The University of Texas Publication
No. 5022 November 15, 1950
THE STATICS RATIO FOR ANALYSIS OF FRAMES
THAT DEFLECT
By
PHIL M. FERGUSON
Professor of Civil Engineering
The University of Texas
and
ARDIS H. WHITE
Assistant Professor of Civil Engineering
University of Southern California
(Formerly Instructor, The University of Texas)
Engineering Research Series No. 45
Bureau of Engineering Research
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THE UNIVERSITY OF TEXAS
AUSTIN
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THE UNIVERSITY OF TEXAS PRESS
The University of Texas Publication
No. 5022: November 15, 1950
THE STATICS RATIO FOR ANALYSIS OF FRAMES
THAT DEFLECT
By
PHIL M. FERGUSON
Professor of Civil Engineering
The University of Texas
and
ARDIS H. WHITE
Assistant Professor of Civil Engineering
University of Southern California
(Formerly Instructor, The University of Texas)
Engineering Research Series No. 45
Bureau of Engineering Research
PUBLISHED BY THE UNIVERSITY TWICE A MONTH. ENTERED AS SECOND·
CLASS MATTER ON MARCH 12, 1913, AT THE POST OFFICE AT
AUSTIN, TEXAS, UNDER THE ACT OF AUGUST 24, 1912
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COPYRIGHT, 1950
BY
THE BOARD OF REGENTS
OF
THE UNIVERSITY OF TEXAS
CONTENTS
PAGE
SYN0 PSIS · ____ 5 SYMBOLS AND SIGNS _________________ _
_________________________________________________________ 5
SIGN CONVENTI0 N__ 6 INTR0 DU:CTI0 N __________________ 6 THE MOD EL FRAME CONCEPT____________________________________________________________ 7 STATICS RATIOS AS A MEASURE OF PROPORTIONALITY________ 8 STATICS RATIOS AS GUIDES TO A GENERAL SOLUTION____________ 10
EXAMPLE I 1O ACCURACY 0 F RESULTS_______________________________________________________________________ 17 Table I: Relative Accuracy Using Different Static Ratios____________ 18 TRUE DEFLECTI0 NS _________ _
19 EXAMPLE II 19 Table II: Relative Accuracy Using Different Static Ratios____________ 26 TYPES OF MOVEMENT TO BE USED____________________________________________________ 27 DEFLECTION PATTERNS FOR TRAPEZOIDAL PANELS____________ 27 EXAMPLE I II ________________________________________ 31 Table III: Relative Accuracy Using Different Static Ratios________ __
_ 40 LOADS BETWEEN PANEL POINTS______________________________________________________ 41
CONCLUSI0 NS 42
THE UNIVERSITY OF TEXA:::>
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____ Director and Dean of the College of Engineering ·
Raymond F. Dawson ~Associate Director
B. E. Short Mechanical Engineering
V. L. Doughtie ________ :________________________ __
_____________ __
______ Mechanical ,Engineering
G. H. Fancher Petroleum Engineering
A. W. Straiton Electrical Engineering
M. J. Thompson Aeronautical Engineering
The Statics Ratio for Analysis of Frames that Deflect
Synopsis
A general method for analyzing frames subject to several degrees of freedom of movement or deflection is proposed for practical use. The method operates upon an identical frame as a model. The model frame is arbitrarily deflected by trial increments until it is reasonably similar in its deflection pattern to that of the loaded frame. The arbitrary deflections are represented by simple moment distribution methods. Statics ratios, across particular sections of the frame, relate the resisting shears
(or moments) on the deflected model to the external shears on the loaded frame. These ratios furnish the chief tool for the analysis. They determine the need for corrective deflections of the model and also the magnitude of these corrections ; and finally they determine the degree of proportionality that results. True moments are determined by dividing the final model moments by the statics ratios. ·The method is shown to apply to trapezoidal panels as well as to rectangular. Special deflection patterns are developed to facilitate the solution of trapezoidal panels.
Symbols and Signs
V1 external shear acting on section 11 of loaded frame.
VAB = shear carried by member AB of model frame.
M AB moment acting on joint A from member AB of model frame (equal numerically to bending moment at A end of member AB of the model frame) .
SR1 statics ratio on section 11, i.e., ratio of resisting shear of the members on section 11 across the model frame to the external shear on the same section through the loaded frame; also formulated in terms of resultant moments or forces instead of shears.
Mr. AB = fixedend moment acting on joint A from member AB. (Some may be accustomed to using MFAB for this quantity.) 6 , 6 ' = deflection of a point. cf> angle between a tower leg and the vertical. 8 angle through which a joint rotates; positive when counterclockwise.
Sign Convention
Joint
~
MfT'l_V
Positive moments will be defined as those which tend to
Member rotate the joint clockwise. Shears carry the standard signs.
All the moments and shears shown in Fig. 1 are positive.
M1.J..1V
@Joint
F IGURE 1
Introduction
When moment distribution as a method of frame analysis began to displace more formal solutions involving simultaneous equations, it retained a certain degree of formality in its approach. A fixed sequence of steps was natural: (r) fixedend moments for each loaded member; (2) distributed moments resulting from unbalance at joints; (3) carryover moments to adjacent joints; ( 4) further cycles of distribution and carryover as needed. Under such procedure the beginner and expert go through the same motions and get the same answers in the same number of steps. Their work differs only in the efficiency and accuracy with which the expert introduces and defines artificial boundaries in order to reduce the number of joints involved; and to some extent in the fact that the expert does not feel bound to a definite sequence of joint releases.
When sidesway or frame deflection is important, the formal approach has not led to such efficient procedures except in the case of one story frames. Frames free to deflect laterally at more than one story level or to deflect vertically at more than one panel point, or with equivalent freedoms in any direction or directions, may be said to have more than one degree of freedom of movement. For these frames, the method of successive corrections and the method of simultaneous equations (based upon influence deflections) are both used. The first of these methods is inherently lengthy. The second method becomes lengthy as the number of degrees of freedom of movement is increased; and the solving of simultaneous equations seems contrary to the general philosophy of moment distribution.
In 1933 Dr. L. E. Grinter proposed1 a "simplified method" for wind stresses which in a limited field used moment distribution in a less restricted manner. He estimated deflections, measured the accuracy of his results with a shear ratio, and corrected them if necessary by writing additional fixedend moments. He stated that the method also had value in analyzing single story bents and Vierendeel trusses. He did not explain in detail the philosophy behind this approach, especially the problem of correction moments, with the result that the method has not received
1"Wind Stress Analysis Simplified," by L. E. Grinter, Member ASCE, Transactions ASCE, Vol. 99 (1934), p. 610.
as much attention as it deserves. Some erroneously considered this method semiempirical, for a special usage where only a fair degree of accuracy was import~nt.
A general method of procedure is here proposed which has a wide field of application for structures involving sidesway or deflection. Dr. Grinter1s "simplified method" could be regarded as a special case within this general field. This general method eliminates the need for simultaneous equations. In many cases, especially complex ones, it is the simplest method yet proposed. Any degree of mathematical accuracy desired can be obtained, but an ordinary slide rule is adequate for most practical usage, and a sixinch slide rule was used for most of the sample calculations that follow. As in other moment distribution calculations, the particular problem can be stopped when any desired degree of accuracy has been achieved. Since the method has some of the elements of a trialanderror process, an operator becomes more skillful with practice; and a skillful operator can shorten the process considerably. Nevertheless, the procedure does not require skill; it automatically points the way for each additional trial. ·
The Model Frame Concept
The idea of a model frame, which will be defined here simply as an identical frame dissociated from the given loading and then arbitrarily deflected or displaced, is a useful one in complex situations. In Fig. 2a, a given system of forces causes a unique set of moments, shears, and deflections. If the corresponding model frame is arbitrarily displaced as in Fig. 2b, the chances are that the deflections 6 1' 6 2 , and 6 3 do not bear any constant ratio to the corresponding deflections of Fig. 2a. How
R A
.. .,..__6""1_.,_____..~_o,
p~t8___t
®
&___,__c__
@®
D
FIGURE 2a FIGURE 2b
ever, if identical ratios happen to exist,* Fig. 2b is a true or correct model of the frame loaded as in Fig. 2a; that is, the stresses of Fig. 2a can be obtained from those of the model simply by dividing by this common ratio. Since proportional moments, shears, loads, deflections, reactions must all go together (within the proportional limit of the materials), any of these may be taken as measures of proportionality or correct model action. Until there is proportionality between a loaded frame and its model, there is no very useful theoretical relationship between them. This paper is concerned with methods of producing and measuring proportionality in cases where there is more than one degree of freedom of motion. However, it is also shown that approximate proportionality is often adequate for ordinary use.
