Complas 2007

COLLAPSE OF MASONRY VAULTS ANDARCHES USING NONLINEAR DISCRETE

NUMERICAL METHODS

Rafael Bravo [email protected]

José Luis Pérez [email protected]

1Department of Structural Mechanics & Hydraulic EngineeringUniversity of Granada, SPAIN

2Department of Continuum and Structural MechanicsPolytechnic University of Valencia SPAIN

6 September 2007

R.Bravo J.L.Perez-Aparicio (UGR-UPV) Hola Vaults Using DDA (Complas 07) 6 September 2007 1 / 30

Contents1 Introduction2 DDA’s Formulation3 Non Linear Frictional law and Algorithm Implementation4 Masonry Bridges5 Experimental and numerical cases6 CASE 1

Collapse loadsNumber of joints vs elastic behaviour

7 CASE 2Description of the problemFailure ModesVariable embankment thickness

Failure AnglesSafety Factor

Point LoadFailure AnglesSafety Factor

8 Conclusions9 Acknowledgements


Introduction I

Relatively new discipline in computational mechanicsNumerical solutions of problems for which constitutive laws arenot availableInteractions of hundreds of blocks emerge physical properties ofpractical importance

Masonry structures discontinuous. Discontinuous DeformationAnalysis (DDA) better suited than Continuum Mechanics


Introduction II

Masonry structures composed of blocks. Stability achieved bycontact & friction

W

C1

C2

qh

qv

2D experiments of masonry vaults (cut stone) at real scaledescribed. Experimental & numerical results compared


Formulation IBased on Newtonian MechanicsHamilton’s principle:

−∂Πi(Ui)

∂Ui= 0 ; i = 1, ...,n

Discretization:

Ui = T Di

Discrete equation of motion:

−∂Π(Di)

∂Di= 0 ; i = 1, ...,n


Formulation IIExpansion provides matrix formulation:

MDi + CDi + KDi = F (Di , t) ; i = 1, ...,n

Initial conditions:

Di(0) = Di0 ; Di(0) = Di0

K11 K12 K13 · · · K1n

K22 K23 · · · K2n

K33 · · · K3n

−Sim− . . ....

Knn

D1D2D3...

Dn

=

F1

F2

F3...

Fn

Off–diagonal terms indicate interaction→ contact


Non Linear Frictional law and AlgorithmImplementation I

Contact law models frictional behavior of rocky materials. (A.Nardin, G. Zavarise, BA. Scherefler (2003))Tangential behaviour. Sliding starts tangential force Ft ≥ Fr =Coulomb friction (regularized) + Softening law H(s) (Non linear)

Fr = Ks · s + a · s2 + b · s + c︸︷︷︸H(s)

if Ft < Fr

Fr = N · tanφ+ a · s2 + b · s + c if Ft ≥ Fr

Ks tangential penalty.a, b and c experimentaldata

Ks = 107N/m2

a = −1.5 · 106

b = 2.0 · 105

c = 0

Coulomb Law (Linear)

Applied Non Linear Law

s

H(s)Fr

SlidingStick


Algorithm Implementation non linear frictional law IIDDA’s displacements ∆s at each time step small→ linearization:

H(s0 + ∆s) = H(s0) +∂H(s0)

∂∆s·∆s

Potential energy:

Π = H(s0) ·∆s +∂H(s0)

∂s·∆s2

Minimization:

∂Π

∂∆s= H(s0) +

∂H(s0)

∂s·∆s

Stiffness matrix and force vector:

K =∂H(s0)

∂sF = H(s0)


Masonry Bridges

Stability through thousands ofsemi–rigid interacting blocks: HighComputational Cost


Initial Data

Experiment of arches performed with properties (Delbeq 1982):

Property CASE 1 CASE 2Brick Density 2500 g/cm3 2500 g/cm3

Young Modulus 1E9 N/m2 1E9 N/m2

Poisson Modulus 0.2 0.2Friction Angle 30◦ 30◦

Cohesion 0 N/m2 0 N/m2

Filling density 2000 kg/m3 2000 kg/m3

Embankment density 1200 kg/m3 1200 kg/m3

Block ultimate stress σY 10 MPa 10 MPa

Two different geometries. Properties uncertain (variability in realmaterials)


CASE 1. Collapse loadsUltimate collapse load with different number of joints

5 4

0.5

110

Filling

Load

6.7

N◦ joints Critical Load Critical Load ErrorExperimental (kN) DDA (kN) %

7 250 280 12.215 206 210 1.625 206 205 -0.859 205 205 0.1

199 205 205 0


Number of joints vs elastic behaviourLow number of blocks bad results in stress and strainsElastic block (area S and gravity centre (x0, y0) puntual load(Fx ,Fy ) at (x,y)

