Design criteria for diagrid tall buildings: Stiffness versus
strengthGiovanni Maria Montuori*, Elena Mele, Giuseppe Brandonisio
and Antonello De LucaFaculty of Engineering, Department of
Structures for Engineering and Architecture (DIST), University of
NaplesFederico II, Naples, ItalySUMMARYThe procedures and
formulations suggested in literature for the design of diagrid
structures start from theassumptionthat diagonal
sizingprocessisgovernedbythestiffnessrequirements,
asusuallyoccursforother, less efcient, structural types, and that
member strength demand is automatically satised by the crosssection
resulting from the stiffness requirements. However, thanks to the
high rigidity of the diagonalizedfaade, strength requirements can
be of paramount importance and even be the governing design
criterion.Inthispaper,
stiffnessandstrengthdesigncriteriafordiagridstructuresareexaminedandtranslatedinsimpliedformulaeforquickmembersizing.
Theapplicationofthetwoapproachesforthedesignofa100-storey building
model, carried out for different diagrid geometrical patterns,
gives the opportunity
ofdiscussingtherelativeinuenceofstiffnessandstrengthonthedesignoutcomes,
intermsofresultingdiagonal cross sections and steel weight, as well
as on the structural performance. Copyright 2013 JohnWiley &
Sons, Ltd.Received 29 April 2013; Revised 23 July 2013; Accepted 16
October 2013KEYWORDS: diagrid; steel structures; tube structures;
design criteria; stiffness requirements; strengthrequirements1.
INTRODUCTIONStructural congurations best addressing the traditional
requirements of strength and stiffness for tallbuildings are the
ones employing the tube concept, whose efciency is strictly related
to the involvedshear-resisting mechanism, and in fact, the
historical evolution of the tube concept has been marked bythe
attempts of reducing the occurrence of efciency loss due to shear
deformations. In this perspec-tive, diagrid structures, which can
be considered as the latest mutation of tube structures, show
extraor-dinary efciency, related to the adopted geometrical
pattern; thanks to the triangle tessellation of thefaadestructures,
internal axial forcesarelargelyprevalent inthestructural members;
thus, shearlag effects and racking deformations are
minimized.Themainreasonunderlyingthehighefciencyofdiagridstructuresisthereforethegeometricalpattern:
geometry is in fact the key-word for interpreting not only the
structural efciency of diagridsbut also the form versatility and
the aesthetical features.Triangle tessellation indeed allows to
adapt diagrid congurations to whatever curved and/or
non-extrudedform, wrappingaroundalmost anyshape: examples of tall
buildings characterized bycomplex forms and utilizing diagrid as
lateral load resisting system are almost uncountable, going
fromtherst applications, such as the Swiss Re building and the CCTV
Headquarters, to the most recentexamples of the Guanzhou West
Tower, the Capital Gate and the Bow Tower.Further, the geometric
nature of the diagrid strongly connotes the building aesthetics,
transmittinganoptical affect oflightnessandcrystallinitythat
remainsconsistent withinanyspaceit denes*Correspondenceto: Giovanni
MariaMontuori, Department
ofStructuresforEngineeringandArchitecture(DIST),University of
Naples Federico II, Naples, Italy.E-mail:
[email protected] STRUCTURAL DESIGN OF TALL AND
SPECIAL BUILDINGSStruct. Design Tall Spec. Build. 23, 12941314
(2014)Published online 11 November 2013 in Wiley Online Library
(wileyonlinelibrary.com/journal/tal). DOI:
10.1002/tal.1144Copyright 2013 John Wiley & Sons,
Ltd.(Moussavi, 2009). The inherent geometrical exibility of the
diagrid pattern can be perfectly tuned tomatch the varying
structural and/or architectural demands, revealing a wide range of
additional opticalaffects, including latticing, gradation,
differentiation and diagonality. (Moussavi, 2009)With specic
reference to the structural aspects, possible optimization
strategies of the congurationin fact correspond to variations of
the geometrical attributes of the triangular base unit as it
tessellates,with size, scale, angle and/or depth varying along the
building faades. Diagrid with variable angle
hasbeenproposedandstudiedby(Moon, 2008a, 2008b; Zhanget al.,
2012)asmoreefcient designsolutions than uniform-angle congurations
for very slender buildings. Further, an actual example ofdiagrid
with variable density is provided by the impressive CCTV
Headquarters, where the base unitis differently scaled throughout
the building faades, for responding to changes in local
stresses.Despite the large number of applications and proposed
projects, design criteria are not yet consoli-dated as in the case
of more traditional structural types, and also building codes do
not provide explicitguidelinesandprovisionsfor diagridstructures.
Amajor researchcontributionhasbeengivenbyMoon, with a series of
papers starting from 2007 (Moon et al., 2007), where a
stiffness-based method-ology for the preliminary structural design
of diagrids is suggested and applied to different buildingmodels;
adjustmentsoftheformulationarealsogiventoaddressbothuniformandvariableanglediagrids
(Moon, 2008a), as well as for non-prismatic building forms (Moon,
2011).The procedures and formulations suggested by Moon start from
the assumption that diagonal sizingprocess is governed by the
stiffness requirements, as usually occurs for other, less efcient,
structuraltypes, and that member strength demand is automatically
satised by the cross section resulting
fromthestiffnessrequirements. However, inarecent paper
nalizedtotheassessment ofthestructuralbehaviour of diagrid
buildings (Mele et al., 2012), the authors have pointed out that,
thanks to the highrigidity of the diagonalized faade, strength
requirements can be of paramount importance and even
bethegoverningdesigncriterion. Asalsoreportedin(Munro,
2004)withreferencetotheSwissRebuilding,
thesizingofthesteelelementsisgovernedbystrengthcriteriathetotalswaystiffnessof
the diagrid is sufcient to limit the wind sway to 50 mm over the
full 180-m height and providesa very good level of overall dynamic
performance.Therefore, the major question addressed in this paper
could be formulated as follows: to what extentstiffness and
strength criteria affect the design of diagrid structures?Providing
a comprehensive answer to such question is a non-trivial issue.For
thispurpose,
thestiffness-baseddesigncriterionproposedbyMoonisrstlyreviewedandappliedfor
thedesignof a100-storeybuildingmodel, byadoptingdifferent
diagridgeometricalpattern, i.e. different values of the diagonal
angle. Simple formulae for deriving strength demand inthe members
are dened and applied for alternative strength-based design
solutions.The two sets of diagrid structural solutions, the ones
designed according to stiffness
requirementsandtheonesdesignedaccordingtostrengthrequirements,
arethencomparedintermsofresultingdiagonal cross sections and steel
weight. Further, structural analyses of the two sets of diagrid
struc-tures under gravity plus wind load are carried out, allowing
for a complete assessment of the structuralresponse. Discussion in
terms of different response parameters, i.e. top displacement,
interstorey drift,member strength demand to capacity ratio (DCR),
is presented also considering the unit steel weight ofeach single
solution, and design implications are emphasized.2.