Statics Ratios as a Measure of Proportionality
In a frame of the type of Fig. 2, a convenient measure of proportionality lies in the statics ratio, that is, the ratio of the resisting shear of the members at any level in the model to the external shear at the same level in the loaded frame. These resisting shears at the several levels in the model would be sufficient to define the holding forces Q1' Q2, and Q3• The number of such shears required is a simple matter of statics. Three shears (on independent sections) can define three holding forces; n shears are required for a frame having n degrees of freedom of motion and hence n holding forces. Thus the minimum number of statics ratios to be set up must equal the number of degrees of freedom of motion of the frame.
In Fig. 2b the simplest sections to use are the
horizontal sections shown across the vertical panels.
Theoretically a vertical section down the middle of _ _,;iithe frame might be used with vertical shears in lieu
I
Q)
_ _:J__
(i)
of one of the horizontal sections; but such a section
I
involves practical difficulties of considerable magni
I
tude because of the statistically indeterminate reactions. If the reactions of this frame are statistically ®I determinate, as in Fig. 3, the fourth section is ;
@ readily available for a statics ratio. It should be _ 111 kept in mind, however, that if the ratio is identical on any three of these sections, the laws of statics @indicate that it must be the same for every possible section. In other words, the shear on section 44 is automatically established for this frame when
the shears on sections 11, 22, and 33 are fixed.
FIGURE 3
*The matter of signs is not usually a problem because the general shape of the deflected frame is obvious in most cases. However, some may prefer a more formal determination of signs such as the alternate process indicated near the start of Example III.
The same is true of Fig. 4a. Nevertheless, some practical advantages often accrue from the use of an extra section, as in Example I.
External shears have been discussed for the statics ratio purely as a matter of convenience. In the frame of Fig. 4a, the difference between shears on sections 11 and 22 is equivalent to the sum of the horizontal forces on the free body AB of Fig. 4b (here simplified by showing only
FIGURE 4b
®
I
CD1I
1
I
I I I
+11 I
I I
I
t
VJ
"'"
~
<+
<">.
"'
""
~
::::::
""'
""·
0
0
~
:i:..
;::
~
0'
Cr.>
""·
Cr.>
0
"'l
::::5
._,
~
('>
"'
1.3
;:::,..
˘;,
<+
t:l
('>
";:::t
"'
<+
"'
"'
C;,J
The University of Texas Bulletin
carryover moments) and the first trial statics ratios are calculated as follows:
SR _ + 336 + 284 + 308 + 256 _ + 1184 _
1 4•93
240 240 +
+ . +
SR = 24 + 80 5 + 65 = + 116 = 2.42
2 + 48 + 48 +
152 142 135 130 559
SR3 =~~~~~~~~~~=+3.88
144 144
105172109161 547
SR4 = == + 3.80
144 144
~M5_5 = 336308 260251 + 72 + 70 +247 + 239 +172 +161 =194
It i~ necessary to bring these statics ratios into better agreement, but one considerable advantage of this procedure is that it does not matter upon what value they converge. A statics ratio of 3.80 looks like as simple a value as any to attempt next. This obviously indicates the addition of negative moments (pattern #1) to section 11, the addition of positive moments (pattern #1) to section 22, only slight, if any, changes on sections 33 and 44, and the addition of positive moments (pattern #2) to section 55. The operator can guess at the amounts to use, or he can use preliminary calculations, either rough or refined, involving: the amount of the ratio change desired; the existing (last) carryover moments not yet included in the summation; the distribution factors involved; and the indirect effect of other additions in adjacent panels. Exact consideration of all these items would involve troublesome simultaneous equations, but approximate estimates are a powerful tool in the hands of a designer experienced in moment distribution. Simpler methods will be illustrated here, involving the desired change in the statics ratio, the existing unbalanced carryover moment, and only· a guess at something extra to allow for "shrinkage" due to distribution. Ordinarily approximate mental arithmetic would be used here, but the following table is added to show the method in more detail.
Change Change Unbal. Total
Needed in SR in ~M CarryM
SR Ratio Numerfor Over to Allowing for Shrinkage
No. Change ator Sect. M Add Add Pattern Total
1 1.13 271 271 +mo 451 150 units #1 676
2 + 1.38 66 66 52 14 0 ()
+ + + +
3 0.08 + 12 + 12 86 + 98 30 #1 + 136
+
4 () () () 167 40
+ 167 + #1 + 180
5 + 194 + 21 + 173 + 20 #2 + 151
No tabulation is needed once the general idea is clearly grasped.
The Statics Ratio for Analysis of Frames That Deflect 15
These added moments, after two cycles of distribution, lead to new statics ratios. The detail of these distributions has been omitted from Fig. 9 because of limited room, but the resulting totals and unbalanced carryover are shown. The second trial statics ratios become:
+788
SR1= = + 3.27
+240
+ 191
SR2= + = + 3.97
48
562
SR3= = + 3.90
144
504
SR4 = = + 3.50
144
~M5_5 = + 87
These are in better agreement than the first trials but SR1 and SR4 are not entirely satisfactory. When new statics ratios fail thus to respond nearly as expected, something can ofte:r:i be learned from looking over the distribution calculations. Sometimes errors in signs or arithmetic are thus noted; sometimes, as in this case, the error in one's earlier judgment is revealed. In the case of SR1 the, use of an extra 50 o/o allowance for "shrinkage" was decidedy too much and caused an overcorrection of this ratio. The SR4 correction moments incuded only a small "shrinkage" allowance and the reduction in SR4 just found came from some moments on member EN. New moment additions will now be found to adjust the statics ratios to about 3.50 (3.90 would be equally as logical).
Change Change Unbal. Total
Needed · in SR in~M CarryM
SR Ratio Numerfor Over to Allowing for Shrinkage
No. Change ator Sect. M Add Add Pattern Total
1 + 0.23 + 55 + 55 15 + 70 + 22 units #1 + 99 2 0.47 22 22 7 15 0 3 0.40 + 58 + 58 3 + 61 + 16 #1 + 72 4 0 Q .() 4 4 0
+
5 87 22 65 8 #2 60
One cycle of distribution then leads to the third trial statics ratios and further corrections :
+ 232 + 216 + 214 + 196 +858
SR1 = ==3.57
+ 240 +240
17+1044 +84 _ +167
SR 3.48
2
+ 48 +48
The University of Texas Bulletin
161 106 141 100 508
SR3 = · = = + 3.52
144 144
108 150 107 140 505
SR4 = == + 3.50
144 144
~M5_5 = 232 214 199 192 + 57 + 57 + 214 + 207 + 150 + 140 = 12
Change Change Unbal. Total
Needed in SR in~M CarryM
SR Ratio Numerfor Over to Allowing for Shrinkage
No. Change ator Sect. M Add Add Pattern Total
1 •0.07 17 17 7 10 2 units #1 9 2 +0.02 1 1 9 + 10 2 #1 9
+ + + +
3 0.02 3 3 7 + 10 2 #1 + 9
+ + +
4 0 0 0 1 1 0 0
+
5 + 12 10 + 22 3 #2 + 22
+
Two cycles of distribution bring the model into equilibrium. The fourth (final) statics ratios become:
SR _ + 229 + 213 + 211+194 =+847 = 3.53
1
+ 240 +240
16 + 105 4 +84 +169
= = 3.52
+ 48 +48
162105 140 101 508
= = 3.53
1« 1« .
