Folded Plate Theory and its Application in the Design oj Plated Structures

2.3 Folded ,Plates with Diaphragms RestrainingFree Cross-Sectional Warping

ln praetiee, diaphragms are usual1yassumed to be rigid in their own plane andperfectly flexible out of this plane. This means that the diaphragm preventsdeformation of the shape of the eross-seetion at whieh it is placed, but does notinduce any bimoment effeets. These assumptions are in accordanee with therestrietions ofthe c1assiealfolded plate theory presented in Section 2.1, but oftenthey are far from reality. The problem of end diaphragms which are deformablein their own planes was diseussed in Seetion 2.2. The analysis of the strueturalperformanee of folded plates with thiek or massive diaphragms, whieh exhibita eorresponding out-of-plane stiffness, is presented in this Seetion. ~

A diaphragm (and also a diaphragm with an opening, a frame eross-bracingor a truss) is a planar figure having its in-plane and out-of-plane stiffnesses.Thediaphragm is eonnected to the folded plate strueture, either at the joints only oralong the whole perimeter of the eross-seetion. The diaphragm properties aredeseribed by the stiffness matrix, whieh relates the displaeements of points onthe edge of the diaphragm to the forees aeting at those points.

Suppose the diaphragm thickness (in relation to its other proportions) permitsus to eonsider it as a plate within the meaning of the theory of elastieity. Then,assuming a linear distribution ofwarping ofthe girder eross-section between twoadjaeent eomers (Fig. 2.20a), the middle surfaee of the diaphragm will aequirethe form of a hyperbolie paraboloid deseribed by the equation

u(", ') = -4u ",Ih '

(2.60)

in whieh I, h are the diaphragm dimensions, and ", , the loeal eoordinates (Fig.2.20a). This pattem of deformation of the diaphragm eorresponds, according tothe plate theory, to the loading of the plate by longitudinal forees R (shown inFig. 2.20a by dotted lines in the direction of their aetion on the diaphragm),having a value

I Et3 a2u I 2Et3u 4Gt3uR = 6( 1 + li) a" a, = 31h( 1 + li) = 3th = ku,(2.61)

where G is the shear elasticity modulus of the diaphragm material and k is thestiffness of the diaphragm for a plate bending effeet.

Due to the warping of the box girder eross-seetion, the hyperbolie paraboloiddeformation pattem of a plate according to Eq. (2.60) induces either reaetionsR aeting in the plate eomers (Fig. 2.20a, formula (2.61)) or specifie torsional

50

Folded Plates with Diaphragms Restraining Free Cross-Sectional Warping

reaction moments of an intensity Et3/12(1 + li) . &u/a" a, acting along thecontour of the plate (according to the classical plate theory both systems ofreactions are equivalent). In this state of stress, only torsional moments (equalin value to the torsional reaction moments acting on the plate edges) appear inthe plate; these moments induce only shear stresses t", parallel to the middleplane of the plate. The distribution of these stresses along the plate thickness is,in this particular simple case, linear. This is trne not only if the classical theory

(a)

R

Cd)

51

Folded Plate Theory and its App!ication in the Design oj Plated Structures

is employed, but also if use is made of more precise theories (such as thecomponent theory, Reissner's or Hencky's theories, etc.); consequently, one hasr,,, = E/(l + li) . (&u/a" a() . X, where x is the coordinate measured perpen­dicular to the middle plane of the plate. Therefore, the relations derived are alsosuitable for diaphragms with somewhat greater thickness, which for anotherkind of plate-type loading cannotbe considered as plates.

When the thickness of the diaphragm is increased, the end diaphragmsacquire the character of massive blocks (schematically shown in Fig. 2.20b). Itwill be assumed in the analysis that the cross-sectional shapes of this block arenot deformed, but that only twisting of the block and warping of its cross­section take place. The structure is supported in the section connecting the boxgirder with the block (Fig. 2.20b); this support is able to transfer only thesupport torsional moment and not the bimoment effectswhich in this case mustbe taken over by the end block, representing an elastic fixing. This fixingrestrains the warping of the extreme box girder cross-section. It is assumed thatthe length of the block t is comparable with its cross-sectional proportionsheight H and width A.

If the end block is taken to be a mem~er of approximately solid cross-section,its behaviour may be described. by two·:.differential equations

BIJ" - 0[(1" + I,) f - (I" - Ir;)rp'] = O,

0[(1" - I,) f' - (I" + I,) rp"] = mk, (2.62)

where f is the warping rate of the cross-section, rp is the twisting angle and mk

the torsionalload acting on the block.Then

1I" = -AH3;

12

1I, = -HA3;

12

In view of the type of support (Fig. 2.20b), the block is loaded only by abimoment at the connecting section with the box girder. Otherwise it is notloaded and therefore mk == O in this case. Relations (2.62) may be rearrangedinto one differential equation

- r< 41,,1,EIJ'" - v~-f' = O.I" + I,

(2.63)

Considering furthermore that the bimoment is determined by the relation

52

B = BIJ', (2.64)

Folded Plates with Diaphragms Restraining Free Cross-Sectional Warping

Bq. (2.63) may be further adjusted to the form

where

(2.65)

b= G 4I"I~----=E Iw(I" + I,)

(2.66)

The solution of this equation, using a new longitudinal coordinate ~ with theorigin at the contact section between the end block and the box girder, is

B(~) = CI cosh b~ + C2 sinh b~ . (2.67)

If forces R act on the block at contact points with the corners of the box girdermedian line according to Fig. 2.20c (that is at points with coordinates~ = O, " = ± a/2, ( = ±h/2), they induce a bimoment with the value of

_ ahB(O) = 4R - - = Rah,

22(2.68)

which forms one boundary condition for the determination of constants in thegeneral solution (2.67). The second condition, at the free end of the block~= t where no bimoment is acting, is

B(t)=O. (2.69)

The bimoment then develops along the block length according to the relation

B(~) = Rah(cosh b~ - cotgh bt sinh be}. (2.70)

Longitudinal displacements of the block points - in this case, warping - maybe described by the relation

(2.71 )

By substituting Eq. (2.64) into Eq. (2.71), using Eq. (2.70) and integrating,whi1e noting that on the block axis ~, O, O, no longitudinal displacements takeplace, the following equation is obtained

Rahu(~, ", () = -_- (sinh b~ - cotgh bt cosh b~),,( . (2.72)

E1wb

53

Folded Plate Theory and its Application in the Design oj Plated Structures

ln the place of contact with the upper point of the left box girder web (thatis at a point with coordinates e = O " = a/2 , = - h/2), this results in

(a h) Ra2h2u O,-, - - = -_- cotgh bt .

2 2 4EImb(2.73)

This displacement must equal the longitudinal displacement ji of the comerpoint of the end box girder cross-section. From this the following relation isderived

(2.74)

which is an analogue of Eq. (2.61), where k again means the stiffness of theblock.