DEFINITIONSGEOMETRY AND LOADSInorder toapplythe designmethodologies
providedinthis paper, the diagridpatternmust
bepreliminarilydenedfromthegeometricpoint ofview. Theunit cell
ofthepatternisthediagridmodule, usually extending over multiple
oors, which repeats horizontally along the buildingperimeter and
stacks vertically along elevation. The main geometrical parameters
of the module
andoftheglobalpatternarethefollowing(Figure1):thediagonalangle(),
thediagonallength(Ld),themoduleheight (h),
thenumberofmodulesalongelevation(nm), thenumberofmodulesalongthe
perimeter (nk) and the number of diagonals on each faade (nx and
ny).Thegeometrical attributes of themoduleandof theglobal
patternareinterrelatedparameters,functionofthediagridmoduleasit
appliestoglobal buildingdimensions, i.e. theplandimensionDESIGN
CRITERIA FOR DIAGRID TALL BUILDINGS 1295Copyright 2013 John Wiley
& Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314
(2014)DOI: 10.1002/talalongXandY, respectively, LxandLy,
andthebuildingheight, H. Quitetrivially, thefollowinggeometrical
relationships can be established:nk 2nw nf ; h Hnm; Ld hsen(1)Lx;y
nx;yLd cosx;y2; x;y arctgH nx;y2 Lx;ynm (2)The division of the
building shaft into diagrid modules is also usefully employed for
the structuraldesign and assessment. Since the module usually
extends over multiple oors, loads are transferred tothe module at
every oor level, and load effects vary along the diagonal length.
However, consideringthat a single cross section is adopted for the
diagonal along the global module height, i.e. diagonal sec-tion
only varies from one module to another, the loads utilized for the
design of a specic diagonal arethe ones obtained at the base of the
relevant module.The gravity load (Qm), the shear force (Vm) and the
overturning moment (Mm) for the m-th modulecan be derived as
follows:Qm Xm0i1Qi; Vm Xm0i1Vi; Mm Xm0i1Vizi(3)where i = 1 and i
=m0 correspond, respectively, to the top level of the building and
the base level of them-th module, Qi is the i-th oor gravity load
(unit gravity load multiplied by the total oor area), Vi isthe
shear force due to wind loads at the i-th oor and zi is the
vertical distance between the top of thebuilding and the m0
level.3. STIFFNESS-BASED DESIGNThe rst design procedure examined in
this paper is based on stiffness requirements and has been
orig-inallyproposedbyMoon(2007,2008b).TheprocedureisbasedontheresultsofwideparametricFigure
1. Geometrical parameters of the diagrid module.1296 G. M. MONTUORI
ET AL.Copyright 2013 John Wiley & Sons, Ltd. Struct. Design
Tall Spec. Build. 23, 12941314 (2014)DOI: 10.1002/talanalyses that
emphasized the major role of the geometrical attributes of the
diagrid pattern, primarily ofthediagonal angle, onthestructural
efciency. Inthefollowing, it isbrieyreviewed, andtheformulae for
deriving the diagonal cross sections are
established.Bymakingtheclassicalassumptionthatthebuildingstructureunderlateralloadsbehavesasanideal
cantileveredtube, i.e. neglectingtheshearlageffects,
uniformtensileandcompressiveforcedistributionsariseintheleewardandwindwardfaces,
respectively,asaconsequenceoftheglobaloverturningmoment,
whilethefacesparallel tothewinddirectionaresubjecttoshearforces.
Thelateralstiffnessofthestructure,whichcounteractstheseglobalactions,isgivenbythesumoftwocomponents,
i.e. exural and shear. Moon suggested simplied criteria for
specifying the optimal ratiobetween exural and shear stiffness
components as a function of the building slenderness H/B; once
atarget valueof thebuildingtopdisplacement isset,
preliminarydesignformulaefor derivingthediagonal member size, on
web andange sides, are provided.The limit value of the horizontal
displacement at the top of the building (uH) is a design
parameter,usually expressed as a percentage of the building height
(a typical value is H/500). As mentioned before,it is possible to
consider uH as the sum of contributions due to bending and shear
deformations:uH H H22(4)where and are, respectively, the shear and
the bending deformations. Once the limit value of uH isxed,
e.g.uH=H/500, thedesignvaluesofthedeformationcomponentsinEquation
(4) (appointedas *and
*)arenotunivocallydetermined,sincethesametargetdisplacementcanbeobtainedbydifferent
shares of the bending and shear deformations. However, depending on
the building slender-ness, optimal values of the relative bending
to shear deformation ratio can be established, accordingto Moon
(2007, 2008b).In particular, dening the s factor as the ratio of
the bending to shear deformation at the top of thebuilding, i.e.s
H22 H H2(5)The following empirical formulations for the optimal
value of s, using the diagrid weight as term ofcomparison, are
suggesteds1 HB 2 forHB6 and 60 < < 70 (6)s2 HB 3 forHB5 and
60 < < 70 (7)From which the design values as *and *for the
shear and bending deformations can be derived: 11 s 500(8)DESIGN
CRITERIA FOR DIAGRID TALL BUILDINGS 1297Copyright 2013 John Wiley
& Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314
(2014)DOI: 10.1002/tal 2sH 1 s 500(9)Considering that the required
shear and bending stiffness of the module (KT*and K*) are given
byKT Vmh(10)K Mmh(11)On the basis of purely geometrical
considerations, the cross-sectional areas of the diagonal memberon
the web andange plane, in the m-th module, Ad,m,w and Ad,m,f,
respectively, are obtained:Am;d;w VmLd1 s 5002nwEh cos2(12)Am;d;f
2MmLdH1 s 500nfL2Eh sin2 2s(13)where, in addition to the parameters
already dened, E is the Young modulus of the diagonal
structuralmaterial, andLis the plandimensionof the buildingparallel
tothe consideredwinddirection(Figure 1), which, inthecase of
rectangular oor shape, shouldbeset bothL = LXandL = LY,depending on
the wind direction.3.1. Alternative denition of the parameter
sMaking reference to a prismatic cantilever beam under uniform
distributed load, w, according to
theEulerBernoulliandTimoshenkobeamtheories,
thetransversaldisplacementduetoshear(u)andbending (u) exibilities
are, respectively, given byu w H48 EI(14)u w H22 GAs(15)whereG is
the transversal elasticity moduli of the structuralmaterial,I and
As are, respectively,themoment of inertia and the shear area of the
beam cross section.The factor s can be dened as the ratio of the
above expressions for u and u, i.e.s uu 14GAsH2E I(16)1298 G. M.
