Document Text Contents
Page 2
Stresses in Shells
By
Wilhelm Fliigge
Dr.·lng .• Professor of Engineering Mechanics
Stanford University
With 244 Figures
Springer-Verlag Berlin Heidelberg GmbH
Page 255
5.4 LOADS AT THE EDGES c/J = const 243
two damping exponents is then Xl = 5.1, and a disturbance which
begins with the value 1 at the edge cf> = 0 will have decayed to
exp (-5.1 X 2.094) = 0.23 X 10-4 at cf> = 120°. For higher harmonics,
n> 1, the decay will be even greater. If one requires a decay to only
0.01, one may allow lfa to be as great as 2.6.
Quite to the contrary, the barrel vault, fig. 10 b, may have lfa = 2.5,
hence A. = 1.26 for the first harmonic. Fig. 9 yields Xl = 2.23, and
over an angle of 60° = 1.047 an edge disturbance will decay from 1
to exp( -2.23 X 1.047) = 0.097 = 10%. This, of course, is still very
much and will have to be considered when the boundary conditions
at the other edge are formulated. There will be even less decay if the
shell is thicker, but, still, for the higher harmonics, n = 4, 5, ... , the
values of A. and hence of Xl are high enough to make the corresponding
edge disturbances more or less local.
5.4.1.2 One Boundary Only
In those cases where the disturbance is localized at one edge, the
general solution may be considerably simplified. If we measure the
angular coordinate from the edge under consideration, then the terms
j = 5, 6, 7, 8 in the preceding formulas evidently are unsuitable, since
they describe just the contrary of a local disturbance: stresses and
displacements that increase exponentially the farther away we go
from the edge cf> = o. Therefore, these terms must be dropped, and
we are left with the following formulas:
Un = e-",<i>[(A I + A!) COSt-tlcf> + i(AI - A 2) Sint-tlcf>]
+ e-".<i>[(A3 + A4) COSt-t2cf> + i(A3 - A4) sint-t2cf>]'
Vn = e-",<i>[(B l + B 2) COSt-tlcf> + 'i(B l - B2) sin t-tlcf>]
+ e-"·<i> [(Bs + B 4) COSt-t2cf> + i(Bs - B 4) sint-t2cf>]'
Wn = e-",<i>[(Ol + O2) COSt-tlcf> + i(Ol - O2) sint-tl cf>]
+ e-".<i>[(03 + 0 4) COSt-t2cf> + i(03 - 04,) sint-t2cf>]·
Obviously, the coefficients appearing in these formulas must all have
real values, and they may be expressed by eqs. (28), (29) on pp. 229-230,
substituting, of course, for the IX'S and {J's the quantities defined on p. 241.
When we put the expressions for Un, Vn , Wn into the formulas (34)
(omitting the summation) and then go back to the elastic law (9) and
the equations (ld, e) we see that the n-th harmonics of all displace-
ments and stress resultants assume the form:
1= c[ e-",<i> (al 01 + a2 ( 2 ) COSt-tl cf> + (a1 02 - a2 ( 1) Sint-tl cf» (40)
... (- - - - cos ;. x + e-"'" (a3 Os + a4 04) COSt-t2cf> + (a304 - a4 Os) sint-t2 cf>)] . -. sIn a
16*
Page 256
244 CHAP. 5: CIRCULAR CYLINDRICAL SHELLS
Table 2. Oylinder Loaded along a Generator
c
u 1 "'1
--
V 1 PI
W 1 1
w 1 -Ill
---
N+ DJa 1 + k - (Ill PI + I'tll2) - P A (Xl + k(lI; -I'D
N", DJa - A (Xl + P - P(lIlPl + 1'1Pa) + k AD
---
N+",
D(I- p)
- (1 + k)(lIl (\1 + 1'1 (\2) + fpl - kAlIl
2a
N",+
D(I- ,,)
- (Ill (Xl + Pl(i2) +(1 + k)A.Pl + kAlIl 2a
---
M+ KJa2 1 + (llf - I'D - " A2
M", KJa2 - A2 + A (XI + P(II~ -I'D + " (III PI + PIP.)
