American Society of Civil Engineers.

# Transactions of the American Society of Civil Engineers (Volume 81) online

. (page 149 of 167)
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Font size Horizontal GIrde

Reactions of Vertical Glrdor on Horizontal Girders

A Sill immovable, Plotted thuÂ»_

(inches)'
120 000

lao 000
laoooo

120 000
IW 000
140 OOO
136 000
166 000
161 000
161 000
190 000
100 000
190 000
190 000
190 000
1S6 000

rose-section, in of load
square inches. inches.

49'6"GATE FOR 110 -FOOT LOCK

Horizontal Girders Vertical Girder

120 000
UDOOO
120 000
120 000
140 000
140 000
U6 000
1B6 000
161 000

uoooo

Area of Eccentricity i
cross-section, in of load, in U-i
square inches. inches. I

Effective
Moment of Water Pressure

Inertia in on Gate

T-(inches1Â»-

Loads on Horizontal Girders
A Sill immovable. Vertical girder acting. Plotted tbui
It removed, â– â€˘ â€˘*

Reactions of Vertical Girder on Horizontal Girders

Deflection of Miter Post under Water Pressure

removed

-T^

6000 10 000 15 000 20 000 \ Total 70 670 16 000 10 000 6 000 6 000

Pouuds per Linear Foot of Girder Notet-Thisload is Founds per Linear Foot of Girder ^Note:- This reaction

carried entirely by !â€˘ carried by the sill .

Fio, 7. the sill. Fig. 8.

â€˘â– 7r>; *^ â–  '

;0^-OFf HO^ 3TA0VTT

1o taAmo}P

Ot' O'it

â€˘â€˘ . Oil

DISTHIBUTION OF STKESSES IN LOCK-GATES 1641

Substituting for X^^ its value from Equation (17), we have

= 4 365 I (1 593.75 â€” 16.66 .Y^ . . . â€” 2 X^^)

H

d I.

^.5 (1593.75- 16.66 Xo . . . -2X^3)

For the different independent variables, we have then total values
for H:
Xq as variable :

77 = 4 365 (â€” 16.66) (1593.75 â€” 16.66 X^ ... â€” 2 Z.g) L =
(â€” 115 898 930 + 1 211 500 X^-\-l 122 800 Z, + 1 034 000 Z^
+ 945 370 Z, 4- 856 650 Z^ + 775 900 Z. + 694 920 Zg
+ 614 100 Z, + 533 330 Zg -f 452 500 Zg + 371 700 Z^o
+ 290 883 Z^i + 218 163 X^^ + 145 400 Z^g) L

Zj^ as variable:

if = 4 365 (â€” 15.44) (1593.75 â€” 16.66 Z^ ... â€” 2 Z.g) L =
(â€” 107 411 700 + 1 122 800 Z, + 1 040 590 Z, + 958 400 X^
-\- 876 140 .Y3 + 793 920 Z^ + 719 110 Z, + 644 030 Z^
+ 569 090 Z, + 494 280 X^ + 419 330 Zg + 344 460 X^^
+ 269 580 Z,^ + 202 190 Z,^ + 134 790 Z.g) L

and similar expressions when the differentiation is made with reference
to the other independent variables, X^ . . . Z^g.

These values are not copied out in full, as the work is very
voluminous. Similarly, the method is only indicated in the case of
Arch 15.

Arch 15. â€” For this the general expression for H is :

H = 6 802y^' (P,, + X,,) ^^^'l^^^''^ d I

r^ 5 (796.87 + X,^)

= 6 802 / (796.87 + X,^) *- ^'7 '' ^ I-
Jo o X

If we substitute for Z^g the value from Equation (18), we have

If = 6 802 r (796.87 + 15.66 A^,, + 14.44 X^ + 13.22 X^

+ 12 X3 + 10.78 X^ + 9.67 .Y^ + 8.56 .Y^ + '^â€˘^4 Xj
+ 6.33 Xg + 5.22 Xg + 4.11 X^^ + 3 X,, + 2 X^, + .Y13)
g (796.87 + 15.66 X,, . . . X,^) ^
5 X

1642 DISTRIBUTION OF STRESSES IN LOCK-GATES

whicii, for the different independent variables, becomes :

Xq as variable:

IT = 6 802 (16.66) (796.87 + 15.66 Z^ . . . X^^) L = (84 918 190
+ 1 669 518 Z^ + 1 539 272 Z^ + 1 409 026 Z^ + 1 278 780 Zg
+ 1 148 534 Z^ + 1 030 128 Z^ + 911 723 Zg + 793 317 Z^
+ 674 912 Zg + 556 506 Z^ + 438 101 X^^ + 319 695 Z^^
-f 213 130 Z^,2 + 106 565 X^^) L

and similar expressions when the differentiation is made with reference

to the other independent variables, Z^ . . . Z^j.

