Introduction
A series of problems described in A Proposed Standard Set of Problems to Test Finite Element Accuracy by Macneal and Harder (1985) is used to verify the validity of the DKT plate and Shell elements in SACS. These SACS models are available under Sample 16 in the Samples folder of the SACS installation directory. This folder contains model files, run files and expected results.
Each problem consists of a geometry definition, a boundary condition and a theoretical solution. These theoretical solutions will be compared to the SACS results to check the element validity in SACS.
Bending Plate Patch Test
Geometry Definition
Patch Bending Geometry
Width:
Length:
Thickness:
Modulus of Elasticity:
Poisson's Ratio:
Location of Joints:
| Joint | x | y |
|---|---|---|
| A001 | 0.00 | 0.00 |
| A002 | 0.24 | 0.00 |
| A003 | 0.24 | 0.12 |
| A004 | 0.00 | 0.12 |
| B001 | 0.04 | 0.02 |
| B002 | 0.18 | 0.03 |
| B003 | 0.16 | 0.08 |
| B004 | 0.08 | 0.08 |
Boundary Conditions
Joint Displacements:
| Joint | |||
|---|---|---|---|
| A001 | 0.0000 | 0.00 | 0.00 |
| A002 | 0.0288 | 0.12 | -0.24 |
| A003 | 0.0504 | 0.24 | -0.30 |
| A004 | 0.0072 | 0.12 | -0.06 |
Theoretical solution
Surface Stresses:
Plate Stresses:
| Plate | % Error | |||
|---|---|---|---|---|
| AABB | -0.667 | -0.667 | 0.2 | 0.00 |
| AABC | 0.667 | 0.667 | 0.2 | 0.00 |
| AABD | -0.667 | -0.667 | 0.2 | 0.00 |
| AABE | -0.667 | -0.667 | 0.2 | 0.00 |
| AABF | 0.667 | 0.667 | 0.2 | 0.00 |
Patch Bending Stress
Patch Bending Shear Stress
Joint Displacements:
| Joint | % Error | |||
|---|---|---|---|---|
| B001 | 0.00140 | 0.0400 | 0.0500 | 0.00 |
| B002 | 0.01935 | 0.1200 | 0.1950 | 0.00 |
| B003 | 0.02240 | 0.1600 | 0.2000 | 0.00 |
| B004 | 0.00960 | 0.1200 | 0.1200 | 0.00 |
Patch Bending Displacements
Patch Bending Rotations
Membrane Patch Test
Geometry Definition
Patch Membrane Geometry
Width:
Length:
Thickness:
Modulus of Elasticity:
Poisson's Ratio:
Location of Joints:
| Joint | x | y |
|---|---|---|
| A001 | 0.00 | 0.00 |
| A002 | 0.24 | 0.00 |
| A003 | 0.24 | 0.12 |
| A004 | 0.00 | 0.12 |
| B001 | 0.04 | 0.02 |
| B002 | 0.18 | 0.03 |
| B003 | 0.16 | 0.08 |
| B004 | 0.08 | 0.08 |
Boundary Conditions
Joint Displacements:
| Joint | ||
|---|---|---|
| A001 | 0.0000 | 0.00 |
| A002 | 0.24E-3 | 0.12E-3 |
| A003 | 0.30E-3 | 0.24E-3 |
| A004 | 0.60E-3 | 0.12E-3 |
