CORPS O F E N G IN E E R S , U. S. A R M Y

A P R O C E D U R E F O R T H E D E S IG N O F C O N T IN U O U S BEAM S O F R E IN F O R C E D C O N C R E T E , T A K IN G IN T O

A C C O U N T T H E P R O P E R T IE S O F T H E M A T E R IA L S

(EIN W ERKSTO FFG ERECHTES VERFAHREN ZUR BEMESSUNG VON

D U R C H LA U FB A LK E N AUS STAHLBETON)

K . JÄG ER

O ESTERREICHISCHES IN G E N IE U R A R C H IV , V I, 3, PP. 223-236

BY

wag mm

TR A N S LA TE D FROM TH E GERM AN

BY

JAN C. VAN TIEN H O VEN

RESEARCH CENTER

T A7,W34tr no.53-4 1953

W ATERW AYS EX PE R IM E N T STATION

V IC K S B U R G , M IS S IS S IP P I

>N NO. 53-4 JUNE 1953

,.r-v>

( / o f f r ì S-S/f

LIBRARY

J UN 2 1968

Bureau of Reclamation Denver, Colorado

BUREAU OF RECLAMATION DENVER LIBRARY

CORPS OF ENGINEERS. U. S. ARMY

PROCEDURE FOR THE DESIGN OF CONTINUOUS BEAMS OF REINFORCED CONCRETE, TAKING INTO

ACCOUNT THE PROPERTIES OF THE MATERIALS^-~ > (E IN W E R K S T O F F G E R E C H T E S V E R FA H R E N Z U R B E M E SSU N G VON

D U R C H LA U FB A LK EN AUS STA H LBETO N )

BY

5 K. JÄ G E R -3

O E S T E R R E IC H IS C H E S IN G E N IE U R A R C H IV , VI, 3, PP . 223-236

?

<reCbsLÌcal I n t e r i1 „t 'Reclamation01 oradoDenver,

T R A N S L A T E D FR O M T H E G E R M A N

BY

JAN C. VAN TIEN H O V EN

R E SE A R C H C EN TER W ATERW AYS E X PE R IM E N T STA TIO N

V IC K S B U R G , M IS S IS S IP P I

ARMY-MRC VICKSBURG, MISS.

TR A N SLA TIO N 1 ^ 53-47

JU N E 1953 .

A PROCEDURE FOR THE DESIGN OF CONTINUOUS BEAMS

OF REINFORCED CONCRETE, TAKING INTO ACCOUNT

THE PROPERTIES OF THE MATERIALS by

K. Jäger, Vienna

Synopsis. This paper deals with a more accurate determination of the bending moments in statically indeterminate reinforced concrete beams taking into account the variable bending stiffness resulting from the dis tribution of the steel reinforcements, which bending stiffness is defined by the support moment sustainable in each cross section.

1. Introduction

The cross-sectional forces in statically indeterminate reinforced concrete beams are computed in the usual manner on the basis of the elas­tic theory; i.e., under the assumption of a homogeneous, isotropic ma­terial of unlimited elasticity (condition I, full participation of the tension zone of concrete). Even though the afore-mentioned assumptions are sufficiently accurate in the case df very small load ranges, they are not valid for the working load, and even much less in the region of the practical load capacity (maximum permissible load). The n-method, which is still used in most countries for designing, is based on the elimination of the tension zone of the concrete and on the acceptance of

Hooke fs law for concrete and steel with a constant value n = 15 of the moduli of elasticity (condition II) . Hence it is adjusted to the range of the working load and usually cannot furnish information with respect to the ultimate strength and consequently to the safety of the struc­tural member.

2

The reliable precalculation of the ultimate strength of reinforced concrete structures makes it possible above all to obtain a correct eval­uation of test results. It not only is of the greatest economic signifi­cance because of a better utilization of the strength properties of con­crete, but it also has become a necessity in view of the increasing use

of prestressed concrete where the n-method cannot be applied at all. In due recognition of these relationships the results of a number of re­searches have been published during the past fifteen years, their object being to promote the development of new computation procedures with re­spect to test results or to the introduction of the true laws governing the form changes in steel and concrete. Whereas the majority of these publications is merely concerned with the precalculation of the ultimate load~*~, a group of other investigations strives primarily for the repre­sentation of the bearing behavior within the entire load range and, in doing so, it arrives at the important conclusion that not the ultimateload itself, but a load lying below it, the so-called maximum permissible

2 3load , or critical load , must be regarded as a basis for design. In this case, the maximum permissible load corresponds to the critical

R. Saliger: Die neue Theorie des Stahlbetons (The New Theory of Rein­forced Concrete), 1st to 3rd editions. Vienna: F. Deuticke, 19^7 to1950* F. Gebauer: Die Plastizitätstheorie im Stahlbetonbau (ThePlastic Theory in Reinforced Concrete Structures). Vienna: G. Fromme,19^9• Additional list of publications in a comprehensive representa­tion by K. B. Haberstock, entitled: Die n-freien Berechnungsweisen deseinfach bewehrten, rechteckigen Stahlbetonbalkens (Methods of Computing Simply Reinforced, Rectangular Beams of Reinforced Concrete Without Re­sorting to the n-procedure), Issue No. 103 of Deutscher Ausschuss für Stahlbeton, Berlin, 1951*K. Jäger: Festigkeitsnachweis im Stahlbetonbau (Determination of

Strength in Structures of Reinforced Concrete). Vienna: ManzscheVerlagsbuchhandlung, 19^8.