Vertical Girder.
The second term of Equations (15) gives that part of the final equa-
tions due to the work in the vertical girder. If this term, for con-
venience, is called V, its value for the several panels and variables will
be the following:

Panel 1. â€” Zg as variable:

7 = 0.0871 X2 Zo
and the values for the other variables vanish.
Panel 2. â€” Z^ as variable:

7 = + 0.6097 L- Zo -f 0.2175 L'- Z,
X^ as variable:

7 = 0.2176 L2 x^ -f 0.0871 L^ X^
the values for the other variables being equal to 0.

Panels 3-15. â€” The corresponding expressions for the remaining
panels of the vertical girder will be of the same general form as those
deduced for the first two panels.

In obtaining the value of 7 for the bottom panel (15) the variable,
X, was, of course, expressed in terms of Zg . . . Z^g.

Method of Obtaining Final Equations.

The final equations are obtained from the values for H and for 7
above by adding all the terms which correspond to the same independent
variable, X^, X^ . . . Z^g, and equating them to zero.

In the case of no contact at the sill, the values of H for the bottom
girder (15) must be included, as this girder then carries the full load
which comes on it.

For perfect contact, the bottom girder does no work as an arch, and
hence the H's for this girder must be omitted.

DISTRIBUTION OF STRESSES IN LOCK-GATES 1643

If the contact is such that the bottom girder does some work in arch
action, a portion of its H's may be counted. In other respects, the
method of calculation is independent of the degree of sill contact.

It should be noted that L (the length of the leaf, 787 in.) enters
all values of H in the first power and those of V in the second. By
dividing the final equations by L, it will be eliminated from the H
terms, but will remain in the first power in those corresponding to V.

From their method of derivation, the resulting final equations
should be symmetrical as to coefficients, and, in the equations for
the solution of the problem, an average has been used where two
coefficients which should be identical have differed by a slight amount.
There have been no large discrepancies, proving to a considerable extent
the accuracy of the work.

Only five significant figures were retained.

Final Equations of Conditions. â€” The final equations, as thus ob-
tained, are the following :

(Case A). â€” No Contact at Sill.
Zq as variable:

3 007 500 Zo + 2 762 700 Z, + 2 530 200 Z^ + 2 298 200 X^
+ 2 066 800 Z^ + 1 856 900 Z^ + 1 647 600 Z^ + 1 439 300 Z,
+ 1 232 100 Z3 + 1 025 700 Z., + 820 580 X,^ + 616 660 Z,,
+ 434 260 Z,, + 252 900 Z.g â€” 30 032 300 = 0.

Z^ as variable:

2 762 700 Zo + 2 561000 Z, + 2 335 100 Z, -f 2121400 Z3

+ 1 908 200 Z^ + 1 714 800 Z^ + 1 521 700 Z^ + 1 329 500 Z,
+ 1 138 200 Z3 + 947 700 Z^ + 758 250 X^, + 569 920 Z,,
+ 401420 Zj, + 233 890 X^^ â€” 26 344 200 = 0.

Zj as variable :

2 530 200 Zq + 2 335 100 X\ + 2 151900 Z^ + 1944 600 Z3
+ 1 749 600 Z4 + 1 572 600 Z^ + 1 395 800 Z^ + 1 219 700 Z,
+ 1 044 400 Zg + 869 700 Zg + 695 950 Z.^ + 523 170 Z.^
+ 368 580 Z,, + 214 860 Z.g â€” 22 574 900 = 0.