S · E1− ν2

1 ν 0ν 1 00 0 1−ν

2

εxεyγxy

︸︷︷︸

Elastic Stiffness Matrix[K ]

=

(x − x0)Fx(y − y0)Fy

(y − y0)/2Fx + (x − x0)/2Fy

︸︷︷︸

Puntual Load Vector[F ]

εx = x−x0S·E Fx − ν(y−y0)

S·E Fy

εy = −ν(x−x0)S·E Fx + y−y0

S·E Fy

γxy = (1 + ν)(y−y0

S·E Fx + x−x0S·E Fy

)Strain/Stresse Constant over each blockand dependent on (x,y)Averaged by block’s area SNeed to increase number of blocks toobtain accurate results


ExampleVert.P. load Fy = 1kN at (x , y) = (0.5,1.75), L1 = 1m , L2 = 2mMaterial properties E = 105N/m and ν = 0DDA’s reactions→ Contact forces. 3 Punt. loads

L2

L1

(x0,y0)

(x,y)

(x0,y0)

(x,y)

Fy

Fy/2Fy/2

Fy

Fy

10m

8m

DDA Analyticalσv N/mm2 3.75 · 103 σv = Fy/A = 1 · 103

εv 3.75 · 10−2 εv = σv/E = 1 · 10−2

0 1 2 3 4 5 6 7 8450

400

350

300

250

200

150

100

50S tress distribution

Y coordinate (m)

Str

ess

(N

/m2

)

AnalyticalDDA height 0.5mDDA height 0.1m

Increasing the number of blocks more accurate results


CASE 2

Collapse analysis under:2.a) Variable embankment thickness2.b) Point loads

8

Filling material + Embankment

1

0.5

Filling material

Filling

15

Embankment

h

Lower and upper stability limits bounds obtained


CASE 2. Failure ModesInestability

I Vertical < Horizontal Loads (LEFT). Peak’s ElevationI Vertical > Horizontal Loads (RIGHT)

I Formation of alternative hinges

Failure compressionI Tresca Failure Criteria:

σY = (σI − σII)

σI , σII principal stresses, σY yield stress. Other suitable criteriaDruger Pracker (Owen in combined Finite Discrete ElementMethod).


CASE 2.a) Variable embankment. Failure AnglesComparison hinges’ anglesLEFT: excessive horizontal loadsRIGHT: excessive vertical loadsFive hinges for both cases

18°

60°

26°

78°

Numerical results agree well with experimental data

Limit Numerical ExperimentalLower 18◦ 60◦ 90◦ 19◦ 64◦ 90◦

Upper 0◦ 26◦ 78◦ 0◦ 37◦ 78◦


CASE 2.a) Variable embankment. Safety FactorRelation between applied and failure loads (both numerical andexperimental)3 Failure modes:

1 Elevation of peak (low vertical loads)2 Compression failure (intermediate)3 Peak’s descend (high vertical loads)

0

1

2

3

4

5

6

0 2 4 6 8 10 12

Sa

fety

Fa

cto

r

T hickness (m)

DDAE xperimental


CASE 2.b) Point load & failure anglesResponse analysis under 2 variable concentrated loads(symmetric loads). Rest same as CASE 2. Embankmentthickness fixed to 0.5 mLower bound limit same failure mode as case 2Formation of 3 hinges

63°

Limit Numerical ExperimentalLower 18◦ 60◦ 90◦ 19◦ 64◦ 90◦

Upper 63◦ 90◦ 57◦ 90◦


CASE 2.b) Point load & safety factor

Similar failure as that of embankment loadHigh sensitivity in initial branch: comparison not good (bad loadtransmission due to first order formulation?)

0

2

4

6

8

10

0 20 40 60 80 100 120 140 160 180

Sa

fety

Fa

cto

r

Load (kN)

DDAE xperimental


Conclusions

Basic simulation of masonry behaviour under different conditions

Results fit well to experimental data

3 failure modes simulated

Tresca criteria for stress failure

Need to improve higher order DDA’s formulation

Need to introduce statistical variability on input parameters

Contact law applied to DDA


Acknowledgements

Authors gratitude the support offered by the following researchprojects:

80019/A04 Ministerio de Fomento.

E/03/B/F/PP-149.038. Ag. Leonardo da Vinci.


Complas 2007

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