successive approximations
Autor(en):
Maugh, L.C.
Objekttyp:
Article
Zeitschrift:
17.09.2015
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Introduction.
The term, Vierendeel truss or system, has been widely used to designate
the rigid frame type of structure that is illustrated by the accompanying dia
grams (Figs. 14). Since the diagonal members are omitted in these struc
tures, their function must be taken over by the remaining members which are
thereby subjected to flexural and shearing stresses in addition to the ordinary
truss action. The two Vierendeel trusses that are shown in Figs. 1 and 2 are
suitable for bridges, whereas Figs. 3 and 4 illustrate types of rigid frame
Systems that are used in viaduct and building construction. Before proceeding
with an explanation *of the analysis of these structures, however, it seems
desirable to give at least a brief aecount of their history and development,
and of various methods of Solution, particularly the various methods of suc
cessive approximations that are gradually displacing the socalled exact me
thods of analysis.
than a Century ago many static
History and Development: More
used
framed
indeterminate
structures
in the construction of bridges
were
ally
and buildings. A study of some of the bridges that were built before the de
velopment of the modern methods of rational analysis reveals a firm belief
in the inherent strength of multiple system and rigid frame structures. The
fact that several of these bridges are still in good condition apparently justifies the confidence of the builders. After the application of mathematical
analysis to the design of structures, however, there developed a feeling of
distrust and suspicion toward those structures that could not be rigorously
analyzed and, as practical methods of analysis were limited in scope, most
bridges were therefore built of a type that could be readily solved by the
equations of static equilibrium. The type of structure most commonly used
for bridges was the pinconnected or riveted articulated truss. The develop
ment of steel as a structural material also increased the use of this type of
structure and so, at the present time, we find that a large number of the
existing bridges belong to this class.
x) The major part of this paper is taken from a thesis submitted by the writer to
the University of Michigan in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Engineering Mechanics.
L. C. Maugh
334
^ r
___ii
\
u
ir
Fig.
1.
Fig. 2.
_A
t
4
>
Fig. 3.
Fig. 4.
Fig. 14:
Types de systemes Vierendeel. Typen von Vierendeelsystemen.
Types of Vierendeel Systems.
of these structures,
fr
The Vierendeel Truss by Dana Young, Eng. NewsRecord, Aug. 31, 1931.
ber Rahmentrger und ihre Beziehungen zu den Fachwerktrgern. Zeitschrift
Architektur und Ingenieurwesen (1913).
4) History of Bridge Engineering by Tyrrell, P. 183.
h) Fachwerktrger aus Eisenbeton von S. Zipker, Eisen und Beton, 1906.
2)
3)
335
\m
rtt
7^
2
TTr^
fr
rr2
D/2
rr7
"CiL
k.. _
D/2
1/77
*
4
1
Fig. 5.
Methode de calcul Vierendeel.
Untersuchungsmethode von Vierendeel.
Vierendeel's Method of Analysis.
d,
C2
^^ji
r
Horizontal component in vertical
Horizontal component in vertical r.
IJr
are coefficients that are determined from the dimensions of the
r+l
IJr+1
r+l
structure.
Mrr+l
Since one
L. C. Maugh
336
Belgium and France. Some of the objections that have been expressed are:
Uncertainty of the effect of the various assumptions used in the derivations;
the use of numbers with many digits in the equations to obtain an answer
with relatively few; and also, its failure to meet the requirements for variable
conditions in the arrangement of the structure.
Consequently, in both Europe and America, other methods of Solution
have been proposed and used. For instance, the Kinzua viaduct7) built by
the Erie Railroad in 1900, was designed by C. R. Grimm by the method of
least work. By careful arrangement of the equations, the work was considerably reduced and numerical errors eliminated. However, the amount of work
involved is many times greater than for the Solution of the same structure that
is given later.
In 1904 Professor L. F. Nicolai8) in St. Petersburg, Russia, proposed a
Solution for Vierendeel trusses that is based on the assumption that the
NNN,X
i
>2N
*
^x/'x/'x
^
' ____
fb)
=r^C
Fig.
6.
rectangular panel composed of the upper and lower chords and the two
verticals is the fundamental unit and that its relation to the remainder of the
structure can be ignored. The analysis is then made by first assuming that
diagonals are acting in the panels and that the joints are pinconnected as
shown in Fig. 6 a. Then at each Joint an equal external force is applied that
will neutralize the stress in the diagonal (Fig. 6b). The moments in each
panel due to the external forces that are required to balance the stress in the
diagonal are taken as the bending moments in the original structure (Fig. 6 c).
This method, used by Professor Nicolai, was greatly extended and made
more practicable by Professor K. Calisev9) in 1921 when he added to the
primary moments in each panel, as obtained above, a correction in the form
of a rapidly converging series which takes into consideration the restraining
influence of the remaining portion of the structure. Additional corrections
were also made for the change in length of the chords and for application of
the load directly on the chord members. As several basic principles of this
method will be utilized in the analytical Solution that is described in the
following chapters, a detailed description will not now be given.