Z3 as variable :

2 298 200 Zq + 2 121400 Z, + 1944 600 Z^ + 1780 000 Z3
+ 1 591 000 Z^ + 1 430 400 Z^ + 1 269 900 Zg + 1 109 900 Z,
+ 950 550 Zg + 791 700 Z^ + 633 630 Z.^ + 476 420 Z.^
+ 335 730 Z12 + 195 820 Z^3 â€” 18 806 300 = 0.

1644 DISTRIBUTION OF STRESSES IN LOCK-GATES

X^ as variable:

2 066 800 A'o + 1908 200 Z, + 1749 600 Z^ + 1591000 X^
+ 1 441 400 Z^ -f 1 288 300 Z^ + 1 144 000 Z^ + 1 000 100 Z,
+ 856 700 Zg + 713 680 Z^ + 571 330 Zj^ + 429 690 Z,,
+ 302 890 Z,^ + 176 790 Z.g â€” 17 614 700 = 0.

Zg as variable:

1856 900 Zo + 1714 800 Z^ + 1572 600 Z, + 1430 400 Z,
+ 1 288 300 Z^ + 1 167 800 Z^ + 1 029 500 Z^ + 900 250 Z,
+ 771 370 Zg + 642 750 Z^ + 514 660 X,^ + 387 180 Z.^
+ 273 010 Z^2 + 159 470 Z^g â€” 15 000 500 = 0.

Zg as variable :

1647 600 Zq + 1521700 Z, + 1395 800 Z^ + 1269 900 X^
+ 1 144 000 Z^ + 1 029 500 Z^ -f 921 620 Zg + 800 250 Z,
+ 685 900 Zg + 571 700 Zg + 457 910 X^^ + 344 600 Z,,
-f 243 110 Z,2 + 142 120 Z.g â€” 13 993 300 = 0.

Z^ as variable :

1439 300 Zq + 1329 500 Z, + 1219 700 Z, + 1109 900 X^
+ 1 000 100 X^ + 900 250 Z^ + 800 250 Zg + 707 130 Z,
+ 600 490 Zg + 500 700 Zg + 401 200 X^^ + 302 050 Z,,
+ 213 200 Z,2 + 124 790 Z^j â€” 11 394 500 = 0.

Zg as variable:

1232100 Zo + 1138 200 Z^ -f 1044 400 X^ + 950 550 X^
+ 856 700 Z^ + 771 370 Z^ + 685 900 Zg + 600 490 Z,
+ 520 620 Zg + 429 760 Zg + 344 530 Z.^ + 259 540 Z,,
+ 183 340 Zj, + 107 470 X^^ â€” 10 344 600 = 0.

Zg as variable:

1 025 700 Zq + 947 700 Z^ + 869 700 Z^ + 791 700 X^ + 713 680 Z^
+ 642 750 Zg + 571700 Z^ + 500 700 Z, + 429 760 Zg
+ 364 240 Zg 4- 287 820 X^^ + 217 000 Z,, + 153 440 X^^
+ 90 138 Zâ€ž â€” 7 918 740 = 0.

ZjQ as variable:

820 580 Zo + 758 250 Z, + 695 950 Z^ + 633 630 X^ + 571 330 Z^
+ 514 660 Zj + 457 910 Z, + 401200 Z\ -f 344 530 Zg
+ 287 820 Zg 4- 235 500 X,^ + 174 470 Z,, + 123 560 Z,^
+ 72 819 Z,3 â€” 7 076 810 = 0.

DISTRIBUTION OF STRESSES IN LOCK-GATES 1645

X^^ as variable:

016 660 A'â€ž + 569 920 1\ + 523 170 X., + 476 420 X3 + 429 690 X^
+ 387180 X5 + 344 600 Zg + 302 050 X, + 259 540 Z,
+ 217 000 X^ + 174 470 Z.^ + 136 310 Z,, + 93 679 Z,^
+ 55 495 Zi3 â€” 5 158 560 = 0.

Z^2 ^s variable :

434 260 Z, + 401 420 Z^ + 368 580 Z^ + 335 730 Z3 -f 302 890 Z^
+ 273 010 Zg + 243 110 Zg -f 213 200 Z, + 183 340 Z,
+ 153 440 Zg + 123 560 Z,o + 93 679 Z^^ + 71149 Z^j
+ 39 902 A',3 â€” 3 482 040 = 0.

