Appendix A

Geometry of the beam element in space

We consider a 2-node beam element in space equiped with a local coordinate system x'Y' z' placed at its center. We seek to derive the matrix of direction cosines To beetwen the local coordinate x'y'z' and the global coordinate xyz.

The beam element comprises two nodes with global coordinates Xl, YI, Zl and X2, Y2, Z2, respectively. In order to orient the middle surface we must define an extra point P* with global coordinates x*, y*, z* so that points P*, 1, 2, form a plane that passes through the middle plane of the beam element. We define the differences of the global coordinates for nodes 1,2 as

[X2 - Xl] [XI2] Y2 - YI = Yl2 , z2 - zl zl2

(A.l)

and the differences of the global coordinates between point P* and node 1 as

[X* - Xl] Y: - YI = Z - Zl

[Xle] Yle . Zle

(A.2)

In addition

322 Geometry of the beam element in space

[Xl - X*] [xel] Yl - Y: = Yel . Zl - Z Zel

(A.3)

The entries of the first row in To comprise components of the unit vector

[CXIX] 1 [X12]

PX' = Cx'Y = l Y12 ,

CX' Z Z12

(A.4)

where

1 = VX~2 + Y~2 + z~2' (A.5)

The unit vector perpendicular to the beam's surface is defined as

[CZIX] 0.... 0.... 1 [Y12 Zle - Z12Yle] .... 12 X Ie

PZI = Czl y = 20 20 Z12 Xle - X12 Zle ,

CZI z X12Yle - Y12Xle

(A.6)

where 0 is the area of the triangle formed by the two nodes and the extra point defined as

in which

1 0= -JC12 + C2 2 + C3 2 ,

2

[Cl] [Y12Zel - Yel Z12] C2 = Z12Xel - Zel X12 .

C3 X12Yel - XelY12

Finally, the components of the unit vector Pyl are simply given by

[CyIX] [CZIYCXIZ - CZIZCX1Y]

Pyl Cyl y = PZI X Pxl = CZ'ZCX' X - CZ'XCX' Z •

CyIZ CzlxCxly - CzlyCxlx

(A.7)

(A.8)

(A.9)

323

We are now in the position to define the matrix of direction cosines via

[cx, X cx' Y cx' zl

To = cy'x Cy'y cy'z .

Cz'x Cz'y Cz'z

(A.lO)

Note that matrix To is derived in an identical manner for the 3-node shell element by simply replacing the extra point P* with node 1, and the beam nodes 1, 2 with the triangular nodes 2, 3, respectively.

Appendix B

Contents of floppy disk

Included with in the floopy disk accompanying the book is a model com­puter program for static and buckling analyses of isotropic and laminated composite beams, frames and large three-dimensional beam assemblies. The name of the program is beam1.f and is written in standard Fortran 77. It can be compiled and run on any computer with Fortran 77. The pro­gram was originally written and compiled on a Sun Sparcstation 10 using the command £17 beam1.f.

In addition to the source code, four example problems are provided to familiarize the user with the input data and the execution of the computer program. All axamples were described in the text.

The following files are included in the floopy disk:

1. beam1.f: Main computer program.

2. parcb.h: A short file setting the problem parameters.

3. meshb.dat: A file containing the four test problems, namely

• static analysis of an L-shaped isotropic beam {example in Fig. 5.12}

• static analysis of an isotropic frame {example in Fig. 5.13}

• static analysis of a {45/ -45/ -45/45} cantilver composite beam {example in Fig. 5.20}

• buckling analysis of a {45/ -45/0/90}8 composite beam {example in Fig. 5.22}

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Index

accumulation, 217 algebraic operations, 263 angle-ply, 43 angle-ply laminate, 44 anisotropy ratios, 235 antisymmetrical bending mode, 61,

164 antisymmetrical bending terms, 182 antisymmetrical moments, 175 antisymmetrical rotations, 175 antisymmetrical shearing mode, 164 antisymmetrical shearing stiffness,

185 antisymmetrical shearing terms, 186 aspect ratio, 233 aspect ratios, 229 aspect ratios , 234 assembly loops, 313 assembly operation, 264 averaging procedures, 185 axial direct strain, 51 axial strain, 60 azimuth moments, 162

balanced, 42 balanced laminates, 43 beam element, 31 beam shearing coefficient, 63 biaxial symmetry, 231, 233 bidirectional laminate, 233 bifurcations, 267 binder, 37

body force, 2 body forces, 2 boundary conditions, 221, 230 buckling, 99, 119, 217, 234 buckling loads, 217, 261 buckling modes, 217, 240, 261

cantilever beam, 224 cantilever composite beam, 119,

302 cartesian moments, 215 cartesian nodal forces, 105, 216 cartesian strains, 140 central displacement, 218 clamped isotropic plate, 218 classical finite element methods,

25, 263 classical lamination theory, 41 classical plate of solution, 233 close form solutions, 239 CLT, 42 code structure, 301 collapse loads, 267 component stresses, 145, 204 composite, 37 composite beam columns, 119 composite beams, 119 composite cylinder, 240 composite laminate, 230 composite laminates, 42 composite material, 44 composite satellite, 119

336 INDEX

composite shell, 261 composite shells, 136 compressive loads, 119, 234 computational aspects, 261 computational efficiency, 264 computer program, 261, 320 computing time, 264 congruent operations, 171 congruent transformations, 191 convergence, 221, 232 convergence characteristics, 119,224 coordinate transformation, 213 CPU time list, 263 CRAY-C94, 261 critical buckling load, 119 critical load, 127, 239 critical pressure, 250 cross-ply laminate, 44, 233 cross-ply plate, 239 cylindrical arc-length, 267 cylindrical panel, 239 cylindrical roof, 220

deformation plot, 234 deformed beams, 227 deformed geometry, 100 design, viii direct, 2 direction cosines, 85, 167 double sinusoidal load, 230, 233

efficiency, 261 eigenvalue analyses, 261 eigenvalue problem, 217 elastic curve, 61 elasticity solution, 232 elements, 25 engineering shear strains, 10 equilibrium, 6, 85 error, 219

