45095319 Fysical Engineering Model of Reinforced Concrete Frames in Compression

8/6/2019 45095319 Fysical Engineering Model of Reinforced Concrete Frames in Compression

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YSICAL-E GI EGODELOF REI FORCED CO CRETEFRAMES IN COMPRESSION

BIL I . BL U DIlAAD

FJ . . . . . . . . . . .


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RIJKSWATERSTAAT COMMUNICATIONS

FYSICAL-ENGINEERING MODELOF REINFORCED CONCRETEFRAMES IN COMPRESSION

byIr. J. BLAAUWENDRAADChief Engineer, Rijkswaterstaat

Government Publishing Office - The Hague 1973


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Any correspondence should be addressed to

RIJKSWATERSTAAT

DIRECTIE WATERHUISHOUDING EN WATERBEWEGING

THE RAGUE - NETHERLANDS

The views in this arlicIe are the author's own. ISBN 9012001099


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Summary

A calculation of the complementary moments due to second-order effects and theanalysis of the stability of reinforced concrete framed structures can be conceived asfollows. With the aid of a computer a very large number of M-N-x diagrams can beproduced on the basis of the standard specified stress-strain diagrams for concreteand steel. A framed structure is then analysed with an available program which takesaccount of second-order effects. The flexural stiffnesses EI to be adopted are estimatedand corrected with reference to the M-N-x diagrams calculated once before and heldin store for the purpose.The present paper discusses the drawbacks ofthis approach and proposes a method

of analysis which can be fitted into existing programs for framed structures anddispenses with the large number of stored M-N-x diagrams. I t is shown that directuse can be made of the stress-strain diagrams. The results are just as reliable as thoseobtained by the procedure utilising the M-N-/( diagrams. The method can be appliedto total frames and single members in compression as well.

AcknowledgmentsThe FORTRAN programming was carried out by Mr. H. Eimers of the Rijkswaterstaat. In testing the method of analysis we had the benefit of valuable criticism fromIr. A. K. de Groot of the TNO Institute for Building Materials and Structures. Inthe context of his work for Committee A 17 of the Netherlands Committee forConcrete Research (CDR) he analysed a number of structures in actual practice onthe basis of the approach described in this paper. His efforts helped speedily toreveal the initial shortcomings of the method.This paper is essentially the text of a lecture given by the author at the twicerepeated postgraduate course on "Stabili ty of Buildings". This article has been

published before in "HERON", vol. 18, 1972, No. 4 titled "Realistic Analysis ofReinforced Concrete Framed Structures". A slightly modified version of this textappeared in the Dutchjournal "CEMENT". The present translation has been preparedby Ir. C. van A m e r o n ~ e n .

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Fysical-Engineering model of reinforced concreteframes in compression

1 What went before?Since 1967 there has, in the Netherlands, been animated discussion about how thestability of tall buildings can be investigated. Although the term itself is not usuallyemployed, it is wide1y realised that we are here faced with a non-linear problem. Thefact that the framed structures under consideration, composed of bar-type members,do not behave linearly is due to two causes. For one thing, the horizontal displacements become so large that the vertical loading gives rise to additional moments inthe columns. This second-order effect is sometimes formally referred to as geometrienon-linearity. The second cause is the material non-linearity, also known as physiealnon-linearity. Concrete cannot resist tension, and its compressive stress-strain diagram is not linear, but curved. For reinforcing steel this latter phenomenon appliesboth to tensile and to compressive stress. This material non-linearity is usuallyembodied in a moment-curvature diagram dependent on the normal force (M-N-xdiagram).

Since 1969 the discussion took a turn in that a more computer-oriented approachhas now been adopted. This is based on Livesley's [1] publication in which he showshow second-order effects can quite simply be incorporated in the known computerprograms for the analysis of framed structures in accordance with the displacementmethod.The type of program in current use can indeed take account of the developmentof ideally plastic hinges at the ends of the members, but considers each memberotherwise as a prismatic beam with constant flexural stiffness EI and extensionalstiffness EA. In practice the procedure for using a program of this kind is as follows.First an estimate of the expected stiffness va lues EI is made, and on completion ofthe analysis it must be checked, with reference to the moments obtained, whether theassumption for EI was correct. To do this it is, in principle, necessary to have a large

N

Fig. 1.Moment-curvature diagram applicable to theX calculated normal force in the member.

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number ofmoment-curvature (M-x) diagrams at one's disposal (Fig. 1). Ifthe estimateis found to have been significantly incorrect, the calculation will have to be repeated.2 What do we need?Estimating the stiffnessWe should like to replace the procedure for correcting the flexural stiffness, as described above, by a more automated technique. The M-x diagrams could perhaps bestored in the computer. The program can indeed be so arranged that it will itselfseek out the corrected EI values from this collection of diagrams and repeat thecalculation. However, if we consider this idea more closely, we shall soon realise thatthis is not the way to tackle the problem. For one thing, the number of possiblediagrams is very large. We shall wish to provide a wide choice of cross-sectionalshapes for the structural members, and many different systems or methods of reinforcing them. In addition, the diagram for any particular cross-sectional shape ofa member must be established for a sufficiently large number of values of N. Furthermore, there remains the question of choosing the u-e-diagram to serve as our startingpoint. If it is decided in due course to adopt a modified version of this diagram, itwill necessitate revising the whole set of stored M-N-x-diagrams.

I t is these considerations that compel us to abandon the idea of storage of thediagrams. Instead, the solution to the problem must be sought in a procedure wherebythat par t of the moment-curvature diagram which we require at a particular stage is,at the time, rapidly regenerated in a separate subprogram. This means that for eachnew cross-sectional shape we shall prepare a subroutine of its own, here to be furtherreferred to as "DRSN". All these subroutines are based on one and the same convention for the form in which the u-e-diagram is to be utilised. The part of DRSN whichrelates to the chosen u-e-diagram can therefore in turn advisably be accommodatedin a separate subroutine. For reasons which wiU emerge in due course, the latter wiUbe referred to as "EMOD".With this approach the following advantages are achieved:

- For each cross-sectional shape of a structural member we have to produce onesubroutine (DRSN) instead of a large number of M-x-diagrams. This subroutineis not dependent on the convention for the u-e-diagram and can therefore beutilised as long as the need to apply that cross-sectional shape exists.