108154 106 143 511
==3.55
' 144 144
~M5_5 = 229 211 197 190 + 57 + 56 + 213 + 207
+ 154 + 143 = + 3
If all statics ratios were identical the final solution would be given by dividing all moments by this one ratio. Although this condition essentially exists in these ratios, an appropriate procedure is used here that seems to give better results, especially when the statics ratios show more spread. Moments on any member cut by one of the statics ratio sections are found by dividing the final model moments by the particular statics ratio for that section. Moments for members not cut by any section are taken of such size as to balance the joint moments. These final moments are shown as the last line of data in Fig. 9. (It might be noted here that Fig. 9 shows the complete tabulation except for seven omitted lines of distribution and carryover moments.)
Accuracy of Results
There is no real purpose served in most cases by securing such close agreement as above between statics ratios. In Example I, it was desired to show that exact convergence upon some common value was a relatively simple matter. As to accuracy, it can always be said that the last digit recorded in any moment distribution process may be in error by one or even two units. If one wants mathematical accuracy in the unit column, he must tabulate at least one decimal place. Slightly more than the usual error from discarded fractions is possible here because of the separate addition of several small increments; but this difference is not of much practical significance. A separate check, starting again with the total fixedend moments and then distributing until balanced, is a very good prac:tice in order to locate and eliminate any real errors in arithmetic or signs; and such a check eliminates the errors due to many small increments. Accuracy in this example has also been limited by the use of a sixinch slide rule.
Reasonable accuracy seems to result from statics ratios only reasonably in agreement. In practical work, it would seem to be unnecessary in many cases to secure an agreement between adjacent statics ratios closer than 10%; and in many cases a further spread might be adequate. There seems to be no real purpose served in a practical problem by securing agreement between adjacent statics ratios closer than 3 or 4%. Dr. Grinter has already pointed out2 in his closely related study of wind stresses in tall building frames that his criterion ratios (for the above example the same as the inverse of the statics ratios) may differ by 10% in adjacent panels with final moment errors seldom more than 5 or 6%.
Table I for Example I, and Tables II and III for later examples, have been prepared for further study of the accuracy obtained when the solution is stopped at various stages of agreement between statics ratios. The results of check calculations starting with the summation of fixedend moments have been entered in the first line under the designation of "Recap. Values" and have been used as a reference base. (These values are themselves subject to errors of 0.3kf, and an occasional error of 0.6kf, corresponding to cumulative errors of 1 and 2 units, respectively, in the moment distribution process.) Below these values are tabulated final moments that would have been found by using various preliminary sets of statics ratios; and with each of these the per cent variation from the first line of data. For the first trial set of statics ratios (tabulated last in the table), the maximum statics ratio is more than twice the minimum and this occurs in adjacent panels. Yet the maximum error in moment is 22 % except for four values that are numericaly small and not very important. For the second set of statics ratios, with a maximum spread of 19 % (based on the average ratio) and an adjacent spread of
2Grinter, "Theory of Modern Steel Structures," Vol. II, p. 169.
TABLE I ,_.. 00
RELATIVE ACCURACY USING DIFFERENT STATICS RATIOS
A B c~ D E
AF
AB
RECAP ll:ILUES F I AL "4
"3
"2
"1
64.6
64.9 +0.5% 64.9
+0.5%
64.6 0% 68.2
+ 6% F
64.6
+64.9 +0.5%
+64.9 +Q5%
+ 64.6 0%
+68.2
+ 6%
SR
3.53
3.53
3.57
3.27
4.93
BA
+60.4 +60.4
0°/o
+60.4 0%
+60.3 ·
0.2%
+57.5 5%
BG
55.9 55.9 0%
55.5 07% 578
+ 3%
476 15%
G
BG
4.5
4.5 0% 4.9
+ 9%
2.5 44%
9.9 +120%
SR
3.54
3.52
3.48
3.97
2.42
GB
+29.6
+29.8 +0.7%
+29.9 +1.0%
+ 28.1 5%
+33.0
+ 11% CH
+16.I
+
16.1 0%
+
15.7 2.5%
+
170
+
6%
+
6.1 62%
H
GD
45.7
45.9 +0.4%
45.6 0.2%
45.1 1.3%
39.I 14% SR
3.54
3.53
3.52
3.90
3.88
DJ
DE
DC
29.9
29.7 0.7%
30.0 +0.3%
30.7 +2.7%
36.5 +22%
+60.7
+60.I 1.0%
+60.8 +0.2%
+
59.8 1.5%
+64.2
+
6%
J
30.8
30.4 1.3%
30.8 0%
29.1 6%
27.7 10%
SR
3.53
3.55
3.50
3.50
3.80
ED  EN
43.3 +43.3
43.4 +43.4
+0.2% +0.2%
42.8 +42.8
1.2% 1.2%
44.3 +44.3
+2.3% +2.3%
45.3 +45.3
+ 5% + 5%
N
~
(:s"
~
~
....
~
~
"!
171>
~ ~
c
~
~
~ ~
171>
tr;:i
~
......
~
<'oj..
~
JH JD  JN
2a8 +58.5 29.7
28.7 +58.6 29.9
0.3% +0.2% +0.7%
28.6 +59.1 30.5
0.7% +1.0% +27%
29.0 +58.4 29.4
+0.7% 0.2% 1.0%
.33.5 +62.2 28.7
+17% + 6% 3%
NJ NE 
40.2 40.3 +0.2% 39.9 0.7% 41.2 +2.5% 42.3 + 5% +40.2 +40.3 +0.2% +39.9 0.7% +41.2 +2.5% +42.3 + 5%
FA 60.1  FG + 60.1
59.8 0.5% +59.8 0.5%
59.9 0.3% +59.9 0.3%
59.7 0.7% + 59.7 0.7%
62.5 + 4% +62.5 + 4%
GF + 55.0 GB 53.9  GH I.I
+55.0 0% 53.9 0% I.I 0%
+54.8 0.4% 53.7 04% I.I 0%
+55.4 1.0% 554 +2.8% 0 100%
+ 51.8 6% 49.7 8% 2 .1 +91%
HG
HG
HJ
39.5
+24.0
+
15.5
+
15.9
.i.23.8
39.7
0.8%
+0.5%
+2.6%
40.0
+24.1
+ 15.9
+2.6%
+04%
+1.3%
+22.4
+
16.8
+
8%
39.2
0.8%
7%
:_34.7
+26.8
+ 7.9
+I 2'Yo
49%
12'Yo
R. VALUES FINAL"4
"3 "2 "1
19o/o , the maximum moment error is 2.0kf and the maximum percentage error is 7 'lo (except for one moment MBc which is numerically very small). The third set of statics ratios, with a maximum spread of 2.6%, and an adjacent spread of 2.6%, gives a maximum error of 0.8kf which happens to be 2.7 % error for that medium sized moment; one other small moment has a 9 % error due to a 0.4kf difference, but only five values have errprs greater than 0.4kf. Few calculations would warrant more exactness than given by the second set of statics ratios, almost none more than given by the third set. Stopping at either of these points would shorten the calculations considerably.
True Deflections
It should be noted that definite movements are involved in the writing of initial and added fixedend moments. But these are relative movements, not final deflections relative to the original supports (except by chance). If one investigated the total movement of N relative to F, for the data used in solving Example I, he would find that N has been raised above its original position. For true deflections, the whole truss must then be rotated through a small clockwise angle about F to return N to its original level. This would give a correction to both vertical movements and sidesway. (The fixedend moments written in on any vertical member measure the sidesway already introduced.)