If the entire length of the block is supported (Fig. 2.20d), the solution mayagain be found according to Eq. (2.62), where in this case rp = Ois to be taken,because this type of support prevents twisting of the block. From the secondequation (2.62), the intensity of a torsionalload mk may be expressed; this loadis needed to prevent twisting of the block and it must be transferred by thesupport reactions. It may be written as follows:

The first equation then takes the simple form:

EIJ" - G(I" + ItJ f = O,

(2.75)

(2.76)

which, after differentiation and application of relationship (2.64), assumes aform similar to Eq. (2.65). AIso solution (2.67) and all the other relationshipsretain their validity, but the quantity b must be substituted by another quantity

12G(A2 + H2)--EA2H2

(2.77)

If the block has a square cross-section (that is H = A), then both types ofsupport described are equivalent, because a block with a square cross-sectionunder a bimoment loading of the end cross-section does not tend to twist. Thisalso results from relation (2.75), from which mk = Ois obtained and

b ~ li ~ J24(JEA2

54

Folded Plates with Diaphragms Restraining Free Cross-Sectional Warping

ln the general case of a rectangular cross-section (H =F A), the second type of'support (Fig. 2.20d) is stiffer than the first type and, therefore, induces also ahigher degree of restraint against bimoment effects of the box girder.

On the basis of comparative analyses of the effects of thick diaphragms andend blocks, it has. been found that even a relatively thin diaphragm induces acertain degree of restraint against free cross-sectional warping. On the otherhand, it is apparent that the intensity of restraint depends very little on thelength of the end block, l;lndeven a block with its length growing without limitdoes not represent perfect clamping. This small effect of the block length resultsfrom a very high attenuatión rate of the bimoment acting at the contact with theextreme cross-section of the box girder in accordance with St. Venants principle;according to this principle such loading of the block has a local character. Inattempting to achieve a higher degree of restraint, it is not effective to increasethe length of the block (if the width and the depth of the block are increasedn-times, the stiffness of the block - see relations (2.66) and (2.74) - is increasedapproximately n5 -times).

ln bridge construction, cases may be encountered where a box girder isconnected with massive piers; this is schematically shown in Fig. 2.2Ia. With

(O)

(b)

RR

Fig. 2.21. (a) Connection or a box girder to amassivepier, (b) interaction between the girder

and the pier.

55

Fo/deJ P/ate Theory and its AppUcation in the Design oj P/ated Structures

regard to the assumption conceming the deformation pattems of a thin-walledbox girder under torsion, it may be conc1uded that the individual sides of therectangle in which the pier contacts the girder (in Fig. 2.21a these are, forexample, sides 1-2,2-3), remain straight, and the entire contact rectangle wi1lform the surface of a hyperbolic paraboloid. Due to the interaction of the boxgirder with the pier, a bimoment appears in the top cross-section of the pier; thisbimoment may be characterized by forces R acting at the comers of the cross­section (Fig. 2.21b), and it again induces complementary moments acting on thewebs of the box girder. The pier may well be considered as a member with solidcross-section, and relation (2.74) can be used again for the determination offorces R, now as a function of the displacements of the girder webs. Again, thevalue of these forces wil1 depend primarily on the dimensions of the piercross-section (that is on the length of the line segment 2-3, if the width of thepier is determined by the width of the girder); the effect of the pier height wil1be negligible. If there is also a diaphragm witl1plate bending stiffness situatedin the box girder over the pier, its effects are added to the effects of the pier.

Apart from a plate-formed diaphragm, frame bracing (a diaphragm with anopening) is also encountered)n practice (Fig. 2.22).

~ I ;-J J ](~l:....:L::::.. . __ . __ . .__ lj-'2 h~

I~ /(1 .1 Fig. 2.22. Frame bracing or a diaphragm with an opening.

The plate bending effect of a frame bracing is described by the relation

R = 8u = ku,

(2.78)

where Ih2, 1/2 are the cross-sectional moments ofinertia ofvertica1 or horizontalframe bracing members relative to the axes lying in the frame plane (i.e. themoments of inertia participating in the frame stiffness under bending of themembers perpendicular to the frame plane), and Ikh, Ikl denote the moments ofcross-sectional stiffness in torsion of the vertical or horizontal frame members.

It should be noted that the relation (2.78) requires the correct determination ofthe quantities Ih2, 112,Ikh, Ikl (for example, allowing for the respective effectivewidths of the box girder flanges).

56

(C)

Folded Plate Structures with Various End Conditions

Ifthe relationship between the forces and the displacements at the diaphragmcomers is known, the interaction between the box girder and the diaphragm canbe solved. For this purpose, for example, the force method of analysis may beused, in which the bonds between the box girder and the bracings' are stated, andthe redundant parameters acting in these bonds which maintain the continuityof the structure are sought.

2.4 FoIded PIate Structures with VariousEnd Conditions

The classica1 folded plate theory presented in Section 2.1 is restricted to simplysupported structures, provided with end diaphragms which are infinitely rigid intheir own planes and perfectly flexible out of those planes. The methods presentedin Sections 2.2 and 2.3 a!low for ana!ysis of folded plates with deformable enddiaphragms (or without any diaphragms) and diaphragms with an out-of-planestiffness. However, there are some other arrangements of end conditions which caneasily be solved by establishing appropriate substitute structura! models.

For example, a folded plate structure with overhanging ends (Fig. 2.23a) canbe analysed by taking its total length as the span of a substitute, simplysupported structure loaded both by the given extemalloading and by the knownstatically determinate reactions, Fig. 2.23b. Thus a set of self-equilibrating forcesis formed and, consequently, no reactions appear at the end cross-sections of thesubstitute structure. The states of stress of both the structure with overhangingends and the substitute simply supported structure are identical; the displace­ments thus obtained (Fig. 2.23d), however, are to be corrected by rigid bodymotion, realizing that no deflections originate in the actual supports, Fig. 2.23c.

(a)

(Ó)

b~a"ll'l"'l'~·······no r~actions··-···

Fig. 2.23. Representation or a structure with overhanging ends:(a) a structure with overhanging ends, (b) the substitute structure,

(c) deflections or the original structure, (d) deflectionsor the substitute structure.

57

Folded Plate Theory and its App/ication in the Design oj Plated Structures

Similarly, a folded plate structure formed as a cantilever, Fig. 2.24a, (fixed at oneend and free at the other end, with a diaphragm at the free end) may be solved asa simply supported folded plate structure spanning double the length of the cantil­ever (the original cantilever and its mirror image about the clamped end, Fig. 2.24b)provided with a diaphragm rigid in its own plane at mid-span This substitutestructure is loaded by the given externalload acting on the two cantilevers formedin this way and, at the mid-span of the simply supported structure, where the

diaphragm is situated, by force factors acting in the diaphragm plane (e.g. threeforces, or two forces and one moment) whose combined effect double the value ofthe resultant of the reversedly taken given extemalloading of the cantilever. Sincethese additional factors balance all the applied loads, the resulting reactions of thesubstitute structure are then equal to zero (which coresponds to the free end of thecantilever) and the state of stress of the half span corresponds to the actual cantileverstructure analysed. The deformations of the two structures differ only in that themaximum deflection of the original cantilever occurs at the end, as in Fig. 2.24c,whereas the ends of the substitute structure are supported (Fig. 2.24d).

a) Cantilever

b) Substltute beam

~

I ' I ' I~ L =

c)Oe(lection o(orlglna/ beam

d) Oef/ect/on of thesubstltute beum

~

Fig. 2.24. Representation or a cantilever.

This approach, based on the application of a substitute structure, can be shownby a simple example. A cantilever of height H with a flexural stiffness EI is loadedby a horizontal continuously distributed loading q (Fig. 2.25a). The correspondingsubstitute structure is a simple beam spanning L = 2H, loaded by the load q andthe force R = qL = 2qH (Fig. 2.25b). The bending moments of the substitutestructure are shown in Fig. 2.25c. The deflection line is plotted in Fig. 2.25d. It isevident that the deflections y of the actual structure are given by

H4y(x) = y(H) - y(x) ~ ~ - y(x) ,

SEl

where y(x) is the deflection of the substitute structure.