- If a different u-e-diagram is adopted, it will be necessary merely to alter one smaUsubroutine ( E M d ~ ) which is applicable to aU cross-sectional shapes.

- This new proposal will have the effect of reducing costs, because the performanceof computational operations by computer is becoming steadily cheaper. On theother hand, M-x-diagrams would have to be stored permanently accessible inbacking stores, so that thissystem could be relative1y expensive.

- The interchangeability between various computer centres is greatly simplified. Asubroutine written in standard FORTRAN is easier to despatch than a tape or adisc with tabulated data.

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Stiffness varies along memberSince we can now simply automate the procedure of estimating a new stiffness value,we should like also to obviate another imperfection in programs as mentioned in 1.It is not true that in reinforced concrete structures the flexural stiffness EI along oneand the same member is constant. This would be so only if the bending momentacting on the member were of constant magnitude along the whole length of thelatter. As a result of cracking in the tensile zone and plastification in the compressivezone the flexural stiffness is less according as the moment has a higher value. Fig. 2

a

b

Fig.2a.For a linear bending moment diagram thedistribution of EI is arbitrary.

Fig.2b.This distribution is moreover different for adifferent linear bending moment diagram.

illustrates that a linear moment distribution in the member is associated with anarbitrary distribution of EI, which is deducible from the moment-curvature diagramin the manner represented in Fig. 1.

It will be shown that this aspect can very simply be accommodated in the existingdisplacement method programs.Centroidal axis shiftsThe computer programs familiarly employed in structural analysis schematise anelastic framework to a system ofaxes or centre-lines of members which in generalintersect one another at the joints of the structure. Each such centre-lne coincideswith the centroidal axis of the member in question. It must not be confused with theneutral axis of the member, which (in this particular context) is the line at whichzero strain occurs and which will coincide with the centroidal axis only if there is nonormal force acting. The centroidal axis for a member of composite section is calculated with due regard to the different moduli of elasticity of the constituent parts.8


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A problem arises in circumstances where cracking and plastification are liable tooccur. To illustrate this we shall consider a rectangular section provided with sym-metrically arranged reinforcement (Fig. 3). An obvious choice is to choose the axis

Fig.3a.The centroid is iocated on theaxis of the member.

1 1 ; 1 ~ ~ fz h t---j=+I---O'-,*axis of meml:ier is centroid

f n h ; ~ -g -distribution

- - - - - - ' t - . . . . j l - - - - - -+- - - -+ g

g -distribution

Fig.3b.The centroid is not iocated onthe axis of the member.

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of the member as being at mid depth (h t being the depth of the section).The concrete conforms to a bilinear u-6-diagram which is of significance only for thecompressive zone, while the reinforcing steel conforms to a two-branched U-6 diagram which is valid both for compression and for tension. A normal force Nand a moment M act at the section. We shall now consider two possibilities. In Fig.3a the combination of N and M has been so ch05en that the whole section is in compression and the strains 6 which occur are small. In that case all the concrete fibreswill be located on the e1astic branch of the ub-e-diagram and therefore have the samemodulus of elasticity Eb' The reinforcement, too, has remained e1astic and its stiffnessbehaviour is characterised by Ea, which is the same for the top and the bottomreinforcement. For the distribution of Eb and Ea thus obtained, the centroid of thesection is indeed located exactly at mid-depth. Our choice as to the position of theaxis of the member was therefore correct.Now let us consider another combination of N and M. This is so chosen that a

tensile zone develops (Fig. 3b) and plastification occurs. I f we again plot the strainson to the u-e-diagrams, we find that a new distribution of the apparent moduli Eaand Eb has been obtained. In the tensile zone Eb is zero and in the compressive zoneEb is in part constant, but in the region wh ere yielding occurs it decreases for increasingva lues ofthe strain 6. The reinforcing steel undergoes yie1ding even in the compressivezone. As a result of this the apparent modulus of e1asticity Ea is not of the same magnitude at top and bottom. With the distribution now obtained for Ea and Eb thecentroid of the section does not coincide with the axis located at rnid-depth of thesection, i.e., for this case our choice for the posit ion of the axis of the member wasincorrect.Since we are induding the effect of the normal force Non the flexural deformation

in our consideration of the problem, it is necessary correctly to describe the complementary moment due to N. The shifting of the centroidal axis must therefore betaken into account in the computer program. This, too, wiU be found to constituteno more than a minor intervention in the existing programs based on the displacement method.

Summary of desired featuresThe foregoing considerations can be summarised in the following points:1. The existing programs which can cope with geometric non-linearity (second-order

flexural deflection) must be extended to deal with material non-linearity (crackingand plastification).

2. The extension must be simple to perform in any currently used program based onthe displacement method.

3. The storage of large series of M-N-x-diagrams must be avoided because this istoo expensive and makes interchangeability more difficult.

4. Programming must be so contrived that only a minor alteration to the program isneeded if it is decided to adopt a different u-e-diagram.

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5. The variation of the flexural stiffness along the member must find expression inthe calculation.

6. The shift of the centroidal axis associated with second-order flexural deflection isof importance and must therefore be taken into account in the program.

The requirements stated in points 3 and 4 can be fulfilled by a procedure wherebythe information needed at any particular instant is computed at that same instant.To this end, a subroutine DRSN will have to be established for each cross-sectionalshape of the structural members. The part thereof which is common to all cases,namely, the part relating to the u-B-diagram, is accommodated in one subroutineEMD which is valid for all cross-sectional shapes.

3 Stiffness matrix se of a prismatic member in tbe linear tbeoryThe analysis of a framed structure in accordance with the displacement method willalways follow a scheme as envisaged in Fig. 4. In connection with the discussion ofthe procedure it will be assumed that the reader is familiar with the displacementmethod as generally applied [1], [2]. First, the stiffness matrix se is determined forall the individual members of the structure. This can be done with respect to a systemof co-ordinate axes made to coincide with the axis of the member, and then a transformation is performed to the general system of co-ordinate axes which is adoptedfor the structure as a whoIe. In accordance with a fixed procedure these matrices se

tor allmembers

tor allmembers

..... end

Fig. 4.Approximate flowchart for a linear analysis offramed s t r u c t u r e ~ with the displacementmethod.