---
M+",
K(I- ,,)
- 2AII1 - (Ill (Xl + 1'1 (\2) - API 2a2
Mz+
K(I- p)
- A (Ill + PI) a2
---
Q+ KJa 3 III (AD - I - IIf + 31'D + (1 - ,,) A2Pl
---
Qz
K - 2A3 + 2A(1I1 -I'D + 2A2(i1
2a3 + (1 - p)[(111 -I'D iiI + 2111 1'1 iia] + (1 +,,) A(1I1PI + I'IPt)
---
S+
K
I
2111 [-1 + (2 - ,,) A2 - xi + 31';] + (1 - p) A(1I1 (il + 1'1 (X.)
2a3 + 3(1- ,,)A'PI
The factor c and the coefficients a l , ag are given in Table 2. To find a3
and a4 , we use the same formulas, only changing the subscript 1 to 2
for", and fl, and the subscripts 1, 2 to 3, 4 for (X, fJ.
To solve a specific problem, we have to proceed in the following
way: We choose the order n of the harmonic which we want to in-
vestigate and compute from the dimensions of the shell (a, t, l) the
parameters k and A. We then find "'1' fl,1' "'2' fl,2 from eq. (38) and (x,,
p, (j = 1 ... 4) by solving the first two of eq. (37) for A" Bj with
0, = 1. We are then prepared to establish the boundary conditions.
From the table we find the boundary values of those forces and dis-
placements which appear in the boundary conditions, and using the
numerical values of the aj, p" we obtain four linear equationsfor 0 1 ,
Page 510
498 INDEX
N
Negative curvature 78, 181
Neutral equilibrium 408, 410, 411
Nodal lines 437
Nonconvex shell 92
Nonlinear theory 466
Nonregular dome 155
Nonuniform axial compression 451
Normal force 3
Normal point load 62
Numerical integration 73, 97
o
Oblique coordinates 14
Octagonal dome 154
Octagonal tube 162
Ogival dome
deformation 98
membrane forces 30
One-sheet hyperboloid 78
Oscillatory solutions 387
p
Paraboloid of revolution 71, 93, 169
Parallel circle 18
Particular solution
cylindrical shell 221
shell of revolution 353
Phase angle 275, 338
Piano hinge system 307
Pipe of circular profile 138
Plate bending moment 307
Plywood shell 293
Point load
see concentrated force
see concentrated couple
Point moment
see concentrated couple
Pointed shell 29, 72, 75, 98
POISSON'S ratio 86
Polygonal dome 140, 205
Polygonal shell 171
Potential energy 410
Pressure vessel 26, 116, 194, 201,
204
Prime-and-dot notation 88, 208, 313,
397
Principal directions 11
Principal forces 11
Principal sides 186
Principle of virtual displacements 410
Prismatic structure 156
R
Radial line load
on barrel vault 257
on infinite cylinder 277
Radius of curvature, ellipsoid 27
Reciprocity of deformations 104. 376
Redundant edge moment 312
Reference vectors 417
Regular dome 141
Regular load 141
Reinforcing ring 348
Relaxation method 175
Relaxation pattern 176
Relaxation table 177
Residual 175
Ridge beam 148, 183
Rigid testing machine 471
Rigid-body rotation 91
Rigidity 303, 306
Rigidity moment 298
Ring 110, 299, 478
see also foot ring, lantern ring
Ring of radial forces 281
Ring rib 295
S
Secondary sides 186
Self-equilibrating edge load 75
Semi-infinite cylinder 229, 275
Shallow spherical shell 343, 347
Sharp edge 344
Shear and axial compression 439, 448
Shear buckling 436
Shear deformation 139
Shear edge 187
Shear force 4
Shear load 258
Shear modulus 86
Shear strain 85
Short cylinder 449
Sign convention, MOHR's circle 13
Simplified barrel-vault theory 253
Simply supported edge 232
Singularity of stress system
elliptic paraboloid 174
pointed shell 31, 73
polygonal dome 155
toroid 32
see also concentrated couple, con-
centrated force
Sixth equation of eqUilibrium 127, 319
Skew components 81
Page 511
INDEX 499
Skew cylinder 206
Skew fiber force 15
Skew force 14, 166
Skew shearing force 15, 81
Skew vault 207
Slightly dished circular plate 347
Sludge digestion tank 374
Small hole 348
sphere
axisymmetric stress system 24, 321
buckling 472
