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COLLAPSE OF MASONRY VAULTS AND
ARCHES USING NONLINEAR DISCRETE
NUMERICAL METHODS

Rafael Bravo Pareja
rbravo@ugr.es1
José Luis Pérez Aparicio
jopeap@upvnet.upv.es2
1 Department of Structural Mechanics & Hydraulic Engineering
University of Granada, SPAIN
2 Department of Continuum and Structural Mechanics
Polytechnic University of Valencia SPAIN

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

Contents
1 Introduction
2 DDA’s Formulation
3 Non Linear Frictional law and Algorithm Implementation
4 Masonry Bridges
5 Experimental and numerical cases
6 CASE 1
Collapse loads
Number of joints vs elastic behaviour
7 CASE 2
Description of the problem
Failure Modes
Variable embankment thickness
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

Introduction I

Relatively new discipline in computational mechanics
Numerical solutions of problems for which constitutive laws are
not available
Interactions of hundreds of blocks emerge physical properties of
practical importance

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


Introduction II

Masonry structures composed of blocks. Stability achieved by
contact & friction
qv

qh
C1

C2
W

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


Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

Formulation I
Based on Newtonian Mechanics
Hamilton’s principle:

∂Πi (Ui )
− =0; i = 1, ..., n
∂Ui
Discretization:

Ui = T Di

Discrete equation of motion:

∂Π(Di )
− =0; i = 1, ..., n
∂Di

Formulation II
Expansion provides matrix formulation:

¨ ˙
M Di + C Di + KDi = F (Di , t) ; i = 1, ..., n
Initial conditions:

Di (0) = Di0 ; ˙ ˙
Di (0) = Di0

    
ˆ
K11 ˆ
K12 ˆ
K13 · · · ˆ
K1n D1 ˆ
F1

 ˆ22
K ˆ23 · · ·
K ˆ2n  D2  F2 
K    ˆ 

 ˆ
K33 · · · ˆ    ˆ 
K3n  D3  = F3 
.. .  .   . 
   

 −Sim− . .  .   . 
. . .
ˆ
Knn Dn ˆ
Fn
Off–diagonal terms indicate interaction → contact

Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

Non Linear Frictional law and Algorithm
Implementation 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 if Ft < Fr
H(s)

Fr = N · tan φ + a · s2 + b · s + c if Ft ≥ Fr
Ks tangential penalty. Coulomb Law (Linear)
a, b and c experimental Fr
H(s)
data

Ks = 107 N/m2 Applied Non Linear Law
6
a = −1.5 · 10 Stick Sliding
5
b = 2.0 · 10
R.Bravo J.L.Perez-Aparicio (UGR-UPV) Hola Vaults Using DDA (Complas 07)
s 6 September 2007 10 / 30

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

∂H(s0 )
H(s0 + ∆s) = H(s0 ) + · ∆s
∂∆s
Potential energy:

∂H(s0 )
Π = H(s0 ) · ∆s + · ∆s2
∂s
Minimization:

∂Π ∂H(s0 )
= H(s0 ) + · ∆s
∂∆s ∂s
Stiffness matrix and force vector:

∂H(s0 )
K = F = H(s0 )
∂s

Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

Masonry Bridges

Stability through thousands of
semi–rigid interacting blocks: High
Computational Cost


Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

Initial Data

Experiment of arches performed with properties (Delbeq 1982):
Property CASE 1 CASE 2
Brick Density 2500 g/cm3 2500 g/cm3
Young Modulus 1E9 N/m2 1E9 N/m2
Poisson Modulus 0.2 0.2
Friction 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 real
materials)


Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

CASE 1. Collapse loads
Ultimate collapse load with different number of joints

Load

0.5

N◦ joints Critical Load Critical Load Error
5 4 Experimental (kN) DDA (kN) %
6.7 7 250 280 12.2
Filling 15 206 210 1.6
25 206 205 -0.8
59 205 205 0.1
199 205 205 0