3

condition -where the expansion of the steel reinforcements does not exceed

a given maximum value (dangerous crack condition), in which case the per­

tinent strain at the boundary of the bending compression zone would more­

over be limited by the compressive deformation of the concrete at the2 >

beginning of the plastic state (prelude to a loosening of the structure) •

From this brief consideration concerning the prevailing status of

the bases of design it follows that the computation of the cross-sectional

forces, and the design of statically indeterminate reinforced concrete

beams, is at present being conducted under different and conflicting as­

sumptions. This, and the circumstance that the bearing capacity of such

structures is of the utmost importance in the final analysis, induced the

writer to undertake a thorough investigation concerning the bearing be­

havior of statically indeterminate beams, both under working conditionsband under critical conditions , after the fundamental relationships had

2already been discussed in an earlier publication . On this occasion it

was found that the present procedures of computation and design in

QE. Bittner: Das Halbparabelverfahren (The Semi-parabola Procedure).Published by the author, October 19 -8. H. Pusch: .Bruchlast undBruchsicherheitsnachweis bei Biegebeanspruchung von Stahlbeton unter besonderer Berücksichtigung der Vorspannung (Ultimate Load and Deter­mination of Security Against Failure by Bending Stress-of Reinforced Concrete, Particularly in Consideration of Prestressing). Beton- und Stahlbetonbau No. 9 (1950). K. Jager: Die Bedeutung des Traglastver­fahrens, für die Weiterentwicklung des Stahlbetonbaues (The Significance of the Maximum Permissible Load Procedure for the Further Development of Structures of Reinforced Concrete)• Paper read on the occasion of the Austrian "Concrete Day" in Vienna, 1951* Oesterreichische Bauzeitung, Nos. 1 / 3 (1952). K. Jäger: Über die Bemessungsgrundlagen im Stahl­betonbau (Bases of Design for Structures of Reinforced Concrete).

^ Oesterreichische Bauzeitung, Nos. 2 9 /3 0 (l95l).K. Jager: Der wahrscheinlichste Momentenverlauf in statisch unbestimmtenStahlbetonbalken (The Most Probable Progress of the Moments in Statically Indeterminate Beams of Reinforced Concrete). A. Leon-Gedenkschrift, Vienna, 195^.

general warrant neither a simultaneous utilization of steel reinforcements

at the points of maximum moment values, nor the observance of a required

factor of safety. It also could be shown that the progress of the bending

moments depends on the capacity to sustain moments that is present in the determinative cross sections (cross section of the span, or at the fixed end or the support) as a result of a given reinforcement • Hence these

results in no way confirm the prevailing view that the computation of cross-sectional-forces in accordance with condition I is fully justified

by the experience gained from tests. In this connection it should be pointed out that, even though the deviations usually lie within the in­evitable dispersion range of tests so that they can by no means be deter­mined with certainty in this manner without a knowledge of the theoretical

kvalues that are to be expected , their consideration nevertheless makesit possible to obtain a distribution of the reinforcement that correspondsbetter to the actual conditions. And finally it must be emphasized that

the so frequently mentioned "compensation of the maximum value of themoments through the formation of yield hinges” obviously cannot and must

If 2 1not be considered in the present case 7 7 .In connection with the foregoing procedures a method of iteration

is developed, suitable for practical applications. This method is adapted to the properties of compound building materials, and renders it possible

to improve the progress of the moments found under the usual assumptions,

5 K. Jäger: Die Beanspruchung statisch unbestimmter StahlbetonbalkenIn Abhängigkeit von der Verteilung der Stahleinlagen (The Stress in Statically Indeterminate Beams of Reinforced Concrete as a Function of the Distribution of the Steel Reinforcements). Zeitschrift des oesterreichischen Ingenieur- und Architekten-Vereins, Nos. l/2, 3 A (1952).

5

by taking into account the distribution of the steel reinforcements. In view of the scope of this article, the solution of the problem must neces­sarily be limited to basic considerations, and the application must be re­stricted primarily to beams with a span of constant rectangular cross section* For the sake of expediency, we shall use the progressive mo­

ment distribution method of Cross in the determination of the progress of the moments. This method proceeds from beams that are Immovably fixed at one or at both ends, and is assumed to be known in the discussion that follows^.

2. Bases of Computation

b 5In the preliminary investigations mentioned * it was shown that the most probable distribution of the bending moments, both for the working condition and directly below the maximum permissible load may be determined with sufficient accuracy under the assumption of a constant reinforcement throughout the range, which reinforcement, within the range of the positive or negative moments, corresponds to the sustainable mo­

ment M* of the determinative cross section of the span or of the sup­port. If we designate the effective depth of the beam by h, and the distance between the neutral axis and the boundary of the bending com­pression zone (commonly called x) by y^ = T^h, then the reduced bending

This procedure is represented in a more distinct and improved form in Th. Titze: Momentenausgleichverfahren. Berechnung von Durchlauft rage rn undRahmemtragwerken mittels direkten Momentenausgleichs und vergleichsweise nach dem stufenweisen Momentenausgleich der Methode Cross (Method of Com­pensating Moments. -- Computation of Continuous Members and Supporting Frames by Means of Direct Compensation of Moments, and According to the Progressive Compensation of Moments of the Cross Method) . Vienna: Manzsche Verlags- und Universitätsbuchhandlung, 19 -8.