58

Folded Plate Structures on an Elastic Foundation

This idea is applied to the analysis of tall buildings as shown in Section 2.7.The introduction to this Chapter dealing with the analysis of folded plate

structures included the basic requirement of a constant cross-section of thestructure over its entire length. In the special case of a continuous folded platestructure provided at all its supports with diaphragms perfectly rigid in its ownplane and perfectly flexible perpendicularly to this plane (and having no otherdiaphragms), an approximate procedure may be applied. In this procedurefolded plate structures, simply supported as individual spans and themselvesfulfilling the basic assumptions of the folded plate theory, are chosen as theprimary system (compare the analysis of a continuous girder by the three­moment equations). The redundant quantities in this case are represented by aset of longitudinal forces at selected points of the support cross-sections (re­presenting support bending moments and support bimoments in the ordinarytheory). This approach has the advantage of the flexibility matrices being of aband form; moreover, this selection of the primary structure makes it possibleto have different thicknesses of walls in the individual spans, that is, the foldedplate structure analysed may change its cross-section at the supports.

Q.

J.IEJ].-.

o •

~Nl

119(1/)-(a)

R.2iH~ fj2.••{lN·BEZ

Fig. 2.25. Analysis of a cantileverbeam: (a) the actual structure, (b)the substitute structure and its

loading, (c) the bending momentdiagram, (d) deflections. (b) (c) (d)

2.5 Folded Plate Structures on an ElasticFoundation

Cases are encountered in practice where prismatic thin-walled, cellular orsimilar structures (tunnels, foundation structures under high-rise building,pipelines, subways, etc.), are placed on a foundation which may be regarded aselastic, i.e. in which induced reactions are proportional to deflections. The foldedplate theory again provides a simple tool for structural analysis of these cases.

59

Folded Plate Theory and its App/ication in the Design oj Plated Structures

The folded plate structure (Fig. 2.26a) is divided into an appropriate numberof plate-shaped elements (Fig. 2.26b), and the action of the elastic foundationis represented by a set of continuously distributed elastic supports, acting at thejoints. These joint reactions generally have four components, which correspondto interaction with the elastic foundation in the sense of four degrees of freedom(horizontal, vertical, rotational and longitudinal motions). The elastic propertiesof the joint supports are evaluated on the basis of the parameters of the actualelastic foundation supporting the halves of the elements adjacent to a joint.

(a) (b)

Fig. 2.26. (a) A folded plate structure on an elastic foundation, (b) folded plateidealization.

Due to the deformation of the folded plate structure, reactions in the con­tinuous joint elastic supports appear, thus forming the additional joint loadseR}, whose magnitudes are proportional to the joint displacements (or to thejoint rotation, respectively). Hence,

(2.79)

in which [eK] is a diagonal matrix of the individual components of the stiffnessproperti~s of the joint elastic supports, and {t>} are the joint displacements in thefixed coordinate system (Fig. 2.6). It follows that

ek'l'O,O, ... , O,O

O

ekTJ1, O, ... , O,O

I.[eK] = I 'O

(2.80)

O,

O,O, ... , O, ek8N

where, for example, the term ekTJl represents the stiffness of the elastic supportunder joint 1 against vertical displacement (see Fig. 2.~6).

60

Folded Plate Structures on an Elastic Foundation

Since all the joint displacements are expressed in the form of Fourier series,and since the elements of matrix [eK] are constants, Eq. (2.79) may be writtenfor the n-th harmonie in the form

(2.81)

Analogously to Eq. (2.45), the equilibrium of each elastically supported ridgeprism yields the relation

(2.82)

in whieh [Kn] is the structure stiffnessmatrix corresponding to the n-th harmo­nie, and {~n}are the unknown amplitudes ofthe n-th term ofthe seriesexpressingthe displacements and rotations of the joints, see Section 2.1. Equation (2.82)can be written as

(2.83)

or in the form

(2.84)

whieh is formally identical with Eq. (2.45).The matrix

(2.85)

represents the stiffnessmatrix of the folded plate structure on an elastic founda­tion. It can be simply obtained by adding the elastic support characteristicsforming the diagonal of matrix [eK] to the main diagona! elements of thestructure stiffness matrix [Knl

Thus it is clear that the analysis of folded plates on an elastic foundation canagain be carried out by introducing only minor adjustments (replacement of theoriginal stiffnessmatrices [Kn] by matrices [R'n]' Eq. (2.85))into existing foldedplate computer programs, and that the method presented retains all the advan­tages of the classical folded plate theory.

The harmonie analysis in the form presented requires that simple supports beplaced under the end diaphragms at both ends of the structure (Fig. 2.27a). Inpractice, however, arrangements without end supports (Fig. 2.27b) or, in somecases, also without end diaphragms (Fig. 2.27c) are more frequent.

An idea similar to that applied in Section 2.2 for the analysis of folded plateswith deformable end cross-sections can be used. Considering, for example, thecase shown in Fig. 2.27b, it is possible to supplement the given actualloadingby such a set of uniformly or linearly distributed joint loadings (or their com-

61

Folded Plate Theory and its AppUcation in the Design oj Plated Structures

binations of a trapezoidal form) such that no reactions appear at the endsupports (Fig. 2.27d) - despite the fact that the structure is simply supported atthe ends.

The set of trapezoidally distributed loadings which was added to the actualloads in order to bring about the special state without any reactions at the simplysupported ends (Fig. 2.27d) must be applied in the opposite direction withrespect to the structure in its actual form, i.e. without end supports (Fig. 2.27e).These trapezoidalloadings produce only deflections of linear variation, withoutinducing any longitudinal or shear stress in the structure.

The case without supports and end diaphragms (Fig. 2.27c) can also beanalysed in a similar way,i.e. by application of the idea presented in Section 2.2.

(b)

