Types of analysis: Linear static, linear dynamic and non linear static

המועצה לשימור

אתרי מורשת בישראל

Types of analysis:

Linear static, linear

dynamic and non

linear static

Paulo B. Lourenço

[email protected]

www.civil.uminho.pt/masonry



Safety

Assessment of

Existing Buildings

Institute for Sustainability and Innovation in Structural Engineering

3 | Types of analysis: Linear static, linear dynamic and non linear static Paulo B. Lourenço

Non-linear static analysis

Linear static analysis

Sim

plif

ica

tion

Structural Analysis Methods (Static)



Early Structural Analysis

“Ut tensio sic vis” or / E = is the elasticity law established by R.

Hooke in 1676.The theory is so extensively used that its limitations and

deficiencies are often forgotten. This is in opposition with early forms of

limit analysis.

Cantilever beam according to Galileo (1638) and evolution

of the “hypothesis” for the stress distribution at AB

Retaining wall according

to Coulomb (1773)

B

A

P



Modern Structural Analysis

As structural collapse does not generally coincide with the appearance

of the first crack or localized early crushing, it seems that the elasticity

theory is a step back with respect to limit analysis

Full nonlinear analysis (the most advanced form of structural analysis)

covers the complete loading process, from the initial “stress-free” state,

through the weakly nonlinear behavior under service loading, up to the

strongly nonlinear behavior leading to collapse

Interest has been growing since 1970’s but it remains a field for

selected (few) specialists due to complexity (knowledge) and costs

(time) involved

The possibilities are immense and several commercial software

packages include some form of nonlinear behavior, but an incorrect use

can be very dangerous



Modern Structural Analysis The modern use of nonlinear analysis focuses mostly on these three

fields:

Complex / stringent safety requirement structures (e.g. nuclear plants,

dams, bridges)

Virtual laboratory for parametric studies

Existing structures (evaluation, repair, rehabilitation)

Three types of non-linearities may arise:

Material (or physical) nonlinearity

F

Geometrical

nonlinearity

Contact nonlinearity

Steel

(“code model”) Concrete

(“code model”)



Existing Buildings

Settlements

Vertical dead + live load

Vehicles



Modern Earthquake Design

Macro-models (Braga, Liberatore, D’Asdia, Magenes, Lagomarsino, etc.)

“Storey” model (Por)

Tomazevic

Finite element model

(Many authors)

Elastic analysis leads

to excessively

conservative

solutions for

unreinforced,

confined, and

possibly, reinforced

masonry



Example of Analysis of an Arch

The solution: “Ut pendet continuum flexile, sic

stabit contiguum rigidum inversum” – as hangs the

flexible line, so but inverted will stand the rigid arch.

In 1675 Hooke provided the solution for he

equilibrium of an arch by means of an

anagram included in the book "A

description of Helioscopes and some

other Instruments", which was only

deciphered after his death in 1703.

Robert Hooke (1635-1703)- Principle of the inverted catenary

http://images.google.es/imgres?imgurl=http://i24.photobucket.com/albums/c8/aliceb59/RobertHooke.jpg&imgrefurl=http://www.bath.ac.uk/~ab298/Hooke.html&h=300&w=218&sz=16&hl=en&start=80&um=1&tbnid=CBb_u0TxmCQoeM:&tbnh=116&tbnw=84&prev=/images?q=robert+hooke&start=72&ndsp=18&svnum=10&um=1&hl=en&sa=N



Graphic Statics

The arch is first decomposed in a series of real or fictitious voussoirs

separated by a series of planes (the planes do not need to be parallel)

The thrust line is defined as the geometrical locus of the points of

application of the sectional forces (the resulting forces over each plane

between voussoirs) across the arch An arch is stable if it is possible

to find a thrust line contained

between its boundaries



Kinematic Analysis

Charles-Agustin COULOMB (1736-1806) proposed in 1773 the first general

and accurate theory on the stability of masonry arches

The basic assumptions are:

(1) Sliding between voussoirs is unlikely due to the existing frictional forces

(2) Collapse will be caused by the rotation between parts due to the

appearance of a number of hinges. The location of the hinges is a priori

unknown but can be determined by the method of “maxima and minima”



Kinematics of 4-hinge collapse



Correspondence with THRUST LINE theory: a hinge will develop each

time the equilibrium line becomes tangent to an alternate boundary. In

this condition (failure), the thrust line is determined and unique.



