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rules for its practical dimensioning.

Method of Calculation Used in the Present Investigation. — The
method herewith presented was developed by the writer in a less perfect
form, in 1899, in a study on lock-gates made for the Board of Engineers
on Deep Waterways.

The solution is based on the well-known "method of least work."
It takes account of irregular spacing in the horizontal girders, as well
as of variations in their cross-sections, and considers the cross-bending
as well as the direct compression in the girders. In case timber is used
for cushions at the miter and quoin posts, or at the sill, the formulas
obtained can easily be modified, so as to allow for the difference of
material.

Although the method is correct in theory, the unavoidable lack of
homogeneity in the steel, the difficulty of determining the vertical and
horizontal rigidity of the leaf exactly, still more the uncertainty as to
the relative adjustment of the gate leaves and the sill, prevent a very
■close determination of the actual stresses.

It is believed, however, that the results obtained are reliable within
reasonable limits, and will prove of much use in analyzing the strength
and stiffness of existing gates or proposed designs. It should be added
that the formulas when applied to the gates of the Poe Lock, at Sault
Ste. Marie, gave results agreeing quite closely with deflection measure-
ments made by the writer.

Like most applications of the elastic theory to complex structures,
the method of least work cannot precede, but must follow, the complete
design. In other words, it is necessary to adopt a detailed arrangement
of all parts and afterward determine the distribution of the stresses in
the different members.

Effect of Vertical Stiffness. — A gate leaf consisting only of a cer-
tain number of horizontal girders and an absolutely flexible sheathing
would have no vertical stiffness. Such a gate would transfer no load
• Annales des Fonts et Chaussees.



1626 DISTRIBUTION OF STRESSES IN LOCK-GATES

to the sill, except the water pressure which acts on the lower half of the
bottom panel and is carried directly to the sill by the sheathing. Each
horizontal girder would support simply the load which corresponds to
the hydrostatic head due to its position.

In practice it is impossible and undesirable to build a gate without
vertical rigidity. The sheathing, combined with the quoin and miter
posts and the intermediate vertical brace frames and intercostals, forms
a vertical girder of considerable strength. By its resistance to bend-
ing, this girder modifies the loads oh the different horizontals, making
them greater or less than those corresponding to the hydrostatic pres-
sure. As a rule, the vertical girder transfers a part of the total water
pressure to the sill.

For purposes of calculation, the leaf is taken as consisting of a
horizontal and a vertical system of girders crossing each other at right
angles. The horizontal system consists of the several main girders or
arches, spaced as they are in actual construction. The vertical system
is assumed to be equivalent to a single girder extending continuously
over the whole length of the leaf from the quoin to the miter post. Its
flanges are formed by the gate sheathing, and its web is equivalent in
total crpss-section to the web plat^es in the several vertical frames and
end posts. This simplification is justified by the close spacing of the
vertical frames and intercostals, which prevents the skin plates from
buckling.

Sill Contact. — In case there is no contact at the sill, even when full
water pressure acts, the entire load, of course, will be carried by the
horizontal arches to the side-walls.

If, in such a gate, all the horizontals are proportioned to support
the hydrostatic head with exactly the same unit stresses in the steel,
they will all have exactly the same deformations under load. There
will be no bending stress in the vertical girder, so that it will remain
straight, even after the gate is supporting the water pressure. The
loads on the several horizontals will be those corresponding to their
hydrostatic head.

In practice, the horizontals near the top are always stronger than
theory requires, in order to ensure increased safety against accidental
blows and to avoid the use of unduly small rolled shapes. It is also
hardly possible to design all the other horizontals so that they shall
sustain exactly the same imit stresses. There will always be some



DISTEIBDTION OF STEESSES IN LOCK-GATES 1627

variation, therefore, in the deflections of the different horizontal girders.
This will produce a tendency to bend the vertical girder which is
rigidly connected to the horizontals, and its resistance to bending, in
turn, will affect the deflections and modify the loads of the horizontal
frames.

However, the effect of vertical rigidity, when there is no sill contact,
will be very small, except at the extreme top of the leaf.

The case of "no contact" should always be provided for in the
design, for, from various causes, it is likely to occur in all mitering
lock-gates as either a temporary or permanent condition.