Theoretical solution
Plate Stresses:
| Plate | % Error | |||
|---|---|---|---|---|
| AABB | 1333 | 1333 | 400 | 0.00 |
| AABC | 1333 | 1333 | 400 | 0.00 |
| AABD | 1333 | 1333 | 400 | 0.00 |
| AABE | 1333 | 1333 | 400 | 0.00 |
| AABF | 1333 | 1333 | 400 | 0.00 |
Patch Membrane Stress
Patch Membrane Shear Stress
Scordelis-Lo Roof
Geometry Definition
Scordelis-Lo Geometry
Radius:
Length:
Thickness:
Modulus of Elasticity:
Poisson's Ratio:
Loading
Loading:
Boundary Conditions
- The outer straight edge is free
- The outer curved edge is constrained against translation in the y and z directions
- The straight symmetry edge on top of the roof has symmetry edge constraints
- The curved symmetry edge has symmetry edge Constraints
Mesh Density
Number of Elements:
Mesh Size: and
Joint Locations and Fixities:
| Joint | X | Y | Z | Fixity |
|---|---|---|---|---|
| 221 | -2.083 | 0.487 | 110110 | |
| 222 | -1.875 | 0.487 | 100010 | |
| 223 | -1.667 | 0.487 | 100010 | |
| 224 | -1.458 | 0.487 | 100010 | |
| 225 | -1.250 | 0.487 | 100010 | |
| 226 | -1.042 | 0.487 | 100010 | |
| 227 | -0.833 | 0.487 | 100010 | |
| 228 | -0.625 | 0.487 | 100010 | |
| 229 | -0.417 | 0.487 | 100010 | |
| 230 | -0.208 | 0.487 | 100010 | |
| 231 | 0.487 | 101010 | ||
| 242 | 0.145 | -2.083 | 0.482 | 010100 |
| 243 | 0.145 | -1.875 | 0.482 | |
| 244 | 0.145 | -1.667 | 0.482 | |
| 245 | 0.145 | -1.458 | 0.482 | |
| 246 | 0.145 | -1.250 | 0.482 | |
| 247 | 0.145 | -1.042 | 0.482 | |
| 248 | 0.145 | -0.833 | 0.482 | |
| 249 | 0.145 | -0.625 | 0.482 | |
| 250 | 0.145 | -0.417 | 0.482 | |
| 251 | 0.145 | -0.208 | 0.482 | |
| 252 | 0.145 | 0.482 | 101000 | |
| 263 | 0.290 | -2.083 | 0.467 | 010100 |
| 264 | 0.290 | -1.875 | 0.467 | |
| 265 | 0.290 | -1.667 | 0.467 | |
| 266 | 0.290 | -1.458 | 0.467 | |
| 267 | 0.290 | -1.250 | 0.467 | |
| 268 | 0.290 | -1.042 | 0.467 | |
| 269 | 0.290 | -0.833 | 0.467 | |
| 270 | 0.290 | -0.625 | 0.467 | |
| 271 | 0.290 | -0.417 | 0.467 | |
| 272 | 0.290 | -0.208 | 0.467 | |
| 273 | 0.290 | 0.467 | 101000 | |
| 284 | 0.433 | -2.083 | 0.442 | 010100 |
| 285 | 0.433 | -1.875 | 0.442 | |
| 286 | 0.433 | -1.667 | 0.442 | |
| 287 | 0.433 | -1.458 | 0.442 | |
| 288 | 0.433 | -1.250 | 0.442 | |
| 289 | 0.433 | -1.042 | 0.442 | |
| 290 | 0.433 | -0.833 | 0.442 | |
| 291 | 0.433 | -0.625 | 0.442 | |
