28 Roark Flat Plates

Page 28 - 1

ROARK FLAT PLATES

28

EXHAUST STACK

Christy 10:59 3/21/12

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers 89409861.xls.ms_office A

B

C

D

E

F

G

H

I

J

K

L

M

FLAT CIRCULAR PLATES / CONSTANT THICKNESS Roark, 5th Edition Table 24 pg 334 a 2.400 in radius to outer edge Check plate bending with several models b 0.880 in radius to inner edge from Roark and Young page 334 ro 1.600 in radius to location of w APPROXIMATE RADIUS OF BOLT HEAD P 42.90 k For allowable P allowed working calculated in the Bolt Group Pullout template trib length 10.05 in ALONG EDGE OF WASHER a 2.400 w 4267 lb/in point load / circumference line load ro 1.600 2 q_uniform 2371 lb/in uniformly distributed load r_

1.00 in

t v E

1.000 in 0.285 unit 29000 ksi

ro / r

1.600

radial location of evaluated quantity

0.40 0.148945

C_3' C_3'' C_3

1.14 -0.87 0.024992

row 20

b 0.880

thickness of plate poisson ratio Young's modulus of elasticity

CONSTANTS........................... C_1' 0.24 0.42 C_1 0.658320 unit C_2' C_2

N

G_3' G_3 G_6' G_6 G_9' G_9

-1.67 0.0000 0.62 0.0000 -0.58 0.0000

L_3' L_3

0.59 -0.56 0.0050 0.26 0.0426

row 30

Figure 28-1 PLAN VIEW OF PLATE/WASHER row 40

C_4

1.210583

L_6' L_6

C_5

0.432778

L_9'

C_6' C_6

1.14 0.104596

L_9

0.26 0.10 0.2399

C_7

1.084433 unit

L_11' L_11'' L_11

1.79 -4.35 0.0004

C_8' C_8

0.10 0.690564

C_9' C_9'' C_9

0.64 0.15 0.293091

L_14' L_14'' L_14

0.80 -0.72 0.0051

D

2630314 lb-in

row 50

L_17' 0.86 L_17'' 1.52 plate constant L_17 0.0451 29,000 * 1.00^3 / ( 12 * ( 1 - 0.285^2) ) * 1000 lb/k row 60

F_1' F_1 F_2' F_2 F_4' F_4

0.08 0.13 0.97 0.01 0.63 0.72

F_5

0.11

F_6

0.01

F_7

0.12

F_9'

0.08 0.00 0.07

lb/in2 = k=

psi 1000 Lbs.

row 70

F_9 r ro

0 logic

1.000 > 1.600 singularity factor row 80

Page 28 - 2

ROARK FLAT PLATES

28

EXHAUST STACK

Christy 10:59 3/21/12

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers 89409861.xls.ms_office A

B

C

D

E

F

G

H

I

J

K

L

M

N

FLAT CIRCULAR PLATES / CONSTANT THICKNESS -- Continued Roark Table 24, Case 2k: Outer edge free, inner edge simply supported… page 342 INPUTS FROM ABOVE a 2.400 in radius to outer edge b 0.880 in radius to inner edge ro 1.600 in radius to location of w q_uniform

2371 lb/in2

uniformly distributed load

not used in this analysis

thickness of plate Poisson ratio Young's modulus of elasticity

0.88

t v E

1.00 in 0.29 unit 29000 ksi

yb

0

Mra

0 per Roark

Mrb

0

Qa

0

qb'

-0.01149

qb''

0.22204

row 90

2371

qb Qb

-0.00216

=-q*(a^3)/(D*C7) =(C_9/(2*A*B))*(A^2-RO^2) -0.00255

-0.00255 radian 4310 lb/in

shear (Q_uniform / (2 * b)) * ( a2 - ro2 ) ( 2,371 /(2 * 0.880 )) * ( 2.400^2 - 1.600^2 )

ya'

-0.0040

ya''

0.0006

ya''' ya

2.40 Figure 28-2 PLATE LOADING AND DEFLECTION

0.0000 -0.0035 in

deflection unit shear unit applied line moment

row 110 lb / inch of circumference in-lb / inch of circumference

qa'

-0.00309

Q Mo

qa''

0.00099

Mr

unit radial bending

in-lb / inch of circumference

qa'''