7) "The Kinzua Viaduct of the Erie R. R. Company", by C. R. Grimm. Trans, of
A.S. C. E. 1901, Vol. 46.
8) Journal of the Ministry of Ways of Communication, St. Petersburg, Russia.
9) Dissertation presented by K. Calisev at the Polytechnical Institute at Zagrebu,
Yugoslavia for the Doctorate degree. Also, see Bauingenieur 1922, P. 244.
337
Approximations:
The socalled exact methods of analysis that have been so generally deve
loped and used in academic work have always required the Solution of several
simultaneous equations. The number of such equations must, of course, be
equal to the number of unknown quantities that are necessary to completely
determine both the internal and external force Systems. In the Vierendeel
truss there are three redundant quantities in each panel so that 3 n equations
are required in addition to the equations of equilibrium, where n is the
number of panels. Thus, for a six panel truss with vertical loads, it would
be necessary to solve twenty simultaneous equations. This number can be
materially reduced if the structure is symmetrical about some axis, but even
so, the Solution by these methods is very laborious. Consequently, the ten
dency in practice has been to reduce the number of equations by making
various assumptions in regard to the position of the points of inflexion or
the degree of restraint at the ends of the members. For instance, in the
approximate methods for determining the wind stresses in tall buildings,
sufficient assumptions are made to eliminate the use of simultaneous equations
entirely.
At the present time, however, most engineers regard such indiscriminate
use of assumptions with justifiable suspicion and are favoring the use of
more aecurate methods of Solution that are based on the principle of suc
cessive approximations. These latter methods have one common characteristic
in that certain strain conditions are assumed which give results that, at first,
will not satisfy the conditions of equilibrium but, by successive trials, the
computed values can be made to approach nearer and nearer to the true ones.
The final values must, of course, satisfy both strain and equilibrium requirements. A brief description of several of these methods will be presented.
One of the first engineers to apply the method of successive approxi
mation to the analysis of indeterminate structures was Mr. J. A. L. Waddell12)
who, in 1916, used it to determine the secondary stresses in bridge trusses.
In his description of this method Mr. Waddell states that the Solution is not
10) Die Berechnung der Rahmentrger mit besonderer Rcksicht auf die Anwen
dung", by F. Engesser. Zeitschrift fr Bauwesen, 1913.
11) Beitrag zur Berechnung von Vierendeeltrgern", in Beton und Eisen, 1910.
Eisenbau, 1912.
i2) See "Bridge Engineering", Vol.
Abhandlungen
III.
1,
by J. A. L. Waddell.
22
L. C. Maugh
338
original with him, but the writer does not know of any prior use. In this
Solution the moment at the end of any member MN is expressed in the usual
form
Mmn
4FI
(rmn
i tum)
where
%mn
and
%nm
2M
4
Jf
at Joint
at
joint
The effect of this correction on the various moments can be carried over to
other joints before they are corrected, which procedure makes the convergence
very rapid, particularly when the joints with the greatest unbalance are con
sidered first. The final results can be carried to almost any degree of accuracy,
but in general two cycles of corrections will be sufficient. Many variations
of this method can and have been used but they ordinarily differ only in the
arrangement of the numerical calculations.
The above method for the determination of secondary stresses in bridge
trusses was also used by Professor K. Calisev13) who modified the equations
to take into aecount the decrease in the effective length of the member due
to the greatly increased area at the joints. In 1923, Professor Calisev applied
a similar method11) of successive approximations to the Solution of rigid
frame structures. In this method the Solution was divided into two parts;
first, all translation of the joints was neglected so that only rotation was
involved, and second, the effect of translation was then considered so as to
satisfy the necessary strain and equilibrium conditions. When only rotation
of the joints are considered, the end moments can be expressed by equations
of the form Mab
2/( (2 0a \ 0b) j Mfab in which &s represent the Joint
rotation, K
Effl and Mfab is the end moment produced by the transverse
loads for a fully restrained condition, or what is commonly called the fixedend moment. As in the above method for secondary stresses, the proper value
for the various 6's can be determined by successive approximations. If all
joints are held motionless except one, then the rotation of that Joint will be
=j~~t. The
339
rangement, the convergence can be made very rapid. If the members are
hinged at one end then 3/4 K should be used instead of the fll value.
The second part of the problem which involves the correction for the
translation of the joints can be made in a manner similar to the Solution
Fig. 7a.