In ordinary cases there will always be a greater or less reaction at
the sill. If the water on the up-stream side extends to the top of the
gate, and there is no lower pool, the greatest sill pressure theoretically
possible would be equal to two-thirds of the total load supported by the
gate. This maximum can only occur when the adjustment is so inac-
curate that, even with continuous contact along the sill, the two leaves
will touch only at the very top of the miter posts, even when the gate
is subjected to the full head of water. This is an extreme case which
would seriously overstrain the gate, and can be avoided by ordinary care
in adjusting the leaves and sill.

A much smaller reaction may be counted on as a practical maximum.
It seems quite safe to assume what is sometimes called "perfect con-
tact", that is, continuous contact along the quoin and miter posts and
also along the sill when the gate is closed, but before it is subjected to
water pressure.

With both timber and metallic bearings, the actual conditions will
probably correspond to lower sill pressures, as there will rarely be
absolute sill contact in the dry.

Therefore, two conditions of adjustment at the sill were considered
in the computations :

(1) No contact at the sill, even with full head;

(2) Simultaneous contact at the sill, miter, and quoin posts, before
the water pressure is applied.

Let Fig. 1 represent the vertical section and Fig. 2 the plan of a
gate leaf, of length, L, consisting of (n + 1) horizontal arches and
a continuous vertical girder, the stiffness of which is assumed to be
uniformly distributed over the length of the leaf. The sheathing is



1628



DISTRIBUTION OF STRESSES IN LOCK-GATES



supposed to carry the water pressure directly to the horizontals, and
the connections to be such as to permit the transference of horizontal
reactions between the arches and the vertical girder at their inter-
sections. Let the magnitude of these reactions be denoted by
X^, X^, .... Xji per linear horizontal unit of leaf. They will be
both positive and negative in direction, and will act normally against
the arches exactly as water pressure does.

If, further, Pq, P^, .... P„, are the direct water loads on the
several arches per linear unit, their resultant total loads will be:

(P, + X,) LiP, + X^) L . . . . (P„ + X„) L.

With no contact at the sill, all these loads are carried by arch action
to the hollow quoins ; but, if the lowest arch bears against an absolutely
■fixed sill, that arch will carry no load to the side-wall, and (P,i + X„) L
will become the sill reaction.




Fig. 1. Fig. 2.

In any case the only forces acting on the vertical girder will be the

transverse loads, -v t

Xq L, X^ L, . . . Xn L.

The static conditions of equilibrium as applied to this girder give
only two equations:

2X = X,+X, + . . . . X„_i + X,= (1)

2M = X, K + ^^^1 iK -K) + ... x„_i (/*.„ - K_,) = 0... .(2)

for determining the values of X^ . . . X„, although there are always
at least three such unknown quantities.



DISTRIBUTION OF STRESSES IN LOCK-GATES 1629

Similarly, if the gate as a whole is considered, the reactions of the
horizontals against the side-walls are indeterminate, as their number
is in excess of the number of equations which can be obtained from
static conditions.

The stresses in indeterminate structures of this kind, however, can
be found by an application of the method of least work. The principle
on which this depends may be briefly explained as follows :

If a perfectly elastic structure is subjected to external forces, the
fibers in its parts will be deformed until a new condition of equilibrium
is reached. The work done in this deformation (the internal work or
elastic potential) will always be equal in amount to the external work,
that is, the work done by the external forces.

By the principle of least work, the reactions and internal stresses
corresponding to the new condition of equilibrium, in addition to being
consistent with the statical equations, must be such as will make the
total internal work done by the structure, in passing from its original
to its new condition, a minimum.

This principle is sometimes derived from the theory of virtual dis-
placements, but may almost be considered axiomatic, representing the
theory of equilibrium in its most general form as applied to elastic
solids.

The application of the method of least work to this problem consists
in stating the work of internal deformation in terms of the known
external loads and the indeterminate quantities Xg, X^ . . . X„, and
finding those values of the latter, which, besides satisfying the static
Equations (1) and (2), will give a minimum value for the total internal
work done, while the gate passes from a condition of no stress to that
in which the full water pressure is supported.

The internal work in the whole leaf will consist of the following
parts :

(1) That due to arch action in the horizontal girders, involving
generally both direct compression and cross-bending; and

(2) The work of bending the vertical girder.

The work done as the result of the shearing stresses in the arches
and the stresses in the web members of the vertical girder, being rela-
tively quite small, may be neglected.