deformation 90
dome 24
gas tank 64
tank bottom 35,334
thermal stress 339
unsymmetric stress system 49, 380
water tank 33, 339
Spherical zone 327
Splitting condition 363, 378
Splitting of a differential equation 288,
324,361
Square dome 154
Stable equilibrium 407
Statically detElrminate shell 105
Statically indeterminate
cylinder 138, 280
folded structure 307
pressure vessel 341
shell of revolution 105
water tank 280
Strain
cylinder 130, 212
shell of revolution 85, 317
Strain energy 100, 411
Stress discontinuity 190
Stress function 168, 470
Stress resultant' 3
Stress trajectories 11, 52, 57
Stringer 295,299, 304
Surface load 353
Surface of translation 174
T
Tangential line load 258
Tangential point load 54. 60
Tank bottom
conical 38, 375
elliptic 36, 195
spherical 35, 38, 334
Tank of uniform strength 39
Tank on point supports 64
Thermal stress 339
Thick shell 223, 348
Thin shell 330, 348, 363
THOMSON functions 289, 292, 345, 351,
373
Toroidal shell 32, 99
Torsion 436
Torsional rigidity 305
Trajectories 11, 52, 57
Transfer of edge loads 188
Transverse force 4
Triangular shell 170
Twist 214
Twisting moment 6
Two-way compression 423
U
Unstable equilibrium 408, 411
V
Variable wall thickness
cone 378
cylinder 287
shell of revolution 361
Vaulted hip roof 148
W
Water tank 33, 195., 277, 291
Weak- singularity 73
Weight loading 471
Wind load 50
Stresses in Shells
By
Wilhelm Fliigge
Dr.·lng .• Professor of Engineering Mechanics
Stanford University
With 244 Figures
Springer-Verlag Berlin Heidelberg GmbH
Page 255
5.4 LOADS AT THE EDGES c/J = const 243
two damping exponents is then Xl = 5.1, and a disturbance which
begins with the value 1 at the edge cf> = 0 will have decayed to
exp (-5.1 X 2.094) = 0.23 X 10-4 at cf> = 120°. For higher harmonics,
n> 1, the decay will be even greater. If one requires a decay to only
0.01, one may allow lfa to be as great as 2.6.
Quite to the contrary, the barrel vault, fig. 10 b, may have lfa = 2.5,
hence A. = 1.26 for the first harmonic. Fig. 9 yields Xl = 2.23, and
over an angle of 60° = 1.047 an edge disturbance will decay from 1
to exp( -2.23 X 1.047) = 0.097 = 10%. This, of course, is still very
much and will have to be considered when the boundary conditions
at the other edge are formulated. There will be even less decay if the
shell is thicker, but, still, for the higher harmonics, n = 4, 5, ... , the
values of A. and hence of Xl are high enough to make the corresponding
edge disturbances more or less local.
5.4.1.2 One Boundary Only
In those cases where the disturbance is localized at one edge, the
general solution may be considerably simplified. If we measure the
angular coordinate from the edge under consideration, then the terms
j = 5, 6, 7, 8 in the preceding formulas evidently are unsuitable, since
they describe just the contrary of a local disturbance: stresses and
displacements that increase exponentially the farther away we go
from the edge cf> = o. Therefore, these terms must be dropped, and
we are left with the following formulas:
Un = e-",<i>[(A I + A!) COSt-tlcf> + i(AI - A 2) Sint-tlcf>]
+ e-".<i>[(A3 + A4) COSt-t2cf> + i(A3 - A4) sint-t2cf>]'
Vn = e-",<i>[(B l + B 2) COSt-tlcf> + 'i(B l - B2) sin t-tlcf>]
+ e-"·<i> [(Bs + B 4) COSt-t2cf> + i(Bs - B 4) sint-t2cf>]'
Wn = e-",<i>[(Ol + O2) COSt-tlcf> + i(Ol - O2) sint-tl cf>]
+ e-".<i>[(03 + 0 4) COSt-t2cf> + i(03 - 04,) sint-t2cf>]·
Obviously, the coefficients appearing in these formulas must all have
real values, and they may be expressed by eqs. (28), (29) on pp. 229-230,
substituting, of course, for the IX'S and {J's the quantities defined on p. 241.