MONTUORI ET AL.Copyright 2013 John Wiley & Sons, Ltd. Struct.
Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/talShifting the discussion to the building behaviour, and
assuming that shear force is only carried
bythediagonalsonthewebsidesandoverturningmomentbythediagonalsontheangesides,
itispossible to writeAs 2 nwAd;w cos(17)I
nfAd;fsenL2(18)Substituting in Equation (16),s 12GEnwAd;wcos
H2nfAd;fsenL2(19)Considering that nwAd,w should be equal to nfAd,f,
it is possible to writes 12GEcossenH2L2(20)For the steel material,
= 0.3 and the ratio between G and E is equal to 0.38; therefore,
the s value isnally given bys 0:19tanHL 2(21)It is worth observing
that Equation (21) provides s values as functions of both building
slendernessand diagonal angle, while Equations (6) and (7)
establish a dependence of s on building slendernessonly. In the
following, the values of s calculated according to Equation (21)
are appointed as s3. Fig-ure 2(a, b) shows the comparison among the
values s1, s2 and s3, obtained by means of the relation-ships (6),
(7) and (21), respectively.4. STRENGTH-BASED DESIGNThe second
approach considered in this paper is based on strength
requirements. For this purpose, thesame geometrical denitions and
load assumptions discussed in Section 2 are adopted. Following
thegravityandwindloadpathsandconsideringthepredominant
resistingmechanisminthediagridpattern,
theproceduresuggestssimpliedformulationsforderivingcompressionandtensionaxialforces
in the diagonals; thus, cross-sectionl areas result from member
strength and stability checks.The gravity loads give rise to a
global downward force on the generic diagrid module (Figure
3(a)),identied by the subscripts m and k along the building
elevation and perimeter, respectively. Assumingthat the central
core occupies the 25% of the oor area, the perimeter diagrid shares
the 37.5% of theoor gravity load. The gravity downward load on each
module (Fm,k,G) is given byFm;k;Q 0; 375 Qmnk(22)DESIGN CRITERIA
FOR DIAGRID TALL BUILDINGS 1299Copyright 2013 John Wiley &
Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/talThe diagonals under this gravity downward load are both
in compression(Figure3(a)), with axialforce given byNm;k;Q
Fm;k;Gsen2(23)substitutingNm;k;Q 0; 375 Qmnksen2(24)Horizontal wind
load globally causes overturning moment and shear force. The
overturning moment(Mm) acting at the m-th level gives rise to a
vertical force in each k-th module (Fm,k,M), whose
direction(downward or upward) and intensity depend on the plan
position of the module itself (Figure 3(b)), asexpressed in the
following equation:Fm;k;M MmdkXnki1d2iFigure 2. (a) Parameter s
versus building aspect ratio,xing = 69; (b) parameter s versus
diagonalangle,xing H/B=6.Figure 3. Axial forces in diagonals of the
k-th triangular scheme of the m-th module for: (a) gravityloads,
Qm; (b) overturning moment, Mm; (c) global shear, Vm.1300 G. M.
MONTUORI ET AL.Copyright 2013 John Wiley & Sons, Ltd. Struct.
Design Tall Spec. Build. 23, 12941314 (2014)DOI: 10.1002/talwhere d
is the distance of the module from the centroid of the plan shape
(Figure 3(b)). The diagonalsof the module on the windward building
face are both in compression, while the ones on the leewardface are
both in tension. The axial forces in the diagonals of the generic
module are given byNm;k;M Fm;k;Msen2
MmdkXnki1d2isen2(26)Theglobalshearforceactingatthem-thlevel(Vm)givesrisetoahorizontalforceineachk-thmodule(Fm,k,V),whichintensitydependson
the angleofthemodulewiththedirectionofwind ()(Figure 3(c)):Fm;k;V
Vmcos;kXnki1cos;i(27)Underthishorizontalforce, onediagonal
ofthemoduleisincompression, whiletheotherisintension. The diagonal
axial force is given byNm;k;V Fm;k;Vcos2
Vmcos;kXnki1cos;icos2(28)Under both gravity and wind loads, the
axial force in the diagonals of the generic diagrid modulecan be
calculated as the sum of the three contributions above, i.e.Ndg;i;k
Ndg;k;G Ndg;i;k;M Ndg;i;k;V 0; 375
Qmnksen2MmdkXnki1d2isen2Vmcos;kXnki1cos;icos2(29)Oncetheaxial
forceinthediagonalshasbeencomputed, thecross-sectional
areacanbesizedaccording to strength and stability requirements.5.