(C)

~~~~;~~~~~ll~~;;~;~~~~~~~)

ouurrtl 1I -,I I

~i~~~~~~~~~~~~~~~~

62

Fig. 2.27. (a) A simply supported folded platestructure (with support diaphragms) on anelastie foundation, (b) a structure without endsupports, (e)a strueture without end supportsand without end diaphragms, (d) a simplysupported structure with supplementary load­ing, (e) the action of trapezoidally distributedloading on the structure without end supports.

Statica//y Indeterminate Continuous Folded Plate Structures with Interior Diaphragms

2.6 Statical1y Indeterminate Continuous FoldedPlate Structures with Interior Diaphragms

Statically more complicated folded plate structures (continuous, clampedstructures, frames, girders with interior right or skew diaphragms having platestiffness, or with flexible support frame bents, etc.) can be analysed by means ofthe force method. This method consists of removing all redundant constrainingeffects (supports, bracings, etc.) so that a simply supported folded plate structureis obtained. Hs analysis was described in Section 2.1 (or, for the case withdeformable end diaphragms, in Section 2.2), and this structure will now re­present the primary system for the solution by the force method. The removedconstraining effects are replaced by the redundants {11, whose magnitudes areto be determined from the deformation conditions. The primary system (thesimply supported folded plate structure) being analysed by means of a series, thereality may be better expressed by distributing the redundants over a givenlength (such as the width of the bearing). The redundants are to be chosen soas not to induce any longitudinal force in the primary structure as a whole. Themethod is demonstrated on two most frequent cases of statically indeterminatestructures.

2.6.1 Folded plate structure with frame bents

The structure iscontinuous with interior frame bents for supports (Fig. 2.28a).These frame bents are assumed to be perfectly flexible in the direction perpen­dicular to their own planes. If the rigidity in the longitudinal direction, in thecase of the structure analysed, cannot be neglected, it may be simulated ap­proximately by doubling the frame bents, each of them having half rigidity in thetransverse direction, with the flexural rigidity in the longitudinal direction beingsubstituted by the axial rigidity of the columns (rigidity in tension or com­pression), Fig. 2.28b.

To form the primary structure, the frame bents are separated from the foldedplate structure in the support cross-sections as shown in Fig. 2.28c; the actualshape of the frame bents is idealized by planar frames corresponding to theactual arrangement. For almost rigid links, for example, which connect theframe with the nodal points of the folded plate structure (Fig. 2.28d) and whichare actually situated within the support bracing, a very high value of themodulus of elasticity can be introduced.

The interaction forces (redundants) {Y} are represented by a set ofthree forcesconsisting of vertical, horizontal and rotational components in the plane of thetransverse cross-section.

63

Folded Plate Theory and its Application in the Design oj Plated Structures

(a)

(b)

, I .

I :JI; ,

,

64

(c)

Cd)

DDDD

Statically lndeterminate Continuous Folded Plate Structures with lnterior Diaphragms

(e)

Cf)

Fig. 2.28. (a) A frame bent supporting a folded plate structure, .(b) an idealized framebent having a rigidity in the longitudinal direction, (c) the support cross-section ofthestructure, (d) an idealized frame bent and the action ofthe redundants, (e) the effectof the redundants on the folded plate structure, (f) an interior pinned support of a

continuous folded plate structure.

The primary, simply supported single-span strueture is analysed first for thegiven externalloads acting alone (with all redundants removed). A displacementveetor {do} is found for this ease, whieh defines the displacements of the pointswhere the redundants are to aet. By successively applying unit redundant load,sto the system (Fig. 2.28e) and solving it for displacements, the individualeolumns of the strueture flexibility matrix [L1I] are established.

Eaeh of the planar frame 1;>entsis analysed by the direct stiffness method. Thetotal strueture stiffness for the frame bel1t is found, then statie eondensation isearried out to eliminate the degrees of freedom whieh do not eorrespond to theredundant forees. The flexibility ma,trix of the frame bents [L12] eorrespondingto the unit redundant forees is found by inverting the stiffness matriees (s~e also[2.8], [2.3], [2.9]).

Geometrie eompatibility at the interaetion points requires the fulfilment ofthe eonditions

whieh, with the notation

(2.86)

[A] (2.87)

65

Fblded Plate Theory and its Application in the Design oj Plated Structures

can be written in the form

[..1]{l1 + {do} = O.

The solution of this system of linear algebraic equations

(2.88)

(2.89)

gives all of the values of redundants.Thus, the simply supported folded plate structure and the planar frame bents,

subjected to the given extemalloads and the known redundant forces, can nowbe analysed to determine the final stresses and displacements in the actualstatically indeterminate structure.

Another arrangement of the interior supports of a folded plate structure oftenappears in construction practice, where supports carry a diaphragm rigid in itsown plane and perfectly flexible perpendicular to this ph~,ne(Fig. 2.28f); thestructure has the character of a continuous beam. All points of the supportcross-section are prevented, by the rigid diaphragm, from being displaced in thecross-section plane; therefore, the interaction points ofthe folded plate structurealso cannot be displaced horizontally and vertically, nor can they rotate, at thepoints at which the redundants Y (Fig. 2.28e) act. The problem may then besolved in the same way as that of the frame bents, except that [..12] = Ois to beintroduced in Eq. (2.86).

The connection of the support diaphragms with only some interaction pointsof the folded plate structure (if need be, only in the sense of some reactioncomponents) makes it easily possible to simulate various other arrangements ofthe supports encountered in practice.

2.6.2 Folded plate structure with interior (unsupported)diaphragms deformable in their own planes

The intermediate diaphragms (Fig. 2.29a) are assumed to be perfectly flexiblein the directions perpendicular to their own plane, but unlike the diaphragms atthe end cross-sections, they are deformable in their own plane. Since thesediaphragms are not extemally supported they can undergo, when subjected tointeraction forces, three degrees of rigid body motion in their own plane inaddition to the deformation of the diaphragms themselves. Therefore, the in­teraction forces must be in self-equilibrium.

For this reason, the analysis must be conducted in a manner differing some­what from that of the frame bents. Not all of the restraints between thediaphragm and the folded plate structure are released; three restraints, re­presenting the statically determinate support of the diaphragm on the folded plate

66

Statically lndeterminate Continuous Folded Plate Structures with lnterior Diaphragms

(a)

(b)

(C)

Cd)

(f)

Fig. 2.29. (a) An interior unsupported diaphragm of a folded platestructure, (b~ (c), (d) statically determinate supporting of a diaph­ragm, (e) tbc action ofthe redundants, (f) an example ofideaJiza-

tion of a diaphragm by an equivalent frame.

67

Folded Plate Theory and its Application in the Design oj Plated Structures

structure, are retained. The selection of these three initial restraints is arbitrary;they may be represented by three pendulum members (Fig. 2.29b), or by apinned support and a pendulum member (Fig. 2.29c), or by some fixing at onepoint (Fig. 2.29d). From the point of view of numerical computation, it isappropriate to select an initial connection where the reactions, produced by theloading of the diaphragm, do not far exceed the value of the loading in order.It is therefore more advantageous to use, for instance, the variant according toFig. 2.29c than that shown in Fig. 2.29d.

At the points of the released restraints between the diaphragm and the foldedplate structure (in the direction of their components which, as already stated, areless by three than they were in the case of a supported diaphragm or a framebent), there the redundants Yact. They load the folded plate structure as wellas the diaphragms (e.g., in Fig. 2.2ge the redundant li). As the diaphragm isassumed to be initially statically connected to the folded plate system, thediaphragm reactions due to the redundant produce further loading on the foldedplate structure. All forces acting on the diaphragm, the redundant Yand thereactions produced by this redundant, are in a state of self-balance.

