Lesson14

Masonry Structures, lesson 14 slide 1

Seismic design and assessment ofMasonry Structures

Seismic design and assessment ofMasonry Structures

Lesson 14October 2004


The reality of most historical centers: what is “the building”?

(Carocci et al., 1993)


(Giuffré, 1993)A: Existing cellB and C: Added cells

Cells A and B built after C

PROGRESSIVE GROWTH IN PLAN

STOREYS ADDED TO EXISTING BUILDINGS


Damage in most vulnerable buildings:

Tyipically, partial overturning mechanisms of façade walls or corner walls


Typical distribution of damage in historical

centers.


A strong (arguable) statement:

Historical buildings can be thought of as "…made by an assemblage of partial structures, and each of them can be easily singled out. Walls, floors, roofs, are isostatic structures resting the one on the other and, at the same time, joining the one to the other. It can be asserted that always the damage affects the weakest part of the building, and the analysis has the task of pointing out which part".

A. Giuffré, 1989


A weaker (wiser) statement:

Although the building as a whole is a redundant(hyperstatic) structure, simpler subsystems can be identified, which make up the structure and which can be treated in many cases as statically determined.

Focus is on equilibrium and on the compatibility of external and internal forces with the strength of each subsystem.

First step in modelling: understand the response mechanisms of vulnerable subsystems.


Catalog of damage mechanisms due to earthquakes:

DAMAGE DUE TO INSUFFICIENT QUALITY OF DAMAGE DUE TO INSUFFICIENT QUALITY OF MASONRY (AS TYPICAL IN DOUBLEMASONRY (AS TYPICAL IN DOUBLE--LEAF WALLSLEAF WALLS



OUTOUT--OFOF--PLANE INSTABILITY OF PLANE INSTABILITY OF DOUBLEDOUBLE--LEAF WALLSLEAF WALLS



GLOBAL OVERTURNING OF FAGLOBAL OVERTURNING OF FAÇÇADESADES



GLOBAL OVERTURNING OF FAGLOBAL OVERTURNING OF FAÇÇADESADES


Catalog of damage mechanisms due to earthquakes:OVERTURNING OF FAOVERTURNING OF FAÇÇADES ADES CARRYING OVER CORNER “WEDGES”CARRYING OVER CORNER “WEDGES”



PARTIAL OVERTURNING OF FAPARTIAL OVERTURNING OF FAÇÇADESADES


Catalog of damage mechanisms due to earthquakes:OVERTURNING OF FAOVERTURNING OF FAÇÇADESADES


Catalog of damage mechanisms due to earthquakes:PARTIAL OVERTURNING OF PARTIAL OVERTURNING OF FAFAÇÇADES: EFFECT OF OPENINGSADES: EFFECT OF OPENINGS


Catalog of damage mechanisms due to earthquakes:PARTIAL OVERTURNING OF PARTIAL OVERTURNING OF FAFAÇÇADES: EFFECT OF OPENINGSADES: EFFECT OF OPENINGS



PARTIAL OVERTURNING OF PARTIAL OVERTURNING OF FAFAÇÇADES: EFFECT OF OPENINGSADES: EFFECT OF OPENINGS



DAMAGE DUE TO THRUST FROM DAMAGE DUE TO THRUST FROM ROOF STRUCTUREROOF STRUCTURE









RIGID DIAPHRAGMS AND R.C. BEAMS RIGID DIAPHRAGMS AND R.C. BEAMS SOMETIMES NOT EFFECTIVE IN SOMETIMES NOT EFFECTIVE IN PREVENTING DAMAGE OF WALLS PREVENTING DAMAGE OF WALLS



LOCAL DAMAGE IN WALL LOCAL DAMAGE IN WALL ––JOISTS JOISTS CONNECTIONS CONNECTIONS


Catalog of damage mechanisms due to earthquakes:LOCAL DAMAGE DUE TO LOCAL DAMAGE DUE TO POUNDING OF ADJACENT POUNDING OF ADJACENT BUILDINGSBUILDINGS


Mechanical approach to damage mechanisms: limit analysis

•Many of the collapse mechanisms are partial, in the sense that they involve specific sub-structures or components. •Collapse is due, most of the times, to loss of equilibrium rather than to the exceedance of some level of stress. •When horizontal acceleration are high enough to trigger a mechanism, it may be assumed that the different parts can be idealized as rigid bodies. •The lateral force capacity of the subsystem can be related to a corresponding acceleration.

•Static threshold resistance can be evaluated through limit analysis and the application of the principle of virtual work.