6

stiffness of the cross section is defined by the expression

B = M*h (l - tj0 ) (1)

in which M* depends on the cross section F0 of the steel, and on thestress a * in the steel. We also can introduce the cross section F e eof the tensile reinforcement in Equation (l) and thus arrive at the rep­resentation of the bending stiffness as given earlier by the writer, but the concept of the sustainable bending moment M* appears to be more ex­

pedient and clear for the application that follows. If we further desig­nate the regular moment by M^, and the pertinent width of the compression zone of the concrete by y = T ] t h e n the following equation applies to a cross section with a tensile reinforcement only

VM*9

M* < M . (2)

In the case of a compressive reinforcement, i.e., for M* > M^, t] must be set equal to = constant. For the maximum permissible loadprocedure, a support moment that corresponds to a simultaneous full uti­lization of both concrete and steel is defined as the regular moment of a cross section of given dimensions; i.e., the maximum permissible compres­sion occurs at the boundary of the compression zone of the concrete(beginning of the plastic state), and the critical expansion e * in the steel reinforcements, so that the position of the neutral axis is given by the- following equation

r = P ‘Or "(e + e *)p e 1 (3)

For the numerical evaluation, T)^ will be set equal to 0.1+0 as in

libraryJUL 1 3 2011 7Bureau of Reclamation

I4. Denver, Coloradothe earlier investigations . This corresponds for example to = 0.2$(according to the latest tests of H. Riisch , this value would he fairly

constant for all grades of concrete) and to eg* = 0.3$ ("Torstahl Lo").The effect of the position of the neutral axis on the results was dis-

lj.cussed in an earlier publication .

We now shall derive the subsequently required relationships between the bending stiffnesses of two rectangular cross sections with the sus­tainable moments M- *, Mg*, the effective depths h^, hg, and the equal width b (this assumption nearly always proves to be correct in the case of a continuous beam). It follows from Equations (l) and (2) that

Bi « Y hi u - V «1* hlB2 - t y . h2 (1 - „0 2 )

in which

\ i V ;M „ . 2

(*)

(5)

2 kat uniform properties of the material * . If we further assume that

Mp* "S 2* and that one of the two cross sections is fully utilized, and consequently that M^* or Mg* is equal to the regular moment orM^g depending on the relation between the effective,depths, it follows from Equations ( h ) and (5) that

Mi* hi* ÏU**2

2 , /V“lOr hx y Mg* ) Mx*(6)

8

or

^3B,.

Ml " hl 1 - nOr

1 - n,'Or h.

for (7)

with r\_ = O.kO we obtain the values B-,/b ^ of table 1. in which the'Or 1 2 7most extreme differences between the effective depths are also taken into

account.

Beams Fixed at One End

The beam depicted in fig. 1 is free to rotate at A and fixed at

B. It has a span L and a constant effective depth h. In the zone

(L - x o f the positive span moments it possesses the bending stiffness

B^, and in the zone x^ of the negative moments the bending stiffness

B^, corresponding to the sustainable bending mo-

Fig. 1. Bending mo­ments and bending an­gles in a beam fixed

at one end

ments M^* and according to Equation (l).

In order to determine the fixed-end moment m^

we start from beams that are free to rotate at

either end. The bending angles at the left and

right supports resulting from the load on the

span are designated by 0^, 0 and the bending

angles at the supports caused by the moment

ffiB =tion it is at first assumed, in a very general

way, that the beam can sustain a positive as

= 1 by Og, p . In this method of computa-

well as a negative moment in the sections mentioned, but in the final

9

state the moment distribution illustrated in fig. 1 is obtained, and from it the corresponding position of the tensile reinforcement. If we further assume a uniform load g, and designate the distance between the point of

£* izero moment and the fixed end by x^ = C^L, then we obtain4

V b ' I MoL 1 + f ê ' 7 ( l + 3 5 B ) ( l ' 2 B )3

! Æ . !) (! . Jb)3

Table 1

„ 1 t2 ' Mo = gL '

(8)

PROPORTIONAL VALUES B]_/B2 OF THE BENDING STIFFNESSES OF TWO RECTANGULAR CROSS SECTIONS AS A FUNCTION OF THE SUSTAINABLE

MOMENTS M-j*, M * AND THE EFFECTIVE DEPTHS h , h

\ h! \ h9\ 2

M * \M^* \

0.2 0.4 0.6 0.8 1.0 1.5 2.0 3.0 4.0 5.0

1.0 0.13 0.29 0.47 0.71 1.00 1.82 2.67 4.33 6.00 7.670.8 0.11 0.23 0.39 0.60 0.86 1.53 2.19 3.53 4.85 6.180.6 0.08 0.18 0.31 0.49 0.69 1.19 1.69 2.69 3.69 4.690.4 0.05 0.13 O.23 0.36 0.50 0.83 1.17 1.83 2.50 3.160.2 0.03 0.08 0.l4 0.21 0.27 0.43 0.61 0.94 1.27 1.6l

When the beam :is immovably fixed at B, the geometric condition

V b + <4 = 0 applies, which leads to

“b = - ÔÇ = ~ k V b (9)

dependent on the ratio between the bending stiffnesses B^/b ; i.e., the sustainable moments Mg*/M-^* according to table 1 for h^ = h^ = h, and

10

the position £^ of the point of zero moment. £^ and hence m^ canthen be determined by means of Equations (8) and (9)♦ This evaluation

5was carried out in the preliminary investigation and can be simplified with sufficient approximation by assuming that the position of the point of zero moment is a function of the sustainable moments, in ac­

cordance with the equation

V k SBM * / k n2

i ( i - « Br(10)

For the compensation of the moments at the support

beam the bar stiffness at this point is required,corresponds in the usual manner^ to one-quarter ofproduces the bending angle = 1. We obtainB

3 bbSBk (1 - V 3 L

B of a continuous

which bar stiffnessthat moment s_ which B

(ii)

Table 2BEAM FIXED AT ONE END. FIXED-END MOMENT mB FOR VARIABLE BENDING

STIFFNESS Mi*/MB*, in relation to 5% for Mq*/MB* = 1 AND TO BAR STIFFNESS sB AT THE FIXED END B

........ sustainable moment in zone of span,M * ........ sustainable moment in zone of fixed end.

