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·.I;'....:~;..'.:~~...:,.
2956 October, 1961
~ST 7
.,.f':
\ .
II\:,'
1
It
I!
j
1
Journal of the
STRUcrURAL DIVISIONProceedings of the American Society of Civil Engineers
STRING POLYGON ANALYSIS OF FRAMES WITH STRAIGHT MEMBERS
By Jan J. Tuma,1 F. ASCE and John T. Oden.2 A. M. ASCE
SYNOPSIS
A general method for analyzing continuous, complex and multi-story frameswith straight mem~ers is presented. The frame members may be or constantor variable section and their deformation may be caused by transverse loads,applied couples, change in volume or displacement of supports. The stringpolygon method is the extension of the conjugate frame method to multipanelstructures. End-bending moments of each member of the panel are selectedas unknowns. The moment matrix is written in terms of two sets of linearequations: (a) elasto-static equations, and (b) static equations; and solved si-multaneously. The numerical procedure is illustrated by two examples.
INTRODUCTION
The representation of the elastic curve of a straight bar as a differentialstring polygon was introduced by O. Mohr3 in connection with his concept ofelastic weights and conjugate beams. H. F. B. Miiller-Breslau developed theidea of joint load (knoten last) for straight members4 and bent members5 and
Note.-Dlscusslon open until March I, 1962. To extend the closing date one month, awritten request must be filed with the Executive Secretary, ASCE. This paper Is part ofthe copyrighted Journal of the Structural Division, Proccedlngs of the American Societyof Civil Engineers, Vol. 87, No. ST 7, October, 1961.
1 Prof. of Clv. Engrg" Oklahoma State Unlv., Stillwater, Okla.2 Instr. In Clv. Engrg .• Oklahoma State Unlv., Stillwater. Okla.3 "Behandlung der Elastlschen LiDle Als Sellllnle." by O. Mohr, Zeltschr D. Archl-
tekt. u. Ing., Hannover, 1868.4 "Beitrag Zur Theorle Des Fachwerkes. "by H. F. B. Milller-Breslau, Zeitschr D.
Architekt. u. lng•• Hannover. 1885, p. 31-5 aDle Graphische Statlk Der Baukonstruktlonen." b)' H. F. B. Miiller-Breslau, A.
Kroner-Verlag, Vol. 2, Part 2, 2nd Ed .• Leipzig, 1925, p. 337.
63
64 October, 1961 ST7 ST7 STRING- POLYGON 65
applied these joint elastic weights to the computation of elastic curves of manyimportant structures. The formulation of joint elastic weights may be found inworksofJ. Wanke,6 F. Chmelka,7 C. B. Biezeno,8W. Kaufman, 9 and others. Theuse of elemental and area elastic weights in connection with a two-dimensionalconjugate structure was introduced by Hardy Cross10 as "the column analogymethod» and by J. S. Kinneyll and S. L. Lee,12 M. ASCE, as "the conjugateframe method.» The application of the same elastic weights in connection withthree-dimensional structures was reported by Frank Baron, F. ASCE,13,14and Jaan Kiusalaas.15
The extension of the conjugate frame method to the analysis of multipanelframes by means of joint elastic weights is presented herein. In the first part,the derivation of the analytical expression for the change of change in slope ofa polygonal panel at a given joint is shown. The physical meaning of this changeis explained and the joint elastic weight is introduced as a force-vector normalto the plane of the polygon in the second part of the paper. The formulation ofthe static equations, the deformation equations and the preparation of bendingmoment matrix is examined in the third part. The formation of multipanel con-jugate structures, the formulation of the corresponding moment matrix, thenumerical applications and conclusions follow in the last parts of the paper.
The study is restricted to coplanar systems with straight members. Thecustomary assumptions of the rigid frame analysis are introduced and the signconvention of the three moment equation is adopted. The end-bending momentsand end slopes are positive if they cause tension on the dotted side of the mem-ber. The elastic weights are positive if acting in the positive direction of theZ-axis.
Notatiol/.-The letter symbols adopted for use in this paper are definedwhere they first appear, in the illustrations or in the text, and are arrangedalphabetically, for convenience of reference, in the Appendix.
ANGULAR FUNCTIONS
A bent bar ijk of variable cross section acted on by a general system ofloads is considered (Fig. 1). The geometry of each segment is given by itslength (dj, dk), slope (Wj, wk) and variation of the cross section. The bendingmoments at i, j, k are assumed to be known and denoted by Mi, Mj, Mk' re-
6 "Zur Berecbnung Der Formltnderungen Vollwandlger Tragwerke," by J. Wanke,Der Stahlbau, 1939, No. 23, 24.
7 "Nliherung Forrneln," by F. Chmelka, Der Stahlbau, 1940, No. 23, 24,8 "Elastic Problems of Single Machine Elements, B by C. B. Blezeno and R, Gram-
mel, Engineering Dynamics, Vol. 2, Blackle and Son, Ltd., Glasgow, 1956, p. 2.9 "Statlk Der Tragwerke," by W. Kaufmann, J. Springer-Verlag, 4th Ed .• Berlin.
