psi}_{m}$$ + +$$(31)$$ + +G Ψm G∈Ψm′ + +$$\left| \overrightarrow{G}(\overrightarrow{V}) \right| \le c_{\sqrt{A_m(\overrightarrow{V}, \overrightarrow{V})}}, \ \forall \overrightarrow{V} \in \overrightarrow{\Psi}$$ + (32) + +G∈Ψm′ + + (31) (32) + +![](_page_4_Picture_16.jpeg) + +그림 3. 쉘의 변형형상과 유효응력분포에 나타난 경계층(boundary layer) + +3.2 쉘의 층과 특성길이 쉘의 응력/변형률/변위장들은 그 변화가 매끄러운 영역들 (smooth areas)과 그렇지 않은 여러 종류의 층들(layers)로 나누어진다. 층(layer)은 곡률(curvature)이나 두께의 변화와 같은 쉘의 형상의 변화, 적합하지 않은 변위경계조건 (incompatible boundary condition), 불규칙한 하중(irregular loading) 등에 의하여 유발된다(Lee and Bathe, 2002). 층 (layer)에서는 응력/변형률/변위 등이 매우 급하게 변하며 변 형에너지의 집중이 일어난다. 층의 특성길이(Lc, characteristic length)는 쉘 구조물의 두께에 따라 변하며 쉘의 두께(t) 와 전체 쉘 구조물 길이(L)의 함수로 나타나다. + +$$L_c = ct^{1-l}L^l \tag{33}$$ + +여기서 c는 상수이며 l은 양의 실수 이다. + +참고문헌(Lee and Bathe, 2002)는 쉘의 두께가 얇아짐에 따라 나타나는 Scodelis-Lo roof shell problem에서의 경계 층(boundary layer)과 쌍곡포물면(hyperbolic paraboloid) 쉘 구조문제의 내부층(inner layer)을 보여준다. 참고문헌(Bathe, Chapelle and Lee, 2003)에서는 또 다른 형태의 경계층이 쉘의 두께가 얇아짐에 따라 변화하는 것을 보여준다. 그림 3은 쉘의 변형형상(deformed shape)과 유효응력(effective stress)분포에 나타난 경계층(boundary layer)의 예를 보여주 고 있다. + +3.3 점근거동의 해석 이론적인 방법으로 일반적인 쉘 구조물의 점근거동을 알아 내려는 시도는 Lovadina를 비롯한 많은 연구자들에 의해 수 행되었으나 쉬운 일이 아니었다(Lovadina, 2001). 근래에 수 치적인 방법에 의해 점근거동을 알아내는 연구가 진행되었 으며 유한요소해석을 통한 여러 가지 방법이 Lee와 Bathe에 의해 제안되었다(Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003). 본 논문에서는 가장 손쉬운 한가지 방법을 소 개한다. + +쉘 구조물의 ρ(load scaling factor) 값을 구하게 되면 그 구조물의 점근거동을 알아낼 수 있다. 다음은 Lee와 Bathe 에 의해 제안된 ρ의 근사 값을 구하는 식이다. + +$$\rho \cong \frac{\log E(\varepsilon + \Delta \varepsilon) - \log E(\varepsilon)}{\log(\varepsilon + \Delta \varepsilon) - \log \varepsilon}$$ +(34) + +여기서 E는 ε(=t/L)에 대한 유한요소해석으로부터 구해진 쉘 구조물의 변형에너지(strain energy) 값이다. 주어진 ε에 대하여 유한요소해의 정확도가 높을수록 보다 정확한 ρ값을 얻을 수 있다. 계산된 ρ의 값이 1일 경우 구조물은 막지배 거동을, 3일 경우 휨지배거동을, 1과 3의 중간 값일 경우 막과 휨의 혼합지배거동을 한다. + +Lovadina는 보간이론(interpolation theory)을 사용하여 쉘 구조물의 점근거동을 연구하였으며 ρ값과 쉘 구조물에 저장 되는 에너지들과의 관계를 나타내는 흥미로운 식을 제안하 였다(Lovadina, 2001). + +$$\lim_{\varepsilon \to 0} R(\varepsilon) = \frac{\rho - 1}{2} \tag{35}$$ + +여기서 R(ε)는 쉘 구조물의 전체 변형에너지에 대한 휨 변 + +형에너지의 비이다. + +$$R(\varepsilon) = \frac{\varepsilon^3 A_b(\vec{U}, \vec{U})}{\varepsilon^3 A_b(\vec{U}, \vec{U}) + \varepsilon A_m(\vec{U}, \vec{U})}$$ +(36) + +역으로 쉘 구조물의 R(ε)를 계산하면 또한 쉘의 점근거동을 찾을 수 있는 ρ값을 알 수 있다. + +Lee와 Bathe는 쉘 유한요소의 성능을 평가(benchmark)하 기 위한 쉘 문제로 널리 알려져 있는 Scodelis-Lo roof shell problem을 대상으로 하여 ρ값을 계산하였으며 그 쉘 구조물의 점근거동을 보여주었다(Lee and Bathe, 2002). 그 후에 Lovadina는 이론적인 방법을 이용해 같은 문제의 ρ값 을 구하였고 두 결과는 일치하였다. 쉘 구조물의 휨지배거동 과 막지배거동에 대한 보다 자세한 예들을 참고문헌(Lee and Bathe, 2002)에서 볼 수 있으며, 쉘 두께의 변화에 따 라ρ값이 변동(fluctuation)하는 민감한 쉘 구조물의 점근거 동에 대한 해석이 참고문헌(Bathe, Chapelle and Lee, 2003)에 제시되어 있다. + +# 4. 쉘 유한요소의 잠김현상과 균일최적수렴 + +쉘 구조물의 유한요소해석에서 직면한 가장 큰 어려움은 잠김현상(locking phenomenon)이다. 잠김현상은 쉘 구조물의 두께가 얇을 수록 유한요소해의 오차(error)를 매우 빠르게 증가시킨다. 본 장에서는 쉘의 점근거동에 대한 이해를 바탕 으로 쉘 유한요소의 잠김현상과 균일최적수렴에 대하여 살 펴본다. + +4.1 쉘 유한요소의 종류와 잠김현상 쉘 유한요소의 종류는 크게 평면 쉘(flat shell) 유한요소 그룹과 감절점 쉘(degenerated shell or continuum mechanics based shell) 유한요소 그룹으로 나뉘어진다(Ahmad, Iron and Zienkiewicz, 1970; Bathe, 1996; Choi, Lee and Park, 1999; 최창근, 2002; Chapelle and Bathe, 2003). 평면 쉘 유한요소는 평판 유한요소와 평면응력 유한요소의 결합에 의 해 만들어진다. 곡면의 쉘 구조물을 여러 개의 평면으로 나 누어 표현한다는 물리적 개념에서 출발하였다. 장점은 유한 요소 정식화가 쉽고 면내회전자유도(drilling degrees of freedom)를 쉽게 도입하여 절점 당 6개의 자유도를 가질 수 있으며 그로 인해 절점 당 6개의 자유도를 갖는 빔(beam) 과 같은 유한요소들과 결합이 편리하다는 것이다. 그러나, 기 본적으로 요소자체의 형상이 평면이기 때문에 복잡한 곡면 형태의 쉘의 형상과 그 거동을 정밀하게 표현하기 어려워 곡면 쉘 구조물의 해석 시 수렴이 느리며 정확해로의 수렴 에 실패하는 경우가 발생하는 단점을 가지고 있다. 반면 3 차원 연속체역학에 근거한 감절점 쉘 요소는 절점 당 5개의 자유도를 가지고 있어 빔과 같은 유한요소들과의 결합 시 특별한 인위적인 방법이 필요하지만 쉘 고유의 복잡한 곡면 형상과 그 거동을 잘 표현해 낼 수 있고 정확해로의 수렴속 도가 빠르다. 또한, 2장에서 설명된 바와 같이 가장 일반적 인 쉘 수학모델(mathematical shell model)인 기본쉘수학모 델에 근거한다. + +쉘 유한요소를 유한요소의 정식화(formulation)방법에 따라 + +분류하면 크게 변위법(displacement based method)에 의한 + +Ψ0 + +쉘 유한요소와 혼합법(mixed method)에 의한 쉘 유한요소로 나눌 수 있다. 변위법에 근거한 쉘 유한요소(displacement based shell finite element)는 막지배 구조물들의 해석에 있 어서는 이상적인 거동을 보인다. 그러나 요소의 종류와 근사 차수(interpolation order)에 상관없이 두께가 얇은 휨지배 (bending dominated) 또는 혼합지배(mixed) 쉘 구조물들의 해석에 있어서 유한요소해석의 해가 매우 느리게 수렴하는 치명적인 단점을 가지고 있는데 이를 잠김(locking)현상이라 고 한다. 주어진 요소망(mesh)에 대하여 유한요소해의 정확 성이 쉘의 두께가 얇아짐에 따라 급속히 나빠지는 현상을 말하며 최악의 경우 쉘의 두께가 얇아짐에 따라 변위장이 0 에 수렴하게 된다. 잠김현상은 막잠김(membrane locking)과 전단잠김(shear locking)현상으로 나누어질 수 있으며 잠김현상이 일어나는 근본적인 이유는 변위법에 의한 쉘 유한요소가 식 (29)의 순수 휨 변위공간(space) 을 충분히 근사할 수 없기 때 문이다. 즉, 잠김현상은 쉘 구조물의 점근거동과 직접적인 연 관관계를 가지고 있다. 쉘 구조물의 점근거동이 쉘의 형상, 경계조건, 하중 등에 따라 변하기 때문에 잠김현상 또한 쉘 의 형상, 경계조건, 하중에 의존적이다. 막잠김현상은 곡률을 가진 쉘 구조물에서만 발생하고 평면형상의 쉘 구조물에서 는 발생하지 않으나 전단잠김현상은 곡률에 관계없이 발생 한다. 잠김현상은 휨지배 및 혼합지배거동 시 식 (21a)와 (21d) 에 있는 변형률항들로부터 발생한다. 참고문헌(Bathe, Lee and Hiller, 2003)과 (Lee and Bathe, 2005)는 잠김현상이 유한요소해에 어떻게 나타나는지를 보여주고 잠김현상이 제 거되기 어려운 근본적인 이유를 잘 설명하고 있다. 실제 쉘 유한요소해석에 있어서 잠김현상은 과대한 강성 또는 그로 인한 과소한 변위, 응력, 변형률 및 변형에너지로 나타난다. 그러므로 두께가 얇은 쉘 구조물의 설계 시 충분 하게 조밀하지 못한 유한요소망을 사용하여 해석한 결과를 이용할 경우(강성이 과대평가되어) 과소설계의 오류를 범할 수 있다. 이와 같이 유한요소해의 오차 또는 정확도가 쉘 구조물의 두께에 따라 변하는 현상인 잠김현상은 쉘 구조물 의 유한요소해석에 있어서 빈번하게 일어나기 때문에 쉘 구 조물의 해석에 앞서 잠김현상에 대한 이해는 필수적이다. 4.2 유한요소해의 오차측정 신뢰할 만한 쉘 유한요소의 해는 사용된 요소의 수가 증 가함에 따라 2장에서 설명된 쉘 유한요소의 수학모델 (mathematical shell model)의 정확해(exact solution)에 수 렴하여야 하며 잠김현상을 알아보기 위해서는 쉘 유한요소 해의 수렴곡선(convergence curve)을 관찰해야 한다. 여기서 쉘 유한요소의 수렴을 측정하기 위하여 적절한 규준(norm) 을 사용하는 것이 매우 중요하다. 그 규준(norm)은 한 점에 서 특정 물리량 값(point-wise value)의 수렴이 아니라 쉘 구조물 전체 영역에서의 해의 수렴을 고려할 수 있어야 한 다. 일반적으로 한 점에서의 변위/응력/변형률 등을 가지고 수렴을 측정하는 방법이 많이 사용되어왔는데 이는 전체영 역에서의 수렴을 대표하지 않으므로 적절하지 못하다. 변형 + +⋅ s + +$$\left| \overrightarrow{U} - \overrightarrow{U}_h \right|_s^2 = \int_{\Omega} \Delta \overrightarrow{\varepsilon}^T \Delta \overrightarrow{\sigma} d\Omega \tag{37}$$ + +U Uh ε σ + +$$\overleftarrow{\mathcal{E}} = \left[ \varepsilon_{xx}, \ \varepsilon_{yy}, \ \varepsilon_{zz}, \ 2\varepsilon_{xy}, \ 2\varepsilon_{yz}, \ 2\varepsilon_{zx} \right]^{T}$$ +(38a) + +$$\vec{\sigma} = [\sigma_{xx}, \sigma_{yy}, \sigma_{zz}, \sigma_{xy}, \sigma_{yz}, \sigma_{zx}]^T$$ +(38b) + +$$\Delta \vec{\varepsilon} = \vec{\varepsilon} - \vec{\varepsilon}_h = \vec{\varepsilon}(\vec{x}) - \mathbf{B}_h \vec{x}_h) \vec{U}_h \tag{39a}$$ + +$$\Delta \vec{\sigma} = \vec{\sigma} - \vec{\sigma}_h = \vec{\sigma}(\vec{x}) - \mathbf{C}_h(\vec{x}_h) \mathbf{B}_h(\vec{x}_h) \vec{U}_h$$ + (39b) + +에너지의 산술적 차이 또한 규준으로 사용할 수 있지만 변 위법 정식화에 근거한 유한요소가 아닌 경우에는 일반적으 로 사용될 수 없기 때문에 정식화의 방법에 관계없이 일반 적으로 쓰일 수 있는 규준이 필요하다. Hiller와 Bathe에 의해 제안된 s-norm( )은 쉘 구조물 전체의 거동을 반영할 수 있을 뿐만 아니라 물리적인 개념 으로부터 유도되었기 때문에 정식화 방법에 관계없이 사용 될 수 있다(Hiller and Bathe, 2003). 여기서 는 정확해이며, 는 유한요소해이다. 와 는 전체직각좌표계(global Cartesian coordinate system)에서 정 의된 변형률 벡터와 응력 벡터이다. 식 (37)에서 변형률과 응력에 대하여 정확해와 유한요소해의 차이(difference)는 다음과 같이 구해질 수 있다. x xh + +$$\vec{x} = \Pi(\vec{x}_h) \tag{40}$$ + + (37) (38a) (38b) (39a) (39b) 여기서 C는 재료의 응력-변형률 관계 행렬(matrix)이고 B는 변형률-변위(strain-displacement)관계 연산자(operator)이 다. 위치벡터 와 는 각각 원래 쉘 구조물의 영역 (domain)과 이산화된 유한요소 모델의 영역에 대응된다. 두 벡터의 관계는 일대일 사상(injective mapping, Π)에 의하여 정의할 수 있다. (40) 일반적인 쉘 구조문제에 있어서 이론적인 방법을 이용하여 정확해를 찾아내는 것은 거의 불가능하므로 매우 조밀한 유 Uref + +$$\left| \overrightarrow{U}_{ref} - \overrightarrow{U}_h \right|_s^2 = \int_{\Omega_{ref}} \Delta \overrightarrow{\varepsilon}^T \Delta \overrightarrow{\sigma} d\Omega_{ref}, \tag{41}$$ + +여기서 + +$$\Delta \vec{\xi} = \vec{\xi}_{ref} - \vec{\xi}_h = \mathbf{B}_{ref}(\vec{x}_{ref}) \vec{U}_{ref} - \mathbf{B}_h(\vec{x}_h) \vec{U}_h, \qquad (42a)$$ + +$$\Delta \overrightarrow{\sigma} = \overrightarrow{\sigma}_{ref} - \overrightarrow{\sigma}_h = \mathbf{C}_{ref}(\overrightarrow{x}_{ref}) \mathbf{B}_{ref}(\overrightarrow{x}_{ref}) \overrightarrow{U}_{ref} - \mathbf{C}_h(\overrightarrow{x}_h) \mathbf{B}_h(\overrightarrow{x}_h) \overrightarrow{U}_h,$$ +(42b) + +$$\vec{x}_{ref} = \Pi(\vec{x}_h) \tag{42c}$$ + +한요소망을 사용하여 계산된 해를 정확해로 고려하여 snorm을 계산하는 것이 훨씬 실용적이다. 