Mx*

V

*6

“ B

sb lB

0 .2 0 .4 0 .6 0 .8 1 .0 1 . 2 1 .4 1 .6 1 . 8 2 .0

1.42 1.23 1 . 1 2 1 .0 5 1 .0 0 0.97 0 .9 4 0 .9 1 0 .8 9 0 .8 7

0.48 0 .5 6 0 .6 3 0 .6 9 o .75 0.80 0 .8 6 0.93 1 .0 0 1 .0 7B

11

Under the present load condition we obtain, for a beam with a con­

stant bending stiffness Bg = B^, the fixed-end moment 5g = - in ac­

cordance with the conventional computation, whereas the bar stiffness

amounts to Sg = 0.75 B/L# Table 2 contains the proportional values

and the bar stiffnesses s dependent on the ratio M */M * of the mo-

ments sustainable in the determinative cross sections. The range of the

table is determined by the minimum value M^*/Mg* =0.2 (lower limit for

the reinforcement in the span) and the maximum value Mg*/Mg* = 5* The

upper limit is obtained with regard to the diagonal bars bent up from the

span, which bars may be taken into account with at least 50/o of the span

reinforcement. The ideal condition of a simultaneous utilization of the

steel reinforcements in the cross sections of both span and fixed end

which is to be sought for will be reached when the maximum values of the

bending moments that occur are identical with the sustainable moments

(m^ = M-j*, rig = Mg*). Under the present load condition, this will happen

when nig is equal to 1.26 m^; i.e., when the moment exceeds the resultIT

of the conventional computation by 26$ . Because, within the proportional

values of the sustainable moments that occur in practice , the value

tabulated in table 2 depends but little on the type of load,

table 2 can be used not only for uniform loads but also, with sufficient

accuracy, for any load on the span (concentrated loads, etc.).

Beams Fixed at Both Ends

The beam depicted in fig. 2 is immovably fixed at both ends. It

has a span length L and a constant effective depth h. In the regions

and Xg = 2gL of the negative bending moments it .possesses

12

the bending stiffnesses -33. and B^, and in theA .Dregion of the span (L - - x^) the bendingstiffness B^. In order to determine the tvo fixed-end moments m^ and m^ we start again from a beam free to rotate at the supports. We then must determine the two bending angles

resulting from the load on the span, andA -D

Fig. 2. Bending mo­ments and bending an­gles in a beam fixed

at both ends

finally the bending angles a., fL, resultingfrom m^ = 1, or <x } p^ = p^ = p resultingmg = 1. In the case of a uniform load, with

2M = l/8 gL , we obtain 0

from

V a = J MoL 1 + I 5 - 1) (1 - 3 SA)<1 - V 3 +

Ç - Ç l

V b = \ “oL(12)

“A 1 and furthermore

Ba“a = 3 L

V b - 3 L

V “ 5 1

1 +

1+1

1 - I

X V3 +

H

ba babb " B1

bb bbba ' B1bb bbba " B1

"B

BbVd - yFrom the geometric conditions for complete restraint at both ends of

y (13)

the bar

V a + V + = °' mAP + + = °

we obtain the two fixed-end moments

^BP - W ^ - WA (a a - B2) ’ ^ (a a _ B2)1 A B ' K A B 1

<*. A

4 > .

\ \

13

( 1*0

(15)

Moreover, the following equations apply under the load condition in question:

” A = - !* 0 £ft (1 - SB>’ “ B * J|M0*B (1 - Bk>- ( l 6 )

If we substitute Equations (l6) and (13) in Equation (l^), we obtaintwo equations for determining <5 and £ , and hence from Equation (l6)

the two fixed-end moments as a function of the bending stiffnesses. The troublesome computation of the points of zero moment can be avoided by determining them from Equation (l6) in such a manner that J r n ^ j is set

is set equal to M^*. In the"case where a simultaneous utilization of the steel reinforcements is sought, this holds true in all determinative cross sections.

equal to M * (sustainable moment) and

B a = "R = BA B 3. corresponds to the conventional assumption of a homo-geneous beam and, under the load condition in question, the fixed-end moments m^ = m^ = -2/3 M are obtained. Table 3 shows the proportional

values rn /rn and ^p/^B aS a ^unc^^on the ratio of the sustainable moments - 0.2 < < 2 and < 1, in which case it is

also assumed that minimum is equal to 0.5* Bor this delimita­tion of the range of the table, the considerations with respect to an ever

present minimum reinforcement in the cross sections of both span and fixed ends as mentioned and substantiated in section 3 are conclusive.

For the compensation of moments at A and B required in the

14

T a b le 3

BEAM FIXED AT BOTH ENDS

F ix e d -e n d Moments

F ix e d -e n d Moments

m , nig f o r V a r ia b le S t i f f n e s s .