1957, p. 144.10 "The Column Analogy,' by Hardy Cross, Engrg. Experiment Sta, Bulletin, Univ.
of Illinois, Urbana, No. 215, 1930.11 "Analysis of Multl-Span-Frames," by J. S. Kinney, in "Steel Rigid Frames,' by
M. P. Korn, J. W, Edwards, Inc., Ann Arbor, Mich., 1953, p, 50,12 "Conjugate Frame Method and Its Application In the Elastic and Plastic Theories
of Structures," by S. L. Lee, Journal, Franklin Inst., Vol. 266, 1958, p. 207.13 "Laterally Loaded Plane Structures and Structures Curved In Space, "by F. Baron
and J. P. Michalos, Transactions, ASCE, Vol. 117, 1952, p. 279.14 "A Circuit Analysis of Laterally Loaded Continuous Frames," by F. Baron, Pro-
ceedln~ASCE, Vol. 83, No. ST 1, January, 1957. -15 lYSis of Rigid Space Frames by Conjugate Frame Method, "by Jaan Kiusalaas.
thesis presented to Northwestern Univ., at Evanston, lil., In 1960, as a partial fulfill-ment of the requirements for the degree of Master of Science,
.-.~
"'k
Loaded Segment ijk
Bending Moment
Elastic Curve
FIG. I.-DEFORMATION OF SEGMENT IJK
~0fv
kjIdky
d.JY
Mk
l>kx
l>ky
spectively. The bending moment diagram and the elastic curve are shown inFig. 1. The location of a cross section in the segment ij Ok) is given by theordinates u, u' (v, v'), measured from the respective ends. The initial changein slope of bent line ijk (before deformation) at j is
Pj' = Wj - wk (1)
in which Wj and wk are the angles of inclination of members ij and jk respec-tively. The change in slope of the same line after deformation at j is
Pj" = Wj - wk + ¢j .. (2)
Thus, the change of the change in slope of the same line at j caused by elasticdeformation is
Pj" - Pj' = ¢j (3)
If the influence of normal and shearing forces on the formation of the elasticcurve is assumed to be small and is neglected, the change of change in slopein terms of bending moments is
j . k
¢j = f Mu ~M~:du + f Mv ~M:) dv (4)
i j
in which Mu, Mv are the real bending moments at u, v, respectively; (Mu),(Mv) denote the virtual bending moments at u, v, respectively; du, dv refer tothe elemental lengths of segments ij, jk; Iu, Iv are the moments of inertia ofthe cross sections at u, v; and E is the modulus of elasticity. The real loadingand the corresponding bending moment diagram are shown in Fig. 2. The vir-tualloading and the corresponding bending moment diagram are shown in thesame figure.
From these diagrams, the real bending moments are
u' uMu = Mi do + Mj do + BMu (5a)
] ]
iimiB
MUI ~BMVu - VI r-Mu0 ",Mj dj Mj Mj.. IM.IMv
u'M. a. M +-
1 j k Ok
Real Bending Moment Diagram
66 October, 1961 ST7 ST7 STRING- POLYGON
Mi~ r c:LJ) Mj (CI::b 0M
k
~C:Iu' 1 G) vkCI v' l~1J I dj ~ J1 J J dk l J
Real Loads
~CJ u' ~)lM~+~~v' j~dj J dj I, dj dk J dk l dk
Virtual Loads
67
andv' v
Mv = Mj dk + Mk dk + BMv (5b)
in which BMu, BMv are the bending moments due to loads,Similarly, the virtual bending moments are
u (~(Mu) = dj
Mv) = dk . . . . . . . . . . (6)
The deformation Eq. 4 in terms of Eqs. 5 and 6 becomes
u v'
a. __ -r-~(MulL4 < ~ ~(l\\l
Virtual Bending Moment Diagram
FIG, 2.-REAL AND VIRTUAL BENDING MOMENTS
k
f BMv v' dv
dkElv
k
V,2 dv f v v' dv
~+Mk 2 +dk E Iv j elk E Iv
The integral expressions in Eq. 7 may be interpreted as angular functions ofthe respective simple beams as will be shown.
(al Angular flexibility Fji (Fjk) is the end of j of the simple beam ij Ok)due to unit moment applied at j (Fig. 3)
V,2
dv . . . . (8)d~ ......k v
k
Fjk = fj
fj u2 duFji = d2 E lu
i j
.... (7)
BMu u dudj E Iu
j
u2 du f~+d. E T] -u i
juu'du f-+M
d2 E I jj u i
k
Mj f
j
Mif¢j =
68 October, 1961 ST7 ST7 STRING-POLYGON 69
0i9=PtL
:::::J;V\;+ l~®
._ -L---... j Jk dk Gkj 1
FJG. 4.-ANGULAR LOAD FUNCTIONS
FIG. 3.-ANGULAR FLEXIBILITIES AND CARRY-OVER VALUES
(12b)
(12a)
(Ub)
...... (lla)
jBMu u du 1 f BMu Y dydj E lu = sin Wj djy E Iu .. . . . ..
i~
T jiy
jBMu u du 1 f BMu x dxdj E Iu = cos Wj djX E Iu
~T jix
kBMv v' dv 1 f BMv y' dy
dk .E Iv = sin wk djy E Iv
I j IV
Tjky
j
Tji = fi
j
T ji = fi
k
T jk = fj
j_f BMu u duTji - ~ E Iu
i
k k
f BMv Vi dv 1 f BMv x' dxTjk = dk E Iv = cos wk dkx E Iv .
j jI I
TYkxAll symbols of Eqs. 11 are defined by Fig. 5 and the last integral in each equa-tion is defined as the end slope of the horizontal projection of the simple beam.