| 292 | 0.433 | -0.417 | 0.442 | |
| 293 | 0.433 | -0.208 | 0.442 | |
| 294 | 0.433 | 0.442 | 101000 | |
| 305 | 0.574 | -2.083 | 0.407 | 010100 |
| 306 | 0.574 | -1.875 | 0.407 | |
| 307 | 0.574 | -1.667 | 0.407 | |
| 308 | 0.574 | -1.458 | 0.407 | |
| 309 | 0.574 | -1.250 | 0.407 | |
| 310 | 0.574 | -1.042 | 0.407 | |
| 311 | 0.574 | -0.833 | 0.407 | |
| 312 | 0.574 | -0.625 | 0.407 | |
| 313 | 0.574 | -0.417 | 0.407 | |
| 314 | 0.574 | -0.208 | 0.407 | |
| 315 | 0.574 | 0.407 | 101000 | |
| 326 | 0.712 | -2.083 | 0.362 | 010100 |
| 327 | 0.713 | -1.875 | 0.362 | |
| 328 | 0.713 | -1.667 | 0.362 | |
| 329 | 0.713 | -1.458 | 0.362 | |
| 330 | 0.713 | -1.250 | 0.362 | |
| 331 | 0.713 | -1.042 | 0.362 | |
| 332 | 0.713 | -0.833 | 0.362 | |
| 333 | 0.713 | -0.625 | 0.362 | |
| 334 | 0.713 | -0.417 | 0.362 | |
| 335 | 0.713 | -0.208 | 0.362 | |
| 336 | 0.713 | 0.362 | 101000 | |
| 347 | 0.847 | -2.083 | 0.307 | 010100 |
| 348 | 0.847 | -1.875 | 0.307 | |
| 349 | 0.847 | -1.667 | 0.307 | |
| 350 | 0.847 | -1.458 | 0.307 | |
| 351 | 0.847 | -1.250 | 0.307 | |
| 352 | 0.847 | -1.042 | 0.307 | |
| 353 | 0.847 | -0.833 | 0.307 | |
| 354 | 0.847 | -0.625 | 0.307 | |
| 355 | 0.847 | -0.417 | 0.307 | |
| 356 | 0.847 | -0.208 | 0.307 | |
| 357 | 0.847 | 0.307 | 101000 | |
| 368 | 0.978 | -2.083 | 0.243 | 010100 |
| 369 | 0.978 | -1.875 | 0.243 | |
| 370 | 0.978 | -1.667 | 0.243 | |
| 371 | 0.978 | -1.458 | 0.243 | |
| 372 | 0.978 | -1.250 | 0.243 | |
| 373 | 0.978 | -1.042 | 0.243 | |
| 374 | 0.978 | -0.833 | 0.243 | |
| 375 | 0.978 | -0.625 | 0.243 | |
| 376 | 0.978 | -0.417 | 0.243 | |
| 377 | 0.978 | -0.208 | 0.243 | |
| 378 | 0.978 | 0.243 | 101000 | |
| 389 | 1.104 | -2.083 | 0.171 | 010100 |
| 390 | 1.104 | -1.875 | 0.171 | |
| 391 | 1.104 | -1.667 | 0.171 | |
| 392 | 1.104 | -1.458 | 0.171 | |
| 393 | 1.104 | -1.250 | 0.171 | |
| 394 | 1.104 | -1.042 | 0.171 | |
| 395 | 1.104 | -0.833 | 0.171 | |
| 396 | 1.104 | -0.625 | 0.171 | |
| 397 | 1.104 | -0.417 | 0.171 | |
| 398 | 1.104 | -0.208 | 0.171 | |
| 399 | 1.104 | 0.171 | 101000 | |
| 410 | 1.225 | -2.083 | 0.089 | 010100 |
| 411 | 1.225 | -1.875 | 0.089 | |
| 412 | 1.225 | -1.667 | 0.089 | |
| 413 | 1.225 | -1.458 | 0.089 | |
| 414 | 1.225 | -1.250 | 0.089 | |
| 415 | 1.225 | -1.042 | 0.089 | |
| 416 | 1.225 | -0.833 | 0.089 | |
| 417 | 1.225 | -0.625 | 0.089 | |