DESIGN APPLICATIONSWith the aim of comparing the results of the two
proposed design approaches, an application to a build-ing model has
been dened. Three different diagrid patterns have been considered
(Figure 4(b, c, d) forabuildingmodel withrectangularplan65 40 m,
and100storeys, withatotal height of332.3
m(Figure4(a)).Todenecompletelythediagridgeometries,
thediagonalangle( =64, =69and = 79) and the total number of modules
(i.e. nm64= 12.5, nm69=10 and nm= 8) have been set, whilethe others
geometrical parameters have been calculated using Equation (2)
(Table 1).In each geometrical conguration, the height of the
diagrid modules on the two faades is the
same;themodulewidthisslightlydifferent, i.e.13
monthebroadsideand13.3 montheshortside.Ofcourse,
inthethreegeometrical patterns, theheight ofthemoduleisdifferent
(Table1, Figures5and 6), comprising 8, 10 and 20 storeys,
respectively, for = 64, = 69 and = 79.The framing oor structure is
depicted in Figure 7; the dead loadis 7 kN/m2including
thecontributionsof theoor steel structure, internal partitions
andexternal claddings. Theliveloadcontributionhas beenassumedequal
to4 kN/m2. Thehorizontal
loadduetowindpressurehasbeencalculated(Figure8)
accordingtoASCE-7provisions, (ASCE7-05, 2006) onthebasis
ofwindspeedequal to40 m/s(90 mph)(Table2).DESIGN CRITERIA FOR
DIAGRID TALL BUILDINGS 1301Copyright 2013 John Wiley & Sons,
Ltd. Struct. Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/talThethreediagridsolutionsfor thebuildingmodel
havebeendesignedbothwithstiffnessandstrengthapproaches,
asreportedinthefollowingsubsections. Built-upboxsections,
withweldedplates made of steel S275 (fyk=275 MPa, ftk= 430 MPa),
are adopted for the diagonals.Figure 4. (a) Building model; (b)
diagrid pattern with = 64; (c) diagrid pattern with = 69;
(d)diagrid pattern with =79.Table 1. Geometrical parameters for the
three diagrid patterns. [] 64 69 79nm [] 12.5 10 5h [m] 26.6 33.23
66.5Ld [m] 29.6 35.7 67.7nx [] 10 10 10ny [] 6 6 6nk [] 16 16
16Figure 5. Plan variations in the diagrid module for the three
diagrid patterns.1302 G. M. MONTUORI ET AL.Copyright 2013 John
Wiley & Sons, Ltd. Struct. Design Tall Spec. Build. 23,
12941314 (2014)DOI: 10.1002/talFigure 6. Geometry of triangular
schemes for the three diagrid patterns.Figure 7. Typical framingoor
structure.Figure 8. Horizontal loads due to wind action.DESIGN
CRITERIA FOR DIAGRID TALL BUILDINGS 1303Copyright 2013 John Wiley
& Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314
(2014)DOI: 10.1002/tal5.1. Stiffness
approachTheaspectratioforabuildingwitharectangularplanisfunctionoftheconsideredplandirection.Assuming
the coordinate system reported in Figure 7, for the building model,
the aspect ratios are asfollows: H/BX=5.11 and H/BY=8.31. According
to Equations (6), (7) and (21), two values of the s factorhave been
calculated (Table 3); as already observed, only Equation (21)
proposed in Section 3.1 gives svalues depending on the diagonal
angle.The three diagrid structures are divided into stacking
modules along the building elevation (12.5 modulesfor =64, 10
modules for =69 and 5 modules for =79; see Figure 4), and the
diagonals of each mod-ule have been sized according to the
stiffness criterion using Equations (12) and (13). The resulting
box sec-tions vary from10001000mmto 300300mm, with wall thickness
varying between 100mmand 20mm.In order to nd the most efcient
solution in terms of diagrid structural weight, the design
processhas been iterated varying the parameter s between 1 and 10;
the results, diagrid structures all satisfyingthe top drift limit
of H/500 with different steel material consumption, are shown in
Figure 9 for theTable 2. Loads assumed in the analysis.Gravity load
[kN/m2]Dead 7Live 4Horizontal loadwindWind base shear [MN]X
direction 19.0Y direction34.9Wind overturning moment [MN m]X
direction 3346.2Y direction 5978.1Table 3. Values of the s
factor.Equation sXsYs1(Equation (6)) 2.11 5.31s2(Equation (7)) 3.11
6.31s3(Equation (21))For = 64 2.45 6.47For = 69 1.93 5.10For = 79
0.98 2.58Figure 9. Unit steel weight (diagrid only) as function of
s.1304 G. M. MONTUORI ET AL.Copyright 2013 John Wiley & Sons,
Ltd. Struct. Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/talthree different angles in terms of unit steel weight
(i.e. total diagonal weight divided by the gross oorarea). It
isworthnoticingthat theoptimal valueof
sisstronglyaffectedbythediagonal angle,resulting s64= 3.3, s69=
2.53 and s79= 1.27; however, the curves depicted for = 64 and = 69
showawide range of s values correspondingtosolutions
characterizedbysimilar structural weights,namely, s between 2 and 6
for = 64 and s between 2 and 4 for =69. The curve for = 79 showsa
steeper trend, suggesting similar structural weight, i.e. equally
efcient solutions, only for s between1 and 2.Comparing these values
with the ones provided in Table 3, it can be observed that they are
closer tothe sx values (broad side) than to the sy values (short
side). Figure 10 shows the comparison among theoptimal values
derived from Equations (6), (7), (21) and the optimal values
obtained by means of thedesign iterations; the range of s values
for which the structural solutions show almost no scatters is
alsodepicted in the chart.The unit steel weight of the nal
designsolutions obtainedfor thethreediagridpatterns arecompared in
Figure 12; the solution = 69 is the most advantageous, having a
structural weight ofthesteeldiagridequalto0.51
kN/m2.Thesolutionwith =
79,instead,istheheaviestonewithastructuralweight of 0.75 kN/m2.