Ifno fixedend moments in the verticals (Section 55) had been written, a solution would have been entirely possible, although slightly slower in converging. Such a solution of this problem required six increments of fixedend moments instead of the four used here for equal agreement between statics ratios. In such a procedure no sidesway is introduced and N is displaced even further relative to F. The entire sidesway in this case can then be visualized in terms of a rotation angle (of the entire frame about F) to return N to its original position.
Example II
This is an irregular frame (Fig. 10) and the three logical statics ratio sections shown are not entirely independent, each cutting one or more members also cut by other sections. Hence a deflection that increases the shear on one of these sections directly increases the shear on another section and makes it more difficult to foresee the entire effect of a movement. There are many different movements that can be used, in the sense that they are movements for which one can write the fixedend moments without difficulty. Sometimes complex movements are helpful in this process of solution by trial, as will be demonstrated in Example III, but generally simple movements along the sections used in setting up the statics ratios will be the easiest to manipulate.
soKC 25 G
10' (j)25 50K 25 D
8 ______ 25___
10'
@~2.5 E 25 10' @A 
12.5
10'
F
10'
20' 20'
CD
@
12.5 H
8.33 '@
J
t I
FIGURE 10
This frame has only three degrees of freedom of motion and many engineers may find the use of influence deflections and three simultaneous equations more to their liking. Statics ratios show to more advantage when the number of degrees of freedom is larger. This example is included to show the treatment of overlapping sections and members of unequal length.
On section 11, deflections as shown in Fig. lla will be used, represented by fixedend moments in the ratios recorded on the figure. Here the fixedend moment varies as K/ L or l/ L2. Since the larger moment is in member CB, a simple estimate of the starting fixedend moment is based on a 50 kip shear on this member.
Ll.2 C G.
G r1
r......,.. 162.Ll.;·1.00
1.0 1 +Q.25 21
I
I B 1+1.0 10' 20' l BL_ I
I 1 1:+....,.1.=o=o~T..... +4.oo :
I I I
I EI +4.00 H +1.00
1"'"'H'1..+ 0.25
I I
A +1.00
Pattern No. I Pattern No. 2
F IGURE lla FIGURE llb
rr~~
1"'3
B fi::,=D_,, I +i.00
I
I
I
I
+0.444
A +1.00
Pattern No. 3
FIGURE llc
+50x 10
Mt, CB = Mr. BC = = +250 kf
2
Mr, GH = Mr, HG = 0.25Mt, BC = +62.5 kf
On section 33, the deflections shown in Fig. llc lead to larger moments on BA and EF. A starting set of fixedend moments might be based on assigning the entire 100 kip shear to these two members
100 x 20
Mt, BA= Mt, AB= Mt, EF = Mt,EF= 4 = +500 kf
Mr. HJ = Mt, rn = +222 kf
On section 22, the deflections shown in Fig. llb build up moment largely in DE. Some moment has already been put into BA and GH by the other two movements. Additional movement is necessary to build up a resistance to the 100 kip shear. This is taken, almost arbitrarily, as
Mr, DE = Mr.ED = +200 kf
Mr, BA = Mr, AB = Mr, GH = Mr, HG = +50 kf
For ordinary slide rule calculations without recording decimals, it is convenient to retain these relative moments but to double each value. These doubled values are recorded in Fig. 12, each identified by the pattern numbers shown in Fig. 11. Three cycles of moment distribution reduce the carryover moments to reasonable size. Totals (excluding unbalanced carryover moments) are then run for the first trial of the statics ratios.
Statics ratios will be set up as
Resisting shears on model section
SR=
Corresponding external shear on frame
The University of Texas Bulletin
FIGURE 12
McB +MBc + 0.5 (MGH + MHG) 500
MnE + MED + 0.5 (MBA + MAB + MGH + MHG) 1000
MBA + MAB + MEF + MFE + .0.667 (Mm + Mrn) 2000
The first moment distribution totals from Fig. 12 then give the following values:
Trial 1:
+ 114 176 + 0.5 (126 + 102) ! 52 0 1 4
SR
1 .+ . 0
500 ' 500
SR = + 225 91 + 0.5 (775 + 937 + 126 + 102) =1104 = 1.l04
2
1000 1000 +
SR = + 775 + 937 + 729 + 864 + 0.667 (364 + 404)
3 2000
= 3817 = 1.908
2000
This is a very large spread. SR3 will be lowered by adding 500 units of pattern #3 in Fig. llc (half of the original amount of this pattern used, since SR" needs to be halved). When this set of fixedend moments is written in the table, it is noted that SR2 will also be lowered some, but this is temporarily ignored. SR1 must be greatly increased; hence + 500 units of pattern #1, Fig. lla, is added (compared to an initial trial of equal amount). This was not made larger since it was noted from the moment distribution already made that the effect of the original pattern #1 moment was almost cancelled out by large distribution moments at B and H originating from pattern #3 moments written in there. Added pattern #3 moments are now of opposite sign and presumably will have
The University of Texas Bulletin
an opposite effect. Two more cycles of moment distribution (details omitted on Fig. 12) now lead to:
Trial 2: 860
SR1 ==1.720
500
986
SR2 ==0.986
1000
1716
SR3 = = 0.858
2000
Corrections can now be tried somewhat in proportion to the effect of the last step. SR1 was increased 1.62 units by + 500 units of pattern #1. To lower it now ·. to 0.98, or by 0.74 units, requires approximately
0 74
· x 500 = 228 units. Add 220 units of pattern #1. In like fashion
1.62
a proportion for SR3 would lead to approximately + 58 units of pat
tern #3. However, when the values of pattern #1 are written in it is
noted that the large negative unbalanced moment at B and in some
measure at H will cause a positive distribution moment on BA and HJ.
Hence only+ 20 units of pattern #3 is added.
The distribution through two more cycles leads to new statics ratios:
Trial 3:
622
SR1 = = 1.244
500
938
SR2 = lOOO =0.938
1873
SR3 = = 0.936
2000
Again proportions based on the last trial would indicate a need for about 141 units of pattern #1. Some of this is already supplied by unbalanced carryover moments of 26 units on CB and 10 on GH. Hence add only 120 units of pattern #1. It is desirable not to disturb SR2 and SR3, but pattern #1' has already added 60kf on GH. Cancel this general effect by adding + 5 units of pattern #2, which adds a total of
+60 on the three members of SR2 • (Since the members are of various
lengths, this is a rather rough guess.) These values lead to new statics ratios:
Trial 4:
496
SR1 = = 0.993
500
939
SR2 ==0.939
1000
1918
SR3 = = 0.959
2000
These are fairly close together and unbalanced carryover moments might well be closely considered in further estimates. Try to bring ratios together at about 0.960. Again by proportion, add 12 units of pattern #1, temporarily ignoring the 25 units of carryover moment. To balance 18 units of carryover moment and also raise SR2 numerator by 21 units, add +4 units of pattern #2, a total of +48kf of moment. This adds +8 units to GH of SR1 which partially cancels the 25 units of carryover moment on SR1 • When the distribution is complete, the statics ratios become:
Trial 5:
487
SR1 ==0.974
500
956
SR2 = = 0.956
1000
1933
SRa = = 0.966
2000
This is reasonably close. In a system this complex, it is a very good practice to check the results by starting anew with tl?e summation of fixedend moments. This was done and the difference in individual moments in only one case was as much as 3 units, which is about the usual agreement to be expected. This solution was further corrected by adding +2 units. of pattern #2 and 4 units of pattern #1, which gave statics ratios of 0.970, 0.966, and 0.969. This work has not been shown except to list the results as the first line of Table II under the designation "final" for comparative purposes.
When statics ratios are nearly identical, it is not very important to recognize their small differences. However, differences can be recognized by using SR1 on CB, SR2 on DE, SR; on EF and HJ, these all being members cut only by a single section. Likewise the average of SR1 and SR2
The University of Texas Bulletin
TABLE II
RELATIVE ACCURACY USING DIFFERENT STATICS RATIOS
F "5
"4
on GH, and the average of SR2 and SR3 on AB would be logical. The moment on horizontal members would then be found by making joint moments balance.