When we put the expressions for Un, Vn , Wn into the formulas (34)
(omitting the summation) and then go back to the elastic law (9) and
the equations (ld, e) we see that the n-th harmonics of all displace-
ments and stress resultants assume the form:
1= c[ e-",<i> (al 01 + a2 ( 2 ) COSt-tl cf> + (a1 02 - a2 ( 1) Sint-tl cf» (40)
... (- - - - cos ;. x + e-"'" (a3 Os + a4 04) COSt-t2cf> + (a304 - a4 Os) sint-t2 cf>)] . -. sIn a
16*
Page 256
244 CHAP. 5: CIRCULAR CYLINDRICAL SHELLS
Table 2. Oylinder Loaded along a Generator
c
u 1 "'1
--
V 1 PI
W 1 1
w 1 -Ill
---
N+ DJa 1 + k - (Ill PI + I'tll2) - P A (Xl + k(lI; -I'D
N", DJa - A (Xl + P - P(lIlPl + 1'1Pa) + k AD
---
N+",
D(I- p)
- (1 + k)(lIl (\1 + 1'1 (\2) + fpl - kAlIl
2a
N",+
D(I- ,,)
- (Ill (Xl + Pl(i2) +(1 + k)A.Pl + kAlIl 2a
---
M+ KJa2 1 + (llf - I'D - " A2
M", KJa2 - A2 + A (XI + P(II~ -I'D + " (III PI + PIP.)
---
M+",
K(I- ,,)
- 2AII1 - (Ill (Xl + 1'1 (\2) - API 2a2
Mz+
K(I- p)
- A (Ill + PI) a2
---
Q+ KJa 3 III (AD - I - IIf + 31'D + (1 - ,,) A2Pl
---
Qz
K - 2A3 + 2A(1I1 -I'D + 2A2(i1
2a3 + (1 - p)[(111 -I'D iiI + 2111 1'1 iia] + (1 +,,) A(1I1PI + I'IPt)
---
S+
K
I
2111 [-1 + (2 - ,,) A2 - xi + 31';] + (1 - p) A(1I1 (il + 1'1 (X.)
2a3 + 3(1- ,,)A'PI
The factor c and the coefficients a l , ag are given in Table 2. To find a3
and a4 , we use the same formulas, only changing the subscript 1 to 2
for", and fl, and the subscripts 1, 2 to 3, 4 for (X, fJ.
To solve a specific problem, we have to proceed in the following
way: We choose the order n of the harmonic which we want to in-
vestigate and compute from the dimensions of the shell (a, t, l) the
parameters k and A. We then find "'1' fl,1' "'2' fl,2 from eq. (38) and (x,,
p, (j = 1 ... 4) by solving the first two of eq. (37) for A" Bj with
0, = 1. We are then prepared to establish the boundary conditions.