72
3.0
+ 6.0
 72 0
+ 72.0
7.77
.56
.65
.06
.04
+ 74.02
5.30
co
f

+ .70
.20
+ .02
.04
.70
72.0
.06
 08
+ .04
.02
02
77.87
+72.70
I
+ .05
+ 77 65
J5
72.70
7000 Ibs/PZ.
(jj
K=2p
>
.50
.20
+ 72.0
72 0
\72.0
70
7.8
377
333
.377
333
304
+ 1+1
.50
+r
2.0
.75
+ .44 *
.74
7.50
I
+
7.8
^
.77 R
.06
.03
+.77
+ .44
6 Panneaux  PeZder  Panels
0 18 0
780 0
3.0 3.0 20
526 + .77 + 7.43+70.52 + 545743 286 +70 9
7
29
590 + .92
+ 944
.35 .05 +.46 +2 95  .60 77 + .65+472
.74 + .02 .77
.70
.62.70  .22 760
.02 7.47.22 + .07 + .06 .35  OS .37
+ .07
9 06
b
40
70Z6
 59
<o
7075
^707 0)

.04 .04
7 20 4 57
7
87
+74 0
2?
67 7
( 68 0)
.340
45 3
67.8
73.7
56 4
co
>
+77.7
{58 0)
4 22
 .08.62
+ .62
.04
724 .77
+.02+ .77
2 464 78 6 77+6 02
323 C 290
tv
.05
.77
Z2JQ' =72'Q"
20
+
+ .04
+ .05 xi
.35
+
+
7.4
20
<*>
+ .70
.304
397
50.7
(57
7036 +7036
264
65
77.8
29 8
+72.7
77 7,
(775)
78 3,
{^78 3)
7)
z,
/o
*. ../'
VerZrca/es cons/derees comme articulees
des verticates prendre 3Zz des
ici7 (Z3ourk
valeurs ePPecZives pour eviZer les surcharges fies moments dans /es verticales ne sonZpas
366 24
32.52
17
28
7738
Computed Moment l
ai~10Z6
'
 16
f03l
37
274
76
87.7
 J_
82.7
CZ453
_
449
SSO
61.8
+
co
"
.7
67.7
diques
72 7 _
72 33
f65
34 74~
72 7 _
768 ~
.1
Essai
Versdch  TrraZ
29.6
.7
29.9
1(6S)
(44
82.3
29.7
62 6
H/er Annahme von Gelenken Pur die l/ent/kalen (Verwende Kder VerZikalen als 3/2 der
\wirk/ichen Z< Werte um LZberZrag zu vermeiden (Die ZiomenZe /n den VerZika/en
Sind tu vermeiden IZerZicaZs assumed es hinged hene (Use KopvenZrcats as 3/2
\702 Z)
4)
opacZuaZ
>n
Fig. 7b.
Methode de calcul d'apres Cross.
Untersuchungsmethode von Cross.
Cross's Method of Analysis.
described above for secondary stresses except that now both xp and 0 are
unknown. However, for frames of one story, any value of xp can be assumed
and the moments thus computed will be in the same proportion to the true
moments as the computed shear is to the true shear. If the structure has more
L. C. Maugh
340
than one story, this proportionality does not hold and consequently the So
lution becomes much more involved. As Professor Calisev has presented this
method in detail in another paper of this publication it will not be necessary
to describe the numerical Operations here.
Another application of the method of successive approximations to the
analysis of rigid frame structures has recently been presented by Professor
Hardy Cross 15). This method, which is usually called the moment distribution
method, is similar in principle to the Solution used by Professor Calisev, but
in application it is different in that no direct use is made of the angles 0
and \p. The problem is also considered in two steps, that is, first with only
rotation of the joints and second, a correction for whatever translation may
take place. The first step Starts with the usual assumption that the ends of
all members are fixed for which condition the end moments in each member
due to the transverse loads can be computed. These moments are designated
"fixed end moments". As the algebraic sum of these fixed end moments
around a Joint will not equal zero, a correction must be applied to each end
moment so that the Joint is in equilibrium. This Operation is called balancing
the Joint. If the members have a constant moment of inertia and if all joints
except Joint A are kept motionless, then the correction for any member AB
at the Joint A
will
IS
be AMab
will
be AMba
The ratio
>
i\
factor and the value \, the carryover factor. If the member is hinged at the
end then 3/4 K should be used instead of K and the carryover factor is zero.
The application of this method will be illustrated by analyzing the Vierendeel
truss shown in Fig. 7 a and 7 b which is subjected to a uniform load of
1000 lbs./ft. applied on the top chord. If the structure is first considered
supported at each panel point so that no translation of the joints is permitted,
as in Fig. 7 a, then the end moments can be computed by the moment distri
bution method in the following manner. The fixed end moments for the top
chord members will be iW2/12 or 12.0 ft. kips and zero for the lower chord
members. The moments are considered positive when acting on the members
in a clockwise direction. At the end of each member the distribution factor
^r^is recorded. At Joint b, the unbalanced moment is 12 so that a correction of + 6.0 must be given to each member and the amount carried to
joints a and c will be (\) (+ 6.0)
+ 3.0. This value + 3.0 now constitutes
the unbalanced moment at these joints, so that the corrections at Joint c will
be (+3.0) x (distribution factor). One half of this correction is carried over
to joints b, d, and e and the procedure continued for all joints. Each time a
Joint is balanced a horizontal line is drawn under the figures. When the
correction is small, the column of figures can be added and the final moment
obtained. The value of the reactions that are necessary to prevent translation
of the joints can be computed from the shear in the members.