0.00006

Mt

unit tangential bending

in-lb / inch of radius

qa

-0.00216 radian

r Mt

1.00 in -5231 lb-in/in

radial location of quantity being evaluated -- input from above theta * D * (1 - v^2) /r + v * Mr -0.00216 * 2,630,314 * (1 - 0.285^2) /1.600 + 0.285 * 0 row 120

t S fb

1.0000 in 0.16667 in3 -31 k/in2

from input above 1 * t2 / 6 Mt / S

row 130 CANTILEVER PLATE COMPARISON q 2371 psi L 1.96 in M_pl t_pl S_pl fb

4554 1.00 0.16667 27.32

lb-in in in3 ksi row 140

Figure 28-3 CANTILEVERED ANCHOR PLATE row 150

Page 28 - 3

ROARK FLAT PLATES

28

EXHAUST STACK

Christy 10:59 3/21/12

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers 89409861.xls.ms_office A

B

C

D

E

F

G

H

I

J

K

L

M

N

FLAT CIRCULAR PLATES / CONSTANT THICKNESS -- Continued Roark Table 24, Case 1a: Outer edge simply supported, inner edge free … page 335 yb' 0.1406 yb

0.00 in

theta_b' theta_b

-4,267 * 2.40^3 / 2,630,314 * 0.1406 lb/in in^3 / lb-in

0.22 -0.001

Mrb' Mrb

0.00 0.00

Qb' Qb

0.00 0.00 lb/in

y' y'' y''' y'''' y

0.00 0.00 0.00 0.00 0.00

theta' theta'' theta''' theta'''' theta

row 160

row 170

0.00 0.00 0.00 0.00 -0.001 Figure 28-4 PLATE LOADING

Mr' Mr'' Mr''' Mr'''' Mr

-266.80 0.00 0.00 0.00 -266.80 lb-in

row 180

Mt

-1576.45 lb-in tangential moment / inch of circumference MOMENT

S provided fb

0.166667 in3/in 1.0000^3 /6 -9459 lb/in2 /in stress in plate -9.5 ksi

Q theta_a Qa

=B164*b/r_-W*ro/r_*___*_R_RO_

row 190

-2844.89 lb / inch of circumference SHEAR

Figure 28-5 PLATE LOADING

Mt

row 200

Mt

yb

yb bending in twisting on plate

Figure 28-6 PLATE TWISTING

bending in plate / washer

Figure 28-7 PLATE BENDING

row 210

row 220

Page 28 - 4

ROARK FLAT PLATES

28

EXHAUST STACK

Christy 10:59 3/21/12

ENGINEERING with the SPREADSHEET Copyright 2006 American Society of Civil Engineers 89409861.xls.ms_office A

B

C

D

E

F

G

H

I

J

K

L

M

N

FLAT CIRCULAR PLATES / CONSTANT THICKNESS -- Continued Values from above for Roark Table 24, Case 1a page …. 362 a 2.4 in radius to outer edge ro 1.6 in radius to location of w q 2371 lb/in^2 t v E

1 in 0.285 unit 29000000 psi

thickness of plate poisson ratio Young's modulus of elasticity

Roark, Table 24 Case 10a: Outer edge simply supported … page 363 ya 0 per Roark

Figure 28-8 PLATE BENDING DIAGRAM supported edge

Mra

simply supported edge

L_11 L_17 yc

0 0.000 0.045 -0.0005 in

from above from above -q * a4 /(2 * D) * (L_17 /(1 + v) - 2 * L_11) -2,371 * 2.400^4 /(2 * 2,630,314) * ( 0.04514 /(1 + 0.285) - 2 * 0.00044) deflection at center of plate

Mc S fb

616 lb-in 0.166667 in3 3698 psi

q * a2 * L_17

qa Qa

0.0007 radians -1580.50 L/9,374

q/(8*D*a*(1-v)) * (a^2-ro^2)^2 -q/(2*a)*(a^2-ro^2)

row 240

M at center of plate shear force

row 250

Figure 28-9 PLATE SUPPORT Roark Table 24, Case 10b: Outer edge fixed supported … page 363 ya 0 per Roark theta_a 0

supported edge fixed edge

L_11 L_14 L_17 yc Mc Mra

deflection at center of plate M at center of plate M at fixed edge of plate

S_plate M fb

0.000 0.005 0.045 0.000 in 616.37 lb-in -526.832 lb-in L/75,994

0.167 in^3 616 lb-in 4 k/in^2

from above from above from above -q*a^4/(2*D)*(L_14-2*L_11) q*a^2*L_17 -q/(8*a^2)*(a^2-ro^2)^2

row 260

b*t^3/6

row 270 Figure 28-10 VIEW OF PLATE DEFLECTION

row 280

row 290