ET , nig f o r C o n sta n t S t i f f n e s s .

. . . span moment

ma *> V * • •f ix e d - e n d moments.

m.A

S A

L o v e r

M^*

mA ¥A \

V \

0 .2 0.4 0 .6 0 .8 1 .0 1 .2 1.4 1 .6 1 .8 2.0

1.0 1 .2 21 .2 2

1 .1 31.13

1 .0 71 .0 7

1.0 31.0 3

1.001.00

0.980.98

0.960 .96

0.940.94

0.92O.92

0.910.91

0 .8 1 .1 61 .2 6

1.0 81 .1 6

1.0 31 .1 0

0.991.0 6

0.961.0 3

0.941.00

0 .920.98

0 .9 10.96

0 .9 10.93

O .9 I0.91

0 .6 1.0 81.3 0

1.0 21.2 0

0.981 .1 3

0 .941.0 8

0 .921.0 5

0.891.0 2

O.900.99

0.900.96

0 .9 10.93

0.910.91

0 .5 1 . Oil- 1.32

O.981.2 2

0.931 .1 5

0.891 .1 0

0 .8 71.0 6

0.891.0 2

0.900.99

0.900.96

O .910.93

0.910.91

O.h 0.981 .3 6

0.921 .2 5

0.881 .1 7

0.841 .1 2

0 .8 71.0 6

0.891.0 2

0.900.99

O.900.96

O .910.93

0.910.91

0.3 0.901.41

0.841 .2 9

0 .8 11 .2 1

0.841 .1 2

0 .8 71.0 6

0.891.0 2

O.900.99

O.900.96

0 .9 10.93

0.910.91

0 .2 0.791.48

0.751.35

0 .8 11 .2 1

0.841 .1 2

0 .8 71.0 6

0.891.0 2

O.900.99

0.900.96

0 .9 10.93

0.910.91

a p p l ic a t io n t o a co n tin u o u s beam w h ich w i l l f o l lo w l a t e r , we s h a l l need

th e re d u ce d b a r s t i f f n e s s e s and c a r r y - o v e r f a c t o r s d e f in e d by e q u a tio n s :6

SAba

* L(°k - PV_ ß

VB ' a ( I T )B

JV

15

andB.B

db 4 L (aB - pvA) ' A aA(18)

The bar stiffnesses s are contained in table 4, and the carry-over

Table 4

BEAM IMMOVABLY FIXED AT BOTH ENDS

Bar Stiffnesses

Upper Value:sa L

B /Lower Value:

\ Ml*

MX y“ b* \

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

1.0 0.740.73

0.810.81

0.880.88

0.940.94

1.001.00

1.061.06

1.121.12

1.191.19

1.261.26

1.331.33

0.8 0.770.73

0.860.81

0.950.87

l.o 40.93

1.120.99

1.19i.o 4

1.261.11

1.341.18

1*331.25

1.331.33

0.6 0.800.73

0.920.80

1.03 0.86

l . l 40.92

1.25O.98

1.361.03

1.341.10

1.341.17

1.33I.25

1.331.33

o .5 0.840.72

0.990.79

1.130.85

1.270.91

i . 4o0.97

1.361.03

1.341.10

1.341.17

1.331.25

1.331.33

0.4 0.89O.71

1.100.78

1.270.84

1.450.90

i . 4o0.96

1.361.03

1.341.10

1.341.17

1.331.25

1.331.33

0.3 1.020.71

1.280.76

1.510.82

1.450.90

i . 4o0.96

1.361.03

1.341.10

1.341.17

OO

Lf\ CO

CM

• •

H

H 1.331.33

0.2 1.19O.69

1.580.74

1.510.82

1.450.90

i . 4o0.96

1.361.03

1.341.10

1.341.17

1.331.25

1.331.33

factors in table 5 as functions of the proportional values of the sustain­

able moments

16

Table 5

BEAM IMMOVABLY FIXED AT BOTH E3TDS

Carry-over Factors

Upper Value:

Mx*

m A VA \V \

0.2 0.4

1 . 0 o .68 0.68

0.600.60

0.8 0.620.71

0.560.62

0.6 0.570.75

0.530.63

LT\♦o 0.530.78

0.490.64

0.4 0.490.80

0.470.65

0.3 0.440.83

0.42 0.66

0.2 o.4o0.86

0.390.67

0.6 0.8 1.0 1.2

0.55 0.52 0.50 0.480.55 0.52 0.50 0.48

0.53 0.50 0.48 0.460.56 0.52 0.50 0.48

0.50 0.48 0.46 0.450.57 0.53 0.50 0.48

0.47 0.45 0.45 0.450.57 0.53 0.50 0.48

0.45 0.45 0.45 0.450.58 0.54 0.51 0.49

o .4 i 0.43 0.45 0.450.59 0.54 0.51 0.49

o .4 i 0.43 0.45 0.450.59 0.54 0.51 0.49

Lowe r Value : v _B

1.4 1.6 1.8 2.0

0.470.47

0.460.46

0.450.45

0.440.44

0.450.47

0.450.46

0.440.45

0.440.44

0.450.47

0.45 0.46

0.440.45

0.440.44

0.450.47

0.440.46

0.440.45

0.440.44

0.450.47

0.440.46

0.440.45

0.440.44

0.450.47

0.440.46

0.440.45

0.440.44

0.450.47

0.440.46

0.440.45

0.440.44

5. Continuous Beams

As mentioned before in section 3> the preliminary investigation^

showed that, in the great majority of the practical cases, the propor­

tional values m/m of the fixed-end moments obtained from the method of

computation used here, as well as from the conventional method, depend

but very little on the type of load applied to the beam* Hence tables

IT

1 to 5 not only apply to uniform loads but also to any load on the span ■with a sufficient degree of accuracy. The investigations of sections 3 and k are yet to be finally completed by data concerning the bar stiff­nesses and carry-over factors for a beam loaded by end-moments only.