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are combined ioto a stiffness matrix S which is valid for the overall strCtw"@. All theexternal Ioading is brought together in a vector k corresponding to S. The displacements v of the joints can then be solved from the set of equations:

Sv = k (1)We next return to the individual member. From v we obtain the six displacements veof the two ends of the member, enabling us to caiculate the moments M, the shearforce D and the normal force N. We do this for all the members.

Except for minor extensions the scheme represented in Fig. 4, remains valid forthe non-linear analysis. For the type of structures which we envisage here the takingaccount of the non-linear effects has consequences only in so far as the stiffnessmatrix se relating to the co-ordinate axes system for the member is concerned. Thiswe shall more particularly consider. The other operations remain valid unchanged.

The reader will already have perceived that it is our wish to make matters as easyas possible for ourselves. We shall consider only loading applied at the joints andignore any hinged connections that may be present or the formation of any plastichinges. For these conditions the approach and the procedure adopted in, for example,[1] and [3] remain valid unchanged. Here we shall contine ourselves to discussingthe extension with regard to those publications.Separating the elementary deformation problemThe six-by-six matrix se which establishes the relationship between the six displace-

Uj X- - ; .I ro;]y r;VjI - Uj+ Vjy ifJj

ifJj- - - ~ - - -

X1---- - - - - - - - - -+

I ~ H / ]ViIeM'Hj k = H I 0+ MY 1Fig. 5.Displacements v e and member endforces k e in the eth member.

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ments ve and the six end forces ke in e-th member (Fig. 5) can be built up in two steps.We shall first explain this for the linear theory.

-1'. t!i::::1 : - - - - - - - - - : : : : : I l i+III w;tp;;;iI I

deformotions

I t tM',------ - - - - - ~ k , { ~'Jstoticol quontities

Fig. 6.Elementary deformationproblem.

The first step comprises the analysis of the elementary deformation problem of Fig.6. The three statical quantities k. cause three deformations v. Their interrelationshipcan be written als:

s.v. = k. (2)For a prismatic member this can be written out in the following form:

EA 0 0 AZ N-Z0 4EI 2EI ()i Mi (3)Z Z0 2EI 4EI () j M jZ - Z

We shall now perform the second step. While loading indicated in Fig. 6 continuesto act, we first displace the whole member horizontally through a distance Ui. thenlet the support i move downwards a distance Vi. and the support j next move down-wards a distance Vj' The member is then in the condition shown in Fig. 5. Thedisplacements Ui. Vi and Vj are small in relation to the length Zof the member. Thel inear theory presupposes that the rigid body dispZacement that has been performedhas not changed the stresses in the member nor the forces at the ends thereof. Froma comparison of Fig. 5 and Fig. 6 it then follows:

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H i =-NVi = (Mi+M)/1Mi=M iH j = +NVj = -(Mi+M)/1Mj=M j

/)./ = -Ui+Ujei = CfJi-(vj-vJ/Iej = CfJj-(Vj-vi)/1

(4)

On introducing the combination matrix C the following shorter notat ion for (4) canbe written (CT is the transpose ofC):

k e = CTk, v, = Cvewhere: [-I 0 0 1 0

~ = 1/1 1 0 -1/1I/I 0 0 -1//

(5)

(6)

With the aid of (2) and (5) a relation between the six forces k e and the six displacements ve can be derived:

(7)The stiffness matrix se is found to be simple to ca1culate as :

(8)Quantities neglected in the linear theoryIn each of the two steps described above a simplifying approximation is made whichis no longer permissible in the non-linear analysis. In the fint step, the elementarydeformation problem, it is assumed that the rotations ei and e are due only to themoments Mi and M j They are ca1culated as if the normal force N were not present.I f we are to include the second-order effects in our analysis, this approximation, i.e.,neglecting the presence of N, can no longer be permitted. In the linear theory it isfurthermore assumed that the change in length /)./ is caused by the normal force alone.Strictly speaking, a correction should be applied to this if the member is additionallysubjected to bending moments. The resulting deflection of the member causes its endsto move a short distance towards each other (bowing). However, for normal structuressuch as we are considering here, this effect is negligible even in second-order ca1culations.The second step likewise involves an approximation. I t is tacitly assumed that the

magnitude of the bearing reactions (Mi+Mj)/I remains unchanged when the membershown in Fig. 6 undergoes displacements Vi and Vj to the position shown in Fig. 5.In that case we neglect the fact that, because of the slight inclination of the member,the horizontal forces H i and H j wiU in reality produce an additional coupie. In other14


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words: the normal force N causes additional vertical reactions. To take account ofthese a correction I:1Snmust be applied to se.Our conclusion is that two approximations, involving the neglecting of certain

quantities, as adopted in the linear theory will have to be rectified:- The flexural part of S. is dependent also on the normal force N.- A correction I:1sn necessitated by the inclination of the member must be appliedto the matrix se = CTS.CThe stiffness matrix now becomes:

(9)

4 Stiffness matrix se of a prismatic member in tbe non-lnear theoryElementary deformation problemWe must establish a new relation between the two rotations (Ji and (Jj' on the one hand,and the two moments M i and M j , on the other, in a manner whereby the effect ofthe normal force is duly expressed. In theory this can be done in an exact manner byjudiciously solving the relevant differential equation. In actual practice, however, asimple solution can be found only for prismatic members. I r N is a compressive forceof magnitude P, so that N = - P, the relationship is:

EA 0 0 1:1[ N-[-0 EI EI (Ji Mi (10)p- q-[ [0 EI EI () j M j- p-[ [

where:

p f3 sin f3 - f32 cos f32(1- cos f3) - f3(sin f3)q f32 - f3 sin f32(1- cos f3) - f3(sin f3)f32 = [2pEI

In the limit case where P is zero, p does indeed have the value 4 and q the value 2,so that the matrix of (3) is precisely obtained.