정확해로 고려된 유한요소해를 라고 하면 위식의 s-norm은 다음과 같이 나타내어질 수 있다. , (41) , (42a) , (42b) (42c) 다양한 쉘 구조문제와 여러 경우의 쉘 두께에 대하여 유한 요소해의 수렴을 상호 비교 가능하게 하기 위해서는 상대오 + +차(relative error)를 사용하여야 한다. 상대오차( +$$E_h$$ +)는 다음과 +같이 정의된다. +$$E_h = \frac{|\overrightarrow{U}_{ref} - \overrightarrow{U}_h|_s^2}{||\overrightarrow{U}_{ref}||_s^2} \tag{43}$$ + +![](_page_7_Figure_0.jpeg) + +그림 + +![](_page_7_Figure_2.jpeg) + +그림 + +유한요소해의 이론적인 수렴은 다음과 같다. + +$$E_h \cong ch^{2k} \tag{44}$$ + +여기서 c는 상수, h는 사용된 유한요소의 크기, k는 유한 요소의 변위 근사함수(displacement interpolation function) 차수이다. 예를 들면 선형(linear) 근사함수를 사용하는 3절 점 및 4절점 쉘 유한요소에 대하여 k = 1이며, 2차의 (quadratic) 근사함수를 사용하는 6절점 및 9절점 쉘 유한요 소에 대하여 k = 2이다. S-norm은 수치적인 방법에 의해 계 산될 수 있다(Lee, Noh and Bathe, 2007). 을 때-MITC4 쉘 요소 + +4.3 수렴곡선과 균일최적수렴 다음으로 2개의 휨지배 쉘 구조문제들의 예를 통하여 잠 김현상이 s-norm을 이용하여 구해진 수렴곡선에 어떻게 나 타나는지를 보여주고 균일최적수렴(uniform optimal convergence)에 대하여 알아본다. + +판(plate) 구조는 곡률이 영인 쉘 구조물로서 쉘 구조의 가장 단순한 형태이다. 첫 번째 예로 그림 4(a)에 보여진 네 변이 완전히 구속된 평판 휨(plate bending) 구조문제에 대하여 쉘 유한요소해의 수렴곡선을 살펴보자. 사용된 탄성 계수는 1.7472×107 , 포아손비는 0.3, 길이 L=1.0, 하중은 단위 면적당 90이 −Z 방향으로 작용한다. 구조물의 형상, 하중, 변위경계조건의 대칭성으로 인해 그림 4(a)의 색칠된 부분만 해석되었다. 5. 두께의 변화에 따른 4절점 쉘 유한요소들의 수렴곡선: (a) 잠김현상이 일어날 때-QUAD4 쉘 요소 (b) 잠김현상이 일어나지 않 + +주어진 평판 휨 문제를 풀기 위하여 변위법(displacement based formulation)에 의한 4절점 감절점 쉘 유한요소 (QUAD4)와 혼합법(mixed formulation)에 의한 4절점 감절 점 쉘 유한요소(MITC43 )를 사용하였다(Ahmad, Irons and + +MITC(Mixed Interpolation of Tensorial Components) 방법을 이용하여 개발된 n절점 쉘 유한요소를 MITCn 유한요소라고 부른다. 즉, MITC4는 4절점, MITC6는 6절점 쉘 유한요소이다. + +![](_page_8_Figure_0.jpeg) + +그림 + +Zienkiewicz, 1970; Bathe and Dvorkin, 1989; Bathe, 1996). 각각의 경우 두께 변화에 따른 수렴곡선에서 잠김현상이 어 떻게 나타나는지를 살펴보자. + +그림 5는 두 가지 4절점 쉘 유한요소들에 대하여 쉘 두 께의 변화(t/L=1/10, 1/100, 1/1000, 1/10000)에 따른 수렴 곡선들(convergence curves)을 보여준다. 여기서 h는 사용된 유한요소의 크기를 나타내며 상대오차를 계산하기 위하여 snorm을 사용하였다. h와 상대오차에 log를 취함으로써 식 (44)의 이론적인 수렴률과 구해진 수렴곡선의 수렴률을 비교 할 수 있다. 그림 5의 각각의 그래프에 이론적인 수렴곡선 의 기울기(2k)가 두께가 굵은 직선으로 그려져 있다. + +그림 5(a)는 잠김현상이 일어날 때의 전형적인 수렴곡선들 이다. 쉘의 두께가 얇아짐에 따라 상대오차(relative error)가 급증하는 것을 알 수 있다. 반면에 그림 5(b)의 수렴곡선들 에서는 상대오차가 쉘 두께(t)에 영향을 받지 않고 오직 요 소의 크기 h에만 관계함을 알 수 있다. 또한 그림 5(b)에서 는 수렴곡선들이 이론적인 기울기와 같다. 그림 5(b)와 같은 형태의 쉘 유한요소해의 수렴형태를 균일최적수렴(uniform optimal convergence)이라고 한다. 때 - MITC6 쉘 요소 + +두 번째 예제로 그림 4(b)는 한 변이 구속된 쌍곡포물면 (hyperbolic paraboloid) 쉘 구조문제이다. 쉘의 중심면 (midsurface)은 다음과 같이 정의된다. + +$$\begin{pmatrix} X \\ Y \\ Z \end{pmatrix} = L \begin{pmatrix} \xi^1 \\ \xi^2 \\ (\xi^1)^2 - (\xi^2)^2 \end{pmatrix}; (\xi^1, \xi^2) \in \left[ -\frac{1}{2}, \frac{1}{2} \right]^2$$ + (45) + +경계조건은 X=−0.5인 변을 따라 완전히 구속되며 자중(selfweight)이 −Z방향으로 작용한다. 구조물의 형상과 변위 및 경계조건이 Y=0인 면을 따라 대칭이므로 그림 4(b)의 색칠 된 부분만 해석되었다. 사용된 탄성계수는 2.0×1011, 포아손 비(Poisson's ratio)는 0.3, 길이 L=1.0이며 자중은 단위면적 당 80이다. + +주어진 휨지배문제를 풀기 위해 변위법(displacement + +based formulation)에 의한 6절점 삼각형 쉘 유한요소 (QUAD6)와 혼합법(mixed formulation)에 의한 6절점 삼각 형 쉘 유한요소(MITC6)를 사용하였다(Ahmad, Irons and Zienkiewicz, 1970; Bathe, 1996; Lee and Bathe, 2004). + +그림 6에서 t/L이 1/100, 1/1000, 1/10000인 세가지 경우 에 대하여 6절점 삼각형 쉘 유한요소들의 수렴곡선들 (convergence curves)을 보여준다. 각각의 그래프에 이론적인 수렴곡선의 기울기(2k)가 두께가 굵은 직선으로 그려져 있다. 그림 6(a)는 QUAD6 유한요소가 쉘의 두께가 얇아짐에 따 라 계산된 해의 오차가 늘어나는 경향, 즉, 잠김현상을 유발 한다는 것을 보여준다. 그림 6(b)에서는 MITC6 유한요소를 사용했을 때 잠김현상이 완전히 제거되지는 않으나 QUAD6 유한요소에 비하여 상당히 완화되었음을 나타낸다. + +여기서 우리는 두 가지 쉘 구조문제들을 대상으로 쉘 유 한요소해의 수렴곡선과 잠김현상에 대하여 알아보았다. 중요 한 점은 어떤 쉘 유한요소가 몇 가지 쉘 구조문제들에 대하 여 잠김현상을 일으키지 않는다고 해서 다른 쉘 구조문제들 에 대하여도 잠김현상을 유발시키지 않는다고 말할 수 없다 는 것이다. 즉, 몇 가지 쉘 구조문제들에 대하여 균일최적수 렴을 보이며 잠김현상을 일으키지 않는 유한요소도 다른 쉘 구조문제들에 대해서 잠김현상을 보일 수 있으며 모든 쉘 구조문제들에 대하여 잠김현상을 일으키지 않는 쉘 유한요 소를 개발하는 것은 극히 어렵다. 보다 다양한 쉘 구조문제 들에 대한 유한요소해의 수렴곡선에 대한 예는 참고문헌에 나와있는 Lee와 Bathe의 논문들에서 찾을 수 있다. 6. 두께의 변화에 따른 6절점 쉘 유한요소들의 수렴곡선: (a) 잠김현상이 일어날 때-QUAD6 쉘 요소 (b) 잠김현상을 완화시켰을 + +쉘 유한요소의 잠김현상을 알아보기 위해서는 다양한 형상 을 가진 휨 및 혼합지배 쉘 구조문제에 대하여 그림 5와 6 에 보여진 것과 같이 쉘의 두께를 변화시키며 수렴을 시험 하여야 한다. 또한 특정 쉘 요소가 휨지배문제에 좋은 거동 을 보인다고 해서 막지배문제에 대해서도 좋은 거동을 보이 는 것은 아니다. 결론적으로 다양한 형상의 쉘 구조문제들을 고려하여 휨과 막지배라는 두 가지 양단의 거동에서 이상적 인 수렴을 보이는 쉘 유한요소가 가장 바람직하고 이런 쉘 유한요소는 혼합지배 쉘 구조문제에 대하여도 좋은 거동을 보일 것이다. 즉, 이상적인 쉘 유한요소는 여러 가지 점근적 인 거동을 보이는 다양한 형상의 쉘 구조문제들에 대하여 균일최적수렴을 보여 주어야 한다. + +# 5. 쉘 유한요소의 성능평가에 대하여 + +나날이 셀 수 없을 만큼 많은 쉘 유한요소들이 개발되고 있지만 쉘 유한요소의 성능평가는 아직도 고전적인 방법을 통하여 이루어지고 있다. 본 장에서는 이상적인 쉘 유한요소 의 조건과 잠김현상을 완화시키는 방법들을 정리하고 쉘 유 한요소의 성능평가를 위한 방법론을 제시한다. + +5.1 이상적인 쉘 유한요소 유한요소 구조해석 문제는 다음과 같은 변분식(variational form)으로 나타내어질 수 있다. + +Find such that + +$$A_h(\overrightarrow{U}_h, \overrightarrow{V}_h) = \overrightarrow{F}(\overrightarrow{V}_h), \ \forall \overrightarrow{V}_h \in \overrightarrow{\Psi}_h, \tag{46}$$ + +선형식(bilinear form)이고 는 유한요소 변위장의 Sobolev 공간(space)이다. 물론 이 성립한다. Ψh Ψh⊂Ψ + +$$A_{h}(\overrightarrow{U}_{h}, \overrightarrow{V}_{h}) = \overrightarrow{V}_{h}^{T} \left( \int_{\Omega_{h}} \mathbf{B}_{h}^{T} \mathbf{C}_{h} \mathbf{B}_{h} d\Omega_{h} \right) \overrightarrow{U}_{h}$$ + +$$(47)$$ + +여기서 는 유한요소해(finite element solution), 는 유한요소 시험함수, 는 유한요소 변위장의 공간, 는 외력을 나타내는 선형식(linear form)이다. 쉘 유한요소해 석일 경우 식 (46)은 식 (26)의 형태로 표현될 수 있다. Ψh + +일반적인 쉘 구조물의 효과적인 유한요소해석에 쓰일 수 있는 이상적인 쉘 유한요소의 개발은 매우 어려운 일이다. 이상적인 쉘 유한요소의 조건을 다음과 같이 정리할 수 있다. + +여기서 Ah(·,·)는 유한요소법으로 이산화(discretization)된 겹 ● 첫째, 쉘 유한요소는 거짓영에너지모드(spurious zero energy mode)를 갖지 말아야 한다. 변위경계조건이 주어지 지 않은 임의의 형상을 갖는 개개의 쉘 유한요소는 물리적 인 강체운동에 대응하는 6개의 영에너지모드(zero energy mode)만을 가져야 한다. 이 조건을 ellipticity(타원율)라 하 며 다음과 같이 정의된다. Uh∈Ψh Ah( ) Uh, Vh = F V( )h , ∀Vh∈Ψh Ah( ) Uh, Vh = Vh dΩh ⎛ ⎞Uh + +$$\exists \alpha > 0 \text{ such that } \forall \overrightarrow{U}_h \in \overrightarrow{\Psi}_h, \ A_h(\overrightarrow{U}_h, \overrightarrow{U}_h) \ge \alpha |\overrightarrow{U}_h|_1^2$$ + (48) + +여기서 α는 상수이며 는 1차 Sobolev norm4 이다. ⋅ 1 + +이 조건은 쉘 유한요소 강성행렬(stiffness matrix)의 고유 치들(eigenvalues) 중 0인 개수와 그에 대응하는 고유벡터들 (eigenvectors)을 살펴봄으로써 시험될 수 있다. 탄성체는 강 체운동이 아닌 변위에 대하여 변형에너지를 저장한다. 식 (48)의 조건을 만족시키지 못하는 유한요소는 강체운동이 아 닌 변위에 대하여 변형에너지를 저장할 수 없으며 이는 물 리적으로 적합하지 못하다. Uh Vh F( ) ⋅ Uh∈Ψh Ah( ) Uh, Uh α Uh 1 + +● 둘째, 쉘 유한요소는 쉘 수학모델에 근거하였기 때문에 쉘 + +유한요소해석의 해는 주어진 쉘 구조문제에 대하여 사용된 증가함에 따라 쉘 수학모델의 정확해에 수렴해야 한다. 이 조건을 consistency(무모순성 또는 정합성)라 부르며 다음과 같이 정의된다. + +$$\lim_{h \to 0} \overrightarrow{U}_h = \overrightarrow{U} \quad \text{or} \quad \lim_{h \to 0} A_h(\overrightarrow{U}_h, \overrightarrow{U}_h) = A(\overrightarrow{U}, \overrightarrow{U})$$ + (49) + +bilinear form)이며 는 정확해(exact solution)이다. + +이 조건이 만족되지 않을 경우 쉘 유한요소의 해는 이론 해에 수렴하지 못 하므로 신뢰할 만한 결과를 줄 수 없다. + +요소의 크기(h)가 줄어듦에 따라 또는 사용된 요소의 수가 여기서 Ah(·,·)는 쉘 수학모델의 정확한 겹선형식(exact ● 셋째, 쉘 유한요소는 모든 종류의 휨 및 혼합지배 쉘 구조 문제에 대하여 균일최적수렴(uniform optimal convergence) 을 보여야 한다. 이 조건을 만족시키는 쉘 유한요소는 비 로소 쉘의 두께와 상관 없이 전단잠김과 막잠김으로부터 자유로운 유한요소가 된다. 이는 본 논문의 4장에서 설명 된 방법에 의해 시험될 수 있다. 혼합법에 의해 정식화 (mixed formulation)된 쉘 유한요소에 대하여 이 조건을 "inf-sup condition"이라 부른다(Bathe, 1996; Bathe, Iosilevich and Chapelle, 2000b). Uh = U h→0 Ah( ) Uh, Uh = A U( ) , U U + +위에서 언급된 세가지 조건을 모두 만족하는 이상적이 쉘 유한요소를 개발하는 것은 극히 어렵다. 실용적인 쉘 유한요 소해석에서는 보다 완화된 다음의 조건들을 만족시키는 쉘 요소의 사용이 권장된다(Lee and Bathe, 2004). + +- − 거짓영에너지모드(spurious zero energy mode) 없음 (ellipticity조건 만족) +- − Consistency조건 만족 +- − 판 문제의 해석에 있어서 전단잠김 없음 +- − 막지배거동 쉘 구조물에 대하여 균일최적수렴 +- 범위(1/10~1/10000)에서 신뢰할 수 있는 결과 +- − 비선형 해석에 있어서 효율적인 정식화 + +5.2 잠김현상의 제어 지난 수십 년 동안 쉘 유한요소의 잠김현상을 제어하기 위해 많은 방법들이 고안되어왔다. 