In case the end-span of a continuous beam is not subjected to a load, which situation is also present in the leg of a frame free to rotate at one end and fixed at the other (b), the bar stiffness at B amounts to

SB = °-T r ’ (19)when the sustainable moment in the A region is assumed to be half as large as that in the B region of the restraint (M^* = 0.5 M^*). Equa­

tion (19) differs but slightly from the value s_ = 0.75 b /l of the conventional computation (M^* = M^*) •

In case a continuous .beam is not loaded at mid-span (which condi­tion also exists in a frame leg fixed at both ends), the bar stiffnesses and the carry-over factors can be derived from table 6. As in the case of the delimitations established in sections 3 and b, it may be assumed that minimum (M^*/Mg*) is equal to 0.5*

The iteration procedure de­

scribed hereafter will give an idea of the effect of the reinforcement on the moments M in continuous

beams. This procedure starts from the moment curve M correspond­

ing to the conventional computation method (of homogeneous beams). Fig. 3 shows the moment curve M for two adjoining spans 1 and 2 of a continuous

8

Fig. 3* Distribution of moments in a continuous beam

18

Table 6

BAR STIFFNESSES sB AMD CARRY-OVER FACTORS IN THEUNLOADED CENTER SPAN OF A CONTINUOUS BEAM

MA*, Mg* . . . . sustainable moments in the end cross sections

Ma*

V 1.0 0 . 9 0 .8 o . t 0 .6 0 .5

sa lba

1 . 0 0 1 . 0 1 1 . 0 2 1 . 0 3 l . o 4 1 .0 5

sb l

bb1 . 0 0 0.99 0.98 0 .9 7 0.96 0 . 9 5

V A — 0 .50 0 .4 9 0 . 4 8 0 .4 7 0 . 4 5 0 . 4 3

VB 0 .50 0 . 5 1 0 .5 3 0 . 5 5 0 . 5 7 0 .5 9

beam with the maximum moment values MA , Mg and Mg, Mg, M^. if,

in the first approximation (first improvement), we now choose the sus- tainable bending moments M equal to these M values (M* = M) and

assume complete restraint in each span in the manner of the Cross pro­cedure y we then must determine the fixed-end moments m ^ and m ^

corresponding to the proportional values of the bending stiffnesses in span ly

Mi* Mi m a* m a

1 1 ’Mg*

( 20)

by means of table 3> and from a table for mg^ and m ^ (complete restraint in a homogeneous beam of constant cross section, conventional

19

computation) • It is here assumed that M^* < i.e., the largest

moment is always designated by M^*. The pertinent fixed-end moments in any span are obtained in the same manner as a function of the sus­tainable moments, in which case table 2 must be used for an end-span.If we designate the thus determined moments of complete restraint of the first improvement by m 1, we obtain a difference in moments Am1 above each support. By way of example, this difference amounts to

AV = “bi* + “be' (21)at B, in which case moments with a clockwise direction are considered positive. These differences in moments are now distributed over the entire beam in accordance with the moment compensation procedure, and this is continued until the difference in moments at each support dis­appears. For this distribution of moments, the btir stiffnesses s ac­cording to tables h, 2, and 6 and the carry-over factors according to tables 5 and 6 are required. In special cases (different depths of the beam in two adjoining spans, and different capacity to sustain moments left and right of the support) table 1 must also be used. From this first improvement (1st step) moments M l are obtained, the magnitude of which yields information as regards the errors of the conventional com­

putation, since a design based on the moments M was assumed because of Equation (20). In cross sections where M 1 > M, a corresponding decrease in the factor of safety results (overstressing), whereas in cross sections

7 See for example: R. Guldan: Rahmentragwerke und Durchlauftrager(Supporting Frames and Continuous Members). Vienna: Springer-Verlag,19^3. K. Jager: Festigkeitsnachweis im Stahlbetonbau (Determinationof Strength in Reinforced Concrete Structures), p. 2^8.

7

20

where M* < M , the steel reinforcements are not fully utilized (under-

stressing).In order to arrive at a more equal utilization of the reinforcement

in the largest possible number of cross sections of both span and supports, the sustainable moments may be set equal to the values obtained from the

first improvement (M* = M*). We then obtain for span 1

M^* M1‘ V VIVI ' V - V (22)

We now determine the pertinent moments m" of complete restraint, effect the compensation of moments with the prevailing difference in moments which, above support B, amounts to

AV = V + mB2> (23)

and finally obtain the moments M" of the second improvement. This pro­cedure may be repeated until, after the i-th step, the condition

M1 = M* (24)

is reached in as many cross sections of the beam as possible, with a sufficient degree of accuracy. The equal utilization of the steel rein­forcements in each cross section of span and supports can be achieved only if the ratio between the prevailing determinative bending moments in each span remains within the range of the proportional values of the sustainable moments delimited in the tables; i.e., if the moment of the

** Note of translator; In the German text this is written "M’< M it is believed that it should be "M’ < M ."

21

span amounts to at least 20$ of the greatest support moment, or if the

smallest support moment amounts to at least 50$ of the moment of the span. Since the corresponding conditions of reinforcement are practi­cally always present, the foregoing procedure constitutes the best economical solution as regards the distribution of the bending moments.

