Analysiss of mason
nry vaultss as a topo
ology optimization problem Bruggi, Maatteo1; Talierccio, Alberto2
ABSTRACT oach is propo
osed to analyyze 3D maso
onry vaults, assuming maasonry to be
ehave as a An innovvative appro
linear elastic no‐tension materia
al. Masonry is replaced by a suitable equivalentt orthotropic material ng elastic pro
operties andd negligible stiffness in any directioon along whiich tensile with spaatially varyin
stresses must be preevented. An energy‐baseed algorithm is implemen
nted to definne the distrib
bution and he equivalen
nt material ffor a given lo
oad, minimizzing the poteential energgy so as to the orientation of th
achieve a purely compressive c
state of sttress. The algorithm a
iss embeddedd within a numerical procedure that perfo
orms a non‐incremental analysis und
der given loa
ads. The colllapse load of masonry structuraal elementss can also be predicteed running a sequence
e of indepeendent analyses. The capabilitties of the ap
pproach in prredicting thee crack patte
ern in typical masonry vauults are also shown. Keyword
ds: masonry vvaults, linearr elasticity, nno‐tension m
material, FEM
M, topology ooptimization. 1. INTRO
ODUCTION The collapse of maasonry structtures, especcially archess and vaults, usually occcurs far beyond first cracking. Accordingly, non‐linear analyses a re often pre
eferred to co
onventional approaches based on the classsic theory of o elasticity when w
the saafety of this type of stru
uctures has to be assesssed. Limit analysis is widely adopted a
to evaluate th e bearing capacity c
of ductile strucctures. Assu
uming the ng “tensile compresssive strengtth of masonrry to be infinnite, neglectting its tensile strength aand assumin
strains” to be unbou
unded, limit analysis can be applied tto evaluate tthe collapse load and the relevant failure m
mechanism(ss) also for masonry m
struuctures. Thiss approach has h been exxtensively fo
ollowed to investigaate the strucctural behaviour of brickw
work and sto
onework, in particular arrches and vaults [1]. In principlee, a full non
n‐linear analyysis is able to follow th
he complete loading proocess, from the initial stress‐free state to the highly cracked statte precursorry of collapsse. The reliaability of the
e material near analysiss of existing masonry strructures is models aavailable so far is a crucial issue wheen the nonlin
carried o
out, see e.g. [2]. Among the approacches that are
e currently e
employed too analyse ma
asonry‐like solids, the no‐tensio
on model alllows the sttructural beh
haviour to be b evaluatedd assuming the stress tensor to
o be negativve semi‐defin
nite and to ddepend linearly upon the
e elastic partt of the strain, see e.g. [3]. The no‐tension approach is of major im
mportance since it allows the structuural assessm
ment to be bility limit staate and at in
ncipient colla
apse, recoverring conventtional load performed both at the serviceab
multiplieers computeed through limit analyssis. Notwithsstanding the
e apparent simplicity of o the no‐
tension model, the need to tre
eat discontinnuities in the
e stress and displacemeent fields givves rise to pproaches, several numerical isssues. Well‐kknown difficculties arise when dealing with incrremental ap
1
Departm
ment of Civil a
and Environmental Engineeering. Politecnico di Milano (Italy). [email protected] Departm
ment of Civil a
and Environmental Engineeering. Politecnico di Milano (Italy). alberto
to.taliercio@p
polimi.it (Correspo
onding authorr)
2
91
Analysis of masonry vaults as a topology optimization problem
Third International Conference on Mechanical Models in Structural Engineering
University of Seville. 24-26 june 2015.