This procedure has been followed in preparing Table II which compares the results obtained with various sets of statics ratios. It will be noted for every value except MBc that the error is materially less on a percentage basis than is the spread in statics ratios used. (This percentage spread in statics ratios is based on the extreme spread divided by the average of the ratios of that trial.) MBc is a relatively small moment in the vicinity of large moments; its absolute error is not serious; its percentage error is not significantly higher than the spread in statics ratios. In the writers' opinion a design based on trial 3 would be quite satisfactory, one based on trial 4 quite above any criticism.
Types of Movement to Be Used
There is absolutely no theoretical limitation upon the type of movement that is introduced at any stage of this process. To be useful the movement must be one for which correct moments can be written. The writing of these moments is the only way by which the correct continuity of the structure can be maintained. In other words, when new moments are added they must correspond to some possible deflection pattern. One does not need to evaluate this deflection numerically, but it is the key to the relative moments used. In the preceding example only three of many possible patterns were used. Undoubtedly some other pattern could be developed so as to shorten the trial process, but it is questionable whether it is worth while for a single analysis to invest much time in exploring such possibilities when a problem can be made to converge to satisfactory statics ratios by the use of relatively simple fixedend moments such as those of Fig. 11.
In a tower where each panel is a trapezoid, moments corresponding to simple movements are somewhat complex to write and rather involved to use. Since this shape is a fairly common one, a special deflection pattern that has proven helpful will be developed. Such panels can be solved without this pattern and one of the chief reasons for including it here is to indicate the wide degree of freedom which is open in using the statics ratio method.
Deflection Patterns for Trapezoidal Panels
If the entire upper portion of a tower with sloping legs is deflected a distance !:::. with respect to the lower part, without any rotation of joints, the horizontal members in the upper portion are all deformed as shown in. Fig. 13. Since legs CD and C'D' are assumed unchanged in length,
points C and C' move normal to the axis of these legs. This causes some vertical movement of C downward and C' upward and makes 6' = 6 tan 1 + 6 tan 2 = 6 (tan 1 +tan ˘ 2)
FIGURE 13
6' determines the fixedend moment on CC', and also on BB' and AA'.
. 6 sec<1>1
Mr, co = Mr. oc = 6EKco L
CD
6
Mr, C'D' =Mr, D'C' = 6EKc·o· sec 2 =6EKc·o· ~=Mr. CD Kc'D' Lc·o• h2 Keo
6' 6 (tan <1>1 +tan ˘2)
M r, cc• = M r, c·c = 6EKcc· =6EKcc· L ee· Lee·
Kee· h2
Mr, co (tan 1 +tan 2) Keo Lee·
Similarly
KBB' h2
Mr, BB' = Mr, co K L (tan 1 +tan <1> 2)
CD BB'
KsB' Lee'
Mr, BB' = Mr. cc' 
Kee· LBB'
KAA' Lee·
Mr, AA' = Mr. cc• 
Kee• LAA'
·This is a very troublesome pattern. Mr, BB' and Mr. AA' are apt to be relalively large. The movement !::::. as shown is one that might be used to correct a statics ratio cutting across CD and C'D'. The induced moments Mr, BB' and Mr, AA' when later distributed will probably seriously disturb the statics ratios in the higher panels. In other words, this pattern results in moments over too much of the frame. The pattern could be made more satisfactory by holding A and B undeflected as in Fig. 14. This would
A A'
8
I
I
I
I
cl
I I I I
o·
FIGURE 14
induce moments in legs BC and B'C' of the same general order or magnitude as those in CD and C'D'. This involves a direct change in the statics ratio of panel BC as well as the one desired in panel DC. This is better than having panel AB also disturbed. This pattern is a practical one when BCD and B'C'D' are originally straight, but for many cases the following more complex movement will give smaller moments in panel BC and hence cause less disturbance there. Also, BC and B'C' can be different in slope from CD and C'D' without any complication of the following pattern.
This movement will be visualized in two stages for convenience. In Fig. 13 imagine that BC is temporarily pin connected at C and B'C' likewise at C'. When C is deflected !::::. to the right, with A and B free to deflect without artificial restraint, the upper panels will rotate through a counter clockwise angle 6'/Lee' (Fig. 15), as determined by the movement of CC'. No distortion or moment will exist above CC'. The moments in CC', CD, and C'D' before joint rotation is permitted are the same as in Figs. 13 or 14. The ·continuity has been violated, however, by the fact that CB and C'B' have each rotated through the counterclockwise angle 6'/ Lee· relative to joints C and C'. This continuity will now be restored by rotating end C of member CB through a clockwise angle of 6'/Lee' and rotating end C' of C'B' through the same angle, while A, B, A' and B' are held in the position of Fig. 15 without any other rotation. CB and C'B' are thus fixed at B and B' and build up moments as follows when rotated at C and C'.
FIGURE 15
6' 2 KcB
Mr, cB = 4EKcB8c = 4EKcB = + Mr, cc' 
Lee· 3 Kee·
1 Mr BC =Mr CB
. 2 .
6' 2 Kc'B'
Mr, c'B' = 4EKc'B' = + Mr, cc· 
Lee• 3 Kee•
Mr, B'C' = 21 Mr, C'B'
This pattern of frame movement leads to the pattern of moments shown in Fig. 16 (B and B' are fixed only in the sense that movements as outlined cause no moments in members beyond these points. Actually B and B' have been both displaced and rotated as shown in Fig. 15.) In Fig. 16, C and C' are displaced but not rotated. It should be noted that this pattern is independent of the initial slope of BC and B'C'; these members do not have to be straight line extensions of DC and D'C'. On the other hand, in Fig. 14 a change in leg slope in panel BC would complicate the pattern enormously, because B and B' would then deflect vertically.
FIGURE 16
The basic pattern of Fig. 16 is used in the next example, but, when large unbalanced moments show at the deflected joints, rotation of these joints further improves the pattern (see Figs. 21 and 22). This rotation, or any other movement for which the moments can be written, is entirely permissible with this method. The moments as written must satisfy continuity; the statics ratios must be equal to satisfy exactly the conditions of statics. The operator has full freedom within these two conditions to use either simple or complex patterns of movement.
Example III
For the Kinzua Viaduct tower of Fig. 17, it is not easy to write shear equations directly, because horizontal shears involve components of the direct stresses in the legs. However, relatively simple equations can be written in terms of the summation of moments about point "O" where the legs produced would intersect. This is the device used to eliminate chord stresses in the analysis of ordinary sloping chord trusses. Statics ratios will be set up on this basis. Horizontal sections through each panel are logical here and will be numbered from the top for easy reference.
28.51
31.21
621
621
@__:@
"'"
621
@)~· ®
60.3'
FIGURE 17
For the statics ratio on section,11, consider the section cut just above BB' as in Fig. 18. All unknown quantities have been shown as though positive. This is the safest procedure. Due to symmetry of frame:
MBA+ MAB
VBA=VB'A'=
31.6 Resisting moment about "O"
= 60.5 (VBA + VB'A') MBAMB'A'
60.5 M
= 2 ( BA + MAB) 2MBA
31.6
= 2 (l.915MAB + 0.915MBA)
Moment of external forces about "O" = 26.8 x 28.5 = 764 kf
SR =Resisting moment = 2 (l.915MAn + 0.915MnA) 1 External moment 764
MAB+ 0.478MnA
199.5 0
f\
• I \
oil \
0
a,·' \
/\ ~
cv' \
0, I \
I \ I
co· I '
26.BK .. IA \A28.5 '
cv. I ' \
_ I ' I A___ __
I
~''/...........I A
31.2'
\10s' 1A'
Me'A'
FIGURE 1 8
c'
v;J2.