Collapse of an arch brought experimentally to failure



Static Analysis Methods (I)

Linear Elastic Analysis

elastic properties + maximum admissible

stress

Kinematic Collapse Mechanism Analysis

inelastic properties = friction angle +

tensile and compressive strengths

Static Thrust Line Analysis

Non-linear Analysis (Physical and

Combined)

FULL inelastic properties (ft = 0 and ft ≈

0) + elastic properties

5

2.5

3

1.25 10 kN



Sta

tic

An

aly

sis

Me

tho

ds

(II)

Max. 0.64 N/mm2

Linear Elastic

Kin. load factor : 1.8

Failure Mechanism

Geo. load factor : 1.2

Thrust Line

Min. -1.0 N/mm2

Linear Elastic

Min. -5.4 N/mm2

Phys. Non-Linear

Min. -5.4 N/mm2

Comb. Non-Linear



Static Analysis Methods (III)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 5 10 15

Vertical displacement at quarter span (mm)

Lo

ad

facto

r

Limit analysis

ft = 0, Physically non-linear

ft = 0, Physically / Geometrically non-linear

ft = 0.2 N/mm2, Physically non-linear

ft = 0.2 N/mm2, Physically / Geometrically non-linear



Static Analysis Methods (III)

Approach/Analysis type

Semi-circular arch

Allowable stresses (fta=0.2 N/mm2)

0.31

Kinematic limit analysis

1.8

Geometric safety factor

1.2

ft = 0, Physically non-linear

1.8

ft = 0, Physically and geometrically non-linear

1.7

ft = 0.2 N/mm2, Physically non-linear

2.5

ft = 0.2 N/mm2, Phys. and geom. non-linear

2.5

The “safety factors” of a linear elastic analysis and a static limit analysis cannot

be compared with the remaining safety factors.

Physically non-linear analysis and kinematic limit analysis yield the same failure

mechanisms and safety factors?

The consideration of a non-zero, yet low and degrading, tensile strength

increased the safety factors considerably. The post-peak is a key issue.

Different methods of analysis lead to different safety factors and different

completeness of results.



More on Static Analysis Methods…

Safety factor: 124%



Non-linear time history analysis

Non-linear static analysis

Linear elastic time history analysis

Modal superposition

Linear static analysis

Sim

plif

ica

tion

Structural Analysis Methods: “Dynamic”



In the recent years new methods of seismic assessment and design have

been developed, particularly with respect to push-over analysis

Two methods of analysis can be distinguished:

- Traditional force method, combined with control of performance requirements

based on deformation

- Displacement based method, in which the analysis starts by defining a target

displacement (measuring the structural response).

Push-Over Analysis (I)



The dimensions of the structural members are considered

The stiffness of the members is also considered (the codes might consider

elastic stiffness or 30 to 50% of the elastic stiffness)

Periods are based on stiffness (Note: The design forces can be reduced

about 30 to 50% if the stiffness is reduced to the half)

Forces are distributed in the elements according to the stiffness

Traditional Force Method

Push-Over Analysis (II)



Moment-curvature curves for circular columns (D=2 m, fc=35 MPa, fy=450 MPa)

Mom

ent

(kN

m)

Curvature (1/m)

Percentage of reinforcement = 1%

Curvature (1/m)

Percentage of reinforcement = 3%

Mom

ent

(kN

m)

Push-Over Analysis (III)

Stiffness and strength are correlated!!



HA HB HC

C A B

Force Based Design Displacement Based Design

Stiffness: proportional to 1/H3 proportional to 1/H

Shear: proportional to 1/H3 proportional to 1/H

Moment: proportional to 1/H2 equal

Reinforcement: proportional to 1/H2 equal

Ductility: equal (!) proportional to 1/H2

`

F Hc

HA

HB

Push-Over Analysis (IV)



Masonry

Structures With

Box Behavior



Recent test results: Rigid diaphragm

Worst case scenario: Embedded ring beam + Unfilled vertical joints

Moderate damage up to 100% of the design earthquake in Lisbon

Ductile failure for 250% of the design earthquake in Lisbon



Seismic pushover analysis simulates the evolution of the condition of

structures during earthquakes, through application of incremental

horizontal forces until collapse

Assumptions of box behaviour and in-plane response are considered

Experimental results show that

URM possesses considerable

capacity for inelastic

deformations, and then the

application of nonlinear analysis

is obvious



“POR” Storey Mechanism Developed in former Yugoslavia and Italy as a reaction of the Skopje earthquake

in 1963, and implemented in the region Friuli-Venezia Giulia after the Friuli earthquake in 1976 (DT2, 1977)