Principle of virtual work

If a system which is in equilibrium under the action of a set of externally applied forces is subjected to a virtual displacement(velocity), i.e. an infinitesimal displacement (velocity) pattern compatible with the system’s constraints, the total work (power)done by the set of forces will be zero, i.e.

the vanishing of the work done during a virtual displacement is equivalent to a statement of equilibrium

The principle of virtual work (PVW) is particularly useful whenthe structural systmem is complex, involving a number of interconnected bodies, in which the direct equilibration of forces may be difficult


Example of limit analysis using the PVW: out-of-plane analysis of a simple wall

PVW:

Ψ infinitesimal rotation of the lower body

Φ = Ψ× h1/h2 rotation of upper body

λ (P1δ1x +P2δ2x )- (P1δ1y +P2δ2y +S δNy)=0

λ (P1δ1x +P2δ2x )= P1δ1y +P2δ2y +S δNy

Wactive= Wresisting

λ= (P1δ1y +P2δ 2y +Sδ Ny )/ (P1δ 1x + P2δ2x )

find minimum λ

δλ /dx = 0 x λ min

P1

P2

λP1

λP2


By putting: Hx

xhHx

h )1( and 112

−==

1)1)(/(2

−++

=x

xxPSxHBλ

SSPx +

+= 21

By requiring that dλ/dx = 0 we obtain:

then :


In some simple cases it is possible to write directly equilibrium equations and evaluate λ without recurring to PWV:

Mres = Mactive(λ) = λMactive(λ=1)

λ = Mres /Mactive(λ=1)

in this case, for the two mechanisms below, equilibrium about A and about B are easily written, and two different values of λwill be found, λ1 and λ2.

The lowest of the two will be the “true” mechanism.


Factors influencing static threshold

•geometry and restraints of mechanism

•amount, spatial distribution and nature of vertical loads

•friction forces

•forces coming from devices such as tie-rods

•compression strength of masonry


Geometry, restraints of mechanism, spatial distribution and nature of vertical loads

•Geometry can be assumed on the basis of the knowledge of the seismic behaviour of similar structures or can be identified considering the presence of previous cracks;

•moreover, the quality of the connections between walls, the masonry texture (brickwork), the presence of tie-rods, the possible interactions with other parts of the building and with adjacent buildings have to be considered.

•The definition of the geometry and restraints is also strictly related to how vertical weigths and associated horizontal inertia forces are transferred to the walls.


Friction forces


Friction forces

(De Felice and Giannini, 2001)friction along toothed crack


Forces coming from tie rods

α0P1

P1

θ1



The effect of tie rods can be introduced as an external force whose value depends on displacement

α0P1

P1

θ1

F1



The effect of tie rods can be introduced as an external force whose value depends on displacement

α0P1

P1

θ1

F1

d*

a*

a0*

d0*d'0*

a'0

du*=0.4 d'0*

(a)

(b)

strength of anchorage attained

without tie rod

with tie rod

displacement


Effect of compressive strength of masonry

(a) (b)

SHIFTING OF HINGE

Wall subject to overturning:: (a) assuming infinite (high) compression strength; (b) with limited (low) compression strength: centre of vertical reaction

moves inwards


More complex mechanisms

bh

l

L

hs

T

Ts Qr

Qrs

Qr

Qf

Ti

1

2

i

n

φ

Qrs

Qfs

α

(D’Ayala and Speranza, 2002, Restrepo-Velez, 2004)


bh

l

L

hs

T

Qr

L1

L2

φ

Qr

Qf

1

2

i

n

L1 L2

α

(Restrepo-Velez, 2004)


(D’Ayala and Speranza, 2002)


Use of rigid-body analysis for seismic assessment

•Earlier uses of limit rigid-body limit analysis was made essentially on a comparative basis, to evaluate which part of the structure are most vulnerable, and to check the effect of strengthening techniques (e.g. insertion of tie-rods, of rigid diaphragms…) on out-of-plane mechanisms.

•The general concept is to have horizontal load multipliers for out-of-plane mechanisms which are higher than the global base shear coefficient of the building, corresponding to the strength associate to in-plane response of walls.

•A more recent approach (new Italian seismic code) proposes the use of rigid-body analysis within equivalent static assessment procedures which take into account, in an approximate way, the dynamic nature of the response.



1. definition of a s.d.o.f. mechanism and its kinematics, by idealizing the substructure as a set of rigid bodies which can slide/rotate, separated from each other by fracture lines.

2. evaluation of the static horizontal multiplier of vertical weights α0that corresponds to the static threshold resistance.

To this end, the following forces are applied to the system:

-the vertical self weights of the rigid blocks, applied at their centres of mass;

-the vertical loads carried by the walls transmitted by floors, roof, etc. ;

- a system of horizontal forces, proportional to the vertical weights, and to the loads carried by the walls, if the corresponding inertia forces are expected to be transferred to the walls which are part of the mechanism

- if present, external forces (e.g. from tie rods or from friction at boundaries);

- if present, internal forces (e.g. due to friction/interlocking among units).