Due to the successively acting unit redundants Y,relative displacements dki

occur at the points of the released restraints, forming the flexibilitymatrix [A I].Due to the loading of the diaphragm by the redundants Yas well as by the

reactions of the initial connection, the diaphragm is deformed, contributing tothe flexibility matrix [LIl]. The diaphragm can be analysed as a wall. Longerdiaphragms having the characterof a girder can be idealized as an equivalentframe, for example, statically determinate (Fig. 2.29f), where the horizontalmember (elastic axis ofthe diaphragm) takes over the bending, shear and tensileor compressive stresses, and its cross-sectional characteristics represent therespective stiffnessesof the entire diaphragm. The stiffness of the perpendicularlinks depends on the assumed distribution of the strains along the diaphragmheight (assuming that plane diaphragm cross-sections remain plane, totally rigidlinks shoud be introduced).

The externalloading deforms only the simple folded plate structure, while thediaphragm, owing to its statically determinate support, is displaced only as arigid body without any deformations of its own. Relative displacements {do}appear in this way at the points of the released restraints.

The preservation ofgeometric compatibility between the folded plate struc­ture and the diaphragms at the points of released restraints requires the fulfil­ment ofthe conditions expressed by the system of equations

(2.90)

It often happens that the structure is provided with both frame bents andinterior unsupported diaphragms. It is, therefore, advantageous to maintain thesame system of basic redundants throughout the an3;lysis.For this reason, the

68

Statically lndeterminate Continuous Folded Plate Structures with lnterior Diaphragms

procedure eonceming the interior diaphragms ean be the same at the beginningas the procedure eoneerning the frame bents. The primary folded plate strueture,whieh excludes the diaphragm, is analysed under the extemalloading. The jointdisplaeements {do} at the loeation of the diaphragm are ealeulated, selecting thesame system of redundants {Y} as for the solution of the strueture with framebents. Then (again in t~e same way as when analysing the strueture with framebents), the flexibilitymatrix [JI] of the primary folded plate strueture is formed.The displacements of the points of aetion of the redundants on the primaryfolded plate strueture, produced by the total effect of all redundants, are des­eribed by the relation

(2.91 )

Now, a new set ofredundants {V} is defined so that eaeh redundant, while aetingon the diaphragm, will be in self-equilibrium (see [2.9]). Eaeh of these newredundants Yhas four eomponents: the redundant Yand the diaphragm reae­tions whieh occur when the diaphragm is initially eonnected to the folded platestrueture (Fig. 2.29b, e, d, e). The number ofthe redundants Vis therefore equalto the number of the redundants Yand is for eaeh diaphragm less by three thanthe number of the redundants Y.The relation between the original redundantsYand the redundants Yean be expressed as

{Y} = [VI] {Y}, (2.92)

where [VI] is a force transformation matrix, whose number of rows differs bythree from the number of eolumns for eaeh movable diaphragm.

The relative displacements Iof the diaphragm and of tQ.efolded plate strue­ture with the assumed initial eonnection of the diaphragm depend on theabsolute displacements d of the points of the folded plate strueture. The depen­dence is given by the relation

Henee,

{lo} = [VI]T {do},

{1Y,fOI} = [VI r {dy,rol}·

(2.93)

(2.94)

(2.95)

By introdueing relations (2.86) and (2.92) into (2.95), the following equationis obtained

(2.96)

69

Folded Plate Theory and its AppUcation in the Design oj Plated Structures

which, with the notation

[lJI]T [Al] [lJI] = [Jl],can be written in the form

(2.97)

(2.98)

where [Li a is the modified flexibilitymatrix of the simple folded plate structure(excluding the contribution due to the deformation of the diaphragm).

By analysing the sole diaphragm, initially connected to the folded platestructure and successively loaded by the unit values of the redundants, thedisplacements of the interacting points of the redundants are obtained. Thesedisplacements form the flexibilitymatrix [Li2] of the diaphragm, hence the totaleffect of all redundants Y produces a displacement of the interaction points ofthe redundants

(2.99)

The geometrical compatibility between the folded plate structure and thediaphragm requires fulfilment of the conditions

(2.100)

which, after the introduction of relations (2.98) and (2.99) and the notation

can be written in the form

[1] {Y} + {ao} = O,

(2.101)

(2.102)

in which relation (2.92) can be introduced for the sake of uniformity of thesystem of redundants.

The final solution may be obtained by separately analysing the simply sup­ported folded plate structure, the planar frame bents and the diaphragms beingsubjected to known extemalloading and redundants.

It should be noted, from the point of view of the numerical computation, thatthe flexibility matrices [A] should be established with a very high degree ofaccuracy; they may be so ill-conditioned that small changes of the individualcoefficients may alter the magnitude of the redundants completely. Ir followsflOm this that either a large number of Fourier terms has to be used in theanalysis, or fewer coupled sets of redundant force pattems should be selected inorder to reduce the magnitude of the off-diagonal flexibility coefficients. Res­traints whose corresponding redundants act against themselves on the stiff partsof the structure (for example, vertical restraints of both ends of the same web)should be avoided.

70

Ana/ysis oj Skew Structures

The choice of the primary structure is generally poor, because large correctionforces are required to restore compatibility. This deficiency may be partially over­come by estimating the magnitudes of some interaction forces and by analysing thestructure for these eliminated forces in addition to the given extemal loads. Theconnection forces required to restore complete continuity can be considerablyreduced by this procedure. It is suitable for an analysis in which one can intervene

(for example, by combining the computer analysis of the simple folded plate strne­ture as the primary system with the manual computation of the redundants by theforce method); however, when a general program is being established, it is almostimpossible to find a general, and at the same time simple, guide for these estimates.

2.7 Analysis of Skew Structures

Skew bridges of various types are extensively used in the road and railwaynetwork. Short and medium-span bridges are very often assembled of individualgirders which are mutuaUy connected. Such structures have usuaUy no interiordiaphragms between the supports, but are always provided with diaphragmsover the supports (Fig. 2.30a).

The simple supports of the structure, the constant cross-section and theplacing of diaphragms, rigid in their own planes, at the support cross-sectiononly (Fig. 2.30a), aU aUow it to be regarded as a folded plate structure. Modifica­tion of the classical folded plate theory, however, encounters many difficultiesbecause at the joints of the skew or mutuaUy 10ngitudinaUy shifted plate partsthe effects of the individual terms of the series which approximate the solutionmust be coupled together. In this way, the main advantage of the folded platetheory, viz. the possibility of conducting the analysis šeparately for each termand merely adding the results, is 10s1.The analysis of skew structures may indeedbe based on the folded plate theory, but the approach should be modified.

A solution may be conveniently found by means of the force method. Theprimary structure for this method may be formed by dividing the structure, bymeans of longitudinal sections, into separate parts (Fig. 2.30b). In the solution,general interior forces acting at the joints of the individual parts are ca1culatedto ensure the compatibility of the structure after its deformation.

