131

Page

Note .--Th-Is paper Is a part or the copyrighted Journalofthe Structuril.lOlvision,Proceeding or the American Society or Civil Engineers, Vol. 89, No. ST3, June, 1963.

Folded Plate Structures of Light Gage Steel, byArthur H. Nilson. (October, 1961. Prior discussion:February, June, August, 1962. Discussion closed.)

by Arthur H. Nilson (closure) . . . . . . . . . . . . . . . . . . . . . . . 135

Economy of High-Strength Steel Structural Members,by Geerhard Haaijer. (December, 1961. Priordiscussion: August, 1962. Discussion closed.)

by Geerhard Haaijer (closure) 139

Evaporation from Pyramid and Winnemucca Lakes,Nevada, by S. T. Harding. (March, 1962. Priordiscussion: June, September, December, 1962.Discussion closed.)

by S. T. Hard1ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Analysis of Structures by System Theory, by N. C.Lind. (April, 1962. Prior discussion: October,December, 1962. Discussion closed.)

by Hsuan-Loh Su . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Analysis of Structures by System Theory, by N. C.Lind. (April, 1963. Prior discussion: October,December, 1962; June, 1963. Discussion closed.)

by N. C. Lind (closure) 147

Seismic Forces on Engineering Structures, byFrank Neumann. (April, 1962. Prior discussion:October, December, 1962. Discussion closed.)

by Frank Neumann (closure) . . . . . . . . . . . . . . . . . . . . . . . . 153

Use of Orthotropic Plate Theory in Bridge Design.by Kuang-Han Chu and G. Krishnamoorthy. (June,1962. Prior discussion: December, 1962; Feb-ruary, 1963. Discussion closed.)

by Kuang-Han Chu and G. Krishnamoorthy (closure) . . . . . . . . 165

Synthesis of Materl,al and Configuration Selection, byLucien A. Schmit, Jr. and Thomas P. Klcher. (June,1962. Prior discussion: None. Discussion closed.)

by Lucien A. Schmit, Jr. and Thomas P. Klcher (closure)

Analytical Approach to Biaxial Eccentricity, by E.Czerniak. (August, 1962. Prior discussion: De-cember, 1962. Discussion closed.)

by E. Czerniak (closure) .

Safety of Eccentrically Loaded Reinforced ConcreteColumns, by M1l1kTichy and Milos Vorlicek. (October,1962. Prior discussion: April, 1963. Discussionclosed.)

by Bruno Barbarito . . . . . . . . . . . . . . . . . . . . . . . . .by Janusz Murzewski .by C. Berwanger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Specifications for Structural Joints Using ASTM-A325Bolts. Progress Report, Task Committee on Rivetedand Bolted Joints, Committee on Metals, StructuralDivision. F. H. Dill, Chairman. (October, 1962.Prior discussion: February, April, 1963. Discussionclosed.)

by R. S. Loubser .by D. T. Wright .

Direct Design of Plate and Shell Structures, by JohnF. Brotchie. (December, 1962. Prior discussion:None. Discussion closed.)

by G. I. N. Rozvany .

Model Studies of a Concrete Hyperbolic Paraboloid,by Allen G. Thurman and George J. Herman.(December, 1962. Prior discussion: None.Discussion closed,)

by T. J. McClellan .by A. A. Eremin .

Lateral Stiffness of Infilled Frames, by Bryan StaffordSmith. (December, 1962. Prior discussion: None.Discussion closed.)

by A. A. Eremin .Space Frame Analysis by Matrice and Computer. byKurt Eiseman, Lin Woo, and Saul Namyet. (December.1963. Prior discussion: April. 1963. Discussionclosed.)

by J. D. Renton .by William R Spillers .

Deflection and Slope of Beams with Varying 1. byOtakar Ondra. (February, 1963. Prior discussion:None. Discussion closes July 1. 1963.)

by J. T. Oden .

132

I;.