The following hypothesis are considered:

Thickness of the wall is constant in each level

Slabs are rigid in-plane diaphragms

Ends of the piers do not rotate, but only suffer

translation

Behavior of the piers is elastic-perfectly plastic, with a

predefined ductility

Elastic stiffness of each panel remains constant

Panels collapse by diagonal shear according the

Turnsek-Cacovic expression

Tomaževič, Braga & Dolce



Additional Macro-Mechanisms Since the 1980s, observation of damage in masonry

buildings subjected to significant vertical load due to use

of slabs, and constituted by slender piers, introduces a

new trend of research on the combined flexural mechanism

Comb. flexural Diagonal shear Sliding shear Mixed

0 0.5 1 1.5 2 2.5

Slenderness (H/L)

0

0.25

0.5

0.75

1

Diagonal shear

Combined

flexural

Sliding

Dim

ensio

nle

ss

no

rma

l str

ess (σ

/fm

)

0 0.5 1 1.5 2 2.5

Slenderness (H/L)

0

0.25

0.5

0.75

1

Diagonal shear

Combined

flexural

Sliding

Dim

ensio

nle

ss

no

rma

l str

ess (σ

/fm

)



Early Improvements in Italy Initially these methods only had an impact in the scientific community and the

POR persisted as the method most used by Italian designers

1 2 3

RAN PEFV MAS3D

(Raithel & Augenti) (D’Asdia & Viskovic) (Braga et al.)



A Generation of Design Methods in Italy As a consequence of the 2002 Molise Earthquake the new Italian code OPCM

3274/2003(3431/2005) was introduced, and macro-elements methods emerge as

modern and practical tools

Spandrel

Pier

Joint

SAM 3Muri 3DMacro

(Magenes et al.) (Lagomarsino et al.) (Caliò et al.)



Validation Example



Or even use SAP 2000

One-dimensional macro-element

Bi-dimensional macro-element



Program Country Code Approach Web adress

AEDES Italy Italian FEM and SCM www.aedes.it

CMT+L Spain Eurocode FEM www.arktec.com/cmtl.htm

FEDRA Norway Eurocode FEM www.runet-software.com/FEDRA.htm

WIN-Statik MurDim+ Sweden ? ? www.strusoft.com

Por 2000 Italy Italian SCM www.newsoft-eng.it/Por2000.htm

TQS CAD/Alvest Brazil Brazilian ? www.tqs.com.br/v13/alvest.htm

Tricalc.13 Spain Eurocode FEM www.arktec.com/new_t13.htm

Tricalc.17 Spain Spanish FEM www.arktec.com/new_t17.htm

WinMason USA USA Storey Mech. www.archonengineering.com/winmason.html

3DMacro Italy Italian SCM http://www.3dmacro.it/

3Muri Italy Italian SCM www.stadata.com

ANDILWall Italy Italian SCM www.crsoft.it/andilwall

MURATS Italy Italian Storey Mech. www.softwareparadiso.it/murats.htm

Sismur Italy Italian Storey Mech. www.franiac.it/sismur.html

TRAVILOG Italy Italian Storey Mech. www.logical.it/software_travilog.aspx

Tecnobit Italy Italian Storey Mech. www.tecnobit.info/products/murature.php

CDMaWin Italy Italian FEM and SCM www.stsweb.net/STSWeb/ITA/homepage.htm

There is commercial software available for structural UR masonry, particularly

in Italy. Benchmarking was made in two publications: Azores 1998, Eds. C.

Sousa Oliveira et al., (2008) and Marques, R., Lourenço, P.B., Possibilities and

comparison of structural component models for the seismic assessment of

masonry buildings, Computers and Structures, 89 (21-22), p. 2079-2091 (2011)

Commercial Software (I)



Efficient and high level modeling

Commercial Software (II)



Displacement Based Design

Recent methods implement capacity/displacement-based seismic design, by

evaluating the evolution of damage and displacement

Structural

objects

Geometric

definition

Structural

characteristics

DXF/DWG

Automatic

mesh definition

Equivalent

mesh/frame definition

A – Displacement

capacity

B – Displacement

demand

Final analysis

Non-linear

analysis

Seismic

parameters

0

100

200

300

400

500

0 10 20 30 40

Displacement (mm)

Base s

hear (

kN

)