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Correction LlS"The additional vertical reactions due to the normal force N which arise in connection

i jo 0I II III -!iV, II I+ - o ~ N IHj ~ i H.~ l

...!iVj

Fig. 7.Additional vertical reactions due to the normalforce N when the rnernber is in an inclinedposition.

with the change in position of the member when Se becomes se are red simply fromFig. 7. Their magnitude is as follows:

!iV = Vi-V j ' N, I

In matrix notation this becomes:0 0 0 0 0 0 Ui 00 N 0 0 -N 0 LlViI I Vi0 0 0 0 0 0


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th e three components C, Sz an d I1Sn which compose the stiffness matrix se in (9),only Sz is dependent on the stiffness properties of th e material used. The matrix Cis determined entirely by the geometrie quantity I, an d the matrix I1Sn additionallyby the normal force N. The influence of varying stiffness and a shifting centroidalaxis is therefore confined to the three-by-three matrix Sz. Only this matrix is changedwhen cracldng and plastification (deviation from th e linear u-a-diagram) occur.Thanks to this important conclusion we can now confine our attention to this matrix.

5 Stiffness matrix S: for a member with varying stiffness in the non-linear theoryWe shall again investigate th e problem of Fig. 6, bu t now for an elastic memberwith a given arbitrary distribution of the modulus of e1asticity E, which may varyquite arbitrarily across the depth of the section an d also in the linear direction of themember. This, in effect, is the situation that is liable to arise when cracking and plastification occur. We shaU choose a fixed axis fo r th e member to serve as a referencefine which is independent of cracking and suchlike phenomena. Th e position of thisaxis can be freely chosen. In Fig. 8 it is located at mid-depth. On the other hand, the

cracked

B - ht._ N 1. ._= ' '' '- - - - - - - - - -1 -- - - - -4 Yz

} b t isection I-I

Mi centroidal axis+ '-'-'--- ---+'---. o-N . axis of member .------.----..-..........4--.-. ' - . - .

Fig. 8. Irregular distribution of the modulus elasticity as a result of plastification and cracking.

centroidal axis is determined entirely by the distribution of the modulus of elasticityE. We shaU no w consider a section I - I . From th e known variation of E across th edepth of th e section the location of the centroid of this section can be determined.Let y z denote th e distance thereof to the chosen fixed axis.

We now choose the centroid as the origin of th e vertical axis y (see Fig. 9a). Theextensional stiffness EA an d th e flexural stiffness EI are determined by the weU knownstandard formulas:

h, -hEA = b J E(Y)'dy

- h (13)h , - h

EI =b J E(y)y2'dy-h

I f a normal force N and a moment M a r e acting at the centroid, a linear strain17


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distribution develops a G f ~ the depth of the section. This strain can be separated intoan average strain eg which is constant on the whole depth and a curvature portionwhich is of zero value at the centroid and has a linear distribution from - xh tox(ht-h). The relation between eg and x, on the one hand, and N and M, on the other,can be expressed quite simply with EA and EI:

(14)

The behaviour of any section is determined by the three data Yz, EA and EI. On thebasis of this information we shall now proceed to construct a stiffness matrix S.which is defined at the fixed axis of the member. We shall accordingly let the normalforce N from the vector k. act at that axis. I t appears that S. can be worked out intwo ways. The first of these ties up with the jinite element method, which is used forthe analysis of siabs and plates. An assumption is made as to the distribution orpattern of the displacements. This is not a new feature in stability analysis. VanLeeuwen and Van Riel [5] utilised a sinusoidal deflected shape, even though it wasknown that a different shape would occur in reality. The second possibility is equallyinteresting in that it establishes a link-up between the analysis programs for frameworks (composed of bar-type members) and the stability analysis which Van Rieland De Groot performed with the aid of the jinite difference technique quite sometime ago [6]. In the present paper we shall more particularly be concerned with thefurther d ~ v e l o p m e n t of the finite element method as an approach to the problem.

=N+M

stressotal strainy

hht

a) y +-b Ir eg>J

ht 1 ...x;,l, m'U)eg

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For the procedure employing the finite difference technique a short summary willbe given here; for further information the reader is referred to De Groot's publica-tion [7].Whichever procedure is adopted, we shall base ourselves on the usual fundamentalassumptions of flexural theory, namely, that plane sections remain plane (linearstrain function in Fig. 9) and that normals to the axis of the member remain perpendicular thereto after application of loading (no deformation due to shear).Method of assumed displacement fieldWe shall first deal with the simpIer case where Yz is zero. For this case the usualapproach is to assume a linear distribution in terms of x for the displacements u(x)in the direction of the axis of the member and to assume a cubic polynomial for thedisplacement v(x) perpendicular to the axis:

I - x xu(x) = - I - ' u i + l 'U j(15)

The strain Eg and the curvature x are:_ du u). - Ui ! i Ie = -= - - = -g dx i i (16)

The potentia1 energy for this member is:

p = 1 si [EAE2+EIX2+N(dV)2JdX - N f1i-M(J-M .(J.Zog dx I I ) ) (17)The termwith dvjdx in quadratic form under the integral symbol is the second-orderterm in this expression. I t corresponds to the work done by the normal force N indeflecting the member.

On substitution of (16) into (17) P becomes aquadratic expression comprising thethree deformations M, (Ji and (Jj' By equating to zero the derivative ofP with respectto each of these deformations we obtain three equations containing M, (Ji and (Jj'These equations determine the minimum for Pand are written in abbreviated notationas follows:

(18)19


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So we have 0termined the required matrix S. It is found to be composed of twocomponents:(19)

For these two three-by-three matrices it can readily be deduced that (t/J =x/I):HAdt/J 0 0SO - 0 HI(4-6t/J)2dt/J {i EI(4-6t/J)(2-6t/J)dt/J (20a)-

0 HI(4 - 6t/J)(2 - 6t/J) dt/J {i EI(2 - 6t/J) dt/J0 0 0

sn _ 2 1 (20b)15 NI --NI- 301 20 --NI 15 NI30

We can again drawan important conclusion. The matrix S. has one component partin which the stiffness data do not occur, and the other part is the same matrix thatwould have been found for S. in the linear theory.