Zj^g as variable :

252 900 Zq + 233 890 Z, + 214 860 Z, + 195 820 Z3 -f 176 790 Z^
+ 159 470 Z5 + 142 120 Zg + 124 790 Z, -f 107 470 Z^
+ 90138 Zg + 72 819 Z.^ + 65 495 Z,^ + 39 902 Z,^
+ 28 675 Z,3 â€” 1 593 370 = 0.

The method used for solving these equations is given in the Appen-
dix. The values obtained were the following :

Zo = + 154.60 Z5 = â€” 94.82 Z.^ = + 90.18

Z^ = + 69.59 Zg = -f- 21.62 Z,, = + 33.31

X^ = â€” 26.30 Z,=^ â€” 82.31 Z,2 = â€” 11.51

Z3 = â€” 127.50 Zg = + 37.81 Z,3 = + 70.62

X^ = â€” 36.99 Z9 = â€” 84.21

and, from Equations (21) and (22),

Z,,== â€” 135.1
A\5 = + 121.0

These values are in pounds per linear inch of horizontal arch. In
Fig. 3 (B), of Plate XVII, they are given per linear foot. Here, as in all
other cases when X is positive, the vertical girder presses against the
horizontals in a down-stream direction.

(Case B). -^Perfect Contact at Sill.

Zq as variable :

1338 000 Zq H- 1223 000 Z^ -f 1121000 Z, -f 1019 000 Z3
+ 918 300 Z, -f 826 800 Z^ -f 735 900 Z^ + 646 000 Z,
+ 557 200 Zg + 469 200 Zg -f 382 500 Z.^ + 297 000 Z,^
-f 221 100 Z,2 + 146 300 Z.g â€” 115 000 000 = 0.

1646 DISTRIBUTION OF STRESSES IN LOCK-GATES

X^ as variable:

1223 000 Zq + 1142 000 Z, + 1036 000 Z^ + 942 400 X^
+ 849 300 Z^ + Y65 000 Z, + 681100 Z^ + 598100 Z^
+ 516 000 Zg + 434 600 Z^ + 354 300 Z,,, + 275 200 Z^,
+ 204 900 Z,2 + 135 600 Z^g â€” 104 600 000 = 0.

Zj as variable:

1 121 000 Zq + 1 036 000 Z, + 962 800 Z^ + 865 400 Zg + 780 300 Z^
+ 703 200 Zj H- 626 300 Zg + 550 200 Z^ + 474 800 Z^
+ 400 000 Zg 4- 326 200 Z,,, + 253 400 Z,^ + 188 700 Z,^
+ 124 900 Z,3 â€” 94 240 000 = 0.

Zj as variable:

1 019 000 Zq + 942 400 Z^ + 865 400 Z^ + 800 500 Zg + 711 300 Z^
-f 641400 Zg + 571600 Z^ + 502 300 Z, + 433 600 Zg
+ 365 400 Zg + 298 100 Z^^ + 231 500 Z^^ + 172 500 Z.^
+ 114 200 Z,3 â€” 83 850 000 = 0.

Z^ as variable:

918 300 Zo + 849 300 Z, + 780 300 Z^ + 711 300 X^ + 651 200 Z^
+ 579 600 Zg + 516 800 Z^ + 454 300 Z, + 392 400 Zg
+ 330 800 Zg + 269 900 Z^^ + 209 800 Zâ€ž + 156 300 Z,^
+ 103 500 Z,3 â€” 76 030 000 = 0.

Zj as variable :

826 800 Zq + 765 000 Z^ + 703 700 Z^ + 641 400 Z3 + 579 600 Z^
+ 532 200 Z5 + 466 900 Z^ + 410 800 Z, + 354 900 Zg
+ 299 400 Zg + 244 300 Z.^ + 189 900 Z^^ + 141500 Z,^
+ 93 720 Z,3 â€” 67 400 000 = 0.

Zg as variable :

735 900 Zq + 681 100 Z, + 626 300 Z^ + 571 600 Z3 + 516 800 Z\
+ 466 900 Zg + 423 700 Zg + 367 000 Z, + 317 300 Zg
+ 267 800 Zg + 218 700 Z^^ + 170 000 Z,, + 126 700 Z,^
+ 83 930 Z,3 â€” 60 370 000 = 0.