whereas robust energy‐based procedures can be alternatively developed assuming no‐tension bodies to be hyper‐elastic [4]. This contribution extends the formulation originally presented in [5] for two‐dimensional problems to three‐dimensional ones. The real no‐tension material is replaced by an equivalent orthotropic material, exhibiting negligible stiffness in any direction along which a tensile principal stress must be prevented in the body. The elasticities of the equivalent material along its symmetry axes are reduced with respect to those of the real material using a penalization law typical of Topology Optimization [6]. The equilibrium of the no‐tension body is sought by minimizing the strain energy with respect to the distribution and the orientation of the orthotropic material. A couple of case studies are presented to assess the proposed approach in the simulation of the structural response of three‐dimensional no‐tension structures. The collapse load multiplier of an arch subjected to an increasing horizontal force is computed and compared with the results given by limit analysis. The structural behaviour of a groin vault under dead loads is also investigated, comparing a conventional linear elastic analysis with the results of the proposed non‐linear approach. 2. PROBLEM FORMULATION
2.1. Equivalent orthotropic material Consider a 3D solid, consisting of an isotropic linear elastic no‐tension material, occupying a domain . The position of any point    is defined by a triplet of orthogonal Cartesian (global) coordinates, z1, z2, z3. Let , =I, II, III, be the eigenvalues of the stress tensor  computed at , with I ≤ II ≤ III. According to the sign of the principal stresses, the material behaviour at  is different. If III < 0, the material behaves like a conventional isotropic material. If one or two of the principal stresses are positive, the material behaves like an orthotropic material: “cracking strains” c  0 arise along the tensile isostatic line(s), whereas the material behaves elastically along the direction(s) of the principal compressive stress(es). Finally, if I ≥ 0, the material behaves like a “void phase”, allowing for any positive semidefinite “cracking strain”. A suitable equivalent orthotropic material can be defined to match the behaviour of the real no‐
tension material according to the sign of the principal stresses. Let ̃ , i=1,2,3, be the symmetry axes of the orthotropic material, locally coinciding with the principal stress directions z, =I, II, III, at any point of the real solid. The stress‐strain law for the orthotropic material at any point  can be written as , where, using the notation proposed in [7]: ̃
̃
̃
√2 ̃
√2 ̃
√2 ̃
, (1) √2
√2
√2
and 92
Matteo Bruggi and Alberto Taliercio
1/
 /
 /
 /
1/
 /
0
0
0
0
0
0
 /
 /
0
0
0
1/
0
0
1/
0
0
0
0
0
0
0
1/
0
0
0
0
0
0
1/
. (2) In eq. (2), , i=1,2,3, is the Young’s modulus of the equivalent material along the symmetry axis ̃ , i,j=1,2,3, is the shear modulus in the symmetry plane ̃ , ̃ and  , i,j=1,2,3, is the Poisson’s ratio along ̃ under uniaxial tension along ̃ . The equalities  /
=  /
,  /
=  /
and  / =  /
hold. The orientation of the maximum principal stress (hence, of the material symmetry axes) to the global reference system is locally defined by a triplet of Euler angles (1, 2, 3) ‐ see Figure 1. zIII 2 z3
zII
3 zI
1
z1
z2
Figure 1. Euler’s angles defining the orientation of the principal stresses to the global reference system The elastic properties of the equivalent orthotropic material along its symmetry axes can be related to those of the isotropic no‐tension material through a generalization of the so‐called SIMP material model (see e.g. [6]), such that: , , 

, 
,
, 
, 
, 
, 
, 
, (3) , 
, where E and are the Young’s modulus and the Poisson’s ratio of the isotropic material, respectively, i, i=1,2,3, are nondimensional variables ranging between min(> 0) and 1, which can be interpreted as “normalized material densities” along ̃ , ̃ and ̃ , and p is a penalization parameter (usually taken equal to 3). The normalized densities are given a strictly positive lower bound, min, to avoid any singularity in the stiffness matrix of the body, K, when a finite element discretization is adopted. The interpolation in Eq. (3) is conceived so as to provide vanishing stiffness in any direction along which a variable attains its minimum value, and basically matches the anisotropic damage law proposed in [8]. Finally, denoting by  and  the arrays of the stress and strain components in the global Cartesian reference system Oz1z2z3, the stress‐strain law in this system can be written as  = D(1, 2, 3; 1, 2, 3) (4) 93
Analysis of masonry vaults as a topology optimization problem
Third International Conference on Mechanical Models in Structural Engineering
University of Seville. 24-26 june 2015.