~
Mee
FIGURE 19
For the statics ratio on section 22, consider the section cut just above CC' as in Fig. 19. Resisting moment about "O" M.cB +MBc
= 2 ( ) 123.42McB = 2 (1.961MBc + 0.961McB)
62.9 External moment= 764 + 4.8 X 59.7 = 1051 SR = 2 (1.961MBc + 0.961McB) MBc + 0.490McB 2 1051 268
In like fashion on lower sections: SR = Men + 0.662Mnc
3
301
SR = MnE + 0.748MEn
4 363
SR = MEF + 0.803MF"E
5 429
The University of Texas Bulletin
Usually the statics ratios can be written in this fashion without encountering any uncertainties as to signs. Nevertheless, some may prefer a more formal approach for all problems and others may like it for special cases. The use of a formal equilibrium equation is recommended in such cases. For the section shown in Fig. 18, the final solution of the problem must satisfy the statics equation given by ~M0 = 0. This equation (for clockwise moments positive) is: 1
External moment Resisting moment of loaded frame ~Mo = 26.8 X 28.5 60.5(VBA + VB'A') MBA MB'A'=O
+
If the resisting shears and moments are those of a model frame this becomes:
External moment Resisting moment of model frame 1
~M0=26.8X28.5 +[ 60.5(VBA+VB'A')MBAMB'A']=0
SR1
This equation can be solved for SR1 as follows :
60.5(VBA + VB'A') MBA Mn'A'
SR =
1
+764
which reduces to the same equation already found:
SR = M;AB + 0.478MBA
1 +199.5
The signs automatically follow when a formal equilibrium equation is
written in this fashion.
The patterns of Fig. 16 are too complex to carry in mind. Hence they
are worked out separately for a movement of each joint, in this case (for
convenience) sufficient to give a moment of 0.5 at each end of the lower
leg. These patterns are recorded in Fig. 20. Due to symmetry of the
tower, only onehalf is recorded.
It is noted that the pattern for movement of joint A is not very satis
factory since the excessive moment in AA' will distribute so as to alter
MAB greatly. This situation can be further improved by allowing joints
A and A' to rotate until in balance, a type of movement not yet used in
this paper. Due to symmetry of the frame, all horizontal members take
on a reverse curvature. They can be taken as terminal members by
increasing their stiffness by 50% ; and this leaves only half of the frame
to analyze. In Fig. 21a, the pattern of Fig. 20a is modified by balancing
joint A; and in Fig. 21b these results are proportionately adjusted for
convenience to give MAB + M8A = 1.00. It will be noted that the result
ing pattern of Fig. 21 is really for joint A deflected while wholly unre
strained in any manner. This pattern will be used in place of Fig. 20a.
,,0.5 X 944 X .}LŁX0 .33 3
A; 284 9.5 =l8.15
x 0.66 6
=0.343
•OJ•100
_g__ x 0.834 x 28.4 __ 0 686
v 3 23.0.
t 0.50 (a) A displaced
+ 0.5/0.5 x~x g x 0 .333
14.3 19.9 =T LOO :O .834
t0 .50
( b) B displaced
c
I X 0.0979
~2
=0 .0490
yi X 0.4 52
/ _,_z._X 0.610 X ~= 0.1695 D X Wi= 0.0 9 79
c .___ 3 34.3 +~0.5 x 343 xg x o 333 .,. x.6...Z. x 0.333
.....0.750o 5 x ~ 143 406
16 .4 61.3 = 0.452
= 0 610 •T 1.00
} ~ + 1.00
+0.50
t0.50 (d) D displaced
(c) C displaced E
D
tX0.0798 = 0~0399 _g_ x 0.326 x 16.4 =0.0798 E 3 446 ~f5o'o.5X ~X60.3 X 0.333
16.8 1lf9 tl.00 =0.326
F
o.557
= + l.000
F
c
Total FIGURE 2lb
It also appears that some improvement will come to the pattern of Fig. 20b by releasing joint B and letting it rotate. Figure 22a shows this distribution and Fig. 22b is the equivalent with M Bc +McB = 1.00. This pattern will be used in place of Fig. 20b since it gives small moments in AB.
AB 0.343 F
+
0.188 c 0.155 Total
B 23.0 x 3/2 = 34.5 s'
BA BC BB'
B
+0.
5}37
0.6 8 6 0.500 o.834 F +0.376 +0.188 +0.456 D
=+1.000
0.310 +0.688 0.378 Total
+ 0.463
F IGURE 22b
CB
1' 0.500 F
+o .094 c
+0.594 Total
FIGURE 22a
Very little would be gained by similar releases of joints C, D, and E in Fig. 20c, d, e, since the moments in the upper panels are already small. They will be used without modification.
Reasonable initial fixedend moments in the legs can be determined from shears based on assumed points of inflection. If these are arbitrarily assumed at midheight of each panel, approximate leg shears can be easily found by summation of moments about point "O." These shears multiplied by the half leg lengths are the suggested source of approximate fixedend moments, as follows:
Mr, AB = Mr, BA = +135
Mr, BC =Mr, CB = +179
Mr, CD = Mr, DC = +181
Mr, DE =Mr, ED =+ ~08
Mr, EF = Mr, FE = + 240
The Statics Ratio for Analysis of Frames That Deflect 37
There is a complication with regard to establishing any desired leg moments in these trapezoidal panels. The patterns of Figs. 20c, d, e, 2lb, and 22b show that other accompanying moments are necessary, some of them in other legs. In writing initial fixedend moments, the objective will be to have the total in the legs correspond to the above approximate values (multiplied by an arbitrary factor 2 to get larger numbers, as in earlier problems). The suggested procedure is to start with the lowest panel and write in 4 X 240 = + 960 units of pattern L:;,E representing movement of E, as in Fig. 20e. In the next panel DE, 4 X 208 = + 832, but pattern L:;,E has already written in moments on DE of 3877=115 which must be offset or balanced out. Therefore, write in + 832 + 115 = + 947, say, 960 units of pattern L:;,D, Fig. 20d. (These are estimated needs and this de'gree of "accuracy" is not required. The method is correct for any assumed moments that are consistent with the requirements of continuity, i.e., that correspond to some possible deflection pattern.) In panel CD, 4 X 181 = + 724 and pattern L:;,D has already introduced 47 94 = 141; start with + 724 + 141 = + 865, say, +860 units of pattern L:;,C, Fig. 20c. In panel BC, start with 4X179+73
+ 146 = + 935, say, + 940 units of pattern L:;,B, Fig. 22b. In panel AB, start with 4X135+114+227=+881, say, +880 units of pattern l;,A, Fig.. 2lb. These fixedend moments are all recorded in Fig. 23, where the leg slope has been ignored in aligning the calculations.
After two cycles of distribution, statics ratios indicate that these were good preliminary values of fixedend moments, because the statics ratios are in very good agreement.