Z^ as variable :

646 000 Zq + 598 100 Z, + 550 200 Z^ + 502 300 X^ + 454 300 Z^
+ 410 800 Zg + 367 000 Z^ + 330 200 Z, + 279 800 Zg
+ 236 300 Zg + 193 000 Z,,, + 150100 Z^, + 111900 Z,^
+ 74 150 Z,3 â€” 51 750 000 = 0.

DISTEIBUTION OF STRESSES IN LOCK-GATES 1647

Xg as variable:

557 200 Zq + 516 000 Z, + 474 800 Z^ + 433 600 X^ + 392 400 Z^
+ 354 900 Zg + 317 300 Z^ + 279 800 Z, + 247 800 Z^
+ 204 800 Zg + 167 400 Z.^ + 130 300 Z,^ + 97180 Z^^
+ 64 400 Zi3 â€” 44 670 000 = 0.

Zg as variable :

469 200 Zq + 434 600 Z, + 400 000 Z^ + 365 400 X^ + 330 800 Z^
+ 299 400 Z, + 267 800 Z^ + 236 300 Z^ + 204 800 Z^
+ 178 700 Zg + 141 800 Z,^ + 110 400 Z,, + 82 400 Z,^
+ 54 620 Z,3 â€” 36 220 000 = 0.

Z^P as variable :

382 500 Zo + 354 300 Z^ + 326 200 X^ + 298 100 X^ + 269 900 Z^
+ 244 300 Zj, + 218 700 Zg + 193 000 Z, + 167 400 Zg
+ 141800 Zg + 120 500 Z.^ + 90 580 Zâ€ž + 67 630 Z,^
+ 44 850 Z,3 â€” 29 360 000 = 0.

Z^^ as variable :

297 000 Zo + 275 200 Z, + 253 400 Z, + 231 500 Z3 + 209 800 Z^
+ 189 900 Z5 + 170 000 Z^ + 150100 Z, + 130 300 Zg
+ 110 400 Zg + 90 580 Z^^ + 75 090 Z,, + 52 870 Z^^
+ 35 090 Z13 â€” 21 420 000 = 0.

Z^2 as variable :

221 100 Zo + 204 900 Z, + 188 700 Z^ + 172 500 Z3 + 156 300 Z^
+ 141500 Zp + 126 700 Z^ + 111900 Z, + 97180 Zg
+ 82 400 Zg + 67 630 Z.^ + 52 870 Z,, + 43 940 Z,^
+ 26 300 Zj3 â€” 14 320 000 = 0.

Zj3 as variable :

146 300 Z^ + 135 600 Z^ + 124 900 Z, -f 114 200 Z3 + 103 500 Z^
4- 93 720 Z5 + 83 930 Zg + 74150 Z, + 64 400 Zg
+ 54 620 Zg + 44 850 Z.^ + 35 090 Z,, + 26 300 X^^
+ 21 870 Z,3 â€” 7 014 000 = 0.

The values of the variables in these equations are :

Zo = -]- 156.10 .X^ = â€” 22.82 ^10 = + 37.12

Z^ = + 81.23 Zg = + 124.1 Z,, = â€” 198.5

Z2 = + 0.2828 Z, = + 14.93 Z,2 = â€” 460.7

Z3 = â€” 87.82 Z8 = + 135.9 Z,3 = â€” 837.5

Z, = -|- 27.67 ^9=â€” 38.78

1648 DISTRIBUTION OF STRESSES IN LOCK-GATES

and, substituting in Equations (21) and (22),

Z,^ = â€” 1208.0
Z,5=-f 2 276.0
all these values being pounds per linear inch. In lig. 3 (A), oi Plate
XVII, these values are given in pounds per linear foot.