AU contact forces which have a decisive effect on the structural behaviour,must be considered in the analysis. These continuously distributed forces ­redundants - have in general four components: vertical, horizontal, rotationaland longitudinal, Fig. 2.31. It should be noted particu1arly that the longitudinalforces, that is the shear flows in the flanges, exert a substantial influence on thebehaviour of the structure. To neglect them would mean a serious distortion ofthe results, because these shear flows are equal to the shear flows which circulatearound the multi-ceU cross-section under torsion.

71

Folded Plate Theory and its Application in the Design oj Plated Structures

(Q)

l

!II

jI

L J

Slctionh

~XN.,1 \ ' ,., , Sf',

"~ 2 4~:Xi-:':~__ -::--_--,__

~X' I

Fig. 2.30. (a) A typical skew bridge structure, (b) linking points and the shape ofindividual girders.

72

"

AnaJysis oj Skew Structures

Fig. 2.31. Redundants acting between girders.

Since a direct analytical solution would be unsuitable in this case, an ap­proximate approach is presented; it sufficesto satisfy the conditions of continu­ity of the structure at selected points instead of along the entire length of thejoints. Hence, r points (Fig. 2.30b) are chosen at the joints; the conditions ofcontinuity of the structure are written for the points, and redundant forces {Y}(Fig. 2.31a) are made to act at them.

Bearing in mind that the primary structure considered for the solution by theforce method is formed by a system of individua! independent girders (Fig.2.30b), which by themselves are of constant cross-section and have right supportcross-sections, the folded plate theory can be used for the analysis of theseindividual girders. As this method is exact (within the assumptions of the theoryof elasticity), the concentration of redundant forces directly at the individualpoints (Fig. 2.31a) would cause local singularities. It is, therefore, convenient toconsider the redundants as being distributed along a finite length. This lengthis naturally determined as the sum of the halves of lengths of the intervalsadjacent to the point considered. The actual continuous character ofthe internalforces acting at the joint is in this way expressed to an acceptable approximation,it being represented by a series of forces uniformly distributed in parts (in Fig.2.31b this distribution is shown for the redundant quantity N-l,N 2re, loadingthe upper right-hand ftange of the N-th girder at point 2 in the vertical direction).

73

Folded Plate Theory and its Application in the Design oj Plated Structures

Regarding the actual support conditions of the skew structure, the sum of thelongitudinal redundants acting on a joint need not necessarily equal zero.Therefore, in the calculation of the influence of longitudinal redundants, theprocedure explained in Fig. 2.9 is followed.

Further analysis of the structure corresponds with the current principles of theforce method. The primary structure, Le. the individual girders, is loaded by theextemalload of the structure and, successively, by the unit values of the redundantquantities (i.e. li = 1). In this analysis (using the folded plate method), the mutualdisplacements of the chosen points at the contacts of the girder flanges are calculated(in the direction of the redundant quantities {líc}). In this way, the right-hand side{dOk} of the system of linear algebraic equations for the unknown quantities of theredundants {11, and also the elements dki of the matrix [AJ of this system, areobtained. Bearing in mind that all girders in the structure are geometrically identical(Fig. 2.30b) and that the position ofthe points at which the conditions of continuityof the structure are to be satisfied is the same for all girders, it suffices to analyse onlyone girder (regarded as a computation sample) and to determine the values dki as thesums of the corresponding displacements.

The values of the redundants follow from the solution of the system ofequations

[AJ {Y} + {do} = O. (2.103)

The number of the redundants Y depends on the number of girders Nin thestructure and on the number of linking points in the joints. The number ofredundants for the structure shown in Fig. 2.30 is equal to 8r(N - 1). The timeneeded for the solution of the system (2.103), though, depends much more on thenumber of points r at the joints than on the number of girders in the structure,because the number r is decisive for the bandwidth of the system matrix, whilethe number of girders atfects only the number of equations.

On the basis of the redundants {11 calculated in the described manner, thestresses of aIl girders composing the structure can be determined.

2.8 Folded Plate Analysis of Shear Wall Systemsand Frame Structures

Modem high-rise buildings have to be designed to satisfy static as weIl aseconomic requirements. TaIl buildings are, therefore, often composed of a groupof shear waIls (possibly combined with frame structures) tied together at regularintervals by floor slabs (Fig. 2.32a); structures with coupled or multiple waIlsystems are also often used (Fig. 2.32b). Such composite structures are veryetfective in resisting lateral forces imposed by wind or earthquake. Frame-tube

structures (Fig. 2.32c) seem to be economic for steel and concrete taIl buildings.

74

x

H

Folded Plate Analysis oj Shear Wa/J Systems and Frame Structures

b)

C)

á)-~~.:::.:~·\I

...... '\.:........ . ...•.• ' o. ~'''',''..... .:.:,·.i! : •

0'0· • ":'~~~"':'

:.. ~..)'.... .....

.: .......

L~N::

. . ' ..~Y:.·.:.:: .::.I :. ~.:Io'.': .•..

Fig. 2.32. Examples of structural systems of tan buildings.

75

Folded Plate Theory and its App/icatUJn in the Design oj Plated Structures

Planar systems are also used; these comprise columns and walls with pinnedconnections to horizontal beams (Fig. 2.32d).

Extensive work has been carried out on the static and dynamic analysis ofthese structures and several concepts have been used. The finite element methodappears to be suitable, but special computer programs and digital computers ofreasonable size are stili necessary to obtain the desired solution. It has beenfound that a generalized folded plate theory also offers a very efficient tool forthe structural analysis of these arrangements [2.13].

The folded plate theory was originally developed for the analysis of long andnarrow thin-walled prismatic structures, e.g. roofs and bridges (see Section 2.1).It has also been successfully applied to the analysis of continuous structures,structures with interior (orthogonal or skew) diaphragms, to those with ir­regularly situated supports and so on (Section 2.6).

A tall building obviously differs· considerably from a thin-walled bridgestructure. Besides the vertical character of a tall building, relevant differentboundary conditions are

(a) the top cross-section ofa tall building (x = 0, Fig. 2.32a) is entirely freeand has a rigid diaphragm, Le. the roof slab,

(b) the bottom cross-section (x = H, Fig. 2.32a) is fixed to the base.

H

l

(a) (b)

76

Fig. 2.33. (a) A substitute structural system, (b) a special folded plate element.

Folded Plate Analysis oj Shear Wa/l Systems and Frame Structures

As these boundary conditions do not permit a direct application of the foldedplate theory it is necessary, in accordance with the idea presented in Section 2.4,Figs. 2.24 and 2.25, to create a substitute structure to which it may be applied.This can be done by creating a mirror image of the structure about its founda­tion. Thus a substitute structure ofspan L = 2H (Fig. 2.33a) is formed. Let aforce R be applied in the plane of the mid-span cross-section of the substitutestructure, which balances allloads acting on the substitute structure. As perfectclamping to the bottom of the actual structure (Fig. 2.32) is assumed, thecross-section x = H is rigid in its own plane.

The substitute structural system as a whole is in a state of equilibrium.Assume now that the clamping of the bottom of the actual structure is releasedand that the end cross-sections ofthe substitute structure x = Oand x = L(thediaphragms being rigid in their own planes) are complemented by dummystatically determinate supports (dotted bars in Fig. 2.33a).

Folded plate theory can be applied to such a simply supported substitutestructure of span L. No reactions appear in the dumm'y simple supports. Thestates of stress of the substitute and the actual structures are identical, and thedisplacements of both structures differ only in that there is a deftection of thesubstitute structure at its mid-span (x = LI2 = H), whereas the actualstructure is clamped.

The solution of the substitute structure (Figs. 2.33a and 2.34b) using foldedplate theory is carried out in the following way.

Fig. 2.34. (a) Actual structure, (b) substitute structure, (c) folded plate idealization.