1630 DISTRIBUTION OF STRESSES IN LOCK-GATES

The bending in the horizontal girders is due to the eccentricity
of the line of pressures or resultants with reference to the center of
gravity at the different cross-sections. For arches with continuous
curvature, this eccentricity is quite small, but occurs to some extent
in all gates.

If Uu = the internal work in the horizontal, and U^ that in the
vertical girders, then,

U = Uji -\- Uv is the total internal work.

For any elastic solid under purely axial stress (that is, direct com-
pression or tension), the work of deformation will be:

. Yef" r^'^



and, for one subject to cross-bending only (beam action),

" " 7„ 2E I



^.= / -TT-^^T^^^ (4)



in which equations :

Jj = the total length of the member ;
¥ and 7 = the cross-sections and moments of inertia at any point ;

^ = the modulus of elasticity ; and,
T and M = the total axial force and bending moment at any cross-
section.

From Equations (3) and (4) a general expression for the work
equation for all parts of a mitering lock-gate (such as is shown in Fig.
1) may be written:

In this equation, m corresponds to any horizontal arch and also to
the panel of the vertical girder just above the arch so denoted. The
axial thrust and bending moment at any point in a horizontal are
represented by T^ and M,„„, respectively; and ilf,,,„ is the bending
moment at any point in the vertical girder. The first two terms
give U,, (work in all horizontal arches), the integration being for the
length of each individual arch and the summation to include the



DISTRIBUTION OF STRESSES IN LOCK-GATES



1631



(n -\- 1) different arches. The last term gives U^ (work in vertical
girder), the integration being for each separate panel, while the sum-
mation includes all of them.

The cross-sections, F„,, and the moments of inertia, 7;,^ are the
average values for any given arch, and I^^ is taken as uniform for any
given panel of the vertical girder.

As will be shown later,. T„„ Mj,r„, and M^,„, and, hence, U (Equation
5) may be readily expressed in terms of the indeterminates, X^ . . . X„,
and the known quantities. If, in the resulting equation, the condition
of statical equilibrium is introduced by putting, for X„ and X„_^, their
values in terms of the other (n — 1) variables (using Equations 1 and
2), ZJ will be expressed as a function of X^ . . . X^_2-

Differentiating under the integral sign, we readily obtain the
(n — 1) partial derivatives required for finding those values of X
which, consistent with statical equilibrium, will make the elastic poten-
tial a minimum. Using the same notation as before, they will be:



dU



m = n



•^ n



r„, d T..



E F„, 5 X



'-dl-\-



I






m = 1

.Y, ^, J„ EF,„SX,



«-..'«..; ay =



^L r „ s r



^ fZZ-f






hm " -™ftm 7 ^



EIu,nS^l



+



m = n



a„, M S M



^...(6)



wi = 1



d u
sx^ .



X7



■/



L T 8 T

m n

E F„. S X^









dy = Q



In Equations (6) the first two terms correspond to the work done in
the horizontal arches, and the last term to that in the vertical girder,
so that these equations are of the form:

d X dX 8X



(7)



1632 DISTRIBUTION OF STRESSES IN LOCK-GATES

The direct thrust, T, in a lock-gate is nearly constant throughout
the length of the leaf for any given horizontal girder, and may be
expressed very simply.

If r = the radius of a circle which passes through the centers of
bearings at the quoin and miter posts when the gate is
closed (Fig. 2) ; and
P = the load per linear unit on the horizontal girder ; then
T = P r.

For any horizontal frame, m, in a gate, therefore, we can write,

Tm= (i'^ + ZJr (8)

Also, if e^ is the mean eccentricity of the resultant in the horizontal
girder, m, that is, the average distance between the line of pressures
and the center of gravity of the cross-section, we can also write,

Mn„^ = e„,T„, = {Pm + Xm) e^r (9)

The first two terms in Equations (6), which correspond to the work
in the horizontal arches, therefore, may be written in the form :






g?7,_ %r5'l [- 1 ^ ej-] f HP,, + XjS{P^ + X,„)
dX ^ E

m =



Similarly, the last term of Equations (G) and (7), corresponding to
work in the vertical girder, may be easily expressed as follows :

Let M^,n be the bending moment in any point at any panel, m, and
let the distance of this point below the panel point next above = y. We
can then write,

^v. = LKK -i + y) A^o + ( K - 1 - K + y) ^i +

. . . + (K,_,-h,„_, + y)X,„_, + yX,„_,-] (11)

The partial derivatives will be,

Vlf = ^ ^^''" -1 + •^)' Vxf = ^ (''».-!- ^ + 2/)
' ^- =Ly.