If9\-Foot Gate. â€” This gate consists of the nine panels of the Y7i-ft.
gate which are nearest the top, but the bottom girder, (9), is somewhat
modified, so that F^ = 120 sq. in. ; I^ = 140 000 in.* ; eg = 21 in. ; and
Pg = 622.8 lb. per lin. in.
The static equations for equilibrium give:

Zg = â€” (9.4 Zo + 8.3 Z, + 7.2 Z^ + 6.1 Z3 + 5 Z^ + 4 Z,

+ 3 ZÂ« + 2 Z,)
Z, = + (5.4 Zo + 7.3 Z, + 6.2 Z, -f 5.1 Z3 + 4 Z, + 3 Z^
+ 2 Z, + Z,)
The equations of conditions become:

{Case A). â€” No Contact at Sill.
1 152 900 Zq + 996 270 Z, -f 851 980 Z^ + 708 100 Z3 + 564 850 Z^

+ 435 260 Z5 + 306 440 Zg + 178 480 Z, â€” 12 197 513 = 0.
996 270 Zq + 882 620 Z, + 744 830 Z^ + 619 360 Z3 + 494 280 X^

+ 381 050 Zg + 268 440 Z^ + 156 550 Z, â€” 9 481 511 = 0.
851 980 Zâ€ž + 744 830 Z, + 649 790 Z^ + 530 590 Z3 + 423 710 Z^

+ 326 850 Z5 + 230 440 Zg + 134 610 Z, â€” 6 684 281 = 0.
708 100 Zq + 619 360 Z, + 530 590 Z^ + 453 960 Z3 + 353 120 Z^

+ 272 640 Z5 + 192 440 Z^ + 112 660 Z, â€” 3 887 777 = 0.
564 850 Zo + 494 280 Z^ + 423 710 Z^ + 353 120 Z3 + 291 460 Z,

+ 218 430 Zg + 154 440 Zg + 90 720 Z, â€” 3 668 164 = 0.
435 260 Zo 4- 381 050 Z, + 326 850 Z^ + 272 640 Z3 + 218 430 Z,

+ 178 040 Z5 + 119 890 Zg + 70 775 Z, â€” 1 944 087 = 0.
306 440 Zq + 268 440 Z, + 230 440 Z^ + 192 440 Z3 -f 154 440 Z,

+ 119 890 Zj. + 92 153 Zg + 50 828 Z, â€” 1 854 752 = 0.
178 480 Zo + 156 550 Z^ + 134 610 Z, -f 112 660 Z3 + 90 720 Z^

+ 70 750 Z5 + 50 828 Zg -}- 37 683 Z, â€” 159 775 = 0.

The values of the variables become :
Zo = + 148.71 Z3 = â€” 139.92 Zg = + 9.32 Z9 = + 147.72
Zj = + 61.24 Z, = â€” 53.57 Z, = â€” 82.38
Z, = â€” 36.48 Z,= â€” 110.26 Zâ€ž = + 55.72

DISTRIBUTION OF STRESSES IN LOCK-GATES 1649

(Case B). â€” Perfect Contact at Sill.

526 440 X^ + 451 010 Z^ + 388 880 Z, + 327 170 Zg + 266 080 Z^

+ 211 180 Zj -f 157 050 Z^ + 103 790 Z, â€” 58 716 000 = 0.
451 010 Zq + 408 770 Z, + 342 380 Z, + 288 310 X^ + 234 630 Z^

+ 186 320 Z5 + 138 620 Zg + 91 640 Z, â€” 49 938 000 = 0.
388 880 Zq + 342 380 Z\ + 307 990 Z, + 249 430 X^ + 203 190 Z/

+ 161 460 Z5 + 120 180 Zg + 79 476 Z^ â€” 41 019 000 = 0.
327 170 Zq + 288 310 Z^ + 249 430 Z^ + 222 680 Z3 + 171 730 Z^

+ 136 590 Zg + 101 740 Zg + 67 314 Z, â€” 32 131 000 = .
266 080 Zo + 234 630 Z^ + 203 190 Z^ + 171 730 Z3 + 149 180 Z^

+ 111 720 Z5 -|- 83 305 X^ + 55 154 Z, â€” 25 820 000 = 0.
211 180 Zq + 186 320 Z, + 161 460 Z^ + 136 590 Z3 + 111 720 Z^

+ 98 020 Z5 + 66 544 Z^ + 44 099 Z, â€” 18 558 000 = 0.
157 050 Zg + 138 620 Z^ + 120 180 Z^ + 101 740 Z3 + 83 305 Z^

+ 66 544 Z5 + 56 585 Zg + 33 044 Z, â€” 12 931 000 = 0.
103 790 Zo + 91640 Z^ + 79 476 Z^ + 67 314 Z3 + 55154 Z^

+ 44 099 Z5 + 33 044 Zg + 28 791 Z, â€” 5 697 700 = 0.