If the damage evolution can be

used as a measure of seismic

performance, the confrontation

between displacement capacity

and displacement demand is the

rule for safety verification



In a force based method, the non-linear reserve capacity must be considered

For unreinforced masonry

buildings with 2 or more storeys:

EC8:

q = 1.5-2.5 (recommended 1.5)

OPCM 3431:

αu /α1 (OSR) = 1.8

q = q0 x OSR = 3.6

Energy Dissipation Capacity (I)



Energy Dissipation Capacity (II)



É necessário ter em conta a reserva de capacidade não linear das estruturas em ductilidade

Para edifícios em alvenaria

simples com 2 ou mais pisos:

EC8:

q(0) = 2.0

OPCM 3431:

αu /α1 (OSR) = 1.8

q = q0 x OSR = 3.6

Energy Dissipation Capacity (III)



Application



PARAMETRIC STUDY on the NR. of STOREYS (I)

6.5

0

6.50

1.50 1.00 1.50 1.00 1.50

3.0

0

2.0

0

0.30

6.5

0

6.50

1.50 1.00 1.50 1.00 1.50

6.0

0

2.0

01.0

0

0.30

6.5

0

6.50

1.50 1.00 1.50 1.00 1.50

9.0

0

2.0

01

.00

1.0

0

0.30

Dead load = 6.0 kN/m2

Live load = 1.5 kN/m2

Dead load = 6.0 kN/m2

Live load = 1.0 kN/m2

Dl = 7.0 kN/m2; Ll = 2.0 kN/m2

Dl = 6.0 kN/m2; Ll = 1.0 kN/m2

Dl = 7.0 kN/m2; Ll = 2.0 kN/m2

Dl = 7.0 kN/m2; Dl = 2.0 kN/m2

Undamaged Plastic by shear Failure by shear Plastic by flexural Failure by flexural

Specific weight , γ 17.0 kN/m3

Compressive characteristic strength , fk 2.56 MPa

Sliding pure shear characteristic strength, fvk0 0.15 MPa

Normal elasticity module, E 2560 MPa

Tangential elasticity module, G 1024 MPa



agR

(m/s2) S

A

B

A

B

A

B

Soil

1.00

1.35

1.00

1.33

1.00

1.35

1.70

1.10

0.80

2.3

Zone

2.4

2.5

agR

(m/s2) S

A

B

A

B

A

B

A

B

A

B

Soil

1.00

1.20

1.00

1.20

1.00

1.20

1.00

1.30

1.00

1.30

2.50

2.00

1.50

1.00

0.50

1.3

Zone

1.2

1.1

1.4

1.5

agR

(m/s2) S

A

B

A

B

A

B

Soil

1.00

1.35

1.00

1.33

1.00

1.35

1.70

1.10

0.80

2.3

Zone

2.4

2.5

agR

(m/s2) S

A

B

A

B

A

B

A

B

A

B

Soil

1.00

1.20

1.00

1.20

1.00

1.20

1.00

1.30

1.00

1.30

2.50

2.00

1.50

1.00

0.50

1.3

Zone

1.2

1.1

1.4

1.5

Elastic Analysis ac. PT NA to EC8 (q=1.5)

Unsafe Safe in soil type A Safe in soil types A and B

Pushover Analysis

OSRqF

Fqq

el

y

00

Elastic Analysis ac. IT OPCM 3431

agR

(m/s2) S

A

B

A

B

A

B

Soil

1.00

1.35

1.00

1.33

1.00

1.35

1.70

1.10

0.80

2.3

Zone

2.4

2.5

agR

(m/s2) S

A

B

A

B

A

B

A

B

A

B

Soil

1.00

1.20

1.00

1.20

1.00

1.20

1.00

1.30

1.00

1.30

2.50

2.00

1.50

1.00

0.50

1.3

Zone

1.2

1.1

1.4

1.5

agR

(m/s2) S

A

B

A

B

A

B

Soil

1.00

1.35

1.00

1.33

1.00

1.35

1.70

1.10

0.80

2.3

Zone

2.4

2.5

agR

(m/s2) S

A

B

A

B

A

B

A

B

A

B

Soil

1.00

1.20

1.00

1.20

1.00

1.20

1.00

1.30

1.00

1.30

2.50

2.00

1.50

1.00

0.50

1.3

Zone

1.2

1.1

1.4

1.5

Elastic Analysis ac. PT NA to EC8 (q=1.5)

Pushover Analysis

PARAMETRIC STUDY on the NR. of STOREYS (II)