These resultsareconsistentwiththe ones obtainedby Moonetal.(2007)
by means of parametric analyses.5.2. Strength
approachThethreediagridpatternshavealsobeendesignedonthebasisofstrengthrequirements,byusingEquation
(29) and assuming S275 steel grade for the diagonal members.The
comparison between the diagonal cross sections obtained according
to stiffness (black bars)
andstrength(greybars)approachesisshowninFigure11;inparticular,
thecharts(a)and(b)refertodiagridwith = 64, for xandyfaade,
respectively; thecharts(c) and(d) refer todiagridwith = 69, for x
and y faade, respectively; the charts (e) and (f) refer to diagrid
with =79 for x andy faade, respectively. The cross-sectional areas
for the diagonal members are equal to the maximumvalues obtained
considering both X and Y wind directions, therefore considering
each diagrid faadeacting both as web and angeFirst of all, it canbe
stated, simplyobservingthe chart scales, that larger sectionareas
arerequiredforthediagonalsonthebroadsidethanontheshort side.
Further, stiffnessrequirementsgenerallygovernthedesigninthelowest
modules, whilestrengthrequireslarger sectionsintheupper modules.
However, a different trend of the governing criterion along
elevation can
beobservedconsideringthebroadandtheshortsideofthebuilding,
aswellasconsideringthethreedifferent diagridpatterns.Figure 10.
Comparison among the values of s obtained as design results and
according to simpliedformulations.DESIGN CRITERIA FOR DIAGRID TALL
BUILDINGS 1305Copyright 2013 John Wiley & Sons, Ltd. Struct.
Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/talMoreindetail, for thebuildingbroadside,
strengthalwaysprevailsover stiffnessat theuppermodules; the point
where the governing demand switches from stiffness to strength is
approximately120 m, 150 m and 200 m (36%, 45% and 60% of H),
respectively, for = 64, 69 and 79.On the building short side,
strength prevails over stiffness throughout the elevation for the
solutions = 64;on thecontrary,for
=79,stiffnessdemandalwaysgovernsthediagonal
sizingalongtheheight.Inthesolutionwith =
69,thecrosssectionsrequiredforstiffnessandstrengtharealmostthe same
along elevation, suggesting that the two design criteria tend to
converge as the diagonal angleapproaches the optimal value, that is
also the most efcient (i.e. the lightest)
solution.Thecomparisonofthestructural weight
reportedinFigure12suggeststhat
thestrengthdesignsolutionforthediagridwith =
64isheavierthanthestiffnesscounterpart,while,conversely,thestiffnessdesignsolutionfor
= 79isheavierthanthestrengthcounterpart; forthestructurewith = 69,
theweight of thetwosolutionsisonlyslightlydifferent (0.50
kN/m2vs0.54 kN/m2), asexpected on the basis of the previous
observation.Generally speaking, on the basis of the above results,
it can be stated that the geometrical congu-ration, i.e. the
diagonal angle, has a strong effect in determining the prevalent
design criterion and theresulting structural weight. Further
stiffness and strength approaches are both necessary and
unavoid-able; they are not separately sufcient for an exhaustive
sizing process of the diagonal members.Figure 11. Diagonal cross
sections: comparison between stiffness (black bars) and strength
approach(grey bars). (a) and (b) =64; (c) and (d) = 69; (e) and (f)
= 79.1306 G. M. MONTUORI ET AL.Copyright 2013 John Wiley &
Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/tal6. STRUCTURAL ANALYSES AND PERFORMANCE ASSESSMENTThe
structural diagrid solutions considered for the building model, by
varying the diagonal angle andthecrosssections,
asobtainedwiththestiffnessandstrengthapproaches,
areanalysedusingniteelement method numerical models (Figure 13).
Factored gravity and wind loads previously specied(Table2)
havebeenappliedtothemodels;
thewindloadhasbeenappliedbothalongXandYdirections of the building
plan.The results of the numerical analyses are expressed in terms
of some major response parameters, whichjointly allow for a
complete assessment of the structural performance, i.e. horizontal
displacements, u,interstorey drifts, dh, DCR in the diagonal
members; the latter is dened as the tension/compression axialforce
normalized to the yield/buckling capacity of the relevant
member.The results are shown in Figures 14 (for = 64), 15 (for =
69) and 16 (for =79); in each gure,the charts appointed by the
letters (a) and (b) refer to the results in terms of horizontal
displacements(a) (b) (c)Figure 13. FEM models. (a) =64; (b) = 69;
(c) = 79.Figure 12. Unit steel weight (diagrid only) for the three
diagonal anglesstiffness (black bars) versusstrength (grey bars)
approach.DESIGN CRITERIA FOR DIAGRID TALL BUILDINGS 1307Copyright
2013 John Wiley & Sons, Ltd. Struct. Design Tall Spec. Build.
23, 12941314 (2014)DOI: 10.1002/talfor the stiffness and strength
design solutions, respectively; the charts appointed by the letters
(c) and(d)refertotheresultsintermsofinterstoreydrifts,
forthestiffnessandstrengthdesignsolutions,respectively;
thechartsappointedbytheletters(e)and(f)refertotheresultsintermsofdiagonalDCRforthestiffnessandstrengthdesignsolutions,respectively.Ineachchart,boththeresultsforwind
applied along X and Y directions of the building plan are
provided.6.1. Diagrid structural solutionsstiffness designThe rst
consideration to be made by observing all charts of the horizontal
displacements concerns theshapeofthedeformedconguration,
clearlyresemblinganalmost purelyexural behaviour, thusconrming the
great efcacy of diagrid structures in reducing the contribution of
racking
deformation.Thetophorizontaldisplacement(Figures14(a),15(a),16(a))isalwayslargerforwindloadinYdirection
(normal to the broad face) than in X direction. For all three
geometrical congurations, thebuilding top displacement is very
close to the target value H/500 (i.e. 0.67 m), being equal to 0.63
mfor =64, 0.62 m for = 69 and 0.62 m for = 79.An unsatisfactory
performance can be observed in terms of interstorey drift (Figures
14(c), 15(c), 16(c)) and DCR of diagonals (Figures 14(e), 15(e),
16(e), 17(a)).(a) (b)(c) (d)(e) (f)Figure 14. = 64: (a) and (b)
lateral displacements; (c) and (d) interstorey drift; (e) and (f)
DCR indiagonals.1308 G. M. MONTUORI ET AL.Copyright 2013 John Wiley
& Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314
(2014)DOI: 10.1002/talConcerning interstorey drifts, none of the
structural solutions meets the assumed limit dh,lim= 0.5%;in
particular,the maximumvaluesofdhincreasewiththe diagonal angle,
being equalto0.62% for = 64, 0.77%for =69and0.85%for =79.