| 418 | 1.225 | -0.417 | 0.089 | |
| 419 | 1.225 | -0.208 | 0.089 | |
| 420 | 1.225 | 0.089 | 101000 | |
| 431 | 1.339 | -2.083 | 010100 | |
| 432 | 1.339 | -1.875 | ||
| 433 | 1.339 | -1.667 | ||
| 434 | 1.339 | -1.458 | ||
| 435 | 1.339 | -1.250 | ||
| 436 | 1.339 | -1.042 | ||
| 437 | 1.339 | -0.833 | ||
| 438 | 1.339 | -0.625 | ||
| 439 | 1.339 | -0.417 | ||
| 440 | 1.339 | -0.208 | ||
| 441 | 1.339 | 101000 |
Theoretical Solutions
Midpoint Displacement of Free Edge:
| Joint | SACS | % Error | |
|---|---|---|---|
| 431 | 0.3086 | 0.3043827 | 0.014 |
Morley Spherical Shell
Geometry Definition
Morley Spherical Shell Geometry
Radius:
Thickness:
Modulus of Elasticity
Poisson's Ratio
Loading
| Joint | ||
|---|---|---|
| 1 | 0.0 | -0.001 |
| D | -0.001 | 0.0 |
Boundary Conditions
- Horizontal edges are free
- Vertical edges have symmetry edge Constraints
Mesh Density
Number of Elements
Joint Locations and Fixities:
| Joint | X | Y | Z | Fixity |
|---|---|---|---|---|
| 1 | -0.833 | 101011 | ||
| 2 | 0.109 | -0.826 | ||
| 3 | 0.216 | -0.805 | ||
| 4 | 0.319 | -0.770 | ||
| 5 | 0.417 | -0.722 | ||
| 6 | 0.507 | -0.661 | ||
| 7 | 0.589 | -0.589 | ||
| 8 | 0.661 | -0.507 | ||
| 9 | 0.722 | -0.417 | ||
| A | 0.770 | -0.319 | ||
| B | 0.805 | -0.216 | ||
| C | 0.826 | -0.109 | ||
| D | 0.833 | 010101 | ||
| E | -0.829 | 0.087 | 100011 | |
| F | 0.108 | -0.822 | 0.087 | |
| G | 0.215 | -0.801 | 0.087 | |
| H | 0.317 | -0.766 | 0.087 | |
| I | 0.414 | -0.718 | 0.087 | |
| J | 0.505 | -0.658 | 0.087 | |
| K | 0.586 | -0.586 | 0.087 | |
| L | 0.658 | -0.505 | 0.087 | |
| M | 0.718 | -0.414 | 0.087 | |
| N | 0.766 | -0.317 | 0.087 | |
| O | 0.801 | -0.214 | 0.087 | |
| P | 0.822 | -0.108 | 0.087 | |
| Q | 0.829 | 0.087 | 010101 | |
| R | -0.815 | 0.173 | 100011 | |
| S | 0.106 | -0.808 | 0.173 | |
| T | 0.211 | -0.787 | 0.173 | |
| U | 0.312 | -0.753 | 0.173 | |
| V | 0.408 | -0.706 | 0.173 | |
| W | 0.496 | -0.647 | 0.173 | |
| X | 0.576 | -0.576 | 0.173 | |
| Y | 0.647 | -0.496 | 0.173 | |
| Z | 0.706 | -0.408 | 0.173 | |
| 10 | 0.753 | -0.312 | 0.173 | |
| 11 | 0.787 | -0.211 | 0.173 | |
| 12 | 0.808 | -0.106 | 0.173 | |
| 13 | 0.815 | 0.173 | 010101 | |
| 14 | -0.793 | 0.258 | 100011 | |
| 15 | 0.103 | -0.786 | 0.258 | |
| 16 | 0.205 | -0.766 | 0.258 | |
| 17 | 0.303 | -0.732 | 0.258 | |
| 18 | 0.396 | -0.686 | 0.258 | |