'00 •• 00 ••• :::::' speclal

~ element(C)

replJlar elemtnt

. / ...;-":::::::::::1:'--'"

.(1/

(b)

(Q)

77I

I'.II

Folded Plate Theory and its Application in the Design oJ Plated Structures

The substitute structure (Fig. 2.34b) is divided into elements (Fig. 2.34c)which are interconnected along the vertical ridges. Shear walls are regarded asregular folded plate elements. In order to idealize the structural behaviour of theframework panels (or shear walls with openings), special folded plate elementsare introduced (Figs. 2.33b and 2.34c). Such an element consists of a shear wallor a column - its spine - and a number of connecting half beams spaced atintervals of the storey height h (Fig. 2.33b). For this approach to be successful,the tlexural stiffness of the columns must be considerably greater than thestiffnesses of the connecting beams. This condition is satisfied, for example, inthe case of the multiple shear wall system (Fig. 2.32b).

Cross-sectional functions of connecting half beams are assumed to be distri­buted in a vertical direction. Thus, the distributed cross-sectional area of con­necting half beams is

f= AJh, (2.104)

in which As is the cross-sectional area ofthe connecting beams. Their dislributedmoments of inertia are

(2.105)

corresponding to bending out of the frame plane, and

(2.106)

in which sJs and dJs denote respectively the moments of inertia of the cross­section of the connecting beams resisting bending in the frame plane and out ofthe frame plane, and h is the storey height.

The vertical edges of special elements are situated close to the points wherezero bending moments in the connecting beams are expected.

Each folded plate element (Fig. 2.34c) is rectangular and at its vertical edgesis subjected to the following forces (measured per unit height of the structure,Fig. 2.35a) ~

(a) edge bending moments M,(b) edge transverse shear forces Q,(c) edge vertical shear forces T,(d) edge normal forces P

(compare with Fig. 2.3),which form an edge forces vector

(2.107)

78

Folded Plate Analysis oj Shear Wa/l Systems and Frame Structures

Each joint has four degrees of freedom (Fig. 2.35b): joint rotation 8, trans­verse displacement w perpendicular to the plane of the element, vertical displace­ment u in the plane of the element and transverse displacement v in the planeof the element. The edge displacement vector is

(2.108)

It is evident that relations (2.107)and (2.108)and also the remaining featuresof the solution are fully identical with the general procedure for the classicalfolded plate theory outlined in Section 2.1.

Hence, the relationship between the edge forces and displacements is again(see Eq. (2.1))

{S} = [k] {c5} , (2.109)

where [k] is an 8 x 8 stiffness matrix of the element. Owing to the simplesupports at both boundaries ofthe substitute structure (Fig. 2.33a), the analysisean again be performed using harmonie functions, for any distribution ofapplied load. Also Eqs. (2.2), (2.3) and (2.4) remain valid.

~epulor element

a)

Fig. 2.35. Edge forces and dis­placements of a regular element

and a special element. b)

~~r!t..,;,,, .. ,": ...::.... :.:i

79

Folded Plate Theory and its App/icatum in the Design oj Plated Structures

The eoefficients in the stiffness matrices [dknJ and [SknJ of regular plateelements ean be evaluated using Eqs. (2.20) and (2.32), or according to Eqs.(2.19)* and (2.31)**.

Using the previous assumptions, stiffness matrices for special frame foldedplate elements (Fig. 2.33b) ean be derived. For one-way aetion only, the out-of­plane stiffness is expressed by

4b2, 2b2, 6b, 6b

2b2, 4b2, 6b, 6b

Eid(2.110)[dknJ = [dk J = I 6b, 6b, 12, 12 b3 •

6b,

6b,12,12

ln Eq. (2.110), the stiffness eoefficients are independent of the harmonienumber n. The bending stiffness matrix can therefore be used without ehange forthe eomplete analysis.

Tbe in-plane (membrane) stiffness is given by

c + d,c - d,-e,e

[SknJ = I c - d,

c + d,e,-eIE,-e,

e,g + r,g - r

e,-e,g - r,g + r

L. in whieh12A i ,,21[2

esC =48L 2i + b3A n21[2 's c

g = f/b,

80

(2.111) ,

(2.112a)

(2.112b)

(2.112e)

(2.112d)

(2.112e)

(2. 112f)

Folded Plate Analysis oj Shear Wa// Systems and Frame Structures

These relations, which are derived on the assumption that the clear span of theconnecting beams equals the shear wall axis distances b, can easily be rederivedto achieve greater accuracy for infinitely rigid connecting beams inside the walls.

ln some structural arrangements another form of the special element, insteadof the skeleton-shaped element as in Fig. 2.33b, 2.34c and 2.35, can be moreconvenient to fit the actual struetural performanee. This element (Fig. 2.36)consists of vertieal eolumns (whieh may degenerate as shown in Fig. 2.36c) anda system of eonneeting beams. The stiffness of sueh an element ean again beeasily determined in the way shown in Fig. 2.35 and expressed by formulaesimilar to Eqs. (2.112).

(a)

"'special element

(C)

Fig. 2.36. (a) Actual structure, (b) substitute structure, (c) folded plate idealization.

Having established the substitute folded plate strueture and the elementstiffnesses, the remaining proeedure again follows the clasieal folded plateanalysis 'presented in Section 2.1. Beeause in a folded plate strueture eaehelement is in an inclined position with respeet to the other elements, it isnecessary to transform the edge forces {S} and displaeements {t5}, related to eachelement, into a fixed eoordinate system which is valid for the whole structure.The edge forees in the fixed system are given by

where (Kn] is the stiffness matrix of tRe element, related to the n-th harmonie,in the fixed eoordinate system, and {~n}is the displaeement vector in the samefixed system (in more detail, see Seetion 2.1, Eqs. (2.35) to (2.38)).

81

Folded Plate Theory and its AppJication in the Design oj Plated Structures

When proceeding in accordance with Section 2.1, it is assumed that an edgeof one element is identical with an edge of another element, and that at the jointthe continuity of deformation requires the ridge displacements and rotations ofboth elements to be equal. Thus, the structure stiffness matrix [Kn] can bearrived at by assembling the stiffness matrices [kn] of the elements. Using thestiffness matrix, the amplitudes {gn} of the terms of the series for ridge loadingand the unknown amplitudes {dJ of the n-th terms of the expansions of the ridgedisplacements and rotations can be related by

[Kn] {ón} = {Rn} (compare Eq. (2.45)).

Once the ridge displacements {dn} are known, the amplitudes ofthe edge forces{S'n} are eva1uated using equation (2.45) and the corresponding stresses in theelements are determined. (In regular plate elements, elasticity theory is used(Eqs. 2.20 and 2.32); in special elements, an element (Fig. 2.33b) is solved as astatically determinate frame loaded by known edge forces).

The procedure presented makes it possible to analyse a structure formed asa system of vertical walls with bands of openings, but having a roof slab only.An actual tall building, however, has ftoor slabs as well as vertical walls and aroof slab (Fig. 2.32). In folded plate analysis, ftoor slabs (both rigid and deform­able in their own planes) can be regarded as diaphragms of the substitute foldedplate system. An analysis of folded plates with interior diaphragms was presen­ted in Section 2.6.

The force method is applied. This consists of removing all redundant con­straining effects, so that the primary simply supported folded plate structure isobtained, without any interior restraints but fitted at both boundaries with theroof slabs which are assumed to be rigid in their own plane and ftexible in theperpendicular direction. The redundants are then determined using compatibi­lity equations (see Section 2.6). The primary structure can then be analysed,subject to the loading and the redundant forces being known, to determine thefinal stresses and displacements in the structure.

A wide range of structural systems can be analysed using the approachpresented. For example, a structural system of horizontal beams and verticalwalls with pinned connections (Fig. 2.32d) can be regarded as decreasing theftexural stiffness of connecting beams imin the plane of the element, or eliminat­ing vertical degrees of freedom in the element stiffness by static condensation.