For horizontal loading
and
(b) Angular carry-over value Gij (Gig) is the end slope of the simple beamij Ok) at i (k) due to unit moment applied at j (Fig. 3)
j k
f uu' du f vv' dvGij = 2" ET Gkj = T ET (9)i dj u j dk v
(c) Angular load function T ji (Tjk) is the end slope of the simple beams ij(jk) at j due to loads (Fig. 4)
k
f BMv v' dvT jk = .. '" T •••••• (10)
j
and
If instead of normal loads (Fig. 1), a system of vertical or horizontal loads is·considered, the anguiar load functions of projected simple beams may be used(Fig. 5)
For vertical loading
d.JY
Y'
y'
dy
~®CD
~ JlX ~M.
~, ,
@~ 1J d. J1J
FIG. 5.-ANGULAR LOAD FUNCTION FOR HORIZONTAL AND VERTICAL LOADING All symbols of Eqs. 12 are defined by Fig. 5 and the last integral in each equa-tion is defined as the end slope of the vertical projection of the simple beam.
70 October, 1961 ST7 ST7 STRING-POLYGON 71
Substituting Eqs. 8 through 10 and Eq. 7 gives
/Pj = Mi Glj + Mj !:Fj + Mk GJcj + !:Tj (13)
in which
Thus, the elemental changes in slope are represented by a set of elasticweights in a state of static equilibrium. The line of action of each elasticweight is normal to the plane of the frame and the sense of the elastic weight
and
SINGLE PANEL STRING POLYGON
EFj = Fji + Fjk (14a)
The load functions 7 ji and 7 ik for the most common load conditions reduceto the expressions shown in Table 1.
3'Tij = ~
7 L3'Tji = ~
'T .. = wL3
IJ 'RlIT
:T _ wL3ji - 24 EI
t Lm I M Lm' I.e .r-=: T
w
iu:o::q1 L I
,~I L --Ii2PL m m' (1+m')
6 EI
PL2mm'(1+m)6 EI
T.,IJ
T .. = PL2
IJ TIfET
T .. = PL2
Jl TIfET
PL2 m mt
'Tij = 2 EI2= ML (1 - 3m' )
'Tij 6 EI
2 I 2_ PL m m' _ ML (1 - 3m )Tji - 2 EI Tji - 6 EI
'T •.Jl
TABLE I.-LOAD FUNCTIONS
r Lm f Lm' !i= -- J
j L I
,Cl PL:tI I L Lm' !
.L ~ L-~-1j L l
is governed by the sign of the bending moment Ms' The sign convention forbending moments is stated in the introduction to this paper. The line connect-ing the points of application to all elastic weights is known as the conjugate
':'.'
......•....... (17a)fMS ds --=!:P =0E Is s
fMs x ds....L = E Ps x = 0 (17b)
fMS Y ds -E Is =!: Ps Y = 0 (17c)
16 "Calcul d 'une Poutre Elastlque Reposant Urerernent sur Desappuis InegalmentEspace's, " by B. P. E. Clapeyron, Cornptes Rendus. Paris, 1857.
17 "Uber EiDlge Aufgaben der Stadt; Welche Auf Glelchungen der ClapeyronschenArt FUhren, • by H. MQller-Breslau, Zeitschrift fur Bauwesen, Berlin, 1891.
E 7j = 7 jl + T jk (14b)
Eq. 13 represents the change of the change in slope of the bent line Ijk at jand may be (as any angle change due to elastic deformation) represented by anelastic weight. The similarity of the right side in Eq. 13 with the well knownthree moment equation is apparentl6,17 and it may be said that the three mo-ment equation is a special case of the Eq. 13 with t/lj equal to zero.
If the cross section of each member is different but constant between twojoints, the following simplifications are possible (Eqs. 8 and 9)
~ LkFji = 3 E Ij Fjk = 3 E Ik
........... (15)
G'j -~ ('n~-~1 - 6 E Ij -Kj - 6 E ~
Elemelltal Elastic Weights.-The change in slope of an element ds of a closedpanel frame (Fig. 6) due to bending moment Ms may be treated as a force-vector and denoted as an elemental elastic weight
Ms dsPs = EI (16)
sThen the deformation equations for this panel become
and
72 October, 1961 ST7 :''1' '( ':)J.nu~V-ru ...J.vu!'
conjugate structure (Figs. 8 and 9). The total elastic weight of segments ij andjk are
+x
®
+x
® +x
®
®1\
+z
FIG. 9.-CONJUGATE FRAME-JOINT ELASTIC WEIGHTS
FIG. 7.-CONJUGATE STRUCTURE-ELEMENTAL ELASTICWEIGHTS
L,'O
FIG. 8.-CONJUGATE BEAMS-ELEMENTAL ELASTIC WEIGHTS
:;..