with (5) In Eq. (5), ,
and is a transformation matrix which can be written as ,
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
√2
(6) with (7) being ci = cosi, si = sini, i = 1,2,3 (see also [7]). 2.2. Energy‐based analysis of no‐tension 3D solids The equilibrium of any linear elastic no‐tension solid can be solved distributing the equivalent orthotropic material defined in Sec. 2.1 over the body, according to the sign of the principal stresses ‐ see also [5]. In view of a displacement–based numerical solution, the continuous formulation of the problem can be stated as follows: min
,
s. t.

,
,
| ̃
,
,
|
,
,


,
;
,

,
 ,

, ̃
|
(8) , ̃
0,
,
,
0,
0
1
In the above equation, the objective function is the elastic strain energy computed through the displacement field u over the 3D domain , whereas the minimization unknowns are the fields of the “normalized material densities” i, i=1,2,3. The boundary of the domain,  = t  u, usually consists of two different parts: the former is subjected to given tractions t0, whereas the latter undergoes prescribed displacements u0. Eq. (8.2) requires that the displacement field fulfils the elastic equilibrium in  and along , whereas Eq. (8.3) prescribes alignment of the symmetry axes of the 94
Matteo Bruggi and Alberto Taliercio
equivalent orthotropic material to the principal stress directions. Finally, Eq. (8.4) requires the normalized densities to define a compression‐only stress state all over the domain. Discretizing the body by a mesh of N constant strain tetrahedral finite elements, the proposed formulation reads as follows: min
1
,
1
2
2
,
3
,
,
3
1
,
2
,
3
1
1
,
3
|
2
,
3
,
1
,
2
(9) 1
1
,
2
,
0
,
1
,
0,
,
2
,
3
1,
1…
The objective function of the above optimization problem is the strain energy, computed over the N elements of the mesh from the stiffness matrices, Ke, and the arrays of the nodal displacements, Ue, of each element. f denotes the array of the equivalent nodal loads. The three sets of element unknowns x1e, x2e, x3e (resp. t1e, t2e, t3e) correspond to the material densities along the symmetry axes (resp. to the Euler’s angles defining the orientation) of the equivalent orthotropic material in any finite element e. The above minimization problem is solved through mathematical programming [9]. Details on the numerical implementation of the problem in Eq. (9) can be found in [5]. By repeatedly calling the minimization algorithm for different values of the loads, the collapse load of the structural element can also be estimated as the value at which the slope of the curve relating the load multiplier to the displacement of a control point becomes lower than a prescribed tolerance. 3. NUMERICAL APPLICATIONS
3.1. Semi‐circular arch A semi‐circular arch is first considered (Figure 2). Under plane strain conditions, or with suitable transversal constraints, the arch can be representative of any segment of a barrel vault. The arch has an internal radius ri=0.4 m, whereas the external radius re=0.5 m. A three‐dimensional model is analysed through a tetrahedral‐based discretization made of 8329 elements, assuming the arch to have a thickness of 0.1m. The elastic properties of the material are E = 10,000 MPa and ν = 0.1. A radial pressure of 0.1 MPa acts along the external surface of the arch; a horizontal live load λF, with F = 1 kN and increasing with the multiplier λ, is applied at approximately half of the height of the arch. This structure was originally investigated in [10] to compare the value of the collapse multiplier computed through a no‐tension complementary energy formulation to the result achieved by a conventional limit analysis suggesting a four‐hinges collapse mechanism and an ultimate load multiplier λc = 0.272. The solid line in Figure 3 shows the diagram of the horizontal displacement of the point loaded by the horizontal force versus the load multiplier λ, as computed in the original reference [10] through a plane strain discretization. A much finer 3D mesh is herein implemented to capture the behaviour of the arch at incipient collapse with increased accuracy. The dotted line in 95
Analysiss of masonry vaults as a topology
t
optitimization pro
oblem
Third Intternational Conference on Mechanica
al Models in Structural
S
En
ngineering
Universitty of Seville. 24-26 june 2015.