The values of the variables become:

Zâ€ž = + 310.1 ^3=â€” 96.30 Zg = â€” 301.4 Z9 = + 1615.6

Z, = + 184.4 Z, = â€” 74.07 Z,=â€” 593.4

X^ = -\- 52 A Z5 = â€” 225.5 Zg = â€” 871.9

Discussion of Results.

The values of the variables, X^, X^, etc., previously obtained, give
the reactions between the horizontal and vertical girders for the two
gates and different conditions of sill contact, and (P^ -\- X^), (P^ -\- Z^),
etc., are the resultant girder loads per linear unit of leaf. Plate XVII
gives these results in graphic form for the 77 ft. 6-in. and 49 ft. 6-in.
gates, respectively.

On this plate. Figs. 1 and 6 show cross-sections of the leaf with its
dimensions, moments of inertia, etc., also the total water pressure acting
against the gate, and Figs. 3 and 8 give the reactions between the hori-
zontals and the vertical girder, in pounds per linear foot of gate leaf, B
being for no contact and A for perfect contact at sill.

Figs. 2 and 7 of Plate XVII give the resultant loads per linear foot
on the different horizontals for the three different cases : A, vertical

1650 DISTRIBUTION OF STRESSES IN LOCK-GATES

stiilness and perfect sill contact; B, vertical stiffness but no sill con-
tact; and C, no vertical stiffness at all.

Figs. 4 and 9 of Plate XYII show the deflections of the miter posts,
that is, the distance they move down stream parallel with the axis of the
lock for the three cases just mentioned. These last curves are in general
agreement with those showing the resultant girder loads.

Loads on Horizontals. â€” The curves show that, with no contact at the
sill, the values of X (reactions of vertical against horizontal girders)
are quite small, so that, except at the very top, the deviations from a
purely hydrostatic loading are inconsiderable and due to accidental
causes. With "perfect" contact, the girders in the lower portion of the
leaf (from one-third to one-half the height) were relieved of a large part
of their hydrostatic load, the horizontals higher up receiving a propor-
tionately greater loading. The girders closest to the top showed, in
all cases, the largest proportional increase. In the 77 ft. 6-in. gate,
there was also an increase for the horizontals in the middle third of the
height. As a whole, however, the effect of the vertical stiffness was
decidedly greater for the 49 ft. 6-in. gate.

In proportioning the Panama gates, it was decided to use a load
corresponding to a head of 20 ft. for all girders within 20 ft. of the top.
For those lower down, the hydrostatic head was taken, but, in gates
more than 75 ft. high, the girders in the middle third of the height had
this load increased by from 5 to 10 per cent.

For smaller gates, the effect of vertical stiffness would probably be
greater than for the very high and rather thin Panama gates.

There is no reason, however, to doubt that the common assumption
of hydrostatic loading, except for a few girders near the top, will give
safe results. In the lower part of the leaf, the stresses in actual service
will generally be quite small in a gate designed for the hydrostatic
head, as it is very probable that there will be some contact at the sill.
However, the increase in cost involved is not great, and in any event
the miter gate will weigh much less than any form of caisson or
single-leaf gate.

Reaction of Gate Against Sill â€” For the case of no contact, the
reaction, of course, is equal to zero. For "perfect contact", the sill,
pressure is equal to the end reaction, X, at the bottom of the vertical
girder plus the direct water load, P, on the lowest arch.

DISTRIBUTION OF STRESSES IN LOCK-GATES 1651

These values for the two gates are as follows: (See Figs. 2 and 7

of Plate XYII).

For the 77 ft. 6-in. gate,

12 (Pi5 + Xjg) = 9 560 + 27 322 = 36 882 lb. per lin. ft.

36 882
or, = 19.4% of the total load of the gate.

For the 49 ft. 6-in. gate,

12 (Pg + .Yg) ^ 7 474 + 19 387 = 26 857 lb. per lin. ft.

26 857
or, =35.1% of the total load.

76 565

It will be noted that the total sill reaction is proportionately
greater for the lower gate.