The second step in the Solution will be the calculation of the moments
due to forces equal and opposite to the Joint reactions, as shown in Fig. 7 b.
15) "Analysis of Continuous Frames by Distnbuting Fixed end Moments", by Hardy
Cross. Trans, of Am. Soc. Civ. Eng., Vol. 96, 1932.
341
value for
will
displacement A
be equal to Mf
6E
the fixed end moment is 10 K> There is, of course, no fixed end moment
in the verticals. Each Joint is balanced in the same manner as before except
that the symmetry of the structure is used to reduce the amount of wrork.
This was done by considering the verticals as hinged at their mid points,
which would reduce their length by one half and double the value of K=
In other words, if the rigidity factor of the verticals is taken as (3/4) (2K),
then the lower half of the structure need not be used. The numerical work
was started at Joint e and carried through three cycles. After the end mo
ments were obtained, the ratio of the actual panel moments to the computed
moments was recorded. These ratios are 11.28, 12.33 and 2.12 which indicates that the original choice of fixed end moments for the third panel was
not very aecurate. Corrective moments of 3.0 and  8.4 were then placed
in the first and third panels respectively and distributed as before. When the
results of this distribution are added to the original moments, the ratios be
come 9.72, 9.30 and 9.45 which are sufficiently uniform to insure aecurate
results. The true moments are the sum of the values obtained from Fig. 7 a
and 7 b. These are recorded in Fig. 7 b and can be compared with the results
obtained by the slope deflection method which are placed below them in
parentheses.
From the above example it can readily be seen that the method can be
used for any type of rigid frame structure regardless of the Variation in the
factors. However, it is indirect, and frequently a considerable amount of
time must be spent in securing the correct end moments to be used in the
second step. Nevertheless, the general nature of the method makes it a
valuable tool for the analysis of the type of structures considered in this paper.
1933.
E.
L. C. Maugh
342
Notation.
Moment of Inertia
rigidity factor of top and bottom chords

Length
Ki and K2
rigidity factors of verticals in panel.
ratio
ratio
Ki
A2
hi
h2
ratio
hi
Let us first consider the moments in any panel of the truss shown in
Fig. 8 a, such as abcd. This panel will first be considered as hinged to the
remainder of the structure and will therefore act as an independent frame.
The forces acting on such a panel are shown in Fig. 8 b, in which M is the
bending moment in the truss at the section ab, V the external shear in the
panel and M J VL is the moment at section de. The internal moments at
the ends of the members for this force system will be designated the pri
mary moments, and for trusses with chords of equal rigidity K, and
verticals of rigidity Ki and K2, these primary moments can be computed from
the following equations:
s
a(2 + s)]
ccMVL\3 + +
Mad
Mbc
Mda
in which D
Mcb
2
6Jrr\sJra (2aJrasJr2sJr(j)
r,
Al
17>
A2
klirhi>
771
Panel Length.
Eq.
343
Jlk
P2
^i*
*2
V*
<*
Vbc
*fc
Heb
iMtVL
71+VL
h2
h2
(b)
fa)
Fig. 8a und b.
Contraintes primaires dans un panneau.
Primre Spannungen in einem Feld.
Primary action in a Panel.
2 and 3.
J Mda
A Mad
AMda
S(\ +
A4
JM
bc
XM
Ai Mad
J Mcb
1
Mbc
JMtcb
CX)
q~^
s{\
+_a)^
D
771
Eq. 2
m
0 +
Dm
Eq. 3
In these equations m' and m" are given a positive sign when acting
clockwise on the panel. The corrections due to m' and m" can be recorded
on a sketch of the structure for each panel. Eqs. 1, 2 and 3 can be applied to
trusses of the type shown in Figs. 1 and 2 as well as those shown in Fig. 8.
Numerical example: The application of the above equations will be
illustrated by a numerical example. The bent shown in Fig. 10 was analyzed
by Mr. C. R. Grimm in his description of the Kinzua Viaduct so that a com
L. C. Maugh
344
parison can be made with the results obtained by the method of least work,
The first step in the Solution was to tabulate the values of M, V, r, s, a and
D for each panel on a sketch of the structure (Fig. 10). The primary mo
ments were then computed by means of Eq. 1 and recorded in row 1. By
using these primary moments in Eqs. 2 and 3, the first set of corrections were
obtained. These corrections were made by going from top to bottom and
then back as shown by the arrows. This first set of corrections can then be
used to obtain additional corrections although in general these will be very
small. The sum of the primary and secondary moments gives the final result
as recorded in row 3. It will be noted that in this first Solution the rigidity
of the bottom strut is taken equal to infinity which is equivalent to the fixed
end condition that Mr. Grimm assumed. The values of the moments that were
obtained by Mr. Grimm are given in row 4. The moments recorded in row 5
were computed by using the actual rigidity of the bottom strut and by assuming the base of the columns as hinged. Because of the unknown restraint
at the column base, the actual moments will be between these two results.