Trial 1:
SR = 321+0.478 X 51 = 345 = 1.73
1 199.5 199.5
SR = 361 + 0.490 X 244 = 480 = 1.79
2 268 268
SR = 341+0.662 X 281 = 526 = 1.75
s 301 301
SR = 348+0.748X286=~=1.54
4 363 363
SR = 389 + 0.803 X 434 = ~=
5 1.72
429 429
Fixedend moments will be added in an attempt to bring all ratios to
1.72. In this process added moments will be considered in the light of existing carryover moments that have not yet been balanced. To raise SR4 by 0.1'8, requires that the numerator be raised 66 units; the existing unbalanced carryover moments on DE total +7; add +59, say, +60 units
The University of Texas Bulletin
A
0.0197 0.9803 AB
114 AA
+
490 490 D
+
2 + 112 c
58
D
+ I+ 57 II+ 321 321
B
0.368
0.185
0.447 Ac
73
+ 504277
AA + 390 D 117 59141 c + I 13 D + 4+ 2+ 6
AB 227
II + 51
c
0 .1 78 AD AC 146 AB + 436
D 26
+ 361
0.178 47 + 430
26
c 30 26
D + 10 II + 244
D
0 .148 AE AD 94 AC + 430
D 51 c 13
D
+
10
+
.341
0 .1 69
38
+ 480 58
46
+ 9+
II + 281
E
0.164 0.168 0.668
77 + 480 313
+ 480
93 96 381
29
AE AD D
c
D II
tiE + 480 c 48 c + 2
I I + 434
10
+ 348
412
0.644 525 96
+ 36 585
0.683 434 235
+ 40 629
0.0197 0.9803 II
+ 321
321 C+ 2
ll.B + 4
AA 8+ B D
06
I2 + 319 319
0.368
0.185
0447
412 O+ 5
+
51 + 361
+
7
 16 9
+ 7 D+ I
O+ I
+ 52
+ 350
 402
II c AB AA
I2
II
AD AB I2
I I + 434 c + 2 c 3
I2 + 433
Amirikian
I2
c AD AC AB
D I3 Final Amirikian
I2 c AE
ll.D Ac 0.178
0. 178
0.644
+ 244
+ 341
585 c+ I

+ 4 3
14 D+ 2
+ 2+ 8
+ 233
+ 344
 577
0 .164
0 .168
0.668 II
+ 268
+ 389
675 c + 5 AD+ 30 D6 623 I2 + 315
+ 383
 698
D
0 I3
+ 275 Final
+ 159 Amirikian
+158.I
0 .164 I2
+ 315 c
0 AE I
t.D
+ 10 ·c
0
D?
I3
+ 322 Final
+ 186 Amirikian
+187.6
I2 + 433 AE + 5 C I I3 + 437 Final + 254 Amirikian +254.8
FIGURE 23
0.9803 I2
0.0197
+ 319 319
c
0 AB+ I AA
 2 + 2
D
0I I3
+ 318 318
Final + 185
IB5
Amirikian.
+184.B
184.8
0.368
0.185
0.447
I2
+ 52
+ 350
402
c
O+ I
AC AB+ 2
AA D I3
Final
I 0
+
52
+
30
+
31.3
+
233 0
+
2 4
+
I
+
232
+
134 +134.7
0.178
 0 I 5
a
0 .148
+
280
+
I
2 5
+
I 4+ 2
0I
+
349 + 202 +202.5
+
344
+
338 + 196 +195.0
0.178
0 .169
+ 379
 3 0
+ 10
D+ I+ I c
0I
401 232 232.8
0.644 577
+
6
+
I 570 330 329.7
0.683 659
 9
+ 6
o+ I
+
386
+
223
 661
382 +223.2
381.3
0 .168
0.668
+
383
+
53
+
386
+
225
 698
 2i
708
411 +225.8
413.4
of pattern 6D. (This might well have been a little more since the numerator uses only 0.748MEn and there is also some shrinkage from distribution.) This is entered in the calculation form. SRs is only a trifle high and will not be modified yet, since carryover moments and pattern 6D moments just added on CD total only 9 and will not change the ratio seriously. SR2 needs a reduction of 19 units in addition to the cancelling of +6 units already on hand, a total of 25 units; say, add 30 units of pattern L.B. SR1 is about right but add 15 units of pattern 6A to reduce the +13 units already written for AB.
The new statics ratios that result after one cycle of distribution are:
Trial 2:
344
SR1 == 1.72
199.5
465
SR. = = 1.73
268
529
SR3 == 1.755
301
615
SR4 = = 1.69
363
730
SR5 = = 1.70
429
The objective of 1.72 still appears as good as any. For SR5 to increase
0.02, the numerator must increase +8 units; add +10 units of pattern
6E. For SR4 to increase 0.03, the numerator must increase +11 units;
4 units already exist; add +20 units of pattern 6D. For SRa the
numerator should decrease 11 units; 2 units exist; add 10 units of
pattern L.C. For SR2 the numerator should decrease 3 units; +4 units
exist; add 8 units of pattern 6B. For SR1 no change is wanted but +3
units exist; add 3 units of pattern L.A.
After balancing completely, the statics ratios become:
Trial 3:
As run Recap. values
343
343 = 1.719
SR1 = = 1.719
199.5 199.5
463 462 = 1.724
SR2 = = 1.728
268 268
520 520 = 1.728
SRa == 1.728
301 301
A
Ami rikian Recap.
•3 Oiff.
'"2 Oiff.
...,
Di ff.
B
BB'
BA
BC
Amlrlkian
• 31.3
Recap.
+ 31
+ 31
"3
Olff.
0%
"2.
+ 30
Oiff.
3.3%
•I
+ 30
3.3%
Oiff.
c
Amir~kian
+134.7
Recap.
+134
+134
"3
0%
Oiff.
+135
"2
Oiff.
+0.7%
•1
+136
Olff.
tl.5%
0
The University of Texas Bulletin
Relative Accuracy
AB AA S.R.
+184.8
184.8
1.719
+1 85
185
18 5
1.719
+18 5
0%
0%
+185
185
1.72
0%
0%
1.73
+18 6
186
+0.5%
+0.5%
233.8
+202.5
233
+202
233
+202
0%
0%
232
+202
0%
0.4%
232
+202
0%
0.4%
CB
co cc'
+195.0 l329.7 +196
330 +197
~31
+0.5% i+0.3% +196
331 0%
+0.3% +195
'331 ...0.5•;.
+0.3%
DC
OE oo'
S.R.
TABLE III Using Different Statics Ratios Spread in S.R. Mox. Error in Moments
Overall Adjacent
1.0% 1.0%
1.0% 1.0%
3.8% 3.8%
15% 12%
Amirikian
Recap.
1.724 •3 Oiff.
1.728
"'2 Olff.
1.73
"I Oiff.
1.79
Amirikian Recap.
"3
Olff.
"'Z
oiff.
•1 Diff.
Absolute Percent
Assumed Zero 0.5%
2 0.7% !Except 3.3% on Mb0)
5 1.5°/. (Except 3.3% on Mbo)
0
DC
OE oo' S.R.
+158.I
382.4
+224.3
+159 +223
+15 9
+224 0%
+0.4%
+159
+224 0% +0.4%
+161
+226 +1.3%
tl.3%
E
ED
EF +187.6
+225.8 +187
+225
+186
+225 0.5%
0%
+186 +225 0.5%
0%
+186 +226 0.5% i+0.4"1.
F
FE
~
+254.8 +254
+254 0%
+255 +0.4%
+252 0.8%
'
382 1.730
383 +0.3% 1.727
383 +0 .3% 1.69
397 +1.3% 1.54
EE' 413.4 412 S.R. 1.713
411 02% l.'718
4:11 0.2% 1.70
412 0% 1.72
S.R.
1.728
1.728
l.755
1.75 Amirikicin
~ecap
~
Oiff.
"2
Diff.
•1
Oiff.
627
SR4 = = 1.727 628 = 1.730
363 363
738 735
SR5 = = 1.718 =1.713
429 429
The second set of the above values, headed "Recap. values,'' is based on a separate distribution starting with the total units of each pattern used. It is a better set of values because its calculations involved fewer small fragments. However, the differences between the two sets of statics ratios are negligible in this case and only the original solution has been included here. The final moments are shown on the calculation sheet, the various statics ratios having been used for the legs at their respective height and the horizontal member moment being that required to balance the joints. For comparison, values found by Amirikian by a modified slope deflection process3 are also tabulated. (Signs of these values have been made to conform to the notation of this paper and beam moments have been added as required by joint equilibrium.) The agreement is quite satisfactory, the greatest difference being 2 units for Men. It is noted that part of this particular difference is probably due to the fact that Amirikian's moments in this panel are about 0.5 % too low to balance the static forces. Other solutions for this bent (for half these loads) are also available.45
•
Table III compares the accuracy obtained from these several sets of statics ratios, the per cent difference that is shown being based on the final solution. Amirikian's results are also shown in case the reader prefers to use them as a reference. It should be noted that even the first statics ratios lead to good results in this example. It is inherent in this method that even early statics ratios lead to resisting moments of just the right total amount to balance the external forces. Such errors as exist are due to the improper distribution of these resisting moments to the two ends of members cut by a section or improper distribution between two members cut by the same section. Some error in distribution also exists from use of different statics ratios on adjacent sections which is actually a violation of continuity at joints between the sections.