그 방법들은 크게 세가지 로 나누어질 수 있다. + +- − 식 (47)의 변형률-변위 관계 연산자 Bh를 변형하여 전 단 및 막 변형률장의 차수를 줄여주는 방법, (예) reduced integration, ANS method, MITC method +- − 식 (47)의 Bh에 새로운 항을 추가하여 전단 및 막 변 형률장의 공간(space)을 늘려주는 방법, (예) EAS method +- − 식 (46)에서 유한요소 변위장 의 공간( )을 늘려주 는 방법, (예) non-conforming method + +− 휨 및 혼합지배거동 쉘 구조물에 대하여 실용적인 t/L의 쉘 유한요소의 잠김현상을 완화시키기 위한 대부분의 방법 들은 위의 세 분류들(categories)에 속하며 세 방법들을 복합 적으로 이용한 예들도 있다. Uh Ψh + +가장 쉬운 방법은 감차적분(reduced integration)을 사용하 + +$$\|\vec{v}\|_{1}^{2} = \int_{\Omega} \sum_{i=1}^{3} (V_{i})^{2} d\Omega + \int_{\Omega} \sum_{i,j=1}^{3} \left(\frac{\partial V_{i}}{\partial x_{j}}\right)^{2} d\Omega$$ + +1차 Sobolev norm의 제곱은 다음과 같이 정의된다(Bathe, 1996). + +는 방법이다(Bathe, 1996). 그러나 이 방법은 거짓영에너지 모드(spurious zero energy mode)를 유발시키는 치명적인 단점을 지니고 있다. 이점을 극복하기 위하여 각종 안정화 (stabilization)기법이 사용된다. + +비적합모드(non-conforming or incompatible mode)를 추 가함으로써 요소의 휨 모드를 복원하여 쉘 유한요소의 잠김 현상을 완화할 수 있다. 이 방법은 요소간 변위의 적합성 (inter-elemental compatibility)을 만족시키지 못하는 단점과 최종 강성행렬을 구하기 위해 정적응축(static condensation) 을 사용하기 때문에 비선형 해석으로 확장 시 식이 복잡해 지는 단점을 가지고 있다(Choi, Lee and Park, 1999; 최창 근, 2002). + +변위와 변형률을 각각 따로 근사하는 혼합법(mixed formulation)에 근거한 방법들은 쉘 유한요소의 잠김현상을 완화시키는 알려진 방법들 중 가장 우수하다고 평가된다. 그 중 MITC(Mixed Interpolation of Tensorial Components) 방법은 이론적으로 잘 확립되어 있으며 다양한 수치실험으 로 잠김현상의 제어에 매우 효과적임이 입증되었다(Bathe and Dvorkin, 1989; Bathe, 1996; Bathe, Iosilevich and Chapelle, 2000a; Hiller and Bathe, 2003). 최근의 연구들 은 MITC방법에 의해 만들어진 사각형 쉘 유한요소들이 이 상적인 쉘 유한요소에 상당히 접근해 있음을 보여준다(Hiller and Bathe, 2003; Bathe, Lee and Hiller, 2003). + +MITC 방법에서는 변위법에 근거한 쉘 유한요소의 특정한 위치들(tying points)에서 공변변형률들(covariant strains)을 이용하여 원래 공변(covariant)변형률의 근사차수보다 낮은 차수로 변형률장을 근사(interpolation)한다. 일반적으로 근사 함수는 낮은 차수일수록 잠김현상 제어에 더 효과적이지만 너무 낮은 근사차수는 막지배거동을 하는 쉘 구조문제를 풀 때 유한요소해가 이론해에 수렴하지 못하는(즉, consistency 조건을 만족시키지 못하는) 현상을 유발시킨다. 심한 경우에 는 거짓영에너지모드를 발생시켜 ellipticity조건까지 만족시 킬 수 없게 만든다. 그러므로 MITC방법의 핵심은 휨 및 혼합지배거동 시 잠김현상을 줄여주면서 막지배거동 시 수 렴성을 유지하는(즉, consistency를 만족시키는) 균형 잡힌 변형률의 근사장을 찾아내는 것이다. + +대체변형률장(assumed strain field)을 사용하는 ANS (Assumed Natural Strain)방법은 MITC방법과 유사하며, EAS(Extended Assumed Strain)방법은 ANS 또는 MITC방 법에 추가적인 변형률장을 도입하여 변형률장이 표현 가능 한 형태들(patterns)의 수를 늘려주는 방법이다. EAS방법은 비적합모드를 사용하는 방법과 비슷하게 추가된 변형률장의 자유도를 제거하기 위해 정적응축을 필요로 한다는 단점을 가지고 있으며 기존의 ANS나 MITC방법에 비하여 쉘 유한 요소의 수렴성을 개선시킬 수 있지만 그 효과가 크지는 않 다고 알려져 있다. + +다른 여러 가지 방법을 사용하여 휨 및 혼합지배거동 쉘 구조문제에 대하여 보다 유연(flexible)한 거동을 하는 쉘 유 한요소를 개발하는 것은 어렵지 않디. 그러나 그와 동시에 consistency와 ellipticity조건들을 모두 만족시키는 것은 쉽지 않다. 앞으로 여러 종류의 개발된 쉘 유한요소에 대하여 쉘 이론에 바탕을 둔 심도 있는 성능시험(benchmark test)과 연구가 필요하다. + +5.3 쉘 유한요소의 성능평가 일반적으로 유한요소법을 사용하여 쉘 구조물을 해석하는 대부분의 기술자들(engineers)은 구하여진 해의 오차(error)에 대한 평가 없이 해석결과를 받아들이기 때문에 쉘 유한요소 를 개발하는 연구자들은 개발된 쉘 유한요소를 실제 해석에 사용하기에 앞서 성능을 평가하고 그 결과를 보고하여야 한 다. 개발된 쉘 유한요소의 오차특성이 명확하게 알려진다면 사용자들은 주어진 쉘 유한요소를 어떻게 올바르게 사용할 수 있을지를 판단할 수 있게 된다. 쉘 유한요소의 성능평가 는 다음에 열거된 사항들을 고려하여 이루어져야 한다. + +# ● 기본시험 + +쉘 유한요소는 표 2에 정리되어있는 기본시험들(basic tests)을 통과하여야 한다. 기본시험들을 통과하지 못하는 쉘 유한요소의 사용은 바람직하지 못하다. + +# ● 성능평가방법 + +쉘 유한요소의 성능을 평가하기 위해서는 다양한 쉘 구조 문제들이 사용되어야 한다. 쉘 유한요소들의 성능을 비교평 가하기 위하여 오래 전부터 많은 쉘 해석문제들이 제안되어 왔다. 현재까지 가장 널리 쓰이는 성능평가방법은 1985년에 MacNeal와 Harder에 의해 정리된 것으로 두께가 정해진 몇 가지 쉘 구조문제들을 해석하여 정해진 위치에서 구해진 변 위 및 응력/변형률 등의 결과치들을 유한요소망을 조밀화 하 면서 비교하는 것이다. 이미 언급한 바와 같이 몇몇 점들에 서의 해의 수렴을 측정하는 것은 전체 유한요소해의 거동을 올바르게 반영할 수 없다. 변위형상이나 응력/변형률의 분포 는 유한요소해의 전체적인 수렴정도를 보여줄 수 있으며 이 것들을 비교하는 것은 매우 좋은 보완방법이다. 그러나 이 방법으로 과연 유한요소해가 어느 정도 수렴했는지를 측정 하여 그 정도를 한 개의 값으로 보여주기는 대단히 어렵다. 그러므로, 응력과 변형률의 오차분포로부터 구해진 s-norm은 전체 유한요소해를 반영하는 좋은 오차측정의 규준이 된다. 또한 4.3절에서 설명한 바와 같이 두께의 변화를 고려한 성 + +| 표 2. 쉘 유한요소의 기본시험 | | | | +|----------------------------------------------------------------------|--------------------------|------------------------------------|--| +| 시험 | 대상 | 참고문헌 | | +| 영에너지 시험 (Zero
energy mode test) | 사각형 쉘 유한요소
삼각형 쉘 유한요소 | Bathe, 1996 | | +| 조각 시험 (Patch tests)
− Membrane patch test
− Bending patch test | 사각형 쉘 유한요소
삼각형 쉘 유한요소 | Bathe, 1996
Lee and Bathe, 2004 | | +| 요소 등방성 시험 (Element isotropy test) | 삼각형 쉘 유한요소 | Lee and Bathe, 2004 | | + +![](_page_11_Picture_0.jpeg) + +그림 7. Gaussian곡률에 따른 곡면의 종류: (a) Positive Gaussian curvature, (b) Zero Gaussian curvature, (c) Negative Gaussian curvature + +표 3. 쉘 유한요소의 성능평가를 위한 쉘 구조문제의 예들(Bathe, Iosilevich and Chapelle, 2000; Lee and Bathe, 2002; Bathe, Chapelle and Lee, 2003; Bathe, Lee and Hiller, 2003; Chapelle and Bathe, 2003; Hiller and Bathe, 2003; Lee and Bathe, 2004; Lee and Bathe, 2005; Lee, Noh and Bathe, 2007) + +| 쉘 구조문제 (shell
problems) | Gaussian 곡률 | 점근거동 (ρ) | +|----------------------------------------------------|-------------|------------------| +| Fully clamped plate problem | Zero | 휨지배 (ρ = 3.0) | +| Scodelis-Lo roof shell problem | Zero | 혼합지배 (ρ = 1.75) | +| Modified Scodelis-Lo roof shell problem | Zero | 막지배 (ρ = 1.0) | +| Free cylindrical shell problem | Zero | 휨지배 (ρ = 3.0) | +| Fixed cylindrical shell problem | Zero | 막지배 (ρ = 1.0) | +| Clamped hemispherical cap problem | Positive | 막지배 (ρ = 1.0) | +| Monster shell problem | Positive | Not well-defined | +| Partly clamped hyperbolic paraboloid shell problem | Negative | 휨지배 (ρ = 3.0) | +| Free hyperboloid shell problem | Negative | 휨지배 (ρ = 3.0) | +| Fixed hyperboloid shell problem | Negative | 막지배 (ρ = 1.0) | + +능평가방법을 사용하는 것이 바람직하다. ● 층(layer) 쉘의 응력/변형률/변위장들이 급격히 변하는 층(layer)의 폭, 즉, 특성길이(characteristic length)는 쉘의 두께에 따라 변화 한다. 일반적으로 특성길이는 쉘의 두께가 얇아짐에 따라 식 (33)에 의하여 급격히 줄어든다. 층에서의 에너지 집중현상 때문에 이런 층이 발생하는 쉘 구조문제를 풀 때는 균일한 유한요소망을 사용하여 균일최적수렴을 얻기 힘들다. 각각 층의 특성길이를 반영한 유한요소망을 사용하여야 한다. 즉, 층이 발생하는 영역에서 보다 조밀한 유한요소망의 사용이 요구된다(Bathe, Iosilevich and Chapelle, 2000a). 이러한 유한요소망을 "graded mesh"라 부른다. ● 중심면의 곡률 쉘 구조물의 중심면은 곡률(curvature)을 가지고 있다. 곡 면은 Gaussian곡률의 부호에 따라 세가지로 나누어질 수 있 다. 특히 Gaussian곡률이 음인 곡면을 가지는 쉘 구조물은 유한요소해석에 있어서 상당한 어려움이 뒤따른다(Lee and Bathe 2004). 쉘 유한요소의 성능을 평가하기 위한 해석시 험문제들(benchmark test set)은 다양한 곡률을 고려하여 구 성되어야 한다. 이는 어떤 쉘 유한요소가 특정한 곡률을 가 지는 쉘 구조문제에서 좋은 수렴성을 보인다고 하여 다른 곡률을 가지는 구조문제에 대하여도 좋은 수렴을 보이는 것 은 아니기 때문이다. 그림 7은 Gaussian곡률에 따른 곡면의 예들을 보여주고 있다. + +# ● 점근거동의 종류 + +3장에서 우리는 쉘 구조물의 두께가 얇아짐에 따라 나타나 + +는 3가지 점근거동(휨지배, 막지배, 혼합지배거동)을 살펴 보 았다. 각각의 점근거동을 모두 시험할 수 있도록 해석시험문 제들을 구성해 주어야 한다. 특히, 휨지배 및 혼합지배거동 쉘 구조문제들에서는 잠김현상이 일어나는지를 시험하여야 하며 막지배거동 쉘 구조문제들에서는 5.1절에서 언급한 consistency조건이 만족되는지를 살펴보아야 한다. 표 3은 쉘 유한요소의 성능평가를 위한 쉘 구조문제의 예들을 보여주 고 있다. ● 요소망(mesh)의 형태 유한요소해석의 해는 유한요소망을 어떻게 구성하는지에 따라 그 수렴특성이 변화하게 된다. 요소형상의 찌그러짐에 민감하지 않고 좋은 수렴성을 유지하는 쉘 유한요소를 개발 하는 것은 쉽지 않은 일이다. 따라서 해석시험문제들은 다양 + +# 6. 결 론 + +한 유한요소망에 따른 쉘 유한요소의 수렴특성을 반영해 주 어야 한다. 특히, 비등방성(non-isotropic) 삼각형 쉘 유한요 소의 시험에서는 유한요소망의 형태뿐만 아니라 요소의 방 향에 따라 수렴특성이 변하므로 요소의 방향성 또한 고려되 어야 한다(Lee, Noh and Bathe, 2007). 머리말에서 언급된 바와 같이 쉘 구조물의 유한요소해석을 명확하게 이해하기 위해서는 쉘 구조물의 물리적 거동, 수학 모델 및 쉘 유한요소해석에 대한 이해가 동시에 체계적이고 심도 있게 이루어져야 한다. 본 논문에서는 이 세가지 부분 에 대한 이해와 이들이 서로 어떻게 유기적으로 관계를 맺 고 있는지를 최근 주요 연구들을 토대로 정리하여 고찰하였 고 이상적인 쉘 유한요소의 성질과 쉘 유한요소의 성능평가 방법을 제시하였다. + +본 논문에서는 대표적인 쉘 수학모델과 휨지배거동, 막지 배거동, 혼합지배거동 등으로 나누어지는 쉘 구조물의 점근 거동에 대한 기본적인 이론과 그 점근거동을 수치적으로 알 아내는 방법을 알아보았다. 또한 휨지배 및 혼합지배거동에 서 나타나는 쉘 유한요소의 잠김현상을 두께의 변화에 따른 수렴곡선을 통하여 고찰하였다. 마지막으로 이상적인 쉘 유 한요소의 조건과 잠김현상의 제어하는 방법을 알아보았고 쉘 유한요소의 성능평가방법을 제안하였다. + +쉘 구조물의 수학모델과 점근거동은 쉘의 물리적 거동을 이해하는데 핵심사항으로 쉘 구조물을 설계하는 기술자나 쉘 유한요소해석을 연구하는 연구자들이 명확하게 알아야 할 매 우 중요한 부분이다. 유한요소법을 이용하여 쉘 구조물을 해 석하기에 앞서 쉘 구조물의 점근거동과 그와 관련된 쉘 유 한요소의 감김현상에 대한 이해는 필수적이다. 