SX^



DISTRIBUTION OF STRESSES IN LOCK-GATES 1633

That portion of the last term in Equations (6) which corresponds
to the work in Panels (1) to (n — 1) will then be given by the follow-
ing expression :

ni = n — 1



vm

ni = 1
VI — n — 1



rw = 1

+ I K - 1 (K - 1 - h) «™ + (2 \, - X - h,)"^ + ^ ix,

T^ • • • I i "m - 1 2 3 C ™ ~ 1



(12)



and there will ))e similar forms corresponding to tlie other variables,
X, . . . X,_,.

The part of the last term of Equations (6) corresponding to the
work in the bottom panel must be obtained in a somewhat different
way, as it is necessary to express X„ and X„_j in terms of Xq . . . Xn_2-

Let z ^= the distance of any point in the panel from the bottom
of the gate ; then we can write :

^^vn =^ L Z X,i

From Equations (1) and (2), if we take moments about the girder,
(n - 1),

^n = ^ ] 'V - 1 -^'o + (K - 1 — K) ^l



hence,



+ . . . (K-l-K-2)Xn-2\



Lz



+



+ ~ ] K^, X, + (/i„_ 1 - h,) X,
. . . {K^,-K_,)X^_,^ (13)

for which we readily derive,

r- Kn^ Kn ^'% rj^2 Y+/t (h -h)X

+ . . . /'„-i(^-i-^-2)^^^n-2] (14)

which represents the term corresponding to work in the bottom panel of
the vertical girder. We can readily obtain similar forms when A''
. X„ o are the variables.



1634



DISTRIBUTION OF STRESSES IN LOCK-GATES



General Equations of Condition. — Combining Equations (10), (12),

and (13), and omitting the common factor, -— , we have the following

E

general equations.

With Xg as the independent variable :

m = n

m = n — 1
m = 1

+ I K- 1 iK - 1 - K) «„. + (2 /^^ - , - h) ^' + ^' I X,



+



+ l''-T + "th^v,]



L^a.



and with X^ as the independent variable:






(P„, + X,)3(P„, + XJ ^^

5x;^ '^ ^



K..(15)



/ft =1/1 — 1

+ ^' ^-r [ j ^.-1 (^«-i - M «.

m = 1

+ ](^H-i-M'% + 2(/.,„_j-/iO^' + ^'[ X,+ . . .

+ ^[(/^.-l -M^A'l+ • • • + (/'„-! -^)

(^_l-/i,_2)X„_J =

and equations of similar form for A"^ • • • ^n-2 ^^ independent variables.
By making the proper substitutions and summations in Equations
(15), the (n — 2) simultaneous equations may be written, and from
these, the values, Xj to X'„_j, are readily found.



disteibution of stresses in lock-gates 1635

Application of Formulas to Gates of the Panama Canal.
77^-Fooi Gate. — The general arrangement of the gate leaf is shown
on Plate XVI and the photograph, Fig. 8, which represents a somewhat
lower gate before the sheathing is attached.
The principal dimensions are as follows:
Clear width of lock = 110 ft. ;
Height of gate = 77 ft. 6 in. from top of coping to center of bottom

girder ;
Central thickness ^ 7 ft. in. ;
Shape of gate : straight-backed.
There are sixteen horizontals, the spacing of which is shown on
Plate XVI and Fig. 1 of Plate XVII, and is tabulated subsequently.

The cross-sections, moments of inertia, and eccentricities of lines
of pressures, which are shown on Fig. 1, Plate XVII, and are also given
later, are mean values for each of the horizontal arches. The moments
of inertia for the vertical girder are calculated from the skin thickness
of each panel and the average depth of the girder. If t^ and t^ are the
thicknesses of the up-streara and down-stream sheathing, and d^ and d.,
their distances from the center of gravity of the cross-sections of the
girder, the moment of inertia is given by the expression:
I, = L(t^d,' + t,d,^). .
Summarized, the data are as follows :
Length of leaf, L = 787 in.,

Radius of line of pressures, r = 880 in. (See Plate XVII,
Fig. 5.)