1
10


Low number of blocks bad results in stress and strains
Elastic block (area S and gravity centre (x0 , y0 ) puntual load
(Fx ,Fy ) at (x,y)

    
1 ν 0 x (x − x0 )Fx
S·E 
ν 1 0  y = (y − y0 )Fy 
1 − ν2 1−ν
0 0 2
γxy (y − y0 )/2Fx + (x − x0 )/2Fy
Elastic Stiffness Matrix[K ] Puntual Load Vector[F ]

Strain/Stresse Constant over each block
x−x0 ν(y −y0 ) and dependent on (x,y)
x = S·E Fx − S·E Fy
Averaged by block’s area S
−y
y = − ν(x−x0 ) Fx + yS·E0 Fy
S·E Need to increase number of blocks to
−y
γxy = (1 + ν) yS·E0 Fx + x−x0 Fy
S·E obtain accurate results

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

(x,y) (x,y)
DDA Analytical
L2 (x0,y0) (x0,y0) σv N/mm2 3.75 · 103 σv = Fy /A = 1 · 103
v 3.75 · 10−2 v = σv /E = 1 · 10
−2

S tres s dis tributio
n
L1 Fy/2 Fy/2 50
Analytical
DDA height 0.5m
100
Fy
DDA height 0.1m

150

200
S tres s (N/m2)

8m 250

300

350

400
10m
450
0 1 2 3 4 5 6 7 8
Y coordinate (m)


Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

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

Filling material + Embankment
Embankment
h
0.5

Filling material
Filling
8

1 15

Lower and upper stability limits bounds obtained


CASE 2. Failure Modes
Inestability
Vertical < Horizontal Loads (LEFT). Peak’s Elevation
Vertical > Horizontal Loads (RIGHT)

Formation of alternative hinges
Failure compression
Tresca Failure Criteria:
σY = (σI − σII )
σI , σII principal stresses, σY yield stress. Other suitable criteria
Druger Pracker (Owen in combined Finite Discrete Element
Method).

CASE 2.a) Variable embankment. Failure Angles
Comparison hinges’ angles
LEFT: excessive horizontal loads
RIGHT: excessive vertical loads
Five hinges for both cases

78°

60°

18° 26°

Numerical results agree well with experimental data

Limit Numerical Experimental
Lower 18◦ 60◦ 90◦ 19◦ 64◦ 90◦
Upper 0◦ 26◦ 78◦ 0◦ 37◦ 78◦


CASE 2.a) Variable embankment. Safety Factor
Relation between applied and failure loads (both numerical and
experimental)
3 Failure modes:
1 Elevation of peak (low vertical loads)
2 Compression failure (intermediate)
3 Peak’s descend (high vertical loads)
6

DDA
E xperimental
5
S afety F actor

4

3

2

1

0
0 2 4 6 8 10 12
T hicknes s (m)


CASE 2.b) Point load & failure angles
Response analysis under 2 variable concentrated loads
(symmetric loads). Rest same as CASE 2. Embankment
thickness ﬁxed to 0.5 m
Lower bound limit same failure mode as case 2
Formation of 3 hinges

63°

Limit Numerical Experimental
Lower 18◦ 60◦ 90◦ 19◦ 64◦ 90◦
Upper 63◦ 90◦ 57◦ 90◦

CASE 2.b) Point load & safety factor
Similar failure as that of embankment load
High sensitivity in initial branch: comparison not good (bad load
transmission due to ﬁrst order formulation?)

10
DDA
E xperimental
8
S afety F actor

6

4

2

0
0 20 40 60 80 100 120 140 160 180
Load (kN)


Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

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


Contents
1 Introduction
4 Masonry Bridges
6 CASE 1
Collapse loads
7 CASE 2
Failure Modes
Failure Angles
Safety Factor
Point Load
Failure Angles
Safety Factor
8 Conclusions
9 Acknowledgements

Acknowledgements

Authors gratitude the support offered by the following research
projects:

80019/A04 Ministerio de Fomento.

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


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