통합적인 이 해의 바탕이 있을 때 신뢰할 만한 쉘 유한요소의 개발이 이 루어질 수 있으며 쉘 구조물의 유한요소해석을 통하여 얻어 진 결과를 정확하게 이해할 수 있다. + +# 감사의 글 + +글을 맺으며 본 논문에 소개된 기본개념들을 정립하고 정 리하는데 많은 가르침을 주신 MIT(Massachusetts Institute of Technology)의 Klaus-Jürgen Bathe 교수님과 KAIST(한 국과학기술원)의 최창근 교수님께 깊은 감사드립니다. + +# 참고문헌 + +- 최창근 (2002) 유한요소법. 테크노프레스. +- Ahmad, S., Irons, B.M., and Zienkiewicz, O.C. (1970) Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods and Engineering, Vol. 2, pp. 419-451. +- Bathe, KJ. (1996) Finite Element Procedures. Prentice Hall: New Jersey. +- Bathe, K.J., Chapelle, D., and Lee, P.S. (2003) A shell problem 'highly sensitive' to thickness changes. International Journal for Numerical Methods and Engineering, Vol. 57, pp. 1039-1052. +- Bathe, K.J. and Dvorkin, E.N. (1989) A formulation of general + +- shell elements the use of mixed interpolation of tensorial components. International Journal for Numerical Methods and Engineering, Vol. 22, pp. 697-722. +- Bathe, K.J., Iosilevich, A., and Chapelle, D. (2000a) An evaluation of the MITC shell elements. Computers & Structures, Vol. 75, pp. 1-30. +- Bathe, K.J., Iosilevich, A., and Chapelle, D. (2000b) An inf-sup test for shell finite elements. Computers & Structures, Vol. 75, pp. 439-456. +- Bathe, K.J., Lee, P.S., and Hiller, J.F. (2003) Towards improving the MITC9 shell element. Computers & Structures, Vol. 81, pp. 477-489. +- Chapelle, D. and Bathe, K.J. (1998) Fundamental considerations for the finite element analysis of shell structures. Computers & Structures, Vol. 66, pp. 19-36, pp. 711-712. +- Chapelle, D. and Bathe, K.J. (2003) The finite element analysis of shells? fundamentals. Berlin:Springer-Verlag. +- Choi, C.K., Lee, P.S., and Park, Y.M. (1999) Defect-free 4-node flat shell element: NMS-4F element. Structural Engineering and Mechanics, Vol. 8, pp. 207-231. +- Hiller, J.F. and Bathe, K.J. (2003) Measuring convergence of mixed finite element discretizations: an application to shell structures. Computers & Structures, Vol. 81, pp. 639-654. +- Lee, P.S. and Bathe, K.J. (2002) On the asymptotic behavior of shell structures and the evaluation in finite element solutions. Computers & Structures, Vol. 80, pp. 235-255. +- Lee, P.S. and Bathe, K.J. (2004) Development of MITC isotropic triangular shell finite elements. Computers & Structures, Vol. 82, pp. 945-962. +- Lee, P.S. and Bathe, K.J. (2005) Insight into finite element shell discretizations by use of the basic shell mathematical model. Computers & Structures, Vol. 83, pp. 69-90. +- Lee, P.S. Noh, H.C., and Bathe, K.J. (2007) Insight into 3-node triangular shell finite elements: the effects of element isotropy and mesh patterns. Computers & Structures, Vol. 85, pp. 404- 418. +- Lovadina, C. (2001) Energy estimates for linear elastic shells. Computational Fluid and Solid Mechanics (Bathe KJ ed.), pp. 330- 331, Elsevier Science. +- MacNeal, R.H. and Harder, R.L. (1985) A proposed standard set of problems to test finite element accuracy. Finite Element in Analysis and Design, Vol. 1, pp. 3-20. +- Noh, H.C. (2006) Nonlinear behavior and ultimate load bearing capacity of reinforced concrete natural draught cooling tower shell. Engineering Structures, Vol. 28, pp. 399-410. + +(접수일: 2006.7.18/심사일: 2007.1.16/심사완료일: 2007.3.27) \ No newline at end of file diff --git a/docs/Paper/2007쉘구조물의유한요소해석에대하여/2007쉘구조물의유한요소해석에대하여_meta.json b/docs/Paper/2007쉘구조물의유한요소해석에대하여/2007쉘구조물의유한요소해석에대하여_meta.json new file mode 100644 index 0000000..fe1b089 --- /dev/null +++ b/docs/Paper/2007쉘구조물의유한요소해석에대하여/2007쉘구조물의유한요소해석에대하여_meta.json @@ -0,0 +1,1249 @@ +{ + "table_of_contents": [ + { + "title": "\uc258 \uad6c\uc870\ubb3c\uc758 \uc720\ud55c\uc694\uc18c\ud574\uc11d\uc5d0 \ub300\ud558\uc5ec", + "heading_level": null, + "page_id": 0, + "polygon": [ + [ + 179.5458984375, + 96.0821533203125 + ], + [ + 428.49798583984375, + 96.0821533203125 + ], + [ + 428.49798583984375, + 126.5477294921875 + ], + [ + 179.5458984375, + 126.5477294921875 + ] + ] + }, + { + "title": "", + "heading_level": null, + "page_id": 0, + "polygon": [ + [ + 283.26416015625, + 222.4228515625 + ], + [ + 323.35693359375, + 222.4228515625 + ], + [ + 323.35693359375, + 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zKdVo;Tko*O+%i*Tb+pev-R+bEQh6s2Av!lVuR8{qiA%dNyhK(Gx zTa_+1)*R|d>Dg#JCH4w_Xw&SU*dGJ8Sgz=@kuA<8{&{j{^O5Gm!4I46;g}LbEh5|h z4`(Wl4J83?cG%&yQE4DGo1L8r^CwqSU%$d!kb#oT>&doU_o&#;4kObyq!3soq$bOx z#*M+#Gdz5pA6MAJDP4}UOWNaZ&+EK3AP9KDO1GioLbkrXD~Ww!7ER?BxXAQ1_&tws zwCmRgc!fQ~LmD1yp?4k6I~O7!&y=_jTs6UKHo~FPE^L04OiwK;Dk$A@lECU7CY}9C z5)E~VNQsk`D3jqLwR~_s{RxPobB9EMgQT%gQ~mosaEc&yJv)zBj~{iCE5(_>2j5hk zTz)Ag)x>3;Stg#v-d&v+&qrThlWMqo9PdgOU`yR%@sp&nyZF$YLET?`rtPf*@H@8E zH{deTqDa4-LzL-XJJH@Ey--$8zm%J$OlATy5&Qzf3m z_HpaaFin$OcTrSafR4z)U$xIUy~ur`2}p>+JKNH)UyMNR@j*SO0*z-)AAE(L7Sc8q zvZMg~7}6%tM1zS#GY}an{_r(vN$GA@dfKJ{Y^~hqbok|E-z+v*!vbZl*mLj@uyUxj&96f}Z=E3 to formulate a general shell element for geometric and material non-linear analysis. This element has been employed very successfully when used with 9 or, in particular, 16 nodes. However, the 16-node element is quite expensive, and although it is possible to use in some analyses only a few elements to represent the total structure (see later examples) in other analyses still a fairly large number of elements need by employed5 . + +Considering general shell analyses, much emphasis has been placed onto the development of a versatile, reliable and cost-effective 4-node shell element16' 17,22,28 . Such element would complement the above high-order 16-node element and may be more effective in certain analyses. The difficulties in the development of such element lie in that the element should be applicable in a reliable manner to + +thin and thick shells of arbitrary geometries for general non-linear analysis. + +The objective in this paper is to present a simple 4-node general shell element with the following properties: the element is formulated using three-dimensional stress and strain conditions without use of a shell theory; the element is applicable to thin and thick shells and can be employed to model arbitrary geometries; the element is applicable to the conditions of large displacements and rotations but small strains, and can be used effectively in materially non-linear analysis. + +The formulation of the element is quite simple and transparent, and the element has good predictive capability without containing spurious zero energy modes. + +In the next section of the paper we discuss some basic considerations with respect to the assumptions used, and in the following section we present the element formulation for non-linear analysis. The results obtained in numerical solutions that demonstrate the properties of the element are given in the final section. + +## BASIC CONSIDERATIONS + +The formulation of the 4-node shell element represents an extension of the shell element discussed previously2,3 , and we therefore use the same notation as in those references. Also, to focus attention onto some key issues of the formulation, we consider in this section only linear analysis conditions. + +The geometry of the element (see *Figure 1)* is described using2 : + +$${}^{l}x_{i} = \sum_{k=1}^{4} h_{k}{}^{l}x_{i}^{k} + \frac{r_{3}}{2} \sum_{k=1}^{4} a_{k}h_{k}{}^{l}V_{ni}^{k}$$ + (1) + +![](_page_0_Figure_18.jpeg) + +![](_page_1_Figure_1.jpeg) + +where the *hk(rl ,r2)* are the two-dimensional interpolation functions corresponding to node *k;* the *ri* are the natural coordinates; and 1 *xi =* Cartesian coordinates of any point in the element; = Cartesian coordinates of nodal point :=components of director vector at node *k* (which is not necessarily normal to the midsurface of the element); and *ak* is the shell thickness at node *k,* measured along the vector The left superscript is zero for the initial geometry of the element and is equal to 1 for the deformed element geometry. Note that the thickness of the element varies and the element is in general non-flat. + +The displacements of any particle with natural coordinates *ri* of the shell element in the stationary Cartesian coordinate system are: + +$$u_{i} = \sum_{k=1}^{4} h_{k} u_{i}^{k} + \frac{r_{3}}{2} \sum_{k=1}^{4} a_{k} h_{k} (-{}^{0}V_{2i}^{k} \alpha_{k} + {}^{0}V_{1i}^{k} \beta_{k})$$ + (2) + +where the are the nodal point displacements into the Cartesian coordinate directions, and the *ak* and *Bk* are the rotations of the director vector about the and axes (see *Figure 1).