2
Figure 3 connects reesults achieved by the prroposed optiimization pro
ocedure for independent analyses out at increaasing values of the param
meter λ. A go
ood agreeme
ent is found in the first p
part of the carried o
curve, w
whereas the ultimate value of the looad multiplie
er for which the optimizzation algoriithm finds convergeence is not far from the vvalue of the collapse multiplier, λc, predicted by llimit analysiss. Figure 2. Semi‐circula r arch: geomeetry and load conditions Figure 33. Semi‐circula
ar arch: load‐d
displacement ddiagram obta
ained through the proposedd numerical method vs. benchmarkk results [10]. The collapse load multiplieer given by lim
mit analysis is also shown Figure 4 shows the (magnified) d
deformed geeometry com
mputed by the proposed pprocedure at incipient collapse,, which matcches the exp
pected four‐hhinge mechaanism. The h
highest strainns are localizzed within limited rregions of the three‐dime
ensional dom
main, whereas the remaiining parts oof the structu
ure exhibit an almosst rigid respo
onse. 3.2. Gro
oin vault A groin vvault given b
by the interssection of tw
wo barrel vau
ults with inte
ernal radius ri=0.8 m and external radius re=1.0 m is deealt with. Th
he material properties aare assumed to be equa l to those used in the previouss example. A A discretization with 12 456 elemen
nts is emplo
oyed to inveestigate the structural 96
Matteo Bruggi and Alberto Taliercio
responsee of the vaullt under the effect of sellf‐weight, see Figure 5. LLateral confinnement is en
nforced by suitably constrainingg the horizon
ntal displacem
ments of the
e sides of the
e vault. Figure 4. Semi‐circcular arch: maagnified deforrmed shape at incipient collllapse Figu
ure 5. Groin va
ault: views of tthe adopted tthree‐dimensio
onal discretizaation First, thee vault is anaalysed assum
ming a linear elastic behaviour of the material botth in compre
ession and in tensio
on. Figure 6 aand Figure 7
7 show, respeectively, the contour maps of the priincipal stressses and . is not reported here, as its order of f magnitude is much low
wer than thatt of and . As one es exist both at the intrad
dos and at may easily see from the referencced figures, ppositive principal stresse
the extraados, in the upper part o
of the structuure. The no‐ttension mod
del is then adopted to innvestigate th
he structural response oof the vault neglecting the tenssile strength of masonryy. Figure 8 a nd Figure 9 show, respe
ectively, the e contour ma
aps of the principal stresses and , acccording to thhe results of the analysis performed tthrough the proposed algorithm
m. Apparenttly, no tensile‐stress is ffound in anyy region of the t vault, w
whereas the maximum absolutee values of the t principall compressivve stresses increase with respect too those obta
ained by a linear elaastic analysiss. 97
Analysiss of masonry vaults as a topology
t
optitimization pro
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Third Intternational Conference on Mechanica
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Universitty of Seville. 24-26 june 2015.