In case there is some elastic movement of the sill under pressure,
the sill reaction and also the loads on horizontals and the deflections
will be values intermediate between Cases A and B.

As stated previously, it is believed that Case J. is a sufficient
maximum.

For small locks, it is not unusual to make the masonry sill strong
enough to withstand the theoretical maximum of 66Â§% of the total
water pressure acting against the gate. For the large proportions at
Panama, it would have been difficult to make the sill walls strong
enough to carry this maximum, and it seemed entirely unnecessary.
It was deemed quite safe to assume a pressure of 50 000 lb. per lin. ft.
of sill or about one-quarter of the whole load on the gate.

Stresses in the Vertical Bracing. â€” This bracing corresponds to the
vertical girder in the computations, the loads acting on it being the
forces, Xq, X^, etc., applied transversely at distances corresponding to
the spacing of the horizontal girders.

The chord stresses are readily obtained from the values shown on
Plate XVII. The unit stresses were foiind to be not more than 4 500 lb.
per sq. in. in any part of the bracing.

The shears for proportioning the web thickness and the rivet
connections were obtained in a similar manner from the transverse
forces, Xq, X^, etc.

1652 DISTRIBUTION OF STRESSES IN LOCK-GATES

APPENDIX

Method for the Solution of Simultaneous Equations.

For the solution of simultaneous equations of the first degree, the
method originated by the celebrated astronomer. Gauss, is probably
the best for practical use in the engineer's office. The solution by
determinants is theoretically attractive, but its application is not
satisfactory. Graphical methods are the best in some cases, but can
hardly be used where the coefficients of the different variables differ
widely in magnitude, and where a high degree of accuracy is required.
Gauss' method in a simple form may be stated as follows:
Given, say, three simultaneous equations of the form:

ax-\-'by-^cz-\-m = (1)

hx-\- dy-\- e z-\- n = (2)

cx-\-ey-\-fz + p = (3)

to find the values of x, y, and z.
From Equation (1), we obtain:

li y â– \- c z -\- r)i â–

X =

a

and substituting in Equations (2) and (3), we have:

(,_A,),+ (._1,).+ (â€ž_1â€ž.)=0....(2.)

(e-^6)j,+ (,-|;-c)z+ (,._;-,â€ž) =0....(3')

which may be written,

d,y + e,z-\- n, = (2^)
e,y + f,^+P, =0(3^)

In these two equations the coefficients are the same as in Equations
(2) and (3), with a subscript added.

From Equations (2^) and (3^, we have, in the same way,

(/,-|-e,).+ (p,-^,..)=0 (3')

which may be written,

/,z + p, = (32)

7)

From Equation (3^), we obtain the value of z = r, and the

J 2

value of X and y by substituting in the first and second sets of
equations.

By writing the equation in tabular form, the relation of the suc-
cessive coefficients is made clearer, and a valuable check on the arith-
metical work is obtained at each step.

DISTRIBUTION OF STRESSES IN LOCK-GATES

TABLE 1.

1653

X

j

y z â€” 1

Equation (1)

a
b
c

b 1 r.

m

n

q = a-\-b+c-{-m

r = b-\-d-\-e-j-n

Equation (2)

d
e

e
f

Equation (3)

TABLE 2.

V

z

â€”

Equation (21)

dr

fl

Pi

Sl

Equation ( 3i )

TABLE 3.

Equation (32).

/2

P2

On examining Equations (2^ and (3^, it will be seen that the
coefficients, d^, e^, etc., are always of the form,

if '

in which,

T = the corresponding coefficient in Equations (2) and (3), that

is, the coefficient of the same variable, omitting the

subscript ;
TJ = the coefficient in the top horizontal line, that is, in Equation

(1) vertically above T ;
V =: the last coefficient to the left in the same horizontal line with

T; and
H = the coefficient of x in the top row.

The values of q, r, and s, in the last column of Table 1 are written
down by simply adding the preceding coefficients in each horizontal
row.

In Tables 2 and 3, i\, s^, s, like the coefficients, d^, e^, n^, etc., are

UV . , .

obtained by the formula, T â€” â€” â€” , previously given.

The check consists in adding up each horizontal line; the last term
should be equal to the sum of the preceding ones, that is, for instance,
s should equal e^ + /i + Pi-

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