-tX
......... (19)
...•...... (18)k
Wk = L Fsj
k _ v'Fjk = L Ps dk
j
k _ vPkj = L Ps dk
j
Wj = t Fsi
1_ uPji = t Ps dj
1.. - u'Pij = ~ Ps dj1
FIG. G.-REAL STRUCTURE-REAL LOADS
o=z
y
The respective reactions of the separate beams are
structure (Fig. 7). The application of the elemental' elastic weights requiresconsiderable labor. Bending moments and moments of inertia must be com-puted at the centroid of each strip and the elasto-static equations used are interms of a large number of elastic weights.
Segmental Elastic Weights.-It is, however, always possible to remove in-dividual members of the frames, consider them as separate conjugate beams,compute their reactions and apply these reactions as new loads on the initial
The functions Wj and Wk are the segmental elastic weights defined as thechanges in slope of the elastic curve between the respective ends of each seg-ment.
Joint Elastic Weights. -The segmental reactions develop at the joints of theconjugate structure new elastic loads given by the general formula
Fj = Pji + Pjk (20)
74 October, 1961 ST7 ST7 STRING- POLYGON 75
and denoted as the joint elastic weights. These joint elastic weights representa new set of force-vectors again necessarily in a state of static equilibriumand equivalent to the initial set of elemental elastic weights. Thus
The application of the second procedure leads to a larger number of un-knowns but it offers a more direct solution.
18 'Analysis of Fixed End Frames with Bent Members by the String Polygon Method, "by J. T. Oden, thesis, presented to Oklahoma State Unlv., at Stillwater, in 1960, as par-tial fulfillment of the requirements for the degree of Master of Science.
19 •Analysis of Pinned-End Frames with Bent Members by the String Polygon Meth-od. - by H. C. Boecker. thesis, presented to Oklahoma State Unlv., at Stillwater. In 1960,as partial fulfillment of the requlremonts for the degree of Master of Science.
2; Pj y = 0 .•................. (23)
The joint elastic weight Pj is the change of change in slope of the polygonal lineijk at j. Thus, the identify of Eq. 13 with Eq. 20 becomes apparent.
Pj = ¢j = Mi Gij + Mj 2; Fj + Mk Gkj + 2; T j (24)
The joint elastic weight takes the form of the three moment equation and isa function of the angular parameters of segments ij and jk, and of the bendingmoments at the joints.
These elastic weights may be now used for two important purposes: (1)Computation of bending moments at joints, and (2) computation of joint dis-placements. Because all quantities are related to the joints of the polygon andthe angular changes are defined as changes in slope of the polygon strings, theapproach is called the string polygon method.
Computation of Bendi1lg Moments. -If a fixed end frame loaded as shown inFig. 10 is considered, two types of solutions are possible.
Solution I (Fig. 11).
1. The cross-sectional elements N, Vand M (normal force, shear and bend-ing moment) at a given fixed end or joint are selected as unknowns.
2. The bending moments at the remaining points of the polygon are ex-pressed by statics in terms of these redundants.
3. Joint elastic weights are written by means of Eq. 24.4. Elasto-static Eqs. 21, 22 and 23 are stated and solved simultaneously for
N, V and M.
The application of this solution in connection with the idea of the elasticcenter was recorded by Oden.18 Solution of two hinged polygonal frames waspresented by H. C. Boecker, A. M. ASCE.19
Solution II (Fig. 12).
1. Bending moments at the fixed ends and joints are selected as unknowns(m = number of unknowns).
2. Joint elastic weights are written by means of Eq. 24.3. Three elasto-static Eqs. 21, 22, and 23 are stated.4. Remaining conditions are stated from statics (m-3 = number of required
static equations).5. Moment matrix is solved.
CD ®FIG. 10.-POLYGONAL FRAME
FIG. H.-SOLUTION I
\;)KIM2~®'~M4
J® °1.J@ ~\-:.
M1,-/ ~M5
FIG. 12.-S0LUTION Il
Compl/tation of Joint Displacements. -Once the bending moments are known,the joint elastic weights are also known as numerical quantities and the dis-placement of a given joint along a given line is equal to the bending moment ofthe conjugate structure at that joint about that line. .
. • (21)
.. (22)
2; Pj = 0 .
2; Pj x = 0 .
and
76 October, 1961 ST7 ST7 STRING- POLYGON 77
.. (25)
It must be noted that all quantities are related to the joints. Displacementsof intermediate points are not directly available.
MULTIPANEL STRING POLYGON
The principles previously outlined may be easily extended to multlpanelstructures. The extension is based on two propositions:
(I) For an n-panel frame with m unknown end bending moments there exist3n elasto-statlc equations (three corresponding to each panel) and µ. static-equations (each corresponding to one degree of freedom).