| 19 | 0.482 | -0.629 | 0.258 | |
| 1A | 0.560 | -0.560 | 0.258 | |
| 1B | 0.629 | -0.482 | 0.258 | |
| 1C | 0.686 | -0.396 | 0.258 | |
| 1D | 0.732 | -0.303 | 0.258 | |
| 1E | 0.766 | -0.205 | 0.258 | |
| 1F | 0.786 | -0.103 | 0.258 | |
| 1G | 0.793 | 0.258 | 010101 | |
| 1H | -0.761 | 0.339 | 100011 | |
| 1I | 0.099 | -0.755 | 0.339 | |
| 1J | 0.197 | -0.735 | 0.339 | |
| 1K | 0.291 | -0.703 | 0.339 | |
| 1L | 0.381 | -0.659 | 0.339 | |
| 1M | 0.463 | -0.604 | 0.339 | |
| 1N | 0.538 | -0.538 | 0.339 | |
| 1O | 0.604 | -0.463 | 0.339 | |
| 1P | 0.659 | -0.381 | 0.339 | |
| 1Q | 0.703 | -0.291 | 0.339 | |
| 1R | 0.735 | -0.197 | 0.339 | |
| 1S | 0.755 | -0.099 | 0.339 | |
| 1T | 0.761 | 0.339 | 010101 | |
| 1U | -0.722 | 0.417 | 100011 | |
| 1V | 0.094 | -0.716 | 0.417 | |
| 1W | 0.187 | -0.697 | 0.417 | |
| 1X | 0.276 | -0.667 | 0.417 | |
| 1Y | 0.361 | -0.625 | 0.417 | |
| 1Z | 0.439 | -0.573 | 0.417 | |
| 20 | 0.510 | -0.510 | 0.417 | |
| 21 | 0.573 | -0.439 | 0.417 | |
| 22 | 0.625 | -0.361 | 0.417 | |
| 23 | 0.667 | -0.276 | 0.417 | |
| 24 | 0.697 | -0.187 | 0.417 | |
| 25 | 0.716 | -0.094 | 0.417 | |
| 26 | 0.722 | 0.417 | 010101 | |
| 27 | -0.674 | 0.490 | 100011 | |
| 28 | 0.088 | -0.668 | 0.490 | |
| 29 | 0.174 | -0.651 | 0.490 | |
| 2A | 0.258 | -0.623 | 0.490 | |
| 2B | 0.337 | -0.584 | 0.490 | |
| 2C | 0.410 | -0.535 | 0.490 | |
| 2D | 0.477 | -0.477 | 0.490 | |
| 2E | 0.535 | -0.410 | 0.490 | |
| 2F | 0.584 | -0.337 | 0.490 | |
| 2G | 0.623 | -0.258 | 0.490 | |
| 2H | 0.651 | -0.174 | 0.490 | |
| 2I | 0.668 | -0.088 | 0.490 | |
| 2J | 0.674 | 0.490 | 010101 | |
| 2K | -0.619 | 0.558 | 100011 | |
| 2L | 0.081 | -0.614 | 0.558 | |
| 2M | 0.160 | -0.598 | 0.558 | |
| 2N | 0.237 | -0.572 | 0.558 | |
| 2O | 0.310 | -0.536 | 0.558 | |
| 2P | 0.377 | -0.491 | 0.558 | |
| 2Q | 0.438 | -0.438 | 0.558 | |
| 2R | 0.491 | -0.377 | 0.558 | |
| 2S | 0.536 | -0.310 | 0.558 | |
| 2T | 0.572 | -0.237 | 0.558 | |
| 2U | 0.598 | -0.160 | 0.558 | |
| 2V | 0.614 | -0.081 | 0.558 | |
| 2W | 0.619 | 0.558 | 010101 | |
| 2X | -0.558 | 0.619 | 100011 | |
| 2Y | 0.073 | -0.553 | 0.619 | |
| 2Z | 0.144 | -0.539 | 0.619 | |
| 30 | 0.213 | -0.515 | 0.619 | |
| 31 | 0.279 | -0.483 | 0.619 | |
| 32 | 0.339 | -0.442 | 0.619 | |
| 33 | 0.394 | -0.394 | 0.619 | |
| 34 | 0.442 | -0.339 | 0.619 | |