~~

.'.,;1

~.t,

:1.r:~~\i

"r::~FFl

[,'I

":~;1~:.,II!I

').j

IIIIII

.::11 JJune, Ilfti;1

DISCUSSION

Journal of the

STRUcrURAL DIVISIONProceedings of the American Society of Civil Engineers

J:>oU

;~

.•

~CJ

preparation of a computer program of such scope. They show the computerbeing used to its greatest advantage in the solution of a general problem.

~.

1

216 June, 1963 ST 3 ST 3

DEFLECTION AND SLOPE OF BEAMS WITH VARYING ra

Discussion by J. T. Oden

217

~~t4'J~, ..".'IY'.'~''''.'.' ,•••.•.....': ••• I;--:- •••• ~.. .::>,,<:O: ... ~.~ ...... Y. ••~y••: .....<........ Qi iNca rl

J. T. ODEN,9 A. M. ASCE.-The method presented by Ondra seems wellsuited for slope and deflection analysis of beams with step-wise moment ofinertia variation. It has deflriite shortcomings, however, in the analysis ofbeams with a more general variation in section. The string polygon method,used in complex frame analysis, provides an excellent tool for finding deflec-tions and slopes of beams with any variation in cross section. In most cases,the method results in only a few more calculations than are necessary toconstruct the bending moment diagram of a simple beam. Details of thetheory of the string polygon methOd were presented in an earlier paper,lOand it suffices herein to merely indicate its application to the analysis ofbeam deformations.

The term ·string polygon" refers to a closed polygon of straight lines(string lines) connecting arbitrary points, 1, j, and k, on the elastic curve(Fig. 12). The number of points chosen, as well as their location, is completelyarbitrary. Points i, j, and k may be any finite d1stance apart, and need not bespaced in equal intervals. Angle changes occurring in the string polygon ati, j, and k are denoted ~~ ~,and cJlk,respectively, and become elastic weightsacting on the conjugate beam. The total angle-change of the string polygon atpoint j is

¢j = CPji+.CPjk= (Mi Gij + Mj Fji + Tji) + (Mj Fjk + ~ Gkj + 1jk) .. (18)

in which • ji, CPjkare the angles between the string lines ij, jk and a tangentto the elastic curve at J (these angles are called segmental elastic weights);Mi, Mj, Mk are the bending moments at i, j, k; Gtj, Gji, Fji, Fjk are theangular fiexib1llties of segments ij and jk; and 1ji, Tjk are the angular loadfunctions of the segments.

For constant sections of length L under a uniform load of intensity q,

G =G = Lij j 1 6 E r . . . . . . . . . . . . . .• (I9a)

a February 1963, by Otakar Oodra (Proc. Paper 3409).9 Asst. Prof., School of Clv. Engrg., Oklahoma State Unlv., Stillwater, Okla.10 ·String Polygon AnalYSis of Frames with Straight Members" by J. J. Tuma and

J. T. Oden, Proceedings, ASCE, Vol. 87, No. ST7, pp. 63-89.

t as ;» """... ·brJliPl~#"""",.,##'*· •.•••.• ' .. ' ..

'I . 218 June, 1963 ST 3 ST 3 DISCUSSION 219

F'~:~~ L IBA< ,.B

SIMPLEBEAM,AB

I ~ I I 1 ~ Ik/ ft.2~ --===rI '2'd ,±I' SIMPLE2° 3' 3. BEAM

~ 20' I 10' I 10' ~ 10' Io I 2 3 4

STRINGPOLYGON

120 Ok-'~b BENDINGMOMENTDIAGRAM

i~j¥J1 J1

LJ0j I~fAi fBk'"A+ ¢Ai "'B + ¢Bk

SEGMENT OFPOLYGON

CONJUGATEBEAM

'"

°1 0:! 03

I I I I I"'0 "'4

"'o-1ll1-1ll2

CONJUGATEBEAM

CONJUGATESHEAR

II wA wlj I wi' I wB

I

1 J

FIG. 12

SHEAR DIAGRAM OFCONJUGATE BEAM

MOMENT DIAGRAM OFCONJUGATE BEAM

...~, ~.::.; -.; ~

"'4

FIG. 13

~"'''''/I·'''~~..:..··· .;.-.