On working out the integrals contained in (20a) for constant EA and EI we shallin fact precisely arrive at the matrix in (3).

With (19) the equation (9) now becomes:(21)

Writing f!.snn to denote all the parts dependent on N we obtain:(22)

and (21) thus becomes:

(23)

We see that the assumption of a displaeement field results in an uneoupling of thematerial non-linearity and the geometrie non-linearity. This latter is entirely taken int0account with an additional matrix f!.snn. The material non-linearity affects only thethree-by-three matrix S which is valid also in the linear theory.20


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On now proceeding to consider cracking and plastification, i.e., the cases whereYz =I- 0, we need only investigate how S is altered. As the strains are the derivativesof the displacements, we shall first take a doser look at these. The choice of (15)means that for the displacement u(x, y) at a distance y from the centroidal axis thefollowing expression holds:

( _) _() _dvu x, y = u x - y dx (24)For a line extending parallel to the centroidal axis at some distance therefrom y isconstant, so that there a quadrat ic course for u(x, y) is possible. We shall wish tochoose as our axis for the member such a line which does not coincide with thecentroidal axis, and we must therefore make an assumption as to the functionalbehaviour of u(x) and v(x) in that line (Fig. 9b).At a distance y from this axis of the member the horizontal displacement is:

dvu(x, y) = u(x) - y dx (25)At the actual axis of the member the distance y is zero, so that there u(x) must describethe displacement alone. We have shown that the function for this will be at leastquadratic. We shall accordingly decide to choose a second-degree interpolation foru(x) at the axis of the member. The three displacement parameters adopted for thepurpose are indicated in Fig. lOa. At the intermediate node the additional displacement in relation to a linear function is treated as the unknown. As will appear indue course, we thus obtain the best tie-up with the existing programs.

u(x) quadrotic interpolationfor u(x) at axis of member a.

Ui

v(x)

l k x

cubic interpolation forv(x) perpendicular toaxis of member b.Fig. 10.Assumed displacement field.

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The third degree iBterpelatieB fOf v(X) rema:ins valid irl'eSpeCtive 6f the positiooof the x-axis (Fig. lOb). Our displacement field is therefore determined by:

I - x x 4x( l-x)u(x) = --u +-U + UkI I J 12 (26)

At the axis of the member we may again consider an average strain 6g and the curvaturex (Fig. 9b). The relation with u(x) and v(x) follows directly from (26):du !l.1 41- 8x6 =-=-+--Uk9 dx I 12

X = _ d2v = 41- 6x O, + 21- 6x 0 .dx2 12 I [2 J (27)

The deformation part -t(EA;+Elx 2 ) of the potential energy P as given by (17)(still expressed in g and x) is written as follows in matrix notation:

(28)

With the relation readily deducible from Fig. 9

the deformation part (28) becomes:

(29)

where: D l1 = EAD 2 ! = YzEAD 12 = D2 !D22 = EI+y;EA

The equation (14) expressed the relation between the quantities g, x, N and M asdefined at the centroidal axis. Thus the matrix D from (29) now expresses the relationbetween 6g , x, N and M at the chosen axis of the member.The procedure of rninimalising P follows the same further pattern as before. Wenow use (27) and (29), so that we obtain four equations comprising Uk> !l.I, ei and Oj'22


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S11 S12 S13 S14- Uk 0S22 S23 S24 111 N

S33 S34 ei Mi (30)

symmetric S44 e M- _ JThe terms Smn of the matrix S are integrals which are simple to work out, as in (20a),while the terms of the D matrix take the place of EA and EI. With I/J = xl I the termsare:

1 1 2S11 = 1 ~ ( 4 - 8 t / J ) D11 dl/J1 1

S12 = -l! (4-8t/J)D1l dl/J1 1

S13 = 1 (4 - 8t/J) (4 - 61/J)D 12 dt/J1 1S 14 = 1 (4 - 81/J)(2- 61/J)D 12 dt/J1 1S22 = 1 Dil dt/J1 1

S23 = 1r(4 - 61/J)D 12 dl/J1 1S24 = 1r(2 - 61/J)D 12 dl/J1 1 2S33 = 1 ~ ( 4 - 6 t / J ) D22 dl/J1 1S34 = 1 (2 - 6t/J) (4-61/J)D 22 dt/J11S44 = 1r(2 - 6t/J)2D22 dl/J

For (30) we can write the more compact expression:

(31)

(32)

Here the vector u is the parameter Uk ' and v. contains M, ei and e There are actuallytwo equations stated in (32). From the first of these we can derive a relation between

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f t ftftd ~ E We Cftft theft ifttrodoc-e this relatioft iftto th-e sec-oftd ~ t W n , whicll th6rebybecomes:

(33)

The matrix between square brackets is the required three-by-three stiffness matrix S?for an arbitrary distribution of the stiffness E.The displacement Uk can be calculated from vE according to the first equation of(30):

(34)

Finite difference techniqueIn the approach based on the finite difference method a series of equidistant pointsalong the member is considered. There are m such points between the ends i and j.At all the points we know EA, EI and the distance Yz from the centroid to the chosenaxis of the member. The matrix F of influence coefficients can be determined by asimple procedure. From this we obtain the matrix SE by inversion. The calculationof the three-by-three matrix F is performed as follows. First, we apply at the axis ofthe member a normal force of uni t magnitude. We calculate how much the distancebetween point i and point j is increased (F11 ) and how )11Uch node i and node jrotate (F12 and F13 ) . Next, we consider the case where a unit moment acts at node i;again we calculate the corresponding three deformations: F21 , F22 and FB . Finally,we perform a similar calculation for a unit moment acting at nodej, so that we obtainF 31 , F32 and F33

In each of these three loading cases we first calculate the extension of the actua!centroidal axis and the rotations at the ends thereof. The extension of the axis of themember then follows directly from this by means of a simple transformation. Therotations at the nodes i and jare equal to those which occur at the ends of the centroidal axis.