(II) Static equations are of two types: (a) Joint moment equations corres-ponding to the number of joints free to rotate, and (b) Shear equations corres-ponsing to the number of translatory modes.
Thus there are as many equations as unknowns
m=3n+µ.
+x 0,__\Lj~ @ IIIII I: II® _ ~ III
~ ... oo:;® --====
0= z
FIG. 13.-REAL FRAME
+y
The upper quantity represents the influence of.the member ij. The lowervalue represents the influence of the member jk. The signs of the force-vectorsPji and Pjk are governed by the end bending moments. The values of T'S arepositive if developed on the dotted side of the member. The similarity of the
If the rigid frame shown in Fig. 13 is considered, the procedure of analysismay be summarized in the following steps:
1. End bending moments in all panels are selected as unknowns (Fig. 14).Directions of these bending moments must be selected in such a way that thecompatibility is satisfied.
2. Joint elastic weights are computed for panels in terms of the assumedmoments.
3. Conjugate structures are sketched (Fig. 15) and elasto-static equationsstated.
4. Static equations for joint moments and panel shears are written.5. Moment matrix is solved.
The frame shown requires solution of 16 unknowns from 9 elasto-staticequations, 4 joint moment equations and 3 shear equations. The introductionof three redundants for each panel (Fig. 16) and the formulation of end mo-ments in terms of these redundants reduces the number of unknowns to nineand the solution is simplified to nine elasto-static equations. The introductionof elastic center is also possible.
Because the end bending moments (at a joint of a closed panel in a continu-ous or complex frame) are not necessarily equal, the joint elastic weight ex-pression (Eq. 24) takes a new form as
Pj
Mi' G'j + Mji F .. + Tj'I J 1 )1 1Iv
Pji
Mkj Gkj + Mjk Fjk + T jkI . I
-y
Pjk
........... (26)
,",C: ~ ~ V G ~10110: : lTlr3 .11'"'G 0 0 JJI~ 1Jj=0 c= ~===0
FIG. H.-REAL PANELS
15'43
15"45
FIG. 15.-CONJUGATE PANELS
78 October, 1961 ST 7 ST7 I:lTRlNG-POLYGON '/'d
+y right side expression in Eq. 26 with the less known four moment equation de-veloped by Bleich20,21 is well apparent.
NUMERICAL EXAMPLES
The first numerical example will illustrate the application of the string pol-ygon method to the solution of a one-story fixed-end frame (Fig. 17) while thesecond deals with the solution of a multi-panel frame (Fig. 18).
All values are given in kips, feet, or kip-feet, and moments of inertia areindicated on each member in the respective figure.
ExamPle 1. -An unsymmetrical portal frame with bottom fixed and acted onby a single concentrated load is analyzed (Fig. 17). The end moments of allmembers are required.
(a) Angular Functions F's, G's and T'S must be computed for members AB,BC and CD. Because all members are of constant cross section, the simplifiedexpressions given by Eqs. 15 and Table (1) are used.
Member AB I Member CD
tr~.®
+x
ill
(f/~~o
FIG. 16.-PANEL REDUNDANTS
Pl\ M /P30,,__-J\ ~1 __® -..l~t;: CJ2__:\3 J\ c- __,/i)
: M1...1\iv :--;lJ'=.----:] M(~ivI tl r 2 , t3I 1 II 1 1 I,In II 1 1
@, 1 ,1 ,
_~ I ,
J
o D Z
36'36 6
FeD = FDC = 3 E (21) = EI
36 3GCD = GDC = ~ ,.. (?T\ = EI
T CD = 'TDC = 0
24 8FAB = FBA = 3 EI = EI
24 4GAB = GBA = 6 EI = EI
TAB = TBA = 0
Member BC
48 4FBC = FCB = "l .... (.4n = EI
48 2GBC = GCB = 6 E (41)- EI
(10) (12) (36) (84) 315'TBC = 6 E (48) (41) =TI
(10) (12) (36) (60) 225'T CB = 6 E (48) (41) = Ex
(b) Relative Elastic Weights (in terms of iI) are obtained by means of Eq.
24 as functions of F's, G's, 'T 's and end moments MA, MB' MC' MD.-'
PA =PAB = MA (8) + MB (4)
PB = PBA + PBC = MA (4) + MB (12) + Me (2) + 315
Pc = PCB + PCD = MB (2) + MC (10) + MD (3) + 225
Po = POC = MC (3) + MD (6)
,'r
+x
©
+x
FCD
36'
@
@I,
I'
",I21
48'
41
'Poe
FBC
10k
FIG. 17.-REAL FRAME
P'AB+y
+y
O~Z
+Z
FIG. 18.-CONJUGATE BEAMS20 "Berechnung Statlsch Unbestlmmter Tragwerke Nach der Methode der Viermon-
entensatzes, • by Fr. Bleich, 2nd Ed., J. Springer Verlag, Berlin, 1918.21 "Buckling Strength of Metai Structures." by Fr. Bleich, McGraw-Hill Book Co.,
New York, 1952, p. 200.