* + +A basic problem inherent in the use of the above interpolation of the displacements, and the derivation of the strain-displacement matrices therefrom, is that the element 'locks' when it is thin. This is due to the fact that with these interpolations the transverse shear strains cannot vanish at all points in the element, when it is subjected to a constant bending moment. Hence, although the basic continuum mechanics assumptions contain the Kirchhoff shell assumptions, the finite element discretization is not able to represent these assumptions rendering the element not applicable to the analysis of thin plates or shells2,5,7 . To solve this deficiency, various remedies based on selective and reduced integration have been proposed17,22,23 but there is still much room for more effective and reliable elements for general non-linear analysis. + +Considering our element formulation - because the problem lies in the representation of the transverse shear strains - we proceed to not evaluate these shear strains from the displacements in (2), but to introduce separate interpolations for these strain components. Since we consider non-flat shell elements, the separate interpolations are performed effectively in a convected coordinate system†. + +The choice of the interpolation for the transverse shear strain components is the key assumption in our element formulation, because adequate coupling between the element displacements and rotations must be introduced and the element should not exhibit any spurious zero energy modes. For our element we use (see *Figure 2):* + +$$\tilde{\varepsilon}_{13} = \frac{1}{2}(1+r_2)\tilde{\varepsilon}_{13}^{A} + \frac{1}{2}(1-r_2)\tilde{\varepsilon}_{13}^{C} +\tilde{\varepsilon}_{23} = \frac{1}{2}(1+r_1)\tilde{\varepsilon}_{23}^{D} + \frac{1}{2}(1-r_1)\tilde{\varepsilon}_{23}^{B}$$ +(3) + +Since the kinematic relations for the above shear strains are not satisfied using (3), we impose them using Lagrange multipliers2,27 to obtain, + +$$\Pi^* = \frac{1}{2} \int_{V} \tilde{\tau}^{ij} \tilde{\varepsilon}_{ij} \, dV + \int_{V} \lambda^{13} (\tilde{\varepsilon}_{13} - \tilde{\varepsilon}_{13}^{DI}) \, dV + \int_{V} \lambda^{23} (\tilde{\varepsilon}_{23} - \tilde{\varepsilon}_{23}^{DI}) \, dV - \mathcal{H}^{c} \tag{4}$$ + +where the are the contravariant components of the Cauchy stress tensor13,15 , the are the covariant components of the infinitesimal strain tensor, the are the Lagrange multipliers, the are the transverse shear strains evaluated using the displacement interpolations in (2), and is the potential of the external loads. For the Lagrange multipliers we choose the following interpolations, + +$$\lambda^{13} = \lambda^{A} \delta(r_1) \delta(1 - r_2) + \lambda^{C} \delta(r_1) \delta(1 + r_2)$$ + +$$\lambda^{23} = \lambda^{D} \delta(r_2) \delta(1 - r_1) + \lambda^{B} \delta(r_2) \delta(1 + r_1)$$ +(5) + +where <5(...) is the Dirac-delta function. This represents a weakening of the Lagrange multiplier constraint in (4)10 . Substituting from (5) into (4) and invoking that gives the distinct constrains: + +$$\tilde{\varepsilon}_{13}\Big|_{\text{at A}} = \tilde{\varepsilon}_{13}^{\text{DI}}\Big|_{\text{at A}} \qquad \tilde{\varepsilon}_{13}\Big|_{\text{at C}} = \tilde{\varepsilon}_{13}^{\text{DI}}\Big|_{\text{at C}} +\tilde{\varepsilon}_{23}\Big|_{\text{at D}} = \tilde{\varepsilon}_{23}^{\text{DI}}\Big|_{\text{at D}} \qquad \tilde{\varepsilon}_{23}\Big|_{\text{at B}} = \tilde{\varepsilon}_{23}^{\text{DI}}\Big|_{\text{at B}} \tag{6}$$ + +Hence, the complete element stiffness matrix is calculated using the functional: + +$$\Pi^* = \frac{1}{2} \int \tilde{\tau}^{ij} \tilde{\epsilon}_{ij} \, \mathrm{d}V - \mathcal{H}^{\hat{r}} \tag{7}$$ + +† Note that in refs. 2 and 3, the shell element formulation is discussed in the global stationary coordinate system, because all displacement components are interpolated in the same way. To emphasize that we use here stress and strain measures in the convected coordinate system, we place a tilde (~) over these quantities. + +![](_page_2_Figure_1.jpeg) + +with stress and strain components in convected coordinates and (1) and (2) to evaluate the strain components (3) to evaluate the strain components ; and (6) to express the variables in terms of the nodal point displacements and rotations of (2). + +Considering the representation that we have chosen for the transverse shear strains, we can make the following three important observations: + +- (1) *The element is able to represent the six rigid body modes.* The element contains the rigid body modes because zero strains are calculated in the formulation when the element nodal point displacements and rotations correspond to an element rigid body displacement. This can be verified by using (1) to (6) to evaluate the strains, but more easily we can use the fact that the 4-node shell element of reference 3 satisfies the rigid body mode criterion. Hence, for a rigid body displacement the are zero, from which it follows that also the shear strains in (3) are zero, and the rigid body mode criterion is satisfied. +- (2) *The element can approximate the Kirchhoff-Love hypothesis of negligible shear deformation effects and can be used for thin shells.* Various demonstrative solutions are given in the fourth section. +- (3) *Based on our studies the element does not contain any spurious zero energy modes* (*using a 'full' numerical integration).* We reach this observation by studying the strains along the element sides. If the element were to contain a spurious zero energy mode, the strains along every side should vanish for a displacement pattern (to be identified) other than the displacements corresponding to a true rigid body mode. However, such displacement pattern could not be identified. + +Considering the practical use of the element the interpolation employed for the transverse shear strains shows that is constant with *r*1 and in general discontinuous at (between elements), and similarly is constant with r2 and in general discontinuous at As a consequence, the accuracy with which transverse shear stresses are predicted depends to a significant degree on the mesh used and the geometric distortions of the elements. However, our experience is that the bending stress predictions are relatively little affected by element distortions (see examples). + +To employ (7), we also need to use the appropriate constitutive relations: + +$$\tilde{\tau}^{ij} = \tilde{C}^{ijkl} \tilde{\varepsilon}_{kl} \tag{8}$$ + +where is the fourth-order contravariant constitutive tensor in the convected coordinates r*i* . The constitutive law is known in the local Cartesian system of orthonormal base vectors *i =* 1,2,3, with the condition equal to zero2 , (see *Figure 3).