The change in length of the centroidal axis occurs only in the case where the normalforce of unit magnitude is acting. This change in length is then:

1 1JEA dxo (35)The integration is performed numerically. In general, the rotations of the ends of thecentroidal axis occur in all the three above-mentioned unit cases. In the first casethe external moment M u is equal to the product of the eccentricity yz and the normalforce of unit magnitude. In the second and the third case the external moment M uvaries linearly from unity to zero. The differential equation for the deflection v of thecentroidal axis is as follows (N is posi tive if it is a tensile force):24


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(36)

At the m difference points we shall consider the discrete values of v as unknowns.At each point p there is a known value Elp and a known value M up ; let h denote thespacing of the points. At such a point we can now write for (36):

(37)For N we substitute the value that we expect to obtain as the result. In an iterationprocess each successive stage is performed with the result yielded by the precedingcalculation. Equation (37) is established for all m difference points. For v at node iand node j a zero value is introduced. We obtain a set of equations comprising munknowns v; the matrix of the coefficients is symmetric and has a pronounced bandedstructure. The rotations follow from the solution of the set of equations:

(38)

In this way a completely symmetricmatrix F is established, so tha t S. will also displaypure symmetry. There is a significant difference in relation to the method based on anassumed displacement field. With the finite difference technique it is not possible touncouple the material and the geometric non-linearity. We obtain the matrix S."at one go". The separation into S? and S can now not be done.The six-by-six matrix se therefore follows directly from (compare with 23):

In Section 6 we shall confine ourselves to the "assumed deflected shape" method.The reader can verify for himself how, on similar lines, the problem can alternativelybe dealt with by the finite difference technique.6 How the program worksGeneralIn principle, a non-linear analysis proceeds as indicated in Fig. 11. We choose aninitial estimated value for the normal forces N and the magnitude of the modulus ofelasticity E. For N this value is zero, and for E we adopt the value at the origin of thestress-strain diagram.

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choose cr vatue for tnenormeL torces N end edistribution ot thestittnesses E

....:tI

determine Se trom Eli l end . SNN trom N..Q.I.0E 1.IE

ceLcuLete se in eccordence0L.. with S e : C T S ~ C + .SNN.2

.J.,.. p.. S:1: See

1soLve ., trom

S.,=klo.

li l -.:.IrL..- Q.I trom ve to LLows.0L.. E M,D end NE Q.IE

.J..correct theK desired accuracydistributionot E attained?1 yesceLcuLete support reactions

end.......

Fig. 11.Approximate flowchart for a non-l inearanalysis offramed structures.

For each member of the structure we choose an axis (the axis of that member),after which s can be caIculated in accordance with (33) and snn can be caIculatedwith (22). The total matrix se is then obtained from (23). The further procedure isthe same as that for a program based on the linear theory. When finally ve and Nfor all the members are known, the caIcuIation can be repeated with a better estimatedvaIue for se.

With ve the vaIues of Ui ' Uj , Uk> ()i and ()j respectively are definiteIy estabIished, andtherefore with (27) the average strain eg and the curvature :I{ at each section of themember is Iikewise established. The totaI strain across the depth of each section is26


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then known (see Fig. 9b). The apparent E associated with this strain can then bered from the (J- diagram of the material. This operation of seeking the value of Ecorresponding to a particular strain will always have to be performed, whateverthe shape and constitutionof the section, and is the only stage in the analysis procedureat which the chosen (J--diagram comes into it. As stated earlier on, this part isaccommodated in a separate subroutine (EMOD). The input parameter is the strainand the output parameter is the required E.From the new distribution for E the two-by-two matrix D can be calculated with

(13) and (29). This matrix D is typically deterrnined by the shape and constitution ofthe cross-section of the member and is accordingly produced in a subroutine (DRSN)al ready referred to. For each cross-sectional shape an individual DRSN subroutinewill have to be established. The input parameters are g and x; the output parametersare the terms D l1 , D 12 and D 22 . In the DRSN subroutine a call is made upon theEMOD subroutine for determining the apparent moduli of elasticity E. The integrations in (31) are performed numerically with the aid of the trapezoidal rule.The final step consists in calculating the matrix S for the member from (30), (32)and (33). The calculated matrices D are used for the purpose. Now an integrationover the length of the member has to be performed. Simpson's rule is applied here,as greater accuracy is desirabie. For an elastic prismatic member the exact stiffnessmatrix wiJl then in any case be obtained. Determining the stiffness terms of S isperformed in a subroutine (STYTER). In principle, this subroutine need be programmed only once and is at the disposal of all users. I t will have to be altered onlyif it is felt nessecary in future to assume another displacement field.STANIL programOn the basis of the "philosophy" outlined above, the Data Processing Division(Dienst Informatieverwerking) of the Rijkswaterstaat has prepared a program designated as STANIL, which is made available also to other users. For this purpose aprogram for the analysis of plane framed structures which was already at the disposalof that Division has been modified somewhat. As already stated, this involves mainlythe writing of the three subroutines EMOD, DRSN and STYTER. The FORTRANlists of these routines are appended to this paper. Any one who has a normal programat his disposal can apply the modifications quite simply. The STYTER subroutinewiJl, in principle, always remain valid; EMOD will have to be changed only if theC.E.B. (European Committee for Concrete) sees fi t to revise the recommendedstress-strain diagrams, and DRSN wiJl have to be individually established for eachcross-sectional shape (and is independent of the material properties). The programobtained really is very general. Merely by altering EMOD it can be used also forother materiais, such as aluminium, wood, etc.Example and indication of castFor eccentrically loaded individual reinforced concrete columns De Groot has in-

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1kN

40 kN ,r - - - - - - , - ,

TI TNI1 ~ h 2 II II II II .jI ..