Effect of Change in Length of Chords: In deriving equation 1
the
for
primary moments, the effect of the change in length of the various
members was neglected. This change in length oecurs chiefly in the chord
members as they are subjected to the greatest axial forces. A method for
determining the primary moments in any panel, such as Fig. 8 b, will now
be given in which the deformation due to both bending moments and change
in length of the chords is considered. The internal forces acting in the frame
must be of such magnitude as to make the total strain energy a minimum.
Thus, if an expression for the strain energy U in the frame is written in terms
of Hbc, Vbc, Mbc, the forces at the end of the chord bc, an equation will be
obtained in which
(Hbc, Vbc, Mbc). For a condition of minimum strain
the
is
that
requirement
energy,
0=0
'
Mbc~
'
Hbc
8Vbc
^^(\+s)
(a)
BMbeC^VbeL{\+s)
(b)
c^E^F^ =f
[6
+ 2,+ 5(2)]
^[3 + 5(2)]
(c)
n which
Mbc(l+s)^FM^hl0^
f{l+s)]^(2 + 3s)
C
B a (1 + s)
r+ s
o/, +s) as + 3/A:(cos0> + cos/i)
r (3 + r+s) + a*(l
~
E=
f
^"^i^
F=
3(1
G= 2 +
+s)a(2 +
3s +
3s)
6A:(sin + sin/S)
hxA
sin2/i
/sin2 CD
3K
MIA Vcos (D + cos/i
345
The terms r, s and a are the same as before, while <P and are the slope
angles of the chords whose crosssectional areas are A.
If the terms in the above equations that contain the crosssectional area
A are omitted, then Eq. 4 will reduce to Eq. 1, which neglects the effect of
the direct stress.
The difference between the results of Eq. 1 and Eq. 4 can be shown
1.5.
numerical example. Let r = 1 s
a
by
.6
L=\2it
A1
10ft.
10 sq. in.
Zm
coscP
E^sin
j/5
3.0 in.3.
*,
(W?7*
eXti
TS
2m'
j?
*i
1V5
x2m"
h,
\c
2m'
Il2
/?2
Fig. 9.
Contraintes dues aux panneaux adjacents.
Spannungen von Seite der benachbarten Felder.
Restraining Action of adjacent Panels.
6+r+s+
M
Mtcb
\2LK\
h2A I
24 LK
h2A
VL /_
24LK
(3
2 \ +' r +'
h*A
Eq. 5
If the
L C Maugh
346
.0042 A/ 3.149 V
.0042 M 2.851 V
Mbc
Mcb
2.824 V
Mbc
.0042
M .027 V
Kj340
L
Zc
WS
73
#4/
*>
7 23
03
[rY2
* 2 4+
237
1
/
CN
CO
//
lzrs=34 2
3 Q*
109
.15.8+
.477
618
416
.104
.1670
18+
CS
CO
+ r
c^^cncn
^P\ 3#
1400
418
s= 323
509
*~3 0+
44 3
1
D= 1071
27
^^
.**>.
o)
1.
71= 2563
CN
CO
dL
30+1
KS
*:
4<*5"
pour Solution ]
^ /Pur
2
Losung
&
N/Porso/ution
6032'
/
l
=*f?
\D=22 32
co
i
370
368
338
D 928
248
3915
378
0
245
/^
ZC6
737
11
li
ITT
S
Fig 10.
(Calcul du pont Kinzua d'apres la methode des panneaux
Berechnung der KinzuaBrucke nach der Feldermethode
Analysis of the Kinzua Viaduct by the Panel Method
347
VL
0. Since a
^
H
where
M
//==
h1
S
h
6a 20'
7.0
*fc
2C
< 10
720'
^
//.
2.0
'/
fc
7/r**
3.0
ffV
/
f/
^X
(^
IF
(^
cd
\ <y
1.0
Mcb
H*
^ x
(+>
7.0
2.0
V
Fig.
11.
Lignes d'influence pour moments flechissants dans des poutres de forme differente.
Einflulinien fr Momente in Trgern verschiedener Form. *
Influence diagrams for Moments in various Trusses.
the axis of either chord coincides with the equilibrium polygon for any load
If the applied
system since the slope of the equilibrium polygon is equal to
load is uniformly distributed over the span as is usually the case for the dead
load, the equilibrium polygon is a parabola and, therefore, a parabolic curve
for the top or bottom chords will be the most economical. The Variation in
L. C. Maugh
348
the moments of the chords when the axis of the top chord is moved above or
below a parabola can be readily seen from the influence diagrams shown in
Fig. 11. For truss A, in which the joints of the top chord lie on a parabola,
the positive and negative areas of the influence lines are equal, while for
trusses B and C, they are unequal.