..,. , ::>1' I ::>1 l1..I.!~'-'-r'U"".lUU!' u<
These elastic weights are now applied as force-vectors on the conjugate struc-ture as shown in Fig. 18.
(e) Elasto-Static Equations may be written in many different forms. In thiscase, three moment equiUbrium equations are utilized.
Static moment about AB
(48) (PD + pc) = 0and
2 MB + 13 MC + 9 MD + 225 = 0
Static moment about BC
(24) (PA) + (36) (PD) = 0and
becomes
3 MA - 3 MB - 2 MD + 2 MC = 0
(e) BendingMoment Matrix formed by three elasto-static equations and onestatic equation follows.
[
0 + 2 + 13 + 9] [MA] [- 225]+ 16 + 8 + 9 + 18 MB 0
+ 12 + 16 + 2 0 MC = - 315
+ 3 - 3 + 2 - 2 MD 0
(f) Final Moments, obtained after finding the inverse of the bending momentmatrix, are
16 MA + 8 MB + 9 MC + 18 MD = 0
Static moment about CO
MA=+ 6.49
MC = - 25.39
MB = - 21.38
MD = + 16.43
(48) (PA + PB) = 0and
Example 2. -An unsymmetrical, three panel, trapezoidal frame, with bottomfixed is analyzed (Fig. 20). Uniformly varying load is applied on columns 12
12 MA + 16 MB + 2 MC + 315 = 0
These three deformation equations are in terms of four unknown moments; andfor their solution one additional condition, based on the equilibrium of horizontalforces, must be introduced.
MB.----.......[V
B;[
24' V :AB I
1.1A'----'"
MC
7[co]! 36'IIII VOC-'--"MO
TABLE 2.-RELATIVE ANGULAR FUNCTIONS
Member F G -r
12 8.33 4.17 -431021 8.33 4.17 -412023 9.09 4.55 150032 9.09 4.55 1312
34 = 43 10.00 5.00 ----45 = 54 3.33 1.67 ----56 ,. 65 10.53 5.26 ----36 = 63 6.67 3.33 ----67'= 76 9.58 4.79 ----72 .. 27 10.00 5.00 ----78 = 87 10.02 5.01 ----
(d) Shear equation derived from two free-body sketches shown in Fig. 19,
- VAB + VOC = 0
FlO. 19.-COLUMN SHEARS
in terms of
VAB- MA + MB
24 VDC = MD + MC36
and 23. The cross section of each member is constant for its length. The endbending moments are assumed to be unknown. Again the dotted line indicates
1~he tension SIde of the member. All angular functions are in terms of EI and
this factor is omitted in all computations.
(a) Relative angular functions are computed similarly as in the precedingexample and recorded in Table 2.
(b) Relative elastic weights for the ends 'of each member are computed withhelp of Eqs. 26 in Table 3 and applied on the conjugate structure as shown inFig. 21.
(c) Elasto-Static Equations are written for each panel of Fig. 21:
82 October, 1961 ST7 ST 7 STRING-POLYGON l:l;j
x
P54
PS6
P'67
+x
30'
30'
30'-1 n
2
P'121/ \1P87
®P23
P34
FIG. 20.-MULTIPANEL FRAME
nL
+y
z
P12 8.33 M12 + 4.17 M21 - 431012
P21 8.33 M21 + 4.17 M12 - 4120
P27 10.00 M27 + 5.00 Mn27
P72 10.00 M72 + 5.00 M27
P78 10.02 M78 + 5.01 }I8778
P87 10.02 M87 + 5.01 M78
P23 9.09 M23 + 4.55 M32 + 150023
P32 9.09 M32 + 4.55 M23 + 1312
P36 6.67 M36 + 3.33 M6336
P63 6.67 M63 + 3.33 M36
P67 9.58 Mu7 + 4.79 M7667
P76 9.58 M76 + 4.79 M67
P34 10.00 M34 oj. 5.00 1\14334
P43 10.00 M43 + 5.00 M34
P45 3.33 M45 + 1.67 M5445
P54 3.33 M54 + 1.67 M45
P56 10.53 M56 + 5.26 ::'rI6556
P65 10.53 M65 + 5.26 ~156
Panel 1;
EM12 = 0.-. 30 (Pn + P78) + 50 P87 = 0
1. 11.02 M78 + 6.00 M72 + 3.00 M27 + 13.03 M87 = 0
EM27 = 0 --- 30 (P12 + P87) = 0
2. 8.33 M12 + 4.17 M21 + 10.02 M87 + 5.01 M78 - 4310 = 0
EM18 = 0 --- 30 (P21 + P27 + P72 + P78) = 0
3. 8.33 M21 + 4.17 M12 + 15.00 M27 + 15.00 M72 + 10.02 M78
+ 5.01 M87 - 4120 = 0
TABLE 3.-RELATIVE ELASTIC WEIGHTS
FIG. 21.-CONJUGATE STRUCTURE
84 October, 1961 ST7 ST7 STRING- POLYGON 85
Panel 2;