* Denoting this constitutive tensor by the constitutive tensor for (8) is obtained using the transformation: + +$$\tilde{C}^{ijkl} = (\mathbf{g}^i \cdot \hat{\mathbf{e}}_m)(\mathbf{g}^j \cdot \hat{\mathbf{e}}_n)(\mathbf{g}^k \cdot \hat{\mathbf{e}}_0)(\mathbf{g}^l \cdot \hat{\mathbf{e}}_n)\hat{C}^{mnop} \tag{9}$$ + +where the gi are the contravariant base vectors of the convected coordinates *ri .* These vectors are calculated, using the covariant base vectors gi , where: + +$$\mathbf{g}_i = \frac{\partial^0 \mathbf{x}}{\partial r_i} \tag{10}$$ + +$$g_{ij} = \mathbf{g}_i \cdot \mathbf{g}_j \tag{11}$$ + +$$\mathbf{g}^{i} = g^{ij}\mathbf{g}_{j}$$ + +$$g^{ij} = \frac{D^{ij}}{\Pi^{2}}$$ +(12) + +where *Dij* is the cofactor of the term *gij* in the matrix of the metric tensor and |J| is the determinant of the Jacobian matrix at the point considered. + +# TOTAL LAGRANGIAN FORMULATION + +The large displacement formulation of the shell element is based on the derivation given in ref. 2 (Section 6.3.5), and the concepts and interpolations presented in the previous section. + +The geometry of the element at any time *t* is defined as in (1) but using the nodal point coordinates, and director vectors + +$${}^{t}x_{i} = h_{k}{}^{t}x_{i}^{k} + \frac{r_{3}}{2}a_{k}h_{k}{}^{t}V_{ni}^{k}$$ +(13) + +where we imply summation over *k.* The displacements, and incremental displacements, *ui* , of a particle of the element at time *t* are hence given by: + +$${}^{i}u_{i} = h_{k}{}^{i}u_{i}^{k} + \frac{r_{3}}{2}a_{k}h_{k}({}^{i}V_{ni}^{k} - {}^{0}V_{ni}^{k})$$ + +$$u_{i} = h_{k}u_{i}^{k} + \frac{r_{3}}{2}a_{k}h_{k}(-{}^{i}V_{2i}^{k}\alpha_{k} + {}^{i}V_{1i}^{k}\beta_{k})$$ + +$$(14)$$ + +where the are the nodal point displacements at time *t*, the are the incremental nodal point displacements from the configuration at time *t,* and the variables and *βk* are defined as in (2) but referred to the configuration at time *t.* + +This kinematic description implies the following hy- + +† Note that the superscript *t* on a variable denotes the configuration at time *t* in the incremental solution, and does not imply a dynamic analysis. + +potheses: the director vectors remain straight during the deformations; the 'thickness' of the element measured along the director vectors remains constant during the deformations; hence only small strain conditions are considered. + +Using the assumptions in (13) and (14) the geometric and material non-linear response is analysed using an incremental formulation2 , in which the configuration is sought for time (load step) '*t* +∆*t',* when the configuration for time *t* is known. The basis of this incremental formulation is the use of the virtual work principle applied to the configuration at time *t + ∆t.* In essence, two approaches can be employed leading to the updated Lagrangian and the total Lagrangian formulations. These approaches are, from a continuum mechanics point of view, equivalent, and in the following we develop the governing finite element relations for the total Lagrangian formulation. + +The principle of virtual work applied to the configuration at time *t+At* is: + +$$\int_{0V}^{t+\Delta t} \tilde{S}^{ij} \delta^{t+\Delta t} \tilde{e}_{ij} \,^{0} \mathrm{d}V = {}^{t+\Delta t} \mathcal{R}$$ + (15) + +where the are the contra variant components of the second Piola-Kirchhoff stress tensor at time and referred to the configuration at time 0, and the are the covariant components of the Green-Lagrange strain tensor at time and referred to time 0. Both sets of tensor components are measured in the convected coordinate system / = 1,2,3. The external virtual work is given by and includes the work due to the applied surface tractions and body forces. + +For the incremental solution, the stresses and strains are decomposed into the known quantities, and unknown increments, so that + +$${}^{i+\Delta i}_{\alpha}\tilde{S}^{\bar{i}j} = {}^{i}_{\alpha}\tilde{S}^{\bar{i}j} + {}_{\alpha}\tilde{S}^{\bar{i}j} \tag{16}$$ + +$${}^{t+\Delta t}\circ\tilde{\varepsilon}_{i,i} = {}^{t}\circ\tilde{\varepsilon}_{i,i} + {}^{\circ}\circ\tilde{\varepsilon}_{i,i} \tag{17}$$ + +In addition, the strain increment can be written as a linear part, and a non-linear part, hence + +$$_{0}\tilde{\varepsilon}_{ii} = _{0}\tilde{e}_{ii} + _{0}\tilde{\eta}_{ii} \tag{18}$$ + +Substituting from (16) to (18) into (15) and using the linearized expressions we obtain the linearized equation of motion: + +$$\int_{\delta_{V}} {0} \tilde{C}^{ijkl} {0} \tilde{e}_{kl} \delta_{0} \tilde{e}_{ij} {0} dV + \int_{\delta_{V}} {0} \tilde{S}^{ij} \delta_{0} \tilde{\eta}_{ij} {0} dV$$ + +$$= {0 \choose k} {0} \tilde{S}^{ij} \delta_{0} \tilde{e}_{ij} {0} dV$$ +(19) + +This equation is the basic equilibrium relation employed to develop the governing finite element matrices. For the actual solution of problems it is frequently important to use equilibrium iterations, but the finite element matrices and vectors used in these iterations can be derived directly from the matrices obtained using (19*) 2 .* Note that is now obtained using (9) with the condition =0, which implies the more natural condition only in the small strain case. + +The basic problem of the finite element discretization of (19) lies in expressing the strain terms of (19) in terms of the finite element interpolations. Using the definition of the Green-Lagrange strain components: + +$${}_{0}^{t}\tilde{\varepsilon}_{i,i} = \frac{1}{2} ({}^{t}\mathbf{g}_{i} \cdot {}^{t}\mathbf{g}_{i} - {}^{0}\mathbf{g}_{i} \cdot {}^{0}\mathbf{g}_{i}) \tag{20}$$ + +$${}_{0}\tilde{e}_{ii} = h_{k,i}{}^{t}\mathbf{g}_{i} \cdot \mathbf{u}_{k} + \frac{r_{3}}{2}a_{k}h_{k,i}(-\alpha_{k}{}^{t}\mathbf{g}_{i} \cdot {}^{t}\mathbf{V}_{2}^{k} + \beta_{k}{}^{t}\mathbf{g}_{i} \cdot {}^{t}\mathbf{V}_{1}^{k})$$ + (21a) + +$${}_{0}\tilde{\eta}_{ii} = \frac{1}{2}h_{k,i}h_{p,i}\mathbf{u}_{k} \cdot \mathbf{u}_{p} + \frac{r_{3}}{2}h_{k,i}h_{p,i}a_{p}(-\alpha_{p}{}^{t}\mathbf{V}_{2}^{p} \cdot \mathbf{u}_{k} + \beta_{p}{}^{t}\mathbf{V}_{1}^{p} \cdot \mathbf{u}_{k}) +$$ + +$$\frac{(r_{3})^{2}}{8}h_{k,i}h_{p,i}a_{k}a_{p}(-\alpha_{k}{}^{t}\mathbf{V}_{2}^{k}+\beta_{k}{}^{t}\mathbf{V}_{1}^{k})\cdot(-\alpha_{p}{}^{t}\mathbf{V}_{2}^{p}+\beta^{p}{}^{t}\mathbf{V}_{1}^{p})$$ +(i = 1,2) +(21b) + +$${}_{0}e_{12} = \frac{1}{2} [h_{k,2} \mathbf{g}_{1} \cdot \mathbf{u}_{k} + h_{k,1} \mathbf{g}_{2} \cdot \mathbf{u}_{k} + \frac{r_{3}}{2} h_{k,2} J_{k} (-\alpha_{k}^{i} \mathbf{V}_{2}^{k} \cdot {}^{i} \mathbf{g}_{1} + \beta_{k}^{i} \mathbf{V}_{1}^{k} \cdot {}^{i} \mathbf{g}_{1}) +$$ + +$$\frac{r_3}{2}h_{k,1}a_k(-\alpha_k{}^t\mathbf{V}_2^k \cdot {}^t\mathbf{g}_2 + \beta_k{}^t\mathbf{V}_1^k \cdot {}^t\mathbf{g}_2)]$$ + (22a) + +$$\frac{r_3}{2}h_{k,1}h_{p,2}a_p(-\alpha_p{}^t\mathbf{V}_2^p \cdot \mathbf{u}_k + \beta_p{}^t\mathbf{V}_1^p \cdot \mathbf{u}_k) +$$ + +$$\frac{r_3}{2}h_{k,1}h_{p,2}a_k(-\alpha_k{}^t\mathbf{V}_2^k\cdot\mathbf{u}_p+\beta_k{}^t\mathbf{V}_1^k\cdot\mathbf{u}_p)+$$ + +$$\frac{(r_3)^2}{4}h_{k,1}h_{p,2}a_ka_p(-\alpha_k{}^t\mathbf{V}_2^k+\beta_k{}^t\mathbf{V}_1^k) \cdot (-\alpha_p{}^t\mathbf{V}_2^p+\beta_p{}^t\mathbf{V}_1^p)]$$ + +Further, we obtain for the