Euler formula, 119 exact solution, 221 experimental results, 234 external loads, 86

FEM, vii fiber reinforced composites, 38 fibers, 37 filaments, 37 finite element method, vii, 19, 20 finite element programming, 301 force method, 18 Fortran commands, 302

generalized forces, 166 geometrical forces, 212 geometrical stiffness, 20, 99, 208,

319 global arrays, 309 global cartesian coordinate, 166 global elemental vector, 84 Global equilibrium, 202 global force increments, 104 global load vector, 88 graphical visualization, 314

hemispherical shell, 221 heterogeneous medium, 44 High speed flight, 38 higher-order shear deformation the-

ory, 235 higher-order theory, 229 homogeneous coordinates, 176 hydrostatic pressure, 240

ill conditioning, 314 incremental, 267 inextensional bending modes, 221 initial load vectors, 89 input parameters, 302 integration formula, 182

interpolation matrix, 87 isoparametric, 267 isotropic beams, 114 isotropic column, 119 isotropic frames, 115 isotropy, 152 iterative, 267

jet propulsion, 18

Kirchhoff, 176

lamina, 44 laminated beam element, 52 lamination schemes, 250 large deflections, 233 limit points, 267 load factor, 217 load-displacement curves, 234 loading vector, 313 local coordinate system, 40 local coordinates, 154 locking, 63 lower order theory, 233

macromechanics, 44 material anisotropy ratio, 235 material coordinate system, 40, 52 material properties, 229 material transformations, 263 matrix, 37 matrix displacement method, 19 matrix materials, 39 matrix multiplications, 263 membrane forces, 216 membrane stresses, 100, 217 membrane theory of shells, 135 micromechanics, 44 middle plane, 321 middle surface, 321 minimum, 15

INDEX 337

model problem, 261, 302

natural coordinate system, 28, 191, 212

natural directions, 177 natural forces, 212 natural geometrical stiffness, 106,

216 natural modes, vii, 20, 48 natural stiffness matrix, 30, 57 natural strain energies, 96 natural straining modes, 56 natural thermal load, 91, 200 natural thermal load vector, 89,

198 Newton-Raphson, 268 normal, 2 normalized central displacement,

231 normalized deflection, 227 normalized form, 231 normalized solutions, 221 normalized stresses, 230 normalized values, 224 numerical quadratures, viii

opposite, 41 optimization, viii optimization procedures, 250 orthotropic layers, 230 orthotropic shell, 220

parallelization, 263, 264 partly simplified geometrical stiff-

ness, 108, 110 path, 267 physical lumping, 136 physical problem, 314 physical vectors, 11 Piecewise summation, 172 pinched cylinder, 220

338 INDEX

plane stress triangle, 31 Polymer composites, 37 postprocessing, 95 prestress state, 263 principal of virtual work, 13 principle of stationary energy, 15

quasi-isotropic, 119 quasi-isotropic laminate, 227 quasi-isotropic laminates, 44

rectangular laminate, 234 reference solution, 219 regional discretization, 17 reinforcement, 37 representative volume, 45 rigid body modes, vii, 52 rigid body moments, 101, 208, 212,

215 rigid body rotation, 215 rigid body rotations, 101, 208, 212,

221 rigid diaphragms, 250 rocket-like composite shell, 250 rosettes, 19 rotational geometrical stiffness, 103,

215

sandwich plate, 219, 229 shear, 2 shear correction factor, 78, 219 shear correction factors, 189, 219,

233 shear deformations, 175 shear locking, 48, 166 shear strains, 8 shell element, 32, 163 shell elements, 224 shell structures, 136 simplified geometrical stiffness, 105,

216

sinusoidal loading, 229 skyline, 261 skyline storage, 310 square laminate, 234 stationary, 15 statistics report, 261 stiffening or softening effect, 106 stiffness matrix, vii storage scheme, 310 strain energy, 84, 145 strain energy density, 11 strain modes, vii strain operator matrix, 29 strain rossete, 140 straining modes, vii, 163 stress tensor, 10 stress vector, 1 structural laminate, 41 structural mechanics, 21 surface forces, 2 surface load, 196 symbolic computation, 216 symmetric, 42 symmetric laminate, 42 symmetrical, 8 symmetrical bending mode, 60 symmetrical lamination, 93 symmetry, 220

tangent stiffness, 100 temperature, 89, 240 temperature increase, 239 tensor, 5, 8 tetrahedron element, 32 thermal load vector, 198 thermal strain, 200 thermoelastic coefficients, 90, 198 thermomechanical buckling, 239 three dimensional solution, 230 total natural strain, 144

total natural strains, 140 total potential energy, 15 total strain, 167 traction, 1 transformation matrix, 211 transverse shear deformation, 219 transverse shear strains,. 148 transverse shear stresses, 204 triangle, 18 truss element, 31

uniform pressure, 219 uniform pressure load, 227 unit extension, 57 unit matrix, 212 unit vector, 322 unstable branches, 267

vectorization, 263, 264 vertex rotation, 191 vertical displacement, 221 virtual displacements, 12 virtual work, 12, 13, 181, 196

warping, 224 weight sensitive structures, 38

INDEX 339

Mechanics SOUD MECHANICS AND ITS APPLICATIONS

Series Editor: G.M.L. Gladwell

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