Horizontal Arches. — Mean cross-sections :

F^ =F^ =F^ =F^ =100 sq. in.

F^ =F^ =120 " "

F^ =F^ =F^^ =140" "

F^ =F^ =164 " "

F =F =^F =F =F =200 " "

•^10 -^ 11 12 •'is ^14 ^""

Mean moments of inertia:

I^ =7, =1^ =1, =120 000 in."

7^ = 7^ = 140 000 "

7g =7, =7,5 =156 000 "

73 =7g = 167 000 "

r„ = 7„=7, =7,, = L. =190000 "



1636



DISTRIBUTION OF STRESSES IN LOCK-GATES



Mean eccentricity of line of pressure :



p — p — p — p






.-=26 in.


1 2 ^Z

e. = e






= 21 "


4 ^5

e^ = e„ = e, ,






= 16 "


6 7 ^15








e„ =eQ






= 13 ''


8 9
^10 ^ ^11 ^^^ ^12 ^^^ ^13 ~


= ^14




= 11 "


Vertical Girder. — Moments of inertia:




I^ =T =T






= 1 100 000 in.*


1 2 3

h=h






= 1180 000 "


7e =7,






= 1240 000 "


7^ =L =L.=L^






= 1400 000 "


8 9 10 11

L,=I,,






= 1 580 000 "


12 13

I,, = L.






= 1720 000 "


14 15

Vertical Panels. —








a. ^= a~ == a = a






. = 66 in.


"'1 "'2 "^3 **4 • • • •

^5 = «6 = ^7 = ^8 = ^£


= a


10 = «11


= 60 "


ffl,- = a, „ =: fl, . =^ tt, _






= 54 "


12 13 "14 15








Heights. —








h^= 66 in. Ji^ =


384


in.


/i^^ = 684 in.


/(2 = 132 " h, =


■ 444


ii


h^^ = Td8 "


/i3 = 198 " h^ =


504


a


/H3 = 792 -


/l, = 264 " Ag =


564


a


;ii^ = 846 "


A, = 324 " A,„ =


624


a


h^^ = dOo "



Water Pressure. — The water is assumed to extend to the top of
the coping on the up-stream side of the gate, with a pool 9 ft. 6 in. deep
below. Fig. 1, Plate XVII, shows the total water pressure and Fig. 2,
Column C, of that plate, the load per linear foot on each horizontal
arch.

Reduced to the linear inch of girder, these values are:



Po = 73.34 lb.
p^ = 229.15 "
^2=: 386.07 "
P3 = 544.13 "
p^ = 665.38 "



^5= 768.25 lb.
Pg= 898.45 "
p. = 1028.65 "
Pg = 1158.9 "
Vg = l 289. "



p^^ = 1 419.35 lb
p,, = 1 467.9
Pj2 = 1 500
p,3 = 1 580.7
Pj^ = 1593.75
p,^= 796.87



PLATE XVI.

TRANS. AM. SOC. CIV. ENGRS.

VOL. LXXXI, No. 1402.

GOLDMARK ON
PANAMA LOCK GATES.




.ivx axA- '
.so>r ..)M ,ixxxj .JO"

no XHAMOJOO
.83TAO >IOOJ AMAMA'



._,^.

■c'',




-r^'i






f -






n




! j


1-


SST'


^^lij


4
11







i\\



l-M-ii







M» — -_- -;



■ ' f




3TA0 THDiAHTc



DISTRIBUTION OF STRESSES IN LOCK-GATES 1639

The conditions of static equilibrium, as applied to the vertical
girder, become:

Z,,=- (Z„ + Z, + Z, + Z3 + Z, + X,JrX, + X, + X,

+ Z, + Z,, + Z,^ + Z,3 + Z,3 + Z,J (16)

and,

Z,, = — — (900 Zo + 834 Z, + Y68 Z^ + 702 Z3 + 636 Z, +

576 Z5 + 516 Zg + 456 Z, + 396 Zg + 336 Zg + 276 Z.^
+ 216 Z,, + 162 Z,2 + 108 Z,3)
whence,

Z,, = — (16.66 Zq + 15.44 Z, + 14.22 Z^ + 13 Z3 + 11.78 Z, +
10.67 Zg + 9.56 Zg + 8.44 Z, + 7.33 Zg + 6.22 Z„ +

