##### 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