The convergence of the method is substantially accelerated if the compensation procedure is not started with the M values of the con­

ventional computation (Equation 20), but if at the outset consideration

is given to a correspondingly higher estimated difference in the bending stiffnesses or in the sustainable moments M*. Moreover, an accuracy of approximately 5$ Eoay be sufficient, which will permit us to discontinue the compensation at an earlier moment. In any case, the very first step of this procedure already leads to a significant reduction of the in­evitable errors inherent in the conventional computation, even when

Equation (20) is used, and thus to an improvement in the moment curve.In the case of a varying load, the maximum values of the moments

in the individual spans and at the supports corresponding to the differ­ent load conditions are conclusive for the sustainable moments M* and must be taken as a basis for the foregoing procedure.

6. Further Possibilities of Application

The procedure mentioned is in principle also suitable for the com­putation of floor frames with straight members; that is, it may in gen­eral be applied to such supporting structures as can be treated in ac-

6cordance with the moment-distribution methods of H. Cross or Th. Titze • In order to determine the effect of a variable effective depth within a

22

span/ it 'will be necessary to prepare individual tables for the fixed- end moments, the bar stiffnesses, and the carry-over factors that are to be introduced. Special tables will also be required for determining the effect of other forms of cross sections, such as those of plate girders. These additional details are at present being worked out by the author

and will be published in another article.

7# Numerical Example

The distribution of the bending moments in the doubly statically in­determinate beam depicted in fig. 4 is to be determined. The beam is free

© ©

fii iiHii iim im i nMMiiiiiim m n n iiiiiin iiiii ii ii ii ii ii i it m iTn ïïTïïTr

4.00 m 6.00 m iMoments M of homogeneous beam

(conventional computation)

© ©Moments M' of the first improvement

iimmpr íííilllllllliiíü

Final moments M for uniform safety in all e cross sections of span and supports.j.

Fig. 4. Distribution of moments for different conditions of stiffness (numerical example)

to rotate at A and B, and immovably fixed at C. The span widths

are = 4 m and = 6 m, the load amounts to 4 t/ra**, and therectangular cross section is constant over the entire length of the beam.

a) Conventional Computation as Homogeneous Beam. The support mo­

ments are determined according to the Cross method. The moments with complete restraint of the two spans are:

** Note of translator: t/m = metric tons per meter. (Metric ton =2204.6 pounds.)

23

“Bl1E 4.00 • 4.00 = 8.00 tm, iOgg = -m^g =

O y6.00 = -12.00 tm

Since in this case the bending stiffness is considered to be constant, the

bar stiffness of bar 1 at B is given by s ^ = 0.75 b /l ^, and that of

bar 2 at B by s^g = BLg. Hence we obtain a^/s^g = 1.12, whereas the

carry-over factor to C amounts to = 0.50. The support values of the

bar at B are = 1.12/(1.12 + 1.00) = 0.53 and = 1 - = 0.47.

The difference between the moments at support B amounts to Am^ =

8.00 - 12.00 = -4.00 tm and must be distributed according to the support

values of the bar. We obtain A m ^ = -E^Am^ = 2.12 tm^Am^ = -k^gAm^ =

1.88 tm, and Am^g = - v^Am^g = 0.94 tm. The compensation is represented

schematically in fig. 5 and leads to the support moments = -10.12 tm,

Mç = -12.94 tm, and to the span moments = 3*74 tm, Mg = 6.48 tm.

B C\ %£ V % Sb i vb vc Remarks1.12 0.53 2 .12 0 . 1*7 1.00 0.50 0.50

Compensation for homo­geneous beam, moments ÎÎ8.00

2 .12

0 00

O 00

1 12.000.9lf

10.12 -10.12 12.91*

0.92 0 .W 1.9 2 0 .52 1.00 0.59 Compensation for the 1st improvement, mo­ments M* •

10.081 .2 7

-12 .721.37

13.560 .8 1

11.35 - 11*35 H*.370.93 0 .1*8 1 .9.3 0.52 1.00 0.67

Ultimate compensation, moments M for uni­form safety

10.88 -13.1*1* H*.6l*1 .2 3 1.3 3 — 0.89

12.11 -12.11 15.53

Fig. 5. Compensation of moments for the numerical example

** Note of translator: This result, or the stated equation, appears tobe in error. As stated, the result would be -144.00 tm. The test equa­tion for mg2 = -™C2 appears to have the value l/l2 omitted, although the result is correctly given as -12.00 tm.