Thiseffectmainlyoccursatupperlevels,
wherediagonalsmembersaremoreexibleandslenderthan
inlowermodules;further,itismoreevidentthat the steeper is the
diagonal angle, i.e. the longer is the diagonal member (Ld= 29.6
mfor = 64, Ld= 35.7 mfor = 69, Ld= 67.7for = 79) andthelarger
isthenumber of
oorsinamodule.Thestrengthcheckfordiagonalssizedaccordingtothestiffnessprocedurehasbeencarriedoutconsideringtheeffectsof
bothgravityandwind loads;theresultsareprovidedin terms ofDCR
inFigures 14(e) for = 64, 15(e) for = 69, 16(e) for = 79 and in
Figure 17(a) for a global compar-ison. It can be observed that in
the diagrid solution with =64 several members, almost one fourth
ofthe total number (i.e. 824, 26% of elements), have DCR larger
than 1; most of them are concentrated atthe upper modules and
equally distributed between X and Y sides. Also for = 69, several
membershave DCR larger than 1 (23% of elements), but they are
mainly located on the broad side, in the upperhalf. On the
contrary, only few diagonals (i.e. 8, 0.3 % of elements) are
overstressed for = 79. For = 64 and 69, the largest percentage of
diagonals (around 75%) has DCR between 0.75 and 1, with agood
exploitation of the cross-sectional strength capacity.(a)
(b)(d)(f)(c)(e)Figure 15. = 69: (a) and (b) lateral displacements;
(c) and (d) interstorey drift; (e) and (f) DCR indiagonals.DESIGN
CRITERIA FOR DIAGRID TALL BUILDINGS 1309Copyright 2013 John Wiley
& Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314
(2014)DOI: 10.1002/talRecognizing that the performance assessment
in terms of DCR is strictly related to the steel materialused in
the stiffness-based design, the member strength/stability checks
have been also carried out byadopting a higher steel strength,
namely fy=355 MPa, which corresponds to the European steel S355(a)
(b)(c) (d)(e) (f)Figure 16. = 79: (a) and (b) lateral
displacements; (c) and (d) interstorey drift; (e) and (f) DCR
indiagonals.(a) (b)Figure 17. Results of FEM analyses. DCR in
diagonals, comparison between (a) stiffness and (b)strength
design.1310 G. M. MONTUORI ET AL.Copyright 2013 John Wiley &
Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/taland approximately to the US steel Gr. 50. However, a
large number of diagonals still have DCR largerthan 1, also in
these alternative solutions.6.2. Diagrid structural
solutionsstrength designThehorizontal
displacementsofthediagridstructuresdesignedaccordingtothestrengthcriterion(Figures
14(b), 15(b), 16(b)) are smaller thanthe stiffness
designcounterparts for the solutions = 64and = 69; inparticular,
thetopdisplacement is, respectively, 0.51 mand0.57 mversus0.63 m
and 0.62 m in the analogous stiffness design
solutions.Onthecontrary,
thestrength-baseddesignforthediagridpatterncharacterizedby =
79givesrise toamore exiblestructure
thanthestiffness-baseddesignandexhibits atopdisplacement(H,79=0.80
m)that exceedsthedesignlimit H/500(equal to0.67 m).As already
observed for the stiffness-based solutions, also the structures
designed according to thestrength approach do not respect the
interstorey drift limitation (Figures 14(d), 15(d) and 16(d)),
withmaximum interstorey drift values increasing with the diagonal
angle (dH,64= 0.52%; dH,69= 0.97%;dH,79= 1.99%).The outcomes of the
strength checks (Figures 14(f), 15(f), 16(f) and 17(b)) show no
diagonals, oralmost nodiagonals, withDCRlarger than1,
namely0elements for = 64, 17elements (i.e.0.53%) for = 69, 8
elements (i.e. 0.3%) for =79.7. DISCUSSION OF THE RESULTS AND
DESIGN IMPLICATIONSTheresultsofthestructural analysesshowthat
thetwoproposedprocedurearebothreliable,
butnoneofthemcanbeusedwithouttheotherone. Infact,
withreferencetothespeciccasestudy,a 100-storeybuilding with
rectangular plan and maximumslenderness H/B= 8.31, it is not
possibletopredict ina preliminaryphase, whichwill be the
predominant approach, namely, if eitherglobal
stiffnessdemandormember strengthdemandwill governthedesign.In
particular, the results show that in structures with low values of
the diagonal angle (i.e. = 64),thestrengthdesignismorestringent
anddrivestolarger diagonal
sectionsthanstiffnessdesign,throughouttheshortsideofthebuildingandalongtwothirdofthebroadsideelevation;whiletheoppositeoccursinthecaseofsteeperdiagonal
angles,
wherethestiffnessthoroughlygovernsthedesignofdiagonalsontheshort
side. Inthesolutionwith = 69, thecrosssectionsrequiredforstiffness
and strength are almost the same along elevation, suggesting that
the two design criteria tendto converge as the diagonal angle
approaches the optimal value, that is also the most efcient (i.e.
thelightest)
solution.Itshouldbeunderlinedthattheseresultshavebeenobtainedforasinglecasestudy;therefore,awider
range of building characteristics, wind loads and steel properties
is currently being investigatedby the authors in order to have a
complete and denitive assessment of the problem.It
isworthnoticingthatanoverviewofthestructural behaviourofsomereal
diagridstructures,namely, the Swiss Re Tower, the Hearst Tower and
the Guangzhou West Tower (Mele et al., 2012),has given results that
are well aligned with the ones obtained in this paper: in
particular, the analysesof therst two diagrid structures (Swiss Re
and Hearst Towers) under horizontal plus vertical loadshave
shownhighvaluesofdiagonalDCRsand buildingtop displacement closeto
the designvalue(H/500). Theseresultstestifythat,
thankstotheinherent highrigidityofthediagonalizedfaade,the sizing
of the steel member is mainly governed by strength
criteria.Therefore, on the basis of the above considerations, it
could be guessed that the design of diagridbuildings in the range
of 40 storeys (like Swiss Re and Hearst Towers) is mainly governed
by strength,while for taller buildings, like the one examined in
this paper, both strength and stiffness criteria
shouldbeconsidered, andthepredominanceofoneovertheother
isstrictlyrelatedtothechoiceofthediagonal angle. Quite trivially,
the outcomes of the two design criteria tend to converge in the
optimalgeometrical pattern that allows for the full exploitation of
employed structural material and gives riseto the most efcient
(i.e. the lightest) solution.DESIGN CRITERIA FOR DIAGRID TALL
BUILDINGS 1311Copyright 2013 John Wiley & Sons, Ltd. Struct.