A comparison of the total combined dead and live load moments for the
three trusses of Fig. 11 has been given in Table 1. These moments were
computed from the influence lines of Fig. 11 for a dead load of 1500 lbs.
per linear foot of truss and a live load of the American Ef 20 Standard
highway loading plus 30 % for impact. It can be seen that the maximum
moments for trusses B and C may be 50 to 60 o/o higher than the maximum
moments for truss A. With a reduced dead load and a larger concentrated
Table
Mbc
B
C
48.7
+ 37.5
Mcb
B
C
Mcd
B
C
A
Mdc
25.3
5.8
0
17.6
+ 45.8
0
35.0
+ 68.3
Vn5 \Z^m
+30% In,Pact
Maximum
Combined
140.5
140.5
125 0
154.0
73.0
97 8
53 2
70 8
66.1
71.5
69.4
82.3
64 5
107.6
122 5
113.8
147.2
94.1
101.6
75.8
119.2
87.5
72.5
112.1
Tlce
Truss
146.5

Moment
+70 8
f 77.3
+122 5
93.6
115 3
53 6
+191.5
193.0
93.6
150.3
+180.4
live load, truss B might show a more favorable comparison, but truss C is
unsatisfactory for any practical use.
If all panel lengths in a truss are the same, and if other factors are kept
constant, then Eq. 1 shows that the moments are directly proportional to
the panel length. For example, if the panel lengths in Fig. 11 are changed from
20 ft. to 30 ft., the moments given should be multiplied by 3/2. However, if
the panel lengths are not equal then of course this proportibn no longer holds.
349
the building. They also point out that some analysis of the structure must
be made before the design can be completed, and that certain approximate
methods give sufficiently aecurate results for this purpose. One of the
approximate methods that is commonly used involves the following steps 17):
a) Distribute the total shear in each story to the columns so that the
exterior columns have onehalf the shear of the inferior columns, or in some
m^3+s
K\
Vh
2
a
t7
*i
a'
Ab t
1
l
1
1
Fig.
V*
12 a.
d ml 3+r
D
k
6+n+s
k2
C ir
Vh
JTL
\l
>
ar
y'
^^Afi= sm
k7
'+rrn " 
r ^777"
Fig. 12b.
M,
<
<
aM=
d
v.
sm'rm"
D
Jm*
IS
*2
v.
J
)m'
kj
3\m"
L 1+2S+3\Z7i'
V
A
D'2
3\
zn
L7+2S+3]z2l
\ A W2
A*1+2r+2S+3rS
I1+2r+3\zn" (+7+2r+3\m
Alli
3\ zn'
ur 3\m'
\
A"")
fr
^A'l
Ali
Fig.
12 c
AM
k2
Fig. 12ac.
Formules pour le calcul des contraintes dues au vent dans la construction des btiments
de grande hauteur d'apres la methode des panneaux.
Formeln zur Berechnung der Windspannungen in hohen Gebuden nach der Feldmethode.
Formulae for the Calculation of Wind Stresses in Tall Buildings by the Panel Method.
taken at .65 of the height from the top for former and .6 of the height from
the bottom for the latter.
c) Select the position of the points of contraflexure in the exterior girders
at .55 of their length from the outer ends and at the midpoint of other girders
unless the conditions of symmetry or equilibrium require otherwise.
The above procedure will usually give satisfactory results for buildings
of regulr proportions unless there is a sudden Variation in the rigidity of
7) See
L. C. Maugh
350
the girders at consecutive floors. For the portion of a frame where such a
Variation may occur, the panel method that has just been applied to the
Vierendeel truss can be used, since a building frame with lateral loads may
be considered as a series of vertical trusses. The application of this method
to any portion of a building frame in which the columns have approximately
the same rigidity can be made in the following manner:
1. Each story of the bent, except the lower, is assumed to consist of a
series of separate panels, such as abcd, Fig. 12 a, each one of which must
resist a part of the total shear in the story. Each interior column will there
fore form a chord of two different panels and the sum of the moments that
are computed for the separate panels will be taken as the primary moment
in the column. This moment must, of course, be corrected for the action of
the adjacent panels in the same maoiner that was used in the preceding
problems.
To determine the amount of the shear V that is taken by each panel,
the usual assumption that the top of each panel in the story moves horizontally
the same amount A (Fig. 12 a) will be made. This movement A can be ex
pressed for each panel by the equation,
A
12EJ_
JL \l*!^Il]
A2
\2E L K(6 + r+s)
3 + 2r+ 25 + rs
'_
C
r77Z!