CONJUGATEMOMENT

Application of the string polygon method may be shown by demonstratingits use in evaluating the slope and deflection of point 2 of the beam shown inFig. 13. It is found from Tuma, Lassley and French13 that FlO = 3.2400/EIo.The remaining flexib1lities and load functions follow directly from Eqs. 19as follows:

Tables of angular flexib1lities and load functions for segments of variabledepth are available in a number of places in literature.ll,12,13,14,15,16 Atable of angular load functions for strips of constant depth is given in Tumaand Oden.10

The bending moments of conjugate beams at points corresponding to pointsof the string polygon (i, j, and k) are equal to the "exact" deflection of thereal beam at those points. The shear diagram of the conjugate beam representsslopes of the string line (wA, Wij,and so forth). True slopes at i, j, and k arefound by simply subtracting or adding the proper segmental elastic weightand the slope of the string line. The true slope at j, for example, is

31 . = 1 = ~i] ji 24EI (19c)

\- .I

,

!

I.I.V

and

olWle, .1:10,)

LFij = Fji = 3 E I

9j = wij - rt>ji= wjk + If>jk .

Fl = F - 3.3332 21 - EI

0

F2

= F - 0.9883 32 - ~

0

G12

= G - 26.66721 - EI

0

.:> 1 ,)

(19b)

(20)

F = 26.66734 EI

0

0.494G23 = G32 = ~

0

12.346123 = 132 = ~

0

333.333134 = 143 = ~

0

These constants and the appropriate values of the bending moment aresubstituted into Eq. 18 to yield

259.20 466.67 _ 725.97f/ll = f/l10+ 1f>12= EI + E'I- EI

0 0 0

533.33 185.14 _ 718.471f>2 = 1f>2l+ 1f>23 = EI + ~- ~

0 0 0

f/l = If> + If> = 180.26 + 3,266.70 = 3,446.963 32 34 EI EI EI

000

These elastic weights are applied at points 1, 2, and 3 of the conjugatebeam. The "exact" deflection of point 2 follows by simply evaluating the

35 381bending moment of the conjugate beam at that point. Thus, ~2 = Tx- ft.o

Deflections at other points are found in a similar manner. In order toobtain the "exact" slope at 2, iUs necessary to subtract the segmental elasticweight f/lZlfrom the conjugate shear to the left of this point. Hence,

11 "Frames and Arches,· by V. Leontovich, McGraw-Hill Book Co., Inc., New York,N. Y., 1959, pp. 417-448.

12 "Theory ofStructures," by S. P. Timoshenko and D. H. Young, McGraw-Hill BookCO'f Inc., NewYork, N. Y., pp. 405-411.

3 "Analysis ofConUnuous Beam Bridges, Vol. 1, Carry-Over Procedures,· by J. J.Tuma, T. LassIey, and S. French, School of Civil Enr.eering Research Publication,Oklahoma State University, Stillwater, Okla., No.3, 195 .

14 "Rahmentragwerke and Ourchlauftrager," by R. Guldan, Sprloger Verlag, Vienna,Austria, 1952.

15 "Die Cross Methode,· by R. Guldan, Springer-Verlag, Vienna, Austria, 1955, pp.131-138.

16 "Neure Methoden zur Statik der Rahmentragwerke" by A. Strassner, 4th edition,Vol. I, Berlin, Germany, 1937.

82 = G21 - 1f>21695.39 533.33EI-~

o 0

162.06EI

o

~m:'7.' ••,;-;..,;;-::::::.:-::.;-;.;-,:::.: '.', .... ',', ~.;.:.~... ;"::-,;-;.'.:;..;;.~~);~~.. ; ;;;;::yp ~m~wn."" ·.·,.. - .