The simple approach presented for the analysis of shear wa1l systems ofarbitrary plan geometry, based on generalized folded plate theory, maintains allthe advantages of the folded plate theory. The method is adequate because itconsiders the structure in its actual form as an assemblage of rectangularelements which together form a real spatial system. The solution does notdistinguish between open and c10sed sections or multi-cell structures, nor be­tween horizontal and verticalloadings.

82

Folded Plate Analysis oj Shear Wa/l Systems and Frame Structures

The approach is valid regardless ofwhich method is used to analyse the singleplate or frame element. Again, only a minor adjustment to existing folded platecomputer programs is necessary, Í.e. the inclusion of stiffness matrices of specialelements representing frame-shaped structural parts.

The folded plate approach to the analysis of tan three-dimensional shear wallsystems can easily be generalized to be capable of incorporating soil-structureinteractive behaviour (the base flexibility). The stiffness method can be used forsuch an analysis.

Built-in rigid base conditions are assumed in the first stage of the stiffnessmethod. The method just presented may be used to evaluate both the stresses inthe structure and the distribution of reactions at the rigid base cross-section dueto given extemalloads (Fig. 2.37a).

Fig. 2.37. Folded plateanalysis of talI buildingsaccounting for the base flexi-

bility.

a)

C)

83

Folded Plate Theory and its Application in the Design oj Plated Structures

ln accordance with the principles of the stiffness method, unit deformationimpulses are applied to the base cross-section of the structure (Fig. 2.37b) andto the flexiblefoundation (Fig. 2.37c), using, for example, the elastic foundationconcept or the elastic half-space concept.

Bearing in mind that the rigid body motion ofthe structure (due to a verticaldisplacement and rotations of the base cross-section about two axes) does notinduce any changes of internal forces, the number of the independent deforma­tion impulses equals the number of degrees of freedom in the vertical directionof the base cross-section, less three. To simplify the calculation, the unit de­formation impulses can be orthogonalized and a set of independent deformationpatterns thus introduced into the analysis (Fig. 2.37b,d).

The behaviour of the structure (the stress distribution and deformations)subjected to the deformation impulses applied at the base cross-section (Fig.2.37b) can be analysed by the folded plate method presented in this Section,based on the introduction of the substitute structure (with a mirror image, Fig.2.33). Orthogonalized sets of vertical forces are applied to the cross-section atthe mid-height of the substitute structure (Fig. 2.37d) as further loading cases.As all the force and moment resultants of such a loading are zero, the substitutestructural system as a whole is in a state of equlibrium, no reactions appear atthe ends of the substitute structure, and so the folded plate solution again fulfilsthe boundary conditions at the free top cross-section.

Although the sum of the verticalloading for the entire substitute structure iszero, it is not self-balancing on eachjoint ofthe folded plate system. The analysismust respect this (see the approach presented in Section 2.1, Fig. 2.9).

If follows from the symmetry of the substitute structure (with regard to itsmid-height) and from the antisynunetry of the loading represented by a set ofvertical forces (Fig. 2.37d) that the stresses as well as the deformations of theactual structure and those of its mirror image are identical (but with oppositesigns). ft is, therefore, evident that the vertical forces necessary for the deforma­tion impulses at the base cross-section of the actual structure equal half themid-height forces causing the same impulses on the substitute structure (Fig.2.37d).

Force factors corresponding to the individual unit deformation impulses onboth the base cross-section of the structure (Fig. 2.37b) and the flexible founda­tion (Fig. 2.37c) together represent the stiffness of the structure foundationagainst the deformation pattern considered. Thus, the stiffness matrix of thesystem can be assembled, taking full advantage of its band nature due to theorthogonalized deformation patterns; the size of the stiffness matrix corres­ponds to the number of deformation impulses.

Using the stiffness matrix, it is possible to write the relationship between thedesired values ofbase deformation impuses and reactiori components at the rigidbase cross-section assumed in the first stage ofthe stiffnessmethod (Fig. 2.37a).

84

Folded Plate Analysis oj Shear Wa/l Systems and Frame Structures

This relationship, representing the equilibrium conditions of the base cross­section, has the form of a system of linear algebraic equations for the unknownvalues of deformation impulses.

The analysis is almost complete when the required values of base deformationimpulses have been determined; it remains only to evaluate the stresses in thestructure due to these impulses (Fig. 2.37d) and to add them to the stresses ofthe state when the structure has a built-in rigid base cross-section (Fig. 2.37a).

An analysis of the influence of a prescribed settlement of the structure basecan be carried out using only the second stage of the method (Fig. 2.37d).

The method can be further generalized to take account of the time-dependentsoil-structure interaction.

The use of the folded plate theory is basically restricted to structures with aconstant cross-section. With high-rise building, cases are sometimes encoun­tered in which some parts of horizontally resisting members (e.g. some parts ofbracing cores) are eliminated in the upper part of the building. These possiblechanges in the cross-section of the stiffening members then necessarily violatethe aforementioned assumption of a constant cross-section. It has been proved,however, that the structural performance of tall buildings depends only on thestiffness characteristics of stiffening elements in the lower part of the structure.As a simple example, we can compare the values of the horizontal deflections oftwo vertical, uniformly loaded cores:

(i) A core with a constant flexure stiffness EI along the whole height (Fig.2.38a),

(ii) a core with its stiffness reduced to 50 %, in the upper half of the coreheight (Fig. 2.38b).

h)f06%

a)

fOO'l. Hi1P

,In

IIIHá

II

: 111-EI/2

I II II,I

I

~

I I€I

~IIHh

I I

I.J-a

I I

I III

Fig. 2.38. Comparison of the top deflections of tall structureswith different stiffnesses.

85

Folded Plate Theory and its AppUcation in the Design oj Plate Structures

It is seen that such a dramatic decrease in stiffness for the half of the coreheight merely results ina minor increase in the top deflection. Therefore, it maybeexpected that the method of analysis of tall buildings presented, based on thefolded plate theory, offers a reliable tool for the structural analysis of varioustypes of structures.

REFERENCES

[2.1] GoLDBERG,J. E. and LEVE, H. L.: Theory of prismatie folded plate structures, Publ. Inst.Assoc. for Bridge and Struet. Eng., No. 87, 17, Ziirieh, 1957.

[2.2] DEFRIES-SKENE,A. and ScORDELlS,A. C.: Direct stjffness solution for folded plates, Journalof the Struetural Division, ASCE, Vol. 90, Aug. 1964.

[2.3] KIUSTEK,Y.: Theory of Box Girders, J. Wiley and Sons, Chiehester, 1979.[2.4] HÁJEK, p.: Orthotropy in folded plate theory, Space Struetures - Elsevier Applied Science

Publishers - I, 1985.

[2.5] KIUSTEK,V.: Tapered box girders of deformable eross-section, Joumal of the StrueturalDivision, ASCE, Vol. 96, Aug. 1970.

[2.6] VJ.ASOV,V. Z.: Thin-wal1ed elastie beams, U.S. Department of Commerce, PST Catalogue428, Washington, 1959.

[2.7] KIUSTEK,Y.: Folded plates with deformable end cross-sections. Proc. Inst. Civ. Eng., Part 2,Paper No. 8609, Sept. 1983.

[2.8] ScORDELlS,A. C.: Analysis of eontinuous box girder bridges, SESM Report 67-25, Univer­sity of Ca1ifornia, Berkeley, Nov. 1967.

[2.9] MEYER,C.: Analysis and design of eurved box girder bridges, SESM Report 70-22, Univer­sity of Califomia, Berkeley, Dec. 1970.

[2.10] LIN, C. S. and SCORDELlS,A. C.: Computer program for bridges on tlexible bents, SESMReport 71-24, University of California, Berkeley, Dec. 1971.

[2.11] ScoRDELlS, A. C.: Analytical solutions for box girder bridges, Preoceedings of Conferenceon modem developments in bridge design and construetion, Cardiff, April1971.

[2.12] KIUSTEK,Y.: Box girders of deformable eross-section - some theory of elasticity solutions,Proc. Inst. Civ. Eng., Paper No. 7317, Oet. 1970.

[2.13] KlUsTEK, Y.: Folded plate approaeh to analysis of shear wall systems and frame structures,Proc. Inst. Civ. Eng., Paper No. 8289, Dec. 197~.

86