K(6 + r+s)
1
where
the average rigidity factor of the two columns of the panel. In other
the
value of CV must be constant for each panel in the story, which
words,
gives the relation
C2V2
C1V1
CnVn
shears
V for the panels in the story must be equal to
Also, the sum of the
the total shear in the story, or
Vw
total shear in story.
Q
Vi + V2 +
From these two conditions the shear in each panel can be computed.
2. Determine the moments in the columns by means of the following
and K
steps:
a) Calculate the primary moments m' and m" (Fig. 12 a) for the columns
in each panel by means of the equations
/3 + s\ Vh
/3
+ M' Vh
'
 = and m
tn
2
\ D
b) Calculate the vertical correction AM (Fig. 12b) due to the action of
the panels above and below.
c) Calculate the horizontal correction AM (Fig. 12c) in the outer panel
due to the action of the first interior panel. The horizontal correction in the
other panels will ordinarily be small. If the columns are much more rigid
than the beams, the horizontal correction can usually be ignored.
3. Determine the moments in the beams by assuming the position of the
points of contraflexure as given above, or by distributing the sum of the
column moments at any Joint to the connecting girders in proportion to their
rigidity.
The use of the above methods for calculating wind stresses in fall
building frames will be illustrated by analyzing a portion of a bent of the
351
Buhl Building in Detroit (Fig. 13). From the values of the rigidity factors I
1
that are given, it can be seen that there is a sudden Variation in the rigidity
of the beams at the 23rd floor. The stories above and below this floor will
Q= 20.88*
.
77.6
6.56
1.77
35.2
(37.9)
1.76
27.4
54.8
(29.0)
(58.3)
27.4
(24.4)
54.8
49.7 (S3.0)
(50.4) 57.2 (50.6) 1.76
'S
N
<*>
fc
8.33* 24 F/.
*
7.77
cd
a.33* 23FZ.
268v
+ 702V
 4.9v
2.8h
1.10
&3o)
'222
47.5
70.2V
+ 4.9 V
3.2h
s
V =13.22
<*>
CO
r=
7.10
3.9h
2.13
(er.6)
40.7
C=176
cd
60.2
 5.7 v
+ 717V
 3.3Z)
e.33* ,22FZ.
37.4
(38.0)
50.8
(56.2)
o>
*<
47.5
(47.6)
70.2 v
+ 4.9v
Total
84.8,
175.5 (U2.3)
50.3
+ 5.7 V
77 .7 V
+ 59h
Total
40.
77
97. 7
(90.9)
(89.7)
V=11.10
1
3.49
67.0(587;
cu
S 2.28
C=.208
7V
P=2.28
+ 7.4h
Total
63.5
3.40
S=2.27
C=.264
(80.4)
38.3
 3.0v
 5.7v^ 38.3
+ 3.0v
iz=76.1
A= 74.46
r=4.51
36.7
 9.7v
+ 6.8v
+ 7.6 h
6.62
25.8
+ 9.7
 6.8 V
ToZal
46.7
A =25.7
(5S.7)
SO. 3
+ 5.7v
77.7v
26.6 v
\70.2v
4.9v
7.7 h
r= 4.24
68.0
(67.S)
707.6
(172.0)
V=1361
i
cu
Qj
3.49
86.7
(68
2)
cb
fe
50.8
0.33*
21F7.
707.6
[43.6)
(93.6)
3.40
3.49
CD
vi
Fig. 13.
Calcul des contraintes dues au vent d'apres la methode des panneaux.
Berechnung von Windbeanspruchungen durch die Feldmethode.
Calculation of Wind Stresses by the Panel Method.
therefore be analyzed by the panel method while the first approximate method
will be used for the other stories. The values of r, s, and C were first computed
for each panel by using an average K for the columns. Then for the panels
.264 V2 or V2
.84 Vx and
between the 23rd and 24th floors .2221^
2
Vi
V2
VL
37 54
f^~r
be
13.22*.
V2
11.1*. In the
L. C. Maugh
352
tn"
and
(3
(3
+ 44)
(13^25)
.
_ 4? 5
ft Wps
will be:
10.2 + 4.9
at the top and A M"
5.3 at the bottom. The horizontal
correction is made after the vertical correction are all completed, and for
the bottom of the panel will be:
AM"
and
(0.369 +
0.264)
(+ 0.369 +
0.264)
3.2
f 7.6.
The moments in the girders were computed from the column moments
by distributing the sum of the column moments at each Joint to the girders
in proportion to their rigidity. The figures shown in parentheses (Fig. 13)
are the corresponding values computed by the moment distribution method.
The above panel method will usually give satisfactory results when the
rigidity of the columns in each story is fairly constant and the arrangement
of the members is regulr. For more aecurate results the more laborious
exact methods can be utilized in which only a portion of the structure is used,
or the moment distribution method as already applied to the Vierendeel truss
will be found useful.
Summary.