| 35 | 0.483 | -0.279 | 0.619 | |
| 36 | 0.515 | -0.213 | 0.619 | |
| 37 | 0.539 | -0.144 | 0.619 | |
| 38 | 0.553 | -0.073 | 0.619 | |
| 39 | 0.558 | 0.619 | 010101 | |
| 3A | -0.490 | 0.674 | 100011 | |
| 3B | 0.064 | -0.486 | 0.674 | |
| 3C | 0.127 | -0.473 | 0.674 | |
| 3D | 0.187 | -0.453 | 0.674 | |
| 3E | 0.245 | -0.424 | 0.674 | |
| 3F | 0.298 | -0.389 | 0.674 | |
| 3G | 0.346 | -0.346 | 0.674 | |
| 3H | 0.389 | -0.298 | 0.674 | |
| 3I | 0.424 | -0.245 | 0.674 | |
| 3J | 0.453 | -0.187 | 0.674 | |
| 3K | 0.473 | -0.127 | 0.674 | |
| 3L | 0.486 | -0.064 | 0.674 | |
| 3M | 0.490 | 0.674 | 010101 | |
| 3N | -0.417 | 0.722 | 100011 | |
| 3O | 0.054 | -0.413 | 0.722 | |
| 3P | 0.108 | -0.402 | 0.722 | |
| 3Q | 0.159 | -0.385 | 0.722 | |
| 3R | 0.208 | -0.361 | 0.722 | |
| 3S | 0.254 | -0.331 | 0.722 | |
| 3T | 0.295 | -0.295 | 0.722 | |
| 3U | 0.331 | -0.254 | 0.722 | |
| 3V | 0.361 | -0.208 | 0.722 | |
| 3W | 0.385 | -0.159 | 0.722 | |
| 3X | 0.402 | -0.108 | 0.722 | |
| 3Y | 0.413 | -0.054 | 0.722 | |
| 3Z | 0.417 | 0.722 | 010101 | |
| 40 | -0.339 | 0.761 | 100011 | |
| 41 | 0.044 | -0.336 | 0.761 | |
| 42 | 0.088 | -0.327 | 0.761 | |
| 43 | 0.130 | -0.313 | 0.761 | |
| 44 | 0.169 | -0.294 | 0.761 | |
| 45 | 0.206 | -0.269 | 0.761 | |
| 46 | 0.240 | -0.240 | 0.761 | |
| 47 | 0.269 | -0.206 | 0.761 | |
| 48 | 0.294 | -0.169 | 0.761 | |
| 49 | 0.313 | -0.130 | 0.761 | |
| 4A | 0.327 | -0.088 | 0.761 | |
| 4B | 0.336 | -0.044 | 0.761 | |
| 4C | 0.339 | 0.761 | 010101 | |
| 4D | -0.258 | 0.793 | 100011 | |
| 4E | 0.034 | -0.255 | 0.793 | |
| 4F | 0.067 | -0.249 | 0.793 | |
| 4G | 0.099 | -0.238 | 0.793 | |
| 4H | 0.129 | -0.223 | 0.793 | |
| 4I | 0.157 | -0.204 | 0.793 | |
| 4J | 0.182 | -0.182 | 0.793 | |
| 4K | 0.204 | -0.157 | 0.793 | |
| 4L | 0.223 | -0.129 | 0.793 | |
| 4M | 0.238 | -0.099 | 0.793 | |
| 4N | 0.249 | -0.067 | 0.793 | |
| 4O | 0.255 | -0.034 | 0.793 | |
| 4P | 0.258 | 0.793 | 010101 |
Theoretical Solutions
Displacement:
| Joint | SACS | % Error | |
|---|---|---|---|
| 1 | 0.094 | 0.0924 | 0.017 |
| D | 0.094 | 0.0926 | 0.015 |
Conclusion
The SACS solution very closely matches the theoretical solutions presented by Macneal and Harder. The maximum error of 0.017% is acceptable for a finite element solution. This error can be attributed to the mesh density and may be improved by refining the mesh.