I IIIII

N =240 kNeo=6.6 cm

1. onLy secondorder effect

2. second-order endmaterial behaviour(cracked end yielded J

w= 0.3%

EoooCO

a. b.Fig. 12. State of failure of a reinforced concrete column.vestigated a number of cases with the aid of the finite difference technique. One suchinstance is represented in Fig. 12a. The STANIL program gives practically the samere sult. Fpr equal eccentricity eo the failure load differs by less than ! per cent fromthe result obtained by means of the finite difference technique which here is consideredto be the exact one. In this comparison the column comprises one element. In Fig.12b it is shown how the displacement increases in consequence of cracking in theconcrete. Because of this the largest moment increases by about 50% in relation tothe uncracked state. In actual calculations for the analysis of frameworks similar28


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results may be obtained. No examples of such cases will be given here, but we shall,in conclusion, give some idea of the cost of performing such calculations. A framedsystem comprising 183 joints (nodes) and 240 members of seven different types,largest node number difference of 10, attains equilibrium after four iterations, for agiven loading. The cost of performing the analysis on the CDC6600 computer isapproximately Ft. 85 . - ($ 25.-).

7 Concluding remarksI t is possible to fuiftl in a simple manner the desired features listed in Section 2 ofthis paper. Interchangeability of a quite unexpected simplicity has been achieved.The most important task to be performed consists in writing the DRSN subroutinefor the cross-sectional shapes currently used in structural engineering practice. TheDRSN for rectangular sections is given in an appendix to this paper.The present author hopes that others will write and publish DRSN subroutinesfor T-beams, round columns, square box-shaped members (e.g., as embodied in the

structural cores of tall buildings), etc. I t should thus be possible to avoid unnescessaryduplication of work by engineers all individually producing their own programs.

I t is alternatively possible to apply the finite difference technique. For this theSTYTER subroutine will have to be somewhat modified, but otherwise the schemepresented here remains applicabie.Finally, it should be noted that the method can also be so formulated that the

analysis can be performed with increments of the loading. In that case it is notnecessary to use an apparent modulus of elasticity; the tangent modulus can beintroduced instead. This may be advantageous if it is desired to investigate accuratelythe formation of concentrated ideally plastic hinges.

Translator's note:The acronyms used to denote the subroutines and programs are based on Dutch words:DRSN is derived from "doorsnede" = "section" (of a structural member)EMOD is derived from "elasticiteitsmodulus" = "modulus of elasticity"STYTER is derived from "styfheidstermen" = "stiffness terms"STANIL is derived from "staafconstructies niet-lineair" = "non-linear framed structures".

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S U R R O U T I N ~ S T Y T ~ R ( S I I . S 1 2 . S 1 3 . S 2 2 . S ? 3 . S 3 3 . C l , C ? C 3 1C CC TW1' ~ I H " 9ElE_ l" l !> l"'i: I " " H _ a Y ~ T > < R l l STl f fNESS MATRIX CC O A MEMRFR RY MEANS OF NUMERICAL INTEGRATION. T H THREE- CC 8Y-THREE ~ T I F F N E S ~ MATRIX RELATES Ta THE THREE OEFORMATIONS CC OELTAL. T ~ T 1 AND TET2. THF ROUTINF c:;TARTS FRO"l A "lE"lBER CC WHICH IS nIVIOEO INTO A NUI . I=1ER INIAOT> OF SEGMOJTS. PER CC SEC,MENT THE CROSS-StCTIONAL S H ~ P E MAY BE DIFFERENT. CC THIS PROVTSION HAS 8EFN MADE 8ECAlJSF, INTER ALTA. THE CC R E I N F O R C f ~ ~ N T MAY VARY WITHIN ONE ANO THE SAME MEMBER. CC THE LENGTHS OF THE SEGMENTS ARE: GTVr-;-N IN AN ARRAY ALMOT. CC PER S fGMfNT AN EQUIDISTANT O!VISION INTO STEPS IS MADE. CC THF NUfo.18ER OF STEPS ?FR S F G ~ E N T IC:; ~ V E N , 50 AS Ta ENABLE CC INTEGRATION Ta ~ P E R ~ O R ~ F D PER 5FGMENT wITH STMPSONtS RULE. CC THE N lJMBER OF STEPS FOR TH f VAR!OIJS SFGMENTS OF A M n H ~ E R IS CC PASSEO IN AN ARRAY NSTMOT FROt..1 THF JJlATN PROGRAM Ta THIS CC SURROUTINE. HOW THIS NUMBFR IS ESTAqLlSHEO IN MA IN IS NOT CC OF IMPORTA"JCE HE:RE. IT MAY DIFFER FR OM ONE P R O G R A ~ TO CC ANOTHfR. CePER Sfr,MENT THE NUM8EQ OF ~ E C T t O N C : ; cONSIOERfD TS ONE ~ O R E CC THAN THE NUMRER OF STFPS . THE STIFFNESS N U ~ R E R s 011 . nZI ANn CC 02 ? OF ~ L THE SEcTIONs OF ALL TH f SEGMENTS OF THE MEMRER ARE CC ARRANGfD IN SQUfNCE TN T H THREE ARRAYS 0011 , DD21 A ~ OD22. CC THF S!X TFRMS OF THE lIPPER TRIANGLE OF THE: 5 T I ~ n . I E S S t..1ATRIX. CCARE 511. 512, 513. S2? S? ] ANI) 513. CC THE: THRF:E COEFFICIENTC:; C l. CZ ANO Cl CAN BE USFO FOR CC CALCULATING THE HORIznNTAL DI5PLACEMENT UK HALFWAY ALONG THE CC MEMBER ~ R O I o 1 DELTAL' T ~ T l ~ N TI:T2. THE RELATION Is : CC UK= Cl *OELTAL + C2*TETli + C)*TET2 CC IN THE CHn!CE OF THE ARRAY O I ~ E N S T O N S IT IS ASSU"EO THAT CC THERE ARE (lP TO 5 S E G M E N T ~ PER HEHRER ANO AN AvERAGE OF S CC sECTlON5 PER sEGMENT ( M A X I ~ U M i' S ~ E n I O N S ) . THE ARRAYS CC 50 . Y AND S ARE AUXILTRARY ARRAYS . THE ARRAY vuL IN COM"lON CC RELATE s T a DATA WHICH ARE NOT usEn IN STYTER. CC C