Design Tall Spec. Build. 23, 12941314 (2014)DOI: 10.1002/talIt
should be also underlined that all diagrid structures considered in
this paper showunsatisfactoryperformanceinterms of
interstoreydrift. However, this is aproblemarisinginall
structuretypescharacterizedbyaprimarybracingsystememployingmega-diagonals,
whichspanovermultipleoors,
asinbracedtubesandexoskeletonsystems(Abdelrazaqetal., 1993).Infact,
inthediagridmoduleas well as inthemega-diagonal,
concentratedlateral loads areapplied along the diagonal length, at
the locations where intermediate oors intersect thediagonal member.
Therefore, whiletheoverall lateral stiffnessof
thebuildingstructure, thankstothetriangleconguration,
strictlydependsontheaxial stiffnessofthediagonal members,
onthecontrary,thelateralstiffnesswithinthemodulelengthonlyreliesontheexuralstiffnessofthediagonalsthatcouldnotbeadequate(Nair,
1988). Ofcourse, themagnitudeofthisproblemincrease with the
diagonal angle, or better, with the number of oor comprised in a
singlemodule.For small values of (i.e. few intermediate oors within
each module), a reasonable increase of thediagonal area simply
allows to reduce this effect and to satisfy the interstorey drift
check; however, asthe angle increases, the diagonal itself is no
longer sufcient, and a secondary bracing system shouldbe sized for
this specic purpose. At present, the authors are developing further
studies for addressingthis specic problem, with the aim of
providing design guidelines for a quick, geometry-based assess-ment
of theneedfor secondarybracingsysteminbuildingemployingmultiple-oor
diagrids asprimary lateral load resisting system.8. CONCLUSIVE
REMARKSIn this paper, stiffness and strength design criteria for
diagrid structures are examined and translated insimplied formulae
for quick member sizing.Theapplication
ofthetwoapproachesforthedesignofa100-storeybuildingmodel,carriedoutfordifferent
diagridgeometrical patterns,
givestheopportunityofdiscussingtherelativeinuenceofstiffnessandstrengthinthedesignprocessandonthedesignoutcomes,
expressedintermsofresultingdiagonalcrosssectionsandsteelweight.Havingrecognized
themajorroleofgeometricalpatternattributes, criteriafor
selectingoptimal solutions interms of diagonal
angleandshareofbendingtoshear exibilityareidentied.Theanalytical
assessment of thestructural solutions under designloads allows for
statingthefollowing observations and remarks.Ingeneral,
stiffnessandstrengthapproachesarebothnecessaryandunavoidable;
theyarenotseparatelysufcientforan
exhaustivesizingprocessofthediagonal
members.Takingintoaccountthedesignvariables, i.e. thediagonal
angleandthebendingtoshear exibilityratio, evenfor asingle, specic
case study like the one here examined (100-storey building,
rectangular plan,maximumslenderness H/B= 8.31), it is not possible
to predict in advance, which will be
thepredominantapproach,namely,ifeitherglobalstiffnessdemandormemberstrengthdemandwillgovernthedesign.As
a guideline, it has been observed that in structures with lower
values of the diagonal angle, thestrength design is more stringent,
and resulting diagonal members are larger than according to
stiffnessdesign,whiletheoppositeoccursinthecaseofsteeperdiagonalangles,wherethestiffnessmainlygoverns
the
design.Itshouldbeunderlinedthattheseresultshavebeenobtainedforasinglecasestudy;therefore,awider
range of building characteristics, wind loads and steel properties
is currently being investigatedby the authors in order to have a
complete and denitive assessment of the problem.Unacceptable
performance in terms of interstorey drift has been observed for all
structural solutionsdesigned in this study; in the case of lower
values of the diagonal angle, i.e. in the case of small numberof
multiple oors spannedbydiagonals, aniterative designprocess
canquicklyconverge toasatisfactory solution, while in the case of
steeper angles, i.e. numerousoors comprised in a diagridmodule, the
need for a specic secondary bracing system arises. The authors are
currently developingfurther studies for addressing this specic
problem, with the aim of providing quick, geometry-based,design
procedures.1312 G. M. MONTUORI ET AL.Copyright 2013 John Wiley
& Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314
(2014)DOI: 10.1002/talNOMENCLATUREAm,d,f, area of diagonal on
theange faadeAm,d,w, area of diagonal on the web faaded, distance
of the module from the centroid of the plan shapeE and G, Young and
transversal modules of the structural material of diagonalFm,k,M,
vertical force on the k-th triangular scheme of the m-th module for
overturning momentFm,k,Q, vertical force on the k-th triangular
scheme of the m-th module for gravity loadFm,k,V, horizontal force
on the k-th triangular scheme of the m-th module for global shearH,
building heighth, height of the moduleI and As, moment of inertia
and shear resistant area of the equivalent cantilever cross
sectionK, bending stiffness of structureKT, shear stiffness of
structureLd, diagonal lengthLx, Ly, plan dimension along the X and
Y directionsMm, overturning moment at the basis of the m modulenf,
number of diagonals on the ange faadenk, number of triangular
scheme for eachoornm, number of the considered module starting the
numbering from the top of the buildingNm,k,M, axial load in the
diagonal of the k-th triangular scheme of the m-th module due
tooverturning momentNm,k,Q, axial load in the diagonal of the k-th
triangular scheme of the m-th module due togravity loadNm,k,V,
axial load in the diagonal of the k-th triangular scheme of the
m-th module due toglobal shearnw, number of diagonals on the web
faadenx, ny, number of diagonals on x and y facesQm, gravity load
at the base of the m modules, ratio between the bending and shear
deformation at the top of the buildinguH, horizontal displacement
at the top of the buildingu, displacement due to shear
deformationu, displacement due bending deformationVm, global shear
at the basis of the m modulew, costant distributed horizontal
actionzi, vertical distance between the top of the building and the
base of the m-th module., angle of the triangular scheme with the
direction of wind, shear deformation, angle of diagonal, bending
deformationREFERENCESAbdelrazaq A, Baker W, Hajjar JF, Sinn W.