5.11 Z,„ + 4 Z,, + 3 Z,, + 2 Z,3) (17)

and,

Z,„ = + 16.66 Zo + 14.44 Z, + 13.22 Z^ + 12 Z3 + 10.78 Z, +
9.67 Zj. + 8.56 Zg + 7.44 Z, + 6.33 Zg + 5.22 Z^ + 4.11 Z,„

+ 3Z,, + 2Z,, + Z,3 (18)

In order to introduce the static conditions of equilibrium, the values
of Zj^ and Z^g, in terms of Z^ . . . Z^,, as given in Equations (17) and
(18), will be used as shown below.

The resulting simultaneous equations of condition, corresponding to
ft jj ^ TT

— —r = 0, = 0, etc., will then be reduced to fourteen, with

Xq . . . Xj3 as variables.

Work of Horizontal Arches.

The first terms of Equations (15) correspond to the total work of
the horizontals.

Each arch must be taken up separately, and the values of the

expression, r' (^ + |^) /*^ (P^ + A;j ^ '^^" ^^"^ d I, obtained,

taking X . . . Zjg, successively as variables. It may be convenient to
represent this expression generally by the symbol H. Its values for
the separate arches and variables M'ill be the following :
Arch 0:

Xq as variable :
H=12 106 f\p, + X,) '(Po^X,)dl ^ ^2 ^^^^ ^^^ ^ ^^^^ ^^

= 12 106 (78.34 + XJ L r=r (12 106 Xg + 948 .384) L



16i0



DISTRIBUTION OF STRESSES IN LOCK-GATES



Xj as variable :

H =. 12 106






and, similarly, we should find that H

variables.

Arch 1:

Xr. as variable :



d X^
when X, .



X,, are the



fl = 12 106
X^ as variable :
if = 12 106 r (P, + Xj)



f\p, + x,)

Jo



d (Pi + Xj) d I



S X.



=



d (Pj + Xj) d I



= 12 106 (229.15 + Xi) L



For Xo



X,



sx,

= (12 106 Xi + 2 774 090) L.

In like manner, for the arches, 3, 4 . . . 13, H will = for every
variable but one in the case of each arch. There will be a significant
value, however, in the case of each arch for the one variable which has
the same number as the arch in question.

(P + X)

'0



/I e'^ \ /*^ (P A- X)

These values of if = r^ i^ + j) (^ + ^) ^—^ — - ^ ^



will I


)e the following :


Arch


O.-X,


as indep


Arch


l.-X,


«


Arch


2.-X,


li


Arch


5.-Z3


a


Arch


i.-X.


a


Arch


5.-X,


a


Arch


6.-X,


a


Arch


7.-X,


u


Arch


8.-X,


a


Arch


9 - X,


a


Arch


lo.-x,.


a


Arch


ll.-X,^


a


Arch


12.-X,,


u


Arch


IS.—X,,


it



en dent variable ; i? = 12 106 X^
H = 12 106 Z,
H = 12 106 Z,
5" = 12 106 Z,



H =

H =

H =

E =

H =

H =

H =

H =

H =

E =



8 892 Z^
8 892 Zg
6 802 Ze
6 802 Z,
5 477 Zg
5 477 Zg
4 365 Z\
4 365 Zj
4 365 Zj
4 365 Z,



+ 948 384
+ 2 774 090
-h 4 681 027
+ 6 587 238
-f 5 916 559
+ 6 831279
+ 6 111 257
+ 6 996 877
+ 6 347 295
+ 7 059 853
+ 6195 463
+ 6 407 383
+ 6 547 500
+ 6 899 755



Arch IJf. — For this arch, the general form of E will be
i/=4 3(3o ^(P,, + X,,)



5 X "' '"



= 4 365 r (1 593.75 + X,) ^a^^^^+^u) ,,
Jo *^



PLATE XVII.

TRANS. AM. SOC. CIV. ENGRS.

VOL. LXXXl, No. 1402.

GOLDMARK ON
PANAMA LOCK GATES.



77'6"GATE FOR 1 10-FOOT LOCK



^/////////'//yy/'/'/////M'/y>



Loads on Horizontal Girders




Online LibraryAmerican Society of Civil EngineersTransactions of the American Society of Civil Engineers (Volume 81) → online text (page 148 of 167)