Given a virtual rotation θk to the generic block k, it is possible to establish the corresponding virtual displacements of the points of application of all the forces along the respective directions.The value of α0 can be obtained using the Virtual Work Principle, in terms of displacements:

where n is the number of all the dead loads (weights, vertical forces) applied to the different rigid blocks of the mechanism, m is the number of weight forces not directly applied to the blocks, whose inertia forces will be transmitted to the blocks of the mechanism;o is the number of the external forces applied to the blocks but not related to considered masses;

fi

o

1hhh

n

1iiy,i

mn

1njjx,j

n

1iix,i0 LFPPP =δ−δ−⎟

⎟⎠

⎞⎜⎜⎝

⎛δ+δα ∑∑∑∑

==

+

+==


Pi is the generic weight force (block dead load, applied at its centroid, or other weight);Pj is the generic weight force, not directly applied to the blocks, whose mass generates seismic horizontal forces on the elements of the kinematical chain, because not effectively transferred to other parts of the building;δx,i is the horizontal virtual displacement of the point of application of Pi, assuming as positive the positive direction of the considered seismic action; δx,j is the horizontal virtual displacement of the point of application of Pj, assuming as positive the positive direction of the considered seismic action; δy,i is the vertical virtual displacement of the point of application of Pi, assuming as positive if upwards; Fh is the generic external force (absolute value), applied to the block;δh is the virtual displacement of the point of application of Fh , positive if opposite;Lfi is the work of internal forces.

fi

o

1hhh

n

1iiy,i

mn

1njjx,j

n

1iix,i0 LFPPP =δ−δ−⎟

⎟⎠

⎞⎜⎜⎝

⎛δ+δα ∑∑∑∑

==

+

+==


3. Definition of an equivalent s.d.o.f. system with the following characteristics:

∑

∑+

=

+

=

δ

⎟⎟⎠

⎞⎜⎜⎝

⎛δ

= mn

1i

2x,ii

2mn

1ix,ii

*

Pg

PM *

0*

mn

1ii0

*0 e

gM

Pa

α=

α=

∑+

= ∑+

==

mn

1ii

** P/gMe

effective mass ratio

effective threshold acceleration effective mass

4’. Simplified “linear” static safety check (ultimate limit state):

⎟⎠⎞

⎜⎝⎛ +≥

HZ5.11

qSa

a g*0 with q = 2.0


Non linear static methodology

4’’. Non linear static safety check (alternative to linear)

4’’a. Evaluate the evolution of the horizontal multiplier α by progressively increasing the displacement dk of a control point, chosen as suitable by the designer, until the horizontal multiplier reaches the value of zero;

Note: when vertical forces are constant and horizontal forces are only proportional to vertical weights, the relationship between α and dk is approximately linear and can be expressed as

( )0,kk0 d/d1−α=α

where dk,o is the displacement corresponding to zero horizontal force.


4’’b. Evaluate the displacement of the equivalent s.d.o.f. system as:

an plot the a* – d* curve.

∑

∑+

=

+

=

δ

δ= mn

1iikx,

mn

1ix,ii

k*

P

Pdd

d*

a*

a0*

d0*d'0*

a'0

du*=0.4 d'0*

(a)

(b)

(a) with variable external forces

(b) linear


4’’b. The ultimate displacement du* is evaluated conventionally as the

lesser of:- 40% of the displacement at zero force- the displacement limit corresponding to locally incompatible conditions (e.g. unseating of floor joists…)

d*

a*

a0*

d0*d'0*

a'0

du*=0.4 d'0*

(a)

(b)

(a) with variable external forces

(b) linear


4’’c. Define effective secant period at 0.4 du* on capacity curve as:

*

** 2

s

ss a

dT π=



4’’c. Calculate displacement demand ∆d with the following elastic response spectrum, and compare with displacement capacity:

( )( )

2s

s 1 d s g 2 2s 1

1 s1 s D d s g 2

1 DD s d s g 2

3 1 Z HTT < 1.5T ∆ (T ) = a S 0 54 1 1 T T

1 5T T Z1.5T T < T ∆ (T ) =a S 1 9 2 4H4

1 5T T ZT T ∆ (T ) =a S 1 9 2 4H4

.

. . .

. . .

⎛ ⎞+⎜ ⎟−⎜ ⎟π + −⎝ ⎠

⎛ ⎞≤ +⎜ ⎟π ⎝ ⎠

⎛ ⎞≤ +⎜ ⎟π ⎝ ⎠

where Z is the height, with respect to ground, of the centroid of all inertial masses involved in the mechanismH is the total height of the buildingall other parameters (ag, S, TD) are as specified in design acceleration spectra.

Lesson14

Technology

catalog of damage mechanisms

c masonry structures

isostatic structures

local damage

collapse mechanisms

corner wallsmasonry

specific substructures

typical distribution