2
Figuree 6. Groin vaullt: map of the principal streess (extrado
os and intrado
os, with assumptionn of linear elasstic behavior
) und
der the Figure 7. Groin vaullt: map of the principal stresss (extrado
os and intrado
os, with assumptionn of linear elasstic behavior
) un
nder the Regions where the sstress vanish
hes correspoond to region
ns experienccing crackingg strains. In the upper part of tthe intrados,, the vault iss fully crackeed a two prin
ncipal stressses vanish. A
At the extrad
dos, cracks arise peerpendicularlly to one principal direection (corre
esponding to II) in widde regions along the perimeteer of the vau
ult. 98
Matteo Bruggi and Alberto Taliercio
Figuree 8. Groin vaullt: map of the principal streess (extrado
os and intrado
os, with assumptioon of no‐tensio
on behavior ) und
der the Figure 9. Groin vaullt: map of the principal stresss (extrado
os and intrado
os, with assumptioon of no‐tensio
on behavior ) un
nder the 4. CONC
CLUSIONS Followin
ng a recent proposal forr 2D no‐tenssion structures [5], a numerical meethod is pre
esented to perform the analysiss of no‐tensiion 3D structtural elemen
nts under givven loads acccording to a
an energy‐
based non‐incremen
ntal algorithm. The straain energy of o a body made of an eequivalent orthotropic material is minimized, so as to avvoid tensile sstresses thro
oughout the structure. A
An interpolation typical of topolo
ogy optimizaation [6]is em
mployed to ddefine the ellastic properrties of the eequivalent m
material, so that a negligible stifffness is obtained along the directio
on(s) of the tensile princcipal stress(e
es). In the n, the nondim
mensional “d
densities” that define thee elastic mod
duli of the formulattion implemeented herein
99
Analysis of masonry vaults as a topology optimization problem
Third International Conference on Mechanical Models in Structural Engineering
University of Seville. 24-26 june 2015.
equivalent material and the orientation of the relevant symmetry axes are both sets of minimization variables for the strain energy. The proposed algorithm correctly captures the typical crack pattern observed in masonry vaults. Also, the collapse mechanisms and the collapse load multipliers of these vaults can be estimated without any a‐priori hypothesis regarding the position of the “plastic hinges” (Sec. 3). In the continuation of the work, the simplification adopted so far, according to which masonry is macroscopically isotropic, will be removed to take elastic anisotropy into account. The cracking strains predicted by the proposed approach will be also compared with those obtained by incremental analyses carried out with commercial FE (or XFE) programs. Finally, the possibility of defining optimal reinforcing layouts will be dealt with, extending the formulation adopted here to distribute the equivalent orthotropic material over the vault to define the distribution and the orientation of a tension‐only strengthening layer, see in particular [11]. REFERENCES [1] Heyman, J. (1966). The stone skeleton. International Journal of Solids and Structures, 2, 249‐279. [2] Lourenço, P. (2001). Analysis of historical constructions: From thrust–lines to advanced simulations. In: Historical Constructions, P. Lourenço & P. Roca (Eds.), Guimarães, (P), 91‐116. [3] Benvenuto, E. (1991). An introduction to the history of structural mechanics, II: Vaulted structures and elastic systems. New York: Springer. [4] Angelillo, M., Cardamone, L., Fortunato, F. (2010). A numerical model for masonry‐like structures, Journal of the Mechanics of Materials and Structures, 5, 583‐615. [5] Bruggi, M. (2014). Finite element analysis of no‐tension structures as a topology optimization problem. Structural and Multidisciplinary Optimization, 50(6), 957‐973. [6] Bendsøe, M., Sigmund, O. (1999). Material interpolation schemes in topology optimization. Archives of Applied Mechanics, 69, 635‐654. [7] Mehrabadi, M.M., Cowin, S.C. (1990). Eigentensors of linear anisotropic elastic materials. Quarterly Journal of Mechanics and Applied Mathematics, 43(1), 15‐41. [8] Papa, E., Taliercio, A. (2005). A visco‐damage model for brittle materials under monotonic and sustained stresses. International Journal for Numerical and Analytical Methods in Geomechanics, 29(3), 287‐310. [9] Svanberg, K. (1987). Method of moving asymptotes ‐ A new method for structural optimization. International Journal for Numerical Methods Engineering, 24, 359‐373. [10] Cuomo, M., Ventura, G. (2000). Complementary energy formulation of no tension masonry‐like solids. Computer Methods in Applied Mechanics and Engineering, 189(1), 313–339. [11] Bruggi, M., Taliercio A. (2013). Topology optimization of the fiber‐reinforcement retrofitting existing structures. International Journal of Solids and Structures, 50(1), 121‐136. 100