1993. Column buckling considerations in high-rise buildings with
mega-bracing, inIs your structure suitably braced? Proceedings of
the Structural Stability Research Council Annual Technical Session
andMetting, Milwaukee, Wisconsin, April 67, 1993, SSRC: Bethlehem,
Pennsylvania; 155169.ASCE 705. 2006. Minimum design loads for
buildings and others structures. American Society of Civil
Engineers.Mele E, TorenoM, BrandonisioG, De Luca A. 2012.
Diagridstructures for tall buildings: case studies
anddesignconsiderations. TheStructural Designof Tall andSpecial
Buildingsdoi:10.1002/tal1029.MoonK-S,ConnorJJ,FernandezJE.2007.Diagridstructuralsystemfortallbuildings:
characteristicsandmethodologyforpreliminary design. The Structural
Design of Tall and Special Buildings 16(2):
205230.MoonK-S.2008a.Sustainablestructuralengineeringstrategiesfortallbuildings.TheStructuralDesignofTallandSpecialBuildings
17(5): 895914.Moon K-S. 2008b. Material-saving design strategies
for tall buildings structures. In CTBUH 8thWorld Congress, Dubai.
March35, 2008.DESIGN CRITERIA FOR DIAGRID TALL BUILDINGS
1313Copyright 2013 John Wiley & Sons, Ltd. Struct. Design Tall
Spec. Build. 23, 12941314 (2014)DOI: 10.1002/talMoon K-S. 2011.
Diagrid structures for complex-shaped tall buildings. Procedia
Engineering, Volume 12, 2011; 13431350.Moussavi F. 2009. The
function of form. Actar and the Harvard University Graduate School
of Design, 2009.Munro D. 2004. Swiss Res Building, London. Nyther
om stlbyggnad 3: 3643.Nair RS. 1988. Simple solutions to stability
problems in the design ofce. In: 1988 National Steel Construction
Conference; 38.ZhangC, ZhaoF, LiuY. 2012.
Diagridtubestructurescomposedofstraight
diagonalswithgraduallyvaryingangles. TheStructural Designof Tall
andSpecial Buildings21: 283295(2012).AUTHORS BIOGRAPHIESGiovanni
Maria Montuori graduated in Constructions Engineering with
evaluation of 110/110 andhonors exposing a thesis entitled Domini
di interazione di isolatori elastomerici at Naples
UniversityFederico II. Currently, he is in second year at the
School of Philosophy in Construction Engineering(XXVIIcycle)at
thesameuniversity.
Hecontinueshisresearchactivityonanalysisanddesignof steel
structures with conventional and nonconventional architecture. He
participated in nationaland international conferences presenting
papers on these themes.Elena Mele is a full professor of Structural
Engineering (ICAR/09) since December 2010 at the Schoolof
Engineering of the University of Naples Federico II. The research
activity, as testied by more than170 publications,mainly deals with
seismicbehaviour and design of structures, seismic
isolationofstructures, steelandaluminumstructures(fatigue,
stability, seismicproblems), experimentaltestingof beam-to-column
steel connections, modelling and analysis of historic masonry
buildings and seismicretrot of historic buildings.Giuseppe
Brandonisio is an assistant professor at the Department of
Structures for Engineering andArchitecture (DIST) in the University
of Naples Federico II. His research activities deal with
seismicbehaviour and design of steel structures, monumental masonry
constructions and base isolation system.In such elds, he published
more than 60 scientic papers on international and national journals
and onconferences proceedings.Antonello De Luca is a full professor
of Structural Engineering (Group ICAR/09) since 1991. He
authoredmore than 200 scientic papers, with particular attention to
seismic engineering, masonry structures andmetallic structures. He
is referee of different international journals (among which,
Earthquake Engineeringand Structural Dynamics, Engineering
Structures, Journal of Earthquake Engineering, International
Journalof Architectural Heritage, etc.). In the eld of seismic
behaviour of masonry structures, he has recentlypublished on
refereed journals with impact factor; from Scopus, his h-factor is
8. He has coordinatedimportant research projects: Products and
technologies for reducing seismic effects on
constructions,Consorzio COSMES; Seismic protection of cultural
heritage: development and application
proceduresofspecicallydesignedinnovativesystemsProgrammaNazionaledi
RicercaeFormazioneperilsettore dei Beni Culturali e Ambientali,
Progetto Parnaso (1998); Programma di Ricerca di InteresseNazionale
(2000/2001) on: Seismic retrot of monumental buildings through base
isolation and newmaterials; Programma di Ricerca di Interesse
Nazionale (1997/1999) on: Seismic
protectionofexistingandnewbuildingsthroughinnovativesystems(EUR388.126).
Hehasco-organizedandco-edited with Paolo Spinelli the volumes
Wondermasonry 2, 3 and 4 of the Workshop on
MasonrystructuresheldinLaccoAmenoinOctober2007andOctober2009andinFlorenceinNovember2011.
He is the director of second level Master Design of Steel
Structures now in its fourth edition.He is also the president of
CTA Committees and organized the XXIII CTA Conference held in
LaccoAmeno in October 2011. He is also editor of the proceedings
volume.1314 G. M. MONTUORI ET AL.Copyright 2013 John Wiley &
Sons, Ltd. Struct. Design Tall Spec. Build. 23, 12941314 (2014)DOI:
10.1002/tal