In this paper the more recent methods of successive approximations are
presented, with particular emphasis upon two types; one, such as the moment
distribution method, in which each member is considered as a primary unit
of the structure and, a second type, called the panel method, in which the
various panels are taken as the primary structural units. The first type is
very useful for analyzing those structures in which the joints undergo rotation,
but when the joints have considerable displacement, as in the Vierendeel
Systems, then this type of method gives a rather involved and indirect Solution
for the majority of problems. The panel method, however, is particularly
applicable to many structures of the Vierendeel type and, as the numerical
examples show, it provides a direct Solution with very little numerical work.
Several early methods of analysis are briefly mentioned to show more clearly
their place in the development of the above Solutions.
In the panel method, simple formulae are given for the calculation of the
moments in Vierendeel trusses with chords of equal rigidity and these
formulae are then applied to numerical examples of viaduets, bridges and
buildings. A discussion aecompanied by some numerical data is also given
of the effect of the change in length of the chords, which was neglected in
the first set of equations, and of variations in the truss proportions upon the
bending moments.
353
Resume.
Le but du present memoire est d'exposer les methodes d'approximations
successives les plus recemment proposees, en insistant tout particulierement
sur deux prineipes generaux: le premier, constituant la methode de distri
bution des moments, dans laquelle chaque element est considere comme
element primaire de Fouvrage; le second, constituant la methode des pan
neaux, dans lesquels ce sont les differents panneaux qui sont consideres
comme elements primaires de Fouvrage.
La premiere mtkhode est tres utile pour Fetude analytique des ouvrages
dans lesquels les assemblages sont exposes des rotations; toutefois,
lorsque comme dans les systemes Vierendeel les assemblages ont supporter
des deformations tres importantes, cette methode ne donne dans la majorite
des cas que des Solutions plutt compliquees et indirectes. Par contre la
methode des panneaux est appliquable tout particulierement de nombreux
ouvrages du type Vierendeel; comme le montrent les exemples numeriques,
eile permet d'obtenir une Solution directe au prix de calculs numeriques tres
peu compliques.
L'auteur mentionne egalement brievement plusieurs methodes anterieures
d'analyse, afin de mettre plus nettement en evidence le rle qu'elles ont joue
dans le developpement des deux methodes principales cidessus.
Dans la methode des panneaux, Fauteur donne des formules simples
pour le calcul des moments dans les systemes Vierendeel avec membrures
d'egale rigidite; puis il en montre Fapplication des exemples numeriques
de calcul de viaducs, de ponts et de charpentes. Une etude critique, aecomdes modipagnee de donnees numeriques, porte egalement sur Finfluence
de
dans les
cote
ete
laisse
a
des
membrures, point qui
fications de longueur
dans
les pro
des
variations
equations precedentes, ainsi que sur Finfluence
portions des systemes sur les moments flechissants.
La mise en oeuvre parfaitement judicieuse de ces deux methodes permet
de determiner facilement et avec precision la Solution convenant de
nombreux; types d'ouvrages rigides.
Zusammenfassung.
Die vorliegende Arbeit behandelt die neuesten Methoden der sukzessiven
Annherung unter besonderer Hervorhebung von zwei Typen; erstens der
Momentenverteilungsmethode, nach der jedes Konstruktionsglied als primre
Einheit betrachtet wird, und zweitens die Feldmethode, nach der die ver
schiedenen Felder als primre Konstruktionseinheiten angesprochen werden.
Die erste Methode eignet sich besonders zur Berechnung von Konstruktionen,
bei denen die Knotenpunkte Drehungen erleiden; verschieben sich aber die
Knotenpunkte um ein betrchtliches Ma, wie beispielsweise bei Vierendeelsystemen, dann ergibt diese Methode eine ziemlich verwickelte und indirekte
Lsung fr die meisten Probleme. Die Feldmethode hingegen ist hauptsch
lich fr viele Konstruktionen des VierendeelTypus anwendbar, und, wie die
Abhandlungen
III.
23
numerischen Beispiele zeigen, liefert eine direkte Lsung mit geringem Rechen
aufwand. Einige der frheren Berechnungsmethoden sind kurz erwhnt, um
ihre Bedeutung in der Entwicklung der obigen Lsungen deutlicher zu zeigen.
In der Feldmethode werden einfache Formeln entwickelt zur Berechnung
der Momente in Vierendeeltrgern mit Qurtungen von gleicher Steifigkeit,
die bei der Berechnung von numerischen Beispielen fr Viadukte, Brcken
und Hochbauten angewendet werden.
Im weiteren wurde, erweitert durch numerische Beispiele, die Wirkung
der Lngennderung der Gurtungen betrachtet, was in den frheren Glei
chungen vernachlssigt wurde, sowie der Einflu der nderungen der Trger
abmessungen auf die Biegungsmomente.
Diese beiden Methoden der sukzessiven Annherung stellen, wenn sie
richtig aufgefat werden, eine leichte und genaue Lsung fr viele Typen
von Steifrahmenkonstruktionen dar.