transverse shear strains, using (3) and (6): + +$$\begin{split} &_{0}\tilde{e}_{13} = \frac{1}{8}(1+r_{2})\left[{}^{t}g_{3i}^{A}(u_{i}^{1}-u_{i}^{2}) + \right. \\ &_{2}{}^{t}g_{1i}^{A}(-\alpha_{1}a_{1}{}^{t}V_{2i}^{1} + \beta_{1}a_{1}{}^{t}V_{1i}^{1} - \alpha_{2}a_{2}{}^{t}V_{2i}^{2} + \beta_{2}a_{2}{}^{t}V_{1i}^{2})\right] + \\ &_{3}{}^{t}(1-r_{2})\left[{}^{t}g_{3i}^{C}(u_{i}^{4}-u_{i}^{3}) + \frac{1}{2}{}^{t}g_{1i}^{C}(-\alpha_{4}a_{4}{}^{t}V_{2i}^{4} + \beta_{4}a_{4}{}^{t}V_{1i}^{4} - \alpha_{3}a_{3}{}^{t}V_{2i}^{3} + \beta_{3}a_{3}{}^{t}V_{1i}^{3})\right] \end{split}$$ + +$$\tilde{\eta}_{13} = \frac{1}{32}(1 + r_2) [(-\alpha_1 a_1^{\ t} V_{2i}^1 + \beta_1 a_1^{\ t} V_{1i}^1 - \alpha_2 a_2^{\ t} V_{2i}^2 + \beta_2 a_2^{\ t} V_{1i}^2)(u_i^1 - u_i^2)] + \frac{1}{32}(1 - r_2) [(-\alpha_4 a_4^{\ t} V_{2i}^4 + \beta_4 a_4^{\ t} V_{1i}^4 - \alpha_3 a_3^{\ t} V_{2i}^3 + \beta_3 a_3^{\ t} V_{1i}^3)]$$ +(23b) + +$$\begin{split} & _{0}\tilde{e}_{23} = \frac{1}{8}(1+r_{1})[^{t}g_{3i}^{D}(u_{i}^{1}-u_{i}^{4}) + \\ & \frac{1}{2}^{t}g_{2i}^{D}(-\alpha_{1}a_{1}^{t}V_{2i}^{1} + \beta_{1}a_{1}^{t}V_{1i}^{1} - \alpha_{4}a_{4}^{t}V_{2i}^{4} + \beta_{4}a_{4}^{t}V_{1i}^{4})] + \\ & \frac{1}{8}(1-r_{1})[^{t}g_{3i}^{B}(w_{i}^{2}-w_{i}^{3}) + \frac{1}{2}^{t}g_{2i}^{B}(-\alpha_{2}\alpha_{2}^{t}V_{2i}^{2} + \\ & \beta_{2}a_{2}^{t}V_{1i}^{2} - \alpha_{3}a_{3}^{t}V_{2i}^{3} + \beta_{3}a_{3}^{t}V_{1i}^{3})] \end{split}$$ + +$${}_{0}\tilde{\eta}_{23} = \frac{1}{32}(1+r_{1})[(-\alpha_{1}a_{1}{}^{t}V_{2i}^{1} + \beta_{1}a_{1}{}^{t}V_{1i}^{1} - \alpha_{4}a_{4}{}^{t}V_{2i}^{4} + \beta_{4}a_{4}{}^{t}V_{1i}^{4})(u_{i}^{1} - u_{i}^{4})] + \frac{1}{32}(1-r_{1})[(-\alpha_{2}a_{2}{}^{t}V_{2i}^{2} + \beta_{2}a_{2}{}^{t}V_{1i}^{2} - \alpha_{3}a_{3}{}^{t}V_{2i}^{3} + \beta_{3}a_{3}{}^{t}V_{1i}^{3})(u_{i}^{2} - u_{i}^{3})]$$ + +$$(24b)$$ + +Note that, since we assume the thickness of the shell to be constant, the strain through the element thickness is zero. + +The expressions in (21) to (24) are substituted into (19) which in the standard manner yields the linear strain incremental stiffness matrix the non-linear strain (or geometric) incremental stiffness matrix and the nodal point force vector in the finite element incremental equilibrium relations2 , + +$$\binom{t}{0}\mathbf{K}_{t} + \binom{t}{0}\mathbf{K}_{Nt}\mathbf{u} = t + \Delta t\mathbf{R} - \binom{t}{0}\mathbf{F} \tag{25}$$ + +The element matrices in (25) correspond to five degrees of freedom per node (see *Figure 1)* but in some applications it is convenient to use instead of three rotations about the global coordinate axes (see examples). In this case, we simply transform the matrices of (25) in the standard manner2 . + +# NUMERICAL TESTS AND EXAMPLE SOLUTIONS + +We have implemented our shell element in the ADINA computer program and have performed various numerical tests to study the predictive capabilities of the element. The following solutions were all obtained using 2x2 Gauss integration in the r3 =0 surface of the element, and 2 and 4 point Gauss integration in the r3 direction, for elastic and elastoplastic analyses, respectively. + +## *Some simple tests* + +As a first step to test the element, the eigenvalues of the stiffness matrices of undistorted and distorted elements were calculated. In all cases, as expected, the element displayed the six rigid body modes and no spurious zero energy modes. + +*Patch tests.* For the patch test2,18 the mesh shown in *Figure 4a* was used. In the first analysis *(Figure 4b)* the mesh was loaded with the constant moment indicated and a constant curvature (linear distribution of rotations) was obtained for both plate thicknesses in the two plate directions. The transverse displacements predicted by the model were, as expected, those of Kirchhoff-Love plate theory at nodes 7 and 8. + +In the second analysis *(Figure 4c)* the rotational degrees of freedom were deleted and the mesh was subjected to shear forces. As expected, for both plate thicknesses a linear distribution of transverse displacements was obtained. + +In the third analysis *(Figure 4d)* the mesh was subjected to an external twisting moment. In the thin plate analysis, constant curvatures were obtained in both plate directions and the transverse displacements agreed with the analytical thin plate theory solution. In the thick plate analysis, a slight non-symmetry in the displacement response (the third digit) was obtained due to the unsymmetric representation of the transverse shear deformations. This non-symmetry is not observed, if the shear deformations are suppressed (which corresponds to thin + +![](_page_4_Figure_13.jpeg) + +![](_page_4_Figure_15.jpeg) + +![](_page_4_Figure_17.jpeg) + +![](_page_4_Figure_19.jpeg) + +![](_page_5_Figure_1.jpeg) + +![](_page_5_Figure_3.jpeg) + +plate theory) by choosing a large value for the shear correction factor *k* (or when using rectangular elements in the mesh)2 . + +Finally, it should be noted that the patch test is of course passed for the three membrane stress states (Τ1 1 , *τ22* andτT12 constants). + +*Cantilever linear analyses.* A cantilever of unit width, thickness 0.1 and lengths lOand 100 was subjected to a tip bending moment. The structure was modelled using one single element and two distorted elements as shown in *Figure 5.* The results obtained in these analyses for the displacements and rotations at the cantilever tip and the stresses were those of Bernoulli beam theory. + +Next, the cantilever in *Figure 6a* was analysed for the transverse tip load shown. Using 4 equal size elements to idealize the cantilever, again good results were obtained when compared with beam theoretical results (see *Figure 6b* and *Table 1).* + +Finally, the elements modelling the cantilever were distorted as shown in *Figure 6c* for a thin and a thick cantilever. The results given in *Figure 6d* and *Table 2* show that the transverse displacements and normal bending stresses are almost insensitive to the element distortions. However, the calculated transverse shear stresses (not shown in the Figure) are not accurate. + +*Linear analyses of a simply-supported plate.* A simplysupported plate was considered for a static and a frequency analysis using a consistent mass matrix. To model one quarter of the plate the 4 x 4 mesh of equal elements *(Figure 7a)* was used. *Figure 7b* and *Tables 3* and *4* give a comparison of the numerically and analytically predicted results. The same plate was also analysed using the distorted element mesh also shown in *Figure 7a* and the results of *Figure 7b* and *Tables 3* and *4* were obtained. + +![](_page_5_Figure_13.jpeg) + +![](_page_5_Figure_15.jpeg) + +![](_page_5_Figure_17.jpeg) + +![](_page_5_Figure_19.jpeg) + +*Table 1* Cantilever tip transverse displacement: non-distorted meshes of *N* elements + +| 0.750 | +|-------| +| 0.984 | +| | + +*Table 2* Cantilever tip transverse displacements + +| Thickness | ηIpoint
B | ηIpoint
A | | +|-----------|--------------|--------------|--| +| 0.1 | 0.989 | 0.996 | | +| 2.0 | 1.0013 | 0.995 | | + +*η*) = (*u*3 distorted mesh)/(*u*3 non-distorted mesh) + +![](_page_6_Figure_1.jpeg) + +![](_page_6_Figure_3.jpeg) + +![](_page_6_Figure_4.jpeg) + +*Table 3* Non-dimensional displacements at centre of simplysupported plate: distorted and non-distorted meshes + +| Model | FEM/u3
thin plate
u3
at centre | | | | +|--------------------|-----------------------------------------|--|--|--| +| non-dist.