Masonry

Structures

Without Box

Behavior



Recent Tests: Flexible Diaphragm

“Gaioleiro”-type structure (late 19th century / early 20th century)

Moderate damage for 100% of the design earthquake in Lisbon

Light strengthening and collapse for 150% of the design earthquake in Lisbon



Location: New Delhi (India)

Material: Masonry

Total Height: 72.5 m

Crosss section: shell (3 leaves) + core (2 leaves)

71.4

8

28.9

615.3

912.4

27.7

36.9

8

14.07

3.13Qutb Minar



Beam FEM

UMinho

FEM

UPadova

FEM

CBRI

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 1 2 3 4 5 6 7

Calculated Frequencies [Hz]E

xp

eri

men

tal F

req

uen

cie

s [

Hz]

Experimental

FEM - UMinho

FEM - UPadova

FEM - CBRI

BEAM

REM

Bending

Bending

Bending

Torsion

Bending

Axial

Rigid

Blocks

Calibration

Numerical Modeling



Collapse at the base

Uniform Mass Distribution

Push-Over Analysis

Other Mass Distributions

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Lo

ad

Fa

cto

r

Top Displacement [m]

Beam model

Rigid model

Solid model

D$formada Dinamica minar$t$accct$

Fac. $scala: 5.00000 Paso 1810 Rigid - Plot V3.2.1

0.00

0.05

0.10

0.15

0.20

0.25

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Lateral displacement [m]

Lo

ad

fa

cto

r [

ba

se s

hea

r /

self

-wei

gh

t]

Mass Proportional

Linear Proportional

1º Mode proportional



-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 5 10 15 20 25

Tempo [s]

Des

locam

en

to h

ori

zo

nta

l [m

]

5º balcão

4º balcão

3º balcão

2º balcão

1º balcão

Collapse: 4th

balcony

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0 5 10 15 20

Time [s]

Ac

ce

lera

tio

n [

m/s

2]

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2

Period [s]

Seis

mic

co

eff

icie

nt

Ah

DBERecord 1Record 2Record 3Record 4Record 5Série7Série8Série9

Second mode 0.50

10th mode

0.16First mode 1.26

0

10

20

30

40

50

60

70

0 50 100 150 200 250 300

Maximum excentricity

(Bending Moment / Axial Load )

H [

m]

Record 1Record 2Record 3Record 4Record 5Série6Série7

FEM – Collapse for 0.20g

Five articificial accelerograms

REM

Time History Analysis



Location: Lisboa

Material: Masonry walls and

timber pavements

No. of storeys: 4 to 6

Numerical model

“Gaioleiro” Building



“Gaioleiro” Building



Transversal Direction

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.01 0.02 0.03 0.04 0.05

Deslocamento [m]

Fa

cto

r d

e ca

rga

(λ

)

0

0.05

0.1

0.15

0.2

0.25

0.3

0 0.01 0.02 0.03 0.04 0.05

Displacement [m]

Sei

smic

co

effi

cien

t [α

h]

Pushover_1st ModePushover_MassDynamic

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Deslocamento [m]

Fa

cto

r d

e ca

rga

[λ

]0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Displacement [m]

Sei

smic

co

effi

cien

t [α

h]

Pushover_1st ModePushover_MassDynamic

Longitudinal Direction

Pushover Analysis



Numerical model

Principal strains

(external surface)

ε1 [m/m]

Experimental model Time History Analysis



Design and Assessment = Macro-block analysis?

Limit equilibrium analysis using the principle of virtual work is currently

understood as the “best” analysis technique

Overturning



Conclusions



Conclusions Design and assessment methods based on non-linear analysis should be

used for masonry structures. Linear elastic analysis methods (application of

“equivalent” static forces and modal superposition) are questionable

Adequate models and commercial software, based on pushover analysis,

are available for masonry with box behavior

It was shown that pushover analyses do not simulate correctly the failure

mode of masonry structures without box behavior, meaning that higher

vibration modes have a significant contribution

Pushover analysis proportional to the mass are probably the best solution is

global structural analysis models are used

For design purposes, particularly for strengthening design, macro-block limit

analysis is probably the best analysis tool for practitioners

More research needs to be done in the field of masonry structures without

box behavior and earthquakes



Types of analysis:

Linear static, linear

dynamic and non

linear static

Paulo B. Lourenço

[email protected]

www.civil.uminho.pt/masonry

Types of analysis: Linear static, linear dynamic and non linear static

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