*** Note of translator: In the German text this is written "1 - kgf; itis believed that it should be " 1 - kg-j_."

24

Hence the difference between the moments of span and supports is quite large (fig. 4) so that the difference in the bending stiffnesses cannot be disregarded.

b) First Improvement on the Basis of the Results under a). Let us assume that the design is carried out with the bending moments M.Then these values correspond to the sustainable moments M*, and the moments M* are now determined in consideration of the known proportional values = 3.7V 10.12 = 0.37, M2*/Mc* = 6.1*8/12.91* = 0.50 and

= 10.12/l2.9^ = O .78 of the sustainable moments. Table 2 yields mB1’ =1.26 m ^ = -1.26 • 8.00 = 10.08 tm, s*B1 = 0.55 B ^/ l^. From

table 3 we obtain m^g* = 1*0^ “22 = * 12*00 = -12*72 tm, m *Q2 =1.13 = 1.13 * 12.00 = 13.56 tm. The stiffness of bar 2 at supportB is obtained from table 4 and amounts to s^g* = 0.90 B^g/Lg, whereas the carry-over factor to C is v * = 0.59 according to table 5* Be­cause at support B the reinforcement is equal for both spans, we have B , = BTl0 and s^'/s. . 1 = 0.92. Hence the support values of the bar

Jd JdX jd c l

at B are k^* = O.92/1.92 = 0.48, kgg’ = 1 - 0.48 = 0.52.The difference in moments at B amounts to Am^1 = 10.08 - 12.72 =

-2.64 tm, and the compensation leads to Am^* = 0.48 • 2.64 = 1.27 tm,

AmB2‘ = 0,52 ' 2 ' & i ' = 1,37 t m > AmC2' = °*59 .• 1-37 = 0.81 tm (see fig. 5). Thus we obtain the moment curve M* according to fig. 4 with the support moments M^® = -11.35 tm, Mc® = -14.37 tm, and the span moments M^’ =3*32 tm, Mg* = 5*20 tm. If we compare the results thus obtained with the values of the conventional computation, we have

** Note of translator: Minus sign appears here in German text, but isbelieved to be in error.

25

= 1.12 Mg, = 1 .11 Mc , M ^ = 0 .8 9 Mx , Mg* = 0 .8 0 Mg *

This means that, on the basis of the conventional design, the steel re­inforcements at B and C are 12$ and 11$ overstressed, but that they are 11$ and 20$ understressed in the determinative cross sections of

spans 1 and 2. Thus the safety of the beam with respect to the maxi­mum permissible load (critical state) drops 12$ below the nominal value. The differences between support moments and span moments have increased even more: M^’/ |Mg’| = 0.29, Mg’/ |M^’| = O.36. It would of course bepossible to leave the span reinforcement unchanged according to the moments and Mg and merely increase the reinforcement at the sup­

ports in accordance with the new moments Mg* and M ,*, but even though this would reduce the overstressing at the supports to a certain extent, it would also increase the understressing of the span reinforce­ments, which means that this method would not be economical because it would increase the total amount of steel required. However, an equal utilization of the steel reinforcements in all cross sections decisive for the design can be obtained by further iteration.

c) Ultimate Moments. The following assumption for the proportional values of the sustainable bending moments,

M^/Mg* = 0.26, Mg*/Mc* = 0.27, Mg*/Mc* = O.78,

eventually leads to the desired objective of an equal steel stress in all cross sections.

It follows from table 2 that m ^ = 1.36 = 1.36 * 8.00 =10.88 tm and s^^ = 0.50 From table 3 we obtain m^g = 1 .12 m^g

2 6

-1.12 »12.00 = -13.^4 tm, m ^ = 1.22 n^g = 1.22 • 12.00 = 14.64 tm.Table 4 yields = 0.8l B_>0/Lri, and in table 5 we find the carry-

3d 3d d

over factor to support C; namely, v q = O.67. For Bg^ = Bgg andsT,1/sTl0 = 0.93 the support values of the bar at B amount to

B i. 3d

kg-L = 0.93/1.93 = 0.48 and = 1 - 0.48 = 0.52.The difference in moments at B is Am^ = 10.88 - 13.44 =

-2.56 tm, and the compensation leads to Am^ = 0.48 • 2.56 = 1.23 tm, Amgg = 0.52 • 2.56 = 1.33 tm, Arn^ = O .67 • 1.33 = O .89 tm (see fig. 5).

Thus we obtain the following maximum values of the moments:

Mg = -12 .11 tm, Mc = -15.53 tm, M1 = 3.10 tm, Mg = 4.22 tm.

Hence the proportional Values of the characteristic moments amount to M1/ |Mg| =0.26, Mg/ |MC| = 0.27, Mg/Mc = O.78, and agree exactly with the assumed proportional values of the sustainable bending moments. If the afore-mentioned moments are used as a basis for design (distribution

according to fig. 4), then the steel reinforcements can be fully utilized in all determinative cross sections. The deviations of these moments from the values of the conventional computation amount to

' ' Mg = 1.20 Mg, Mc = 1.20 Mc, M]_ = O .83 Mg = O .65 Mg,

which, in my opinion, is sufficient to warrant the lengthier computation involved. In this case, the over-all amount of steel required will be

smaller than that which would be needed if the conventional computation is used, owing to the not inconsiderable decrease in the span moments, whereas the safety will not be affected.

27

8. Summary

The foregoing procedure, which entails a more accurate computation of bending moments in statically indeterminate beams of reinforced con­crete, takes into account the distribution of the steel reinforcements.

It is based on the moment curve of a homogeneous beam and, by repeated iteration, enables us to confute the variable bending stiffness with an accuracy that can be increased at will. This bending stiffness is represented by the sustainable bending moment M*. The objective of a uniform steel stress or factor of safety is reached when the bending moments computed in this manner agree with the estimated sustainable

moments (M = M*). By this formulation the results are independent of the type of design itself; i.e., both the maximum permissible load pro­cedure and the n-procedure can be used for the design. This method takes into account the properties of the materials and is therefore

more accurate. Moreover, its application would permit us to reduce the factor of safety by 10/o, but even if this is not done, the lengthier static computation will in many instances be rewarded by a saving insteel