Loads Between Panel Points
Loads between panel points can be included directly in this type of analysis only if statics ratios are made to take on the special value of
sAmirikian, "Analysis of Rigid Frames" (1942), p. 121. 4"A Rapid and Concise Method of Analyzing Rigid Viaduct Bents," L. C. Maugh, Eng. NewsRecord, Vol. 114 (Mar. 14, 1935), p. 379. S"The Kinzua Viaduct of the Erie Railroad Company," C. R. Grimm, Member ASCE, Trans. ASCE, Vol. XLVI (1901), pp. 2177.
unity.* It is much better in most cases to set up a preliminary simple moment distribution in which no lateral or vertical movement of joints is permitted. This will require that unknown restraining forces be applied at each point where joint deflection could occur. After the distribution process is complete, these restraints can be evaluated.* A separate analysis for deflection or sway of joints can then be run by the general method here proposed, using as external forces these restraining forces reversed in direction. If the original restraining forces act to the left, the frame tends to move to the right and this is the reason for this reversal in direction. Those who are accustomed to analyzing onestory frames by moment distribution will recognize this as the usual procedure for simple sidesway problems. The total moments in the frame are the sum of the moments for the two analyses, the one for loads between joints without sway or joint deflection, the other for the effect of sway or deflection.
As a comment on the flexibility of this general method, it might be noted that the first analysis above may be made to include any arbitrary sidesway or deflection allowance that may be estimated to represent the effects of the loads, or of settlement, or of expansion, etc. In such a case the second analysis furnishes the additional sidesway or deflection that is needed for complete equilibrium.
Conclusions
The method of statics ratios is a convenient method for analyzing frames involving deflections. It is particularly advantageous as the mimber of degrees of freedom of movement of t4e frame increases. It has never failed to yield a satisfactory solution for any type of frame where the authors have tried it. The Vierendeel truss with unequal chords and chords of varying slope becomes a routine problem under this procedure. It is believed to be a general method for any practical frame and loading.
The apparent accuracy secured is limited solely by the numqer of significant places carried in the tabulations and the closeness with which statics ratios are made'"' to agree. The real accuracy is usually much more limited by assumptions of loading, by neglect of the modified deformations present in the joints, the neglect of change in axial length of members, the neglect of shear deformations, and other factors, all of these
*A special statics ratio value of unity makes the procedure very nearly the same as that commonly known by the name successive approximations.
The calculation of restraining forces is really an unnecessary step in most frame problems. A Circular, "Equilibrium Equations without Restraining Forces for MomentDistributionSway Problems," by Phil M. Ferguson, soon to be published by the Bureau of Engineering Research, indicates how problem solutions are often made somewhat simpler by avoiding this calculation. The same procedure can easily be adapted to statics ratio problems involving loads between panel points.
being limitations that are common to most current methods of frame analysis. In view of these conditions engineers may well shorten the solution of many problems by stopping when only a reasonable agreement of statics ratios is attained. The accuracy thus obtained may be estimated from the comparisons tabulated in this paper. In significant values of moment the percentage errors will usually be considerably smaller than the per cent spread in adjacent statics ratios.
AVAILABLE PUBLICATIONS OF THE BUREAU OF
ENGINEERING RESEARCHt
10. Bulletin No. 1733. Papers on Water Supply and. Sanitation, by R. G. Tyler, Editor. 1917.
13.
Bulletin No. 1759. The Friction of Water in Pipes and Fittings, by F. E. Giesecke. 1917.
14.
Bulletin No. 1771. Tests of Concrete Aggregates Used in Texas, by J. P. Nash. 1917.
18. Bulletin No. 1855. The Strength of FineAggregate Concrete, by F. E. Giesecke, H. R. Thomas, and G. A. Parkinson. 1918.
20. Bulletin No. 2215. Progress Report of the Engineering Research Division of the Bureau of Economic Geology and Technology, by F. E. Giesecke, H. R. Thomas, and G. A. Parkinson. 1922.
22.
Bulletin No. 2712. The Friction of Water in Elbows, by F. E. Giesecke, C. P. Reming, and J. W. Knudson. 1927.
23.
Bulletin No. 2780. Effect· of Various Salts in the Mixing Water on the Compressive Strength of Mortars, by F. E. Giesecke, H. R. Thomas, and G. A. Parkinson. 1927.
:j:24. Bulletin No. 2813. Testing of Motor Vehicle Headlighting Devices and Investigation of Certain Phases of the Headlight Glare Problem, by C. R. Granbercy. 1828.
25.
Bulletin No. 2814. Effect of Physical Properties of Stone Used as Coarse Aggregate on the Wear and Compressive Strength of Concrete, by H. R. Thomas and G. A. ParkinsoIL 1928. ·
26.
Bulletin No. 2825. Preliminary Report on Relation between Strength of Portland Cement Mortar and Its Temperature at Time of Test, by G. A. Parkinson, S. P. Finch, and J. E. Huff. 1928.
27.
Bulletin No. 2922. A Study of Test Cylinders and Cores Taken from Concrete Roads in Texas During 1928, by J. A. Focht. 1929.
28.
Bulletin No. 3114. A Method of Calculating the Performance of Vacuum Tube Circuits Used for the Plate Detection of Radio Signals, by J. P. Woods. 1931.
31.
Bulletin No. 3932. Air Conditioning for the Relief of CedarPollen Hayfever, by Alvin H. Wills and Howard E. Degler. 1939.
32.
Bulletin No. 4031. Wavelength of Oscillations Along Transmission Lines and Antennas, by Dr. Ernest M. Siegel. 1940.
34.
Bulletin No. 4238. The City, the Housing and the Community Plan. Some Basic and Historical Considerations, by Hugo Leipziger. 1942.
35.
Bulletin No. 4240. The Influence of Storage Conditions Upon the Physical Properties of Lignite, by Carl J. Eckhardt, Jr., and Chapin Winston Yates. 1942.
36.
Bulletin No. 4303. Cathodic Protection of Metals in Ice Plants, by Robert W. Warner and A. J. McCrocklin, Jr. 1943.
37.
Bulletin No. 4324. Heat Transfer and Pressure Drop in Heat Exchangers, by Byron E. Short. 1948. (Revision of Bulletin No. 3819.)
38.
Bulletin No. 4343. Basic Models for Engineering Drawing and Descriptive Geometry, by Charles Elmer Rowe. 1943.
40.
Bulletin No. 4432. The Specific Heat of Foodstuffs, by Byron E. Short and Luis H. Bartlett. 1944.
41.
Bulletin No. 4704. An Investigation of "U" Type TileConcrete Beams, by J. Neils Thompson and W. D. Ramey. 1946.
42.
Bulletin No. 4811. Ice Air Conditioning for Intermittent Service, by W. R. Woolrich, Francis G. Winters a J. D. McFarland. 1948. ·
43.
Bulletin No. 4813. Intermittent Service Iced Ventilating Systems, by J. D. McFarland,
H. A. Tankersley and W. R. Woolrich. 1948.
44.
Bulletin No. 4913. Tables of Characteristic Functions Representing the Normal Modes of Vibration of a Beam, by Dana Young and Robert P. Felgar, Jr. 1949.
45.
Bulletin No. 5022. The Statics Ratio for Analysis of Frames that Deflect, by Phil M. Ferguson and Ardis E. White. 1950.
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iny error printed as Bulletin No. 2831.