DIMENSION SO( !O I .Y ( IO .Z5 ) .S ( IO ) CCOMMON VIJL(16) .0011 (2t:;) ,0021 (2S1 . O O ~ 2 ( 2 5 ) ,NMOT.ALMOT(S) . ~ S T M O T ( 5 ) CXI =0 CNN=O C00 10 J= I . IO CS I J I=O .10 CONTINUE.6L=O.DO ZO L=I.NHOTAL:AL+ALMOT(L)

20 CONTINUE00 29 0 L=l.NMOTA L I = A L ~ O T ( l )N=NSTMOT ( l )AL 3=AL 11 na-Na-AL a- AL)FN=NX:ALl/(FNa-AL)XI=XI-XNl=N+I00 100 I= .NINN=NN+lXl:Xl+X1 l=0011 (NN)Z2::::0D21 (NN)73=oon (NNIAl::f)a-XlA2:24a-xlA):36* '0Al:xla-xlR2:16*Al83=4A*RIY


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SUBROUTINE E ~ O O I E P S . E L A . N B . N S )CC NB + NS STRAINS ENTER INTO THE ARRAyC EPS. THE ~ O O U L U S OE ELASTICITYC SELECTED AS CORRESPONOING TO THESEC IS IN TURN GIVEN OUT IN THE ARRAYC ELA. THE EIRST NB PDSTTIONS RELATEC TO CONCRETE AND THE EOLLOWING NSC RELATE TO STEEL. THE SIGMA-EPSILONC DIAGRAM OE BOTH THE CONCRETE ANOC THE STEEL IS BILINEAR. THE CONCRETEC CANNOT RESIST TENSIONC IN THE COMMON AREA ARE:C SVLS YIELD STRESS OE STEELC SVLB YIELO STRESS OE CONCRETEC ESI STEEL STRAIN AT START OEC YIELOINGC ES2 EAILURE STRAIN OE STEELC EBI CONCRETE STRAIN AT STARTC OE YIELOINGC EB? EAILURE STRAIN OE CONCRETEC IEOUTS COUNTER; THls IS INCREASEOC ey ONE UNIT IE ERS2 ISC ANYWHERE EXCEEOEOC IEOUTR = OITTO EOR EB?C THESE COUNTERS ARE USED EORC SIGNALLINGIELAGSI.C

COMMON ~ V L S . S V L B , E S l , f S 2 . f 8 1 . E B 2 ,4IFOUTS,IFOUTBDIMENSION ELAlISI.EPS(15)Nt


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References1. LIVESLEY, M. A., Matrix Methods of Structural Analysis, 1964.2. Betonvereniging, Lecture notes for the course "Moderne Rekenhulpmiddelen en Methoden"(Modern computation techniques and methods), Chapter 4: Enkele matrixmethoden voor hetberekenen van staafconstructies (Some matrix methods for the analysis of framed structures), by

J. BLAAUWENDRAAD.*3. JENNINGS, A. and K. MAJID, An elastic-plastic analysis by computer for framed structures loadedup to collapse. The Structural Engineer, December 1965, No. 12, Vol. 43.4. BLAAUWENDRAAD, J. Moderne rekentechnieken voor stabiliteitsonderzoek (Modern computationtechniques for stability analysis). Syllabus of the course on Stability of Buildings (lectures Nos.

10 and 11) of the Stichting Post-Doktoraal Onderwijs in het Bouwen, October 1970.*5. LEEUWEN, J. VAN and A. C. VAN RIEL, Berekening van centrisch en excentrisch gedrukte constructiedelen volgens de breukmethode (Analysis ofaxially and eccentrically loaded structural com-ponents by the ultimate load method). Heron 10 (1962), No. 3/4.*6. GROOT, A. K. DE and A. C. VAN RIEL, De stabiliteit van kolommen en wanden van ongewapendbeton (The stability of plain concrete columns and walls). Heron 15 (1967), No. 3/4.*7. GROOT, A. K. DE, Rekenmethode voor gewapend-betonkolommen (Method of analysis fo r reinforced concrete columns). Syllabus of the course on Stability of Buildings of the StichtingPost-Doktoraal Onderwijs in het Bouwen, September 1970 (TNO-IBBC Report B170-62).** In Dutch.

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In the series of Rijkswaterstaat Communications the following numbers have been published befare :No. 1.* Tidal Computations in Shallow Water

DI. J. J. Dronkers tand Prof. Dr. Ir. J. C. SchnfeldReport on Hydrostatic Levelling across the WesterscheldeIr. A. Waalewijn

No. 2.* Computation of the Decca Pattern for the Netherlands Delta WorksIr. H. Ph. van der Schaaf t and P. Vetterli, Ing. Dip!. E.T.H.

No. 3. The Aging of Asphaltic BitumenIr. A. J. P. van der Burgh, J. P. Bouwman and G. M. A. Steffelaar

No. 4. Mud Distribution and Land Relamation in the Eastern Wadden ShallowsDr. L. F. Kamps t

No. 5. Modern Construction of Wing-GatesIr. J. C. Ie NobelNo. 6. A Structure Plan for the Southern Ijsselmeerpolders

Board of the Zuyder Zee WorksNo. 7. The Use of Explosives for Clearing lee

Ir. J. van der KleyNo. 8. The Design and Construction ofthe Van Brienenoord Bridge across the River Nieuwe Maas

Ir. W. J. van der Eb tNo. 9. Electronic Computation of Water Levels in Rivers during High Discharges

Section River Studies, Directie Bovenrivieren of RijkswaterstaatNo. 10. The Canalization ofthe Lower Rhine

Ir. A. C. de Gaay and Ir. P. BloklandNo. 11. The Haringvliet Sluices

Ir . H. A. Ferguson, Ir. P. Blokland and Ir . Drs. H. KuiperNo. 12. The Application ofPiecewise Polynomiafs to Problems of Curve and Surface ApproximationDr. Kurt KubikNo. 13. Systems for Automatic Computation and Plotting ofPosition Fixing Patterns

Ir. H. Ph. van der Schaaf tNo. 14. The Realization and Function of the Northern Basin of the Delta Project

Deltadienst of Rijkswaterstaat

* out of print


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Documents

45095319 Fysical Engineering Model of Reinforced Concrete Frames in Compression