I:M23 = 0 .-. 20.cP63 + 1>67) + 30 (1)72 + P76) = 0
4. 6.67 M63 + 3.33 M36 + 16.77 M67 + 19.17 M76 + 15.00 M72
+ 7.5 M27 = 0
I:M36 = 0 .-. 30 (P23 + P27 + 1>72 + P76) = 0
5. 9.09 M23 + 4.55 M32 + 15.00 M27 + 15.00 M72 + 9.58 M76
+ 4.79 M67 + 1500 = 0
I:M27 = 0 .-. 30 (P32 + P36 + P63 + P67) = 0
6. 9.09 M32 + 4.55 M23 + 10.00 M36 + 10.00 M63 + 9.58 M67
+ 4.79 M76 + 1312 = 0
Panel 3;
2.:M34 = 0 ...-. 10 (P54 + 1>56) + 20 (P63 + P65) = 0
7. 1.67 M45 + 24.39 M56 + 26.32 M65 + 13.33 M63
+ 6.67 MS6 = 0
I:M45 = 0 .-. 30 (P34 + P36 + P63 + 1>65) = 0
8. 5 M45 + 10 M36 ,t 10 M63 + 10 M34 + 10.53 M65
+ 5.26 M56 = 0
2.:M36 = 0 ...-. 30 (P45 + P43 + P54 + P56) = 0
9. 15.00 M45 + 15.53 M56 + 5.00 M34 + 5.26 M65 = 0
(d) Equations of statics offer seven additional conditions:
the shear equations are
I:MO = 0 (section mm)
10. 2 (M43 - M34) + 2 (M65 - M56) + M34 - M65 = 0
I:MO = 0 (section nn)
11. 2 (M23 - M76) + 3 (M67 - M32) + 900 = 0
I:MO' = 0 (entire structure)
12. 3 (M87 - M12) + 5 (M21 - M78) + 8,560 = 0
~~<~f-<Z
~~Cl~~~IXlI
....~~IXl<f-<
.... N 0 0 0 0 0
'" .... 0 0 N ..... to
" M It) CD .... M It)
0 .... .... .... .... aoU I I I I
II .... M N 0t- o 0 0 0ao aD M 0 M~ .... .....
0 N N .... 0ao 0 ~ 0 0 0t- .... 0 .... It) aD~ .... .....
I
... 0 M t- oN ~ <':! ..... 0::;: ..... ao .... It)
N t- M 0.... ..... " 0
~ ... ao MI
t- o 0 0 0 0
NIt) 0 ~ ~ ~
~ t- It) .... It) M.... ....I
0 0 0 0 0N 0 0 0 0 0t- aD aD .... aD tD~ ..... .... .....
I
to CD t- ao 0 0
t- t- ... It) ~ ~~ .... CD CD N .....
.....I
t- o ao t- en 0
-l ~ ~ to: to: ~.... en 0 .... M....0 CD It) 0
N 0 0 It) 0M .... .,.; ~ M~
I
M It) CD 0 0N
~ It) 0 0 ~..;. en N ....0 0 t- o M
to ~ 0 <0 0 ~'" 0 .... tD 0 M::;: .... ....
I
0 M 0 0 t-M 0 M 0 0 to<0 0 M .... 0 tD::;: ..... .... .....
I
It) M <0 0 N 0
1It) N ~ ~ ~0 It) ... to ......... N
<0 M 0 en.... N It) 0 MIt) aD aD c-i ~::;: .... N
I
It) 0 0 0 t-.... 0 ~ ~ ~
::;: It) It) N .........
0 0 0 0.... 0 ~ ~ ~M 0 It) .... ....~ ....
I
86 October, 1961 ST7 ST7 STRING- POLYGON 87
M32 + M34 - M36 = 0
and the joint equations are
13. 1:M2:= 0 --- M21 + M23
14. 1:M3 = 0
M27 = 0
The extension of this procedure to the analysis of frames with nonprismaticmembers,22,23 bent members,24 wedged members,25 truss-frames,26 column-beams27 and structures in space28 is possible. The application of this pro-cedure to the elasto-plastic analysis of coplanar frames is also :fc0ssible.29The influence of axial and shearing deformations may be included. 0
15. 1:M6 = 0 --- M65 + M67 - M63 = 0
16. 1:M7:= 0 ..-. M76 + M78 - M72 = 0
(e) Bending moment matrix given by nine elasto-static equations, threeshear equations and four joint moment equations in terms of sixteen unknownend moments is represented by Table 4.
TABLE 5.-FINAL MOMENTS
End Moment End Moment End Moment
43 - 12.60 45 - 12.60 56 5.8134 - 9.26 54 5.81 65 27.56
32 103.26 36 93.98 67 - 138.0623 - 126.80 63 - 110.50 76 - 38.77
21 - .306.80 27 - 435.60 78 488.7412 1003.85 72 - 449.97 87 - 520.30
(f) Final Moments obtained from the bending moment matrix are listed'inTable 5.
CONCLUSIONS
It has been shown that the string polygon method may be used in analyzingplane polygonal frames. The resulting bending moment matrix formed by thestatic and the elasto-static equations yield final design values, the bending mo-ments at the ends of each member. Three visual concepts are utilized:
(a) The joint elastic weight is identical in form to the three or four momentequation.