From the table we find the boundary values of those forces and dis-
placements which appear in the boundary conditions, and using the
numerical values of the aj, p" we obtain four linear equationsfor 0 1 ,
Page 510
498 INDEX
N
Negative curvature 78, 181
Neutral equilibrium 408, 410, 411
Nodal lines 437
Nonconvex shell 92
Nonlinear theory 466
Nonregular dome 155
Nonuniform axial compression 451
Normal force 3
Normal point load 62
Numerical integration 73, 97
o
Oblique coordinates 14
Octagonal dome 154
Octagonal tube 162
Ogival dome
deformation 98
membrane forces 30
One-sheet hyperboloid 78
Oscillatory solutions 387
p
Paraboloid of revolution 71, 93, 169
Parallel circle 18
Particular solution
cylindrical shell 221
shell of revolution 353
Phase angle 275, 338
Piano hinge system 307
Pipe of circular profile 138
Plate bending moment 307
Plywood shell 293
Point load
see concentrated force
see concentrated couple
Point moment
see concentrated couple
Pointed shell 29, 72, 75, 98
POISSON'S ratio 86
Polygonal dome 140, 205
Polygonal shell 171
Potential energy 410
Pressure vessel 26, 116, 194, 201,
204
Prime-and-dot notation 88, 208, 313,
397
Principal directions 11
Principal forces 11
Principal sides 186
Principle of virtual displacements 410
Prismatic structure 156
R
Radial line load
on barrel vault 257
on infinite cylinder 277
Radius of curvature, ellipsoid 27
Reciprocity of deformations 104. 376
Redundant edge moment 312
Reference vectors 417
Regular dome 141
Regular load 141
Reinforcing ring 348
Relaxation method 175
Relaxation pattern 176
Relaxation table 177
Residual 175
Ridge beam 148, 183
Rigid testing machine 471
Rigid-body rotation 91
Rigidity 303, 306
Rigidity moment 298
Ring 110, 299, 478
see also foot ring, lantern ring
Ring of radial forces 281
Ring rib 295
S
Secondary sides 186
Self-equilibrating edge load 75
Semi-infinite cylinder 229, 275
Shallow spherical shell 343, 347
Sharp edge 344
Shear and axial compression 439, 448
Shear buckling 436
Shear deformation 139
Shear edge 187
Shear force 4
Shear load 258
Shear modulus 86
Shear strain 85
Short cylinder 449
Sign convention, MOHR's circle 13
Simplified barrel-vault theory 253
Simply supported edge 232
Singularity of stress system
elliptic paraboloid 174
pointed shell 31, 73
polygonal dome 155
toroid 32
see also concentrated couple, con-
centrated force
Sixth equation of eqUilibrium 127, 319
Skew components 81
Page 511
INDEX 499
Skew cylinder 206
Skew fiber force 15
Skew force 14, 166
Skew shearing force 15, 81
Skew vault 207
Slightly dished circular plate 347
Sludge digestion tank 374
Small hole 348
sphere
axisymmetric stress system 24, 321
buckling 472
deformation 90
dome 24
gas tank 64
tank bottom 35,334
thermal stress 339
unsymmetric stress system 49, 380
water tank 33, 339
Spherical zone 327
Splitting condition 363, 378
Splitting of a differential equation 288,
324,361
Square dome 154
Stable equilibrium 407
Statically detElrminate shell 105
Statically indeterminate
cylinder 138, 280
folded structure 307
pressure vessel 341
shell of revolution 105
water tank 280
Strain
cylinder 130, 212
shell of revolution 85, 317
Strain energy 100, 411
Stress discontinuity 190
Stress function 168, 470
Stress resultant' 3
Stress trajectories 11, 52, 57
Stringer 295,299, 304
Surface load 353
Surface of translation 174
T
Tangential line load 258
Tangential point load 54. 60
Tank bottom
conical 38, 375
elliptic 36, 195
spherical 35, 38, 334
Tank of uniform strength 39
Tank on point supports 64
Thermal stress 339
Thick shell 223, 348
Thin shell 330, 348, 363
THOMSON functions 289, 292, 345, 351,
373
Toroidal shell 32, 99
Torsion 436
Torsional rigidity 305
Trajectories 11, 52, 57
Transfer of edge loads 188
Transverse force 4
Triangular shell 170
Twist 214
Twisting moment 6
Two-way compression 423
U
Unstable equilibrium 408, 411
V
Variable wall thickness
cone 378
cylinder 287
shell of revolution 361
Vaulted hip roof 148
W
Water tank 33, 195., 277, 291
Weak- singularity 73
Weight loading 471
Wind load 50