dist. | 0.995
0.992 | | | | +| | | | | | + +*Table 4* Non-dimensional frequencies *f* (cycles/sec) for a simplysupported plate: distorted and non-distorted meshes + +| Mode shape | FEM/fthin plat e
f | | | | +|------------|-----------------------|--|--|--| +| 1-1 | 1.02 | | | | +| 1-3 | 1.18 | | | | +| 3-3 | 1.17 | | | | + +*Analysis of a rhombic cantilever.* The rhombic cantilever shown in *Figure 8,* fixed at one side and subjected to constant pressure was analysed using a 4x 4 element mesh. In *Table 5,* the results for the transverse displacements at six locations are compared against the solutions obtained using the DKT triangular element6 , experimental measurements1 and using the 16-node isoparametric element (with 4x4x 2 Gauss integration). In all cases a one step geometric non-linear analysis with equilibrium iterations was performed. Good correspondence between the experimental results and the solution obtained using our new 4-node element is observed. + +# *Linear analysis of a cylindrical (Scordelis-Lo) shell* + +The shell structure shown in *Figure 9a* has frequently been used to test the performance of shell elements12 . *Figure 9b* shows the solutions obtained with our elements. In each of the solutions uniform meshes with equal sized elements were employed over one-quarter of the shell. Solutions obtained using the 3-node DKT triangular element25 and the 16-node isoparametric element25 are also shown. + +### *Linear analysis of a pinched cylinder* + +The pinched cylinder problem shown in *Figure 10a* was also frequently analysed to test shell elements. *Figure 10b* and *Tables 6* and 7 show the convergence behaviour obtained with our new element, when comparing the finite element solutions11' 21 . Note that using the isoparametric shell element3 also a fairly large number of degrees of freedom are required to predict the response of the cylinder accurately. + +# *Large deflection analysis of a cantilever* + +The cantilever shown in *Figure 11a* was analysed for its large displacement and large rotation response. This is a typical problem considered to test the geometric nonlinear behaviour of beam and shell elements25 . *Figure 11a* also shows the models used in the analysis. + +The first two models are single element, cubic and parabolic isoparametric degenerate shell element models. Model I predicts the response of the cantilever very accurately, whereas model II yields an accurate response solution in linear analysis but locks once the element is curved in the non-linear response solution. This observation is in accordance with the results reported elsewhere5 . + +The same nodal point layouts were next employed for models IN and IV using our new 4-node shell element. *Figures 11b-11d* give the results obtained with these models. It is seen that model III yields an accurate large displacement response prediction, and even model IV yields quite accurate results up to about 60 degrees of rotation. The computer time required in these analyses were only little different using models I, III and IV. + +Another important result is shown in *Table 8.* As reported earlier5 , the cubic shell element is sensitive to 'in-plane' distortions, and hence it is interesting to study the effect of using a distorted element mesh in the analysis of the cantilever (see *Figures 12a* and *12b). Table 8* summarizes the results obtained using the one cubic element and three 4-node elements with a nodal layout that corresponds to distorting the elements. It is seen that the predictive capability of our new 4-node element is considerably less sensitive to the element distortions. + +![](_page_7_Figure_1.jpeg) + +*Figure 8* Response of rhombic cantilever subjected to constant pressure. q = 0.26066; £=10.5x106 ; thickness = 0.125; v = 0.3 + +*Table 5* + +| Element | | CPU time | Deflection at location | | | | | | | +|---------------|------|-----------------|------------------------|-------|-------|-------|-------|-------|--| +| | Mesh | CPU time of DKT | 1 | 2 | 3 | 4 | 5 | 6 | | +| DKT | 4x4 | 1.00 | 0.293 | 0.196 | 0.114 | 0.118 | 0.055 | 0.024 | | +| 4-node | 4 x4 | approx. 2 | 0.272 | 0.183 | 0.106 | 0.102 | 0.046 | 0.019 | | +| 16-node | 2x2 | approx. 61/2 | 0.266 | 0.182 | 0.110 | 0.105 | 0.048 | 0.019 | | +| Experimental1 | | | 0.297 | 0.204 | 0.121 | 0.129 | 0.056 | 0.022 | | + +![](_page_7_Picture_6.jpeg) + +![](_page_7_Figure_7.jpeg) + +*Geometric non-linear response of a shallow spherical shell* + +*Figure 13a* shows the spherical shell that was also analysed3 with one cubic shell element, modelling onequarter of the shell. To test our new 4-node shell element, the same nodal point layout was used3 , giving a mesh of nine elements. *Figure 13b* shows the response calculated, including the post-buckling response (not reported in ref. 3) with the automatic load stepping algorithm4 . Good correspondence with the analytical solution of Leicester2 . 0 and the solution of Horrigmoe16 was obtained. The solution with the 16-node element was almost twice as expensive as the 4-node element solution (using in both cases the same parameters for the automatic step-by-step solution algorithm). + +*Linear buckling analysis and large deflection response of a simply-supported stiffened plate* + +The stiffened plate shown in *Figure 14a* was analysed for its buckling reesponse. Since we expect the buckling mode to be symmetric26 only one-quarter of the plate is modelled using symmetry boundary conditions. The model consists of nine 4-node shell elements and three 2 node isoparametric beam elements. At the nodes where a shell element connects to a beam element, three rotational degrees of freedom aligned with the global axes are considered for the shell element. In order to avoid locking of the isoparametric beam elements, one point Gauss integration along the beam axes was used. This does not introduce spurious zero energy modes in the model although the bending stiffness of the beam is underestimated. + +The linearized buckling problem was solved as described in reference 4(37) and we obtained: + +$$\frac{\sigma_{\rm cr}(\text{finite element solution})}{\sigma_{\rm cr}(\text{analytical solution})} = 1.02$$ + +84 Eng. Comput., 1984, Vol. 1, March + +![](_page_8_Picture_1.jpeg) + +![](_page_8_Figure_3.jpeg) + +*Table 6* Convergence study for 4-node element: pinched cylinder + +| Mesh for 1/8th of shell | Number of d.o.f. | FEM/Ŵc
analyt
Ŵc | +|-------------------------|------------------|------------------------| +| 5x5 | 130 | 0.51 | +| 10x10 | 510 | 0.83 | +| 20x20 | 2020 | 0.96 | + +*Table* 7 Comparison between displacements for 4-node and 16 node elements: pinched cylinder + +| Element | Mesh for 1/8th | Number of | FEM/Ŵcanalyt | | | +|---------|----------------|-----------|--------------|--|--| +| | of shell | d.o.f. | Ŵc | | | +| 4-node | 20x20 | 2020 | 0.96 | | | +| 16-node | 10x10 | 4530 | 0.98 | | | + +Next, an initial imperfection with the shape of the first buckling mode and a maximum amplitude of 1/5 of the plate thickness was introduced. *Figure 14b* shows the large deflection response of this model as calculated using the automatic load stepping scheme of reference 4 with a tight energy convergence tolerance. + +*Analysis of elastoplastic response of a circular plate* + +The thin circular plate shown in *Figure 15a* was analysed for its elastoplastic response, when subjected to a concentrated load at its centre. The plate is simply-supported with its edges restrained from moving in its plane. + +In a first solution, the plate model shown in *Figure 15a* was used to analyse the plate assuming small displacements (materially-non-linear-only conditions). *Figure 15c* shows that the theoretical collapse load is overestimated, but for the coarse mesh used, the predicted response is quite reasonable. + +In a second solution, large displacements and elastoplastic conditions were assumed and in this case the stiffening behaviour of the plate shown in *Figure 15c* was predicted. In order to have a comparison, also the model of five axisymmetric 8-node elements shown in *Figure 15b* was solved. *Figure 15c* shows that both models predict in essence the same response; however, in this case relatively little plasticity was developed for the range of displacements considered. + +#### CONCLUSIONS + +A new four-node non-flat general non-linear shell element has been presented with the following important element properties: (1) the element is formulated using threedimensional continuum mechanics theory; hence the use of the element is not restricted by application of a specific shell theory; (2) the element is reliable and has good predictive capability in the analysis of thick and thin shells; (3) the amount of computations required to calculate the element stiffness matrix are very closely those that are used in standard isoparametric formulations. The computer time used could be reduced considerably in elastic analysis by using analytical integration through the element thickness. + +In this paper we have presented the formulation and some applications of the element. The solution results obtained are most encouraging, but a formal mathematical convergence study of the element would be very valuable, and we are currently pursuing such research. + +Finally, it should be noted that the element presented here provides a very attractive basic formulation that could be extended to large strain analysis and analysis of composite shells. Also, the concepts applied here to formulate a 4-node element could equally well be employed in an effective manner to formulate higher-order shell elements. + +#### ACKNOWLEDGEMENTS + +We are grateful for the financial support by the U.S. Army contract no. DAAK11-82-K-0005 and the ADINA users group for this work. + +*Note added in proof* — We have just learned — and regret not to have known of it earlier — that R. H. MacNeal [J. *Nucl. Eng. Design,* 70, 3-12 (1982)] proposed a plate element for linear analysis that is very close to the element presented above. + +![](_page_9_Figure_1.jpeg) + +# REFERENCES + +- 1 Adini, A. Analysis of shell structures by the finite element method, *PhD Dissertation,* Department of Civil Engineering, University of California, Berkeley (1961) +- 2 Bathe, K. J. *Finite Element Procedures in Engineering Analysis,* Prentice-Hall, Englewood Cliffs, New Jersey (1982) +- 3 Bathe, K. J. and Bolourchi, S. A geometric and material nonlinear plate and shell element, *J. Comput. Struct.,* 11, 23-48 (1979) +- 4 Bathe, K. J. and Dvorkin, E. N. On the automatic solution of nonlinear finite element equations, *J. Comput. Struct.* 17, (5-6), 871-879 (1983) +- 5 Bathe, K. J.,Dvorkin, E. N. and Ho, L. W. Our discrete-Kirchhoff and isoparametric shell elements for nonlinear analysis - an assessment, *J. Comput. Struct.,* 16, (1-4), 89-98 (1983) + +86 Eng. Comput., 1984, Vol. 1, March + +![](_page_10_Picture_1.jpeg) + +![](_page_10_Figure_5.jpeg) + +![](_page_10_Figure_6.jpeg) + +![](_page_10_Figure_8.jpeg) + +- 6 Bathe, K. J. and Ho, L. W. A simple and effective element for analysis of general shell structures, *J. Comput. Struct.,* 13, 673-382 (1980) +- 7 Bathe, K. J. and Ho, L. W. Some results in the analysis of thin shell structures, *Nonlinear Finite Element Analysis in Structural Mechanics,* (Ed. W. Wunderlich *et al.).* Springer-Verlag, Berlin (1981) +- 8 Batoz, J. L., Bathe, K, J. and Ho, L. W. A study of three-node triangular plate bending elements, *Int. J, Num. Meth. Eng.,* 15, 1771-1812(1980) +- 9 Batoz, J. L. and Ben Tahar, M. Evaluation of a new quadrilateral plate bending element, *Int. J, Num. Meth. Eng.,* 18, 1655-1677 (1982) +- 10 Bercovier, M., Hasbgni, V., Gilon, Y, and Bathe, K, J, On a finite element procedure for nonlinear incompressible elasticity, *Hybrid and Mixed Finite Element Methods,* (Ed, S. M. Atluri *et al.),* John Wiley, New York (1983) +- 11 Flügge, W. *Stresses in Shells,* 2nd edn, Springer-Verlag, Berlin (1973) +- 12 Forsberg, K. and Hartung, R. An evaluation of finite difference and finite element techniques for analysis of general shells, *Symp. High Speed Computing of Elastic Structures,* IUTAM, Liege (1970) +- 13 Fung, Y. C. *Foundations of Solid Mechanics,* Prentice-Hall, Englewood Cliffs, New Jersey (1965) +- 14 Gallagher, R. H. Problems and progress in thin shell finite element analysis, *Finite Elements in Thin Shells and Curved Members,* (Ed. D. G. Ashwell and R. H. Gallagher), John Wiley, New York (1976) +- 15 Green, A. E. and Zerna, W. *Theoretical Elasticity,* 2nd edn, Oxford University Press (1968) +- 16 Horrigmoe, G. Finite element instability analysis of free-form shells, *Report 77-2,* Division of Structural Mechanics, The Norwegian Institute of Technology, University of Trondheim, Norway (1977) +- 17 Hughes, T. J. R. and Liu, W. K. Nonlinear finite element analysis of shells: Part I, Three-dimensional shells, *J. Comput. Meth. Appl. Mech. Eng., 26,* 331-362 (1981) + +![](_page_11_Figure_1.jpeg) + +![](_page_11_Figure_2.jpeg) + +![](_page_11_Figure_3.jpeg) + +- 19 Krakeland, B. Nonlinear analysis of shells using degenerate isoparametric elements, *Finite Elements in Nonlinear Mechanics,* Vol. 1, (Ed. P. G. Bergan *et al*.), Tapir Publishers (Norwegian Institute of Technology, Trondheim, Norway) (1978) +- 20 Leicester, R. H. Finite deformations of shallow shells, *Proc. Am. Soc. Civil Eng.,* 94, (EM6), 1409-1423 (1968) +- 21 Lindberg, G. M., Olson, M. D. and Cowper, G. R. New developments in the finite element analysis of shells, *Q. Bull. Div. Mech. Eng. and the National Aeronautical Establishment,* National Research Council of Canada, Vol. 4 (1969) +- 22 MacNeal, R. H. A simple quadrilateral shell element, *J. Comput. Struct.* 8, 175-183 (1978) +- 23 Noor, A. K. and Peters, J. M. Mixed models and reduced/selec- + +![](_page_11_Figure_9.jpeg) + +tive integration displacement models for nonlinear analysis of curved beams, *Int. J. Num. Meth. Eng.,* 17, 615-631 (1981) + +- 24 Ramm, E. and Sattele, J. M. Elasto-plastic large deformation shell analysis using degenerated elements, *Nonlinear Finite Element Analysis of Plates and Shells,* (Ed. T. J. R. Hughes), AMD-Vol. 48, Am. Soc. Mech. Eng., New York (1981) +- 25 *Report AE 83-5,* ADINA System Verification Manual, ADINA Engineering, Vasteras, Sweden and Watertown, Mass. (1983) +- 26 Timoshenko, S. P. and Gere, J. M. *Theory of Elastic Stability,* 2nd edn, McGraw-Hill, New York (1961) +- 27 Washizu, K. *Variational Methods in Elasticity and Plasticity,* Pergamon Press, Oxford and New York (1968) +- 28 Wempner, G., Talaslidis, D. and Hwang, C.-M. A simple and efficient approximation of shells via finite quadrilateral elements, *J. Appl. Mech.,* 49, 115-120 (1982) +- 29 Zienkiewicz, O. 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