(b) The deformation conditions are obtained from the conditions of elasto-static equilibrium.
(c) The continuity is obtained from the conditions of static equilibrium.
ACKNOWLEDGMENTS
The general theory of the string polygon for straight and bent members waspresented by Tuma in his lectures at the Oklahoma State University, Stillwater,Okla.,31,32 and by Tuma and Oden at the joint meeting of the Texas-OklahomaSociety of Civil Engineers in September, 1960. The application of the stringpolygon in connection with the elastic center was recorded by Oden in a Masterof Science Thesis submitted to the Graduate School of the Oklahoma State Uni-versity, Stillwater, Okla. In preparing the numerical examples the writerswere assisted by D. C. McKee, M. ASCE, Assistant Professor of Civil Engi-neering, Louisiana State University, Baton Rouge, La. G. D. Houser, GraduateAssistant in the School of Civil Engineering, prepared the drawings.
22 "Beam Constants by the String Polygon Method, "by S. L. Chu. thesis presented toOklahoma State Univ .. at Stillwater, In 1959. as partial fulflllment of the requirementsfor the degree of Master of Science.
23 "String Polygon Constants for Members with Sudden Change In Section, " by J, W.Exline, thesis, presented to Oklahoma State UnIv .. at St1llwater, In 1961, as partial ful-fillment of the requlrements for the degree of Master of ScIence.
24 "Slope Deflection EquatIons for SymmetrIcal Bent Members by the String PolygonMethod," by G. D. Houser, thesis presented to Oklahoma State Unlv., at St1llwater, in1961. as partial fulfillment of the requirements for the degree of Master of Science.
25 "Analysis of Wedged Frames with Bent Members by the String Polygon Method, ,.by H. S. Yu, thesis presented to Oklahoma State Univ., at Stlllwater, in 1961, as partialfulfillment of the requirements for the degree of Master of Science.
26 "Analysis of RigId Truss-Frames by the String Polygon Method," by Y. 1. Gonul-son, thesIs presented to Oklahoma State Univ., at Stillwater, In 1961, as partial fulfill-ment of the requirements for the degree of Master of Science.
27 "Column-Beams by the String Polygon and Carry-Over Method, "by J. W, Harvey.thesis presented to Oklahoma State Univ., at Stillwater, In 1960, as partial fulfillment ofthe requirements for the degree of Master of Science.
28 "Theory of Space Structures, n by Jan J. Tuma, Part K 'Strlng Polygon In Space, nLecture Notes, Oklahoma State Univ., Stillwater, 1961.
29 "Plastic Deformation Analysis of Frames at Ultimate Load by the String PolygonMethod," by F. G. Gauger, thesIs presented to Oklahoma State Unlv., at St1llwater, in1961. as partial fulfillment of the requirements for the degree of Master of Science.
30 "The General String Polygon," by C. M, Wu, thesis presented to Oklahoma StateUniv., at Stlllwater, in 1960, as partial fulfillment of the requirements for the degree ofMaster of Science.
31 "Numerical Methods in Structural Analysis," Lecture Notes. by Jan J. Tuma,Oklahoma State Unlv., Stillwater, 1959.
32 "NSF Summer Institute for College Teachers of Structures and Soil Mechanics, "Lecture Notes, by Jan J. Tuma, Oklahoma State Univ., Stillwater, 1960.
00 Uctooer. 1!:H:ll ST7 ST7 STRING-POLYGON 69
APPENDIX.-NOTATION
The following symbols, adopted for use in the paper and for the guidance ofdiscussers, conform essentially with -American Standard Letter Symbols forStructural Analysis" (ASA Z10.8-1949), prepared bya committee of the Amer-ican Standards Association with Society representation, and approved by theAssociation in 1949:
Wj' wk
I/lj
Pi'T ji' T jk
= angle of inclination of members ij and jk;= change in the change of slope at j;= change in slope of Une ijk at j after deformation; and
= angular load functions.
BMu, BMv
dj' dk
djx, djy, (d\oc, dky)du, dv
E
Fji' FjkGij, GkjI
i, j, k
L
m
Mi' Mj, Mkm,m'
Mu, Mv
(Mu), (Mv)N
n
P
Psp..
Jlp.
Ju, u' (v, v')W·Jx, y, z£1
µ.
= bending moments due to loads;= lengths of segments ij and jk;
= horizontal and vertical projections of dj ( dk);= elemental length of segment ij, jkj
= modulus of elasticity;
= angular nexibllities;= angular carry-over values;
= moment of inertia of the cross section;= arbitrary points on the elastic curve;= length of member;= number of unknowns;= moments at i, j, and k;
= length coefficients for computation of load functions;
= real bending moment at u, v;
= virtual bending moment at u, v;
= normal force;
= number of panels;
= applied load;
= elemental elastic weightj
= end shear of segment ij at j of the conjugate structure;= joint elastic weight;
= coordinates of cross sections of member ij (jk);
= segmental elastic weight of segment ij;
= orthogonal coordinates of a point in the structure;= end dlsplacementj
= number of static equations;'
I
