Lesson14

Seismic design and assessment of
Seismic design and assessment of
Masonry Structures
Masonry Structures

Lesson 14
October 2004

Masonry Structures, lesson 14 slide 1

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

(Carocci et al., 1993)


PROGRESSIVE GROWTH STOREYS ADDED TO
IN PLAN EXISTING BUILDINGS

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

Cells A and B
built after C


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
quot;…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 partquot;.
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
MASONRY (AS TYPICAL IN DOUBLE-LEAF WALLS
DOUBLE-



OUT-OF-PLANE INSTABILITY OF
OUT- OF-
DOUBLE-LEAF WALLS
DOUBLE-



GLOBAL OVERTURNING OF FAÇADES
FAÇ


GLOBAL OVERTURNING OF FAÇADES
FAÇ


OVERTURNING OF FAÇADES
FAÇ
CARRYING OVER CORNER “WEDGES”



PARTIAL OVERTURNING OF FAÇADES
FAÇ


OVERTURNING OF FAÇADES
FAÇ


PARTIAL OVERTURNING OF
FAÇADES: EFFECT OF OPENINGS
FAÇ


FAÇ


FAÇ


DAMAGE DUE TO THRUST FROM
ROOF STRUCTURE


ROOF STRUCTURE


ROOF STRUCTURE


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


LOCAL DAMAGE IN WALL –JOISTS
CONNECTIONS


LOCAL DAMAGE DUE TO
POUNDING OF ADJACENT
BUILDINGS


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 when
the 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
λP2
Φ = Ψ× h1/h2 rotation of upper body
P2
λ (P1δ1x +P2δ2x )- (P1δ1y +P2δ2y +S δNy)=0
λ (P1δ1x +P2δ2x )= P1δ1y +P2δ2y +S δNy
Wactive= Wresisting
λP1
P1 λ= (P1δ1y +P2δ 2y +Sδ Ny )/ (P1δ 1x + P2δ2x )
find minimum λ
δλ /dx = 0 x λ min


By putting: h2 = 1 H ( x − 1)
and h1 = H
x x

B 2 x + ( S / P )( x + 1) x
then : λ=
H x −1

By requiring that dλ/dx = 0 we
obtain:

P+S
x = 1+ 2
S


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 λ
will 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

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


Forces coming from tie rods

α0P1
P1

θ1



F1

α0P1
P1

θ1

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



F1
a* strength of anchorage attained

a0* (a)
α0P1 with tie rod
a'0
P1
(b)

θ1
without tie rod
d *=0.4 d' *
u 0 d* d'0* d0*
displacement

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

Q rs Qr

Q rs Qf
Ts Q Q fs
r

T 1

hs
2 φ
b Ti
h
l

i α

L n

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


Qr

Qr Qf
T
1
φ
hs
2
b
h
l i

α
L L1 n

L2 L1 L2

(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 α0
that 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:
⎛ n n+m ⎞ n o
α 0 ⎜ ∑ Pi δ x,i + ∑ P j δ x, j ⎟ − ∑ Pi δ y,i − ∑ Fh δ h = L fi
⎜ i =1 ⎟ i =1
⎝ j= n +1 ⎠ h =1
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;


⎛ n n+m ⎞ n o
α 0 ⎜ ∑ Pi δ x,i + ∑ P j δ x, j ⎟ − ∑ Pi δ y,i − ∑ Fh δ h = L fi
⎜ i =1 ⎟ i =1
⎝ j= n +1 ⎠ h =1

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.


3. Definition of an equivalent s.d.o.f. system with the following
characteristics:
2
⎛ n +m ⎞ n+m
⎜ ∑ Pi δ x,i ⎟
⎜ ⎟ α0 ∑ Pi α 0g n +m
M* = ⎝ n+m ⎠
∑ Pi
i =1 i =1
a* =
0 *
= e* = gM* /
M e*
g ∑ Pi δ 2
x,i
i =1

i =1
effective threshold effective mass
effective mass acceleration ratio

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

a gS ⎛ Z⎞
a* ≥
0 ⎜1 + 1.5 ⎟ with q = 2.0
q ⎝ H⎠


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

α = α 0 (1 − d k / d k ,0 )

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:

a* (a) with variable external forces
n+m
∑ Pi δ x,i a 0*
(b) linear
*
d = dk i =1 (a)
n +m
a'0
δ x,k ∑ Pi
i =1
(b)

du*=0.4 d'0* d* d'0* d0*

an plot the a* – d* curve.


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…)

a* (a) with variable external forces

a0* (a) (b) linear
a'0

(b)

du*=0.4 d'0* d* d'0* d0*


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

d s*
Ts* = 2π *
as



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

Ts2 ⎛ 3 (1 + Z H ) ⎞
Ts < 1.5T1 ∆ d (Ts ) = a g S ⎜ − 0.5 ⎟
4π ⎜ 1 + (1 − Ts T1 )
2 2
⎟
⎝ ⎠
1.5T1Ts ⎛ Z⎞
1.5T1 ≤ Ts < TD ∆ d (Ts ) =a g S ⎜ 1.9 + 2.4 ⎟
4π 2 ⎝ H⎠
1.5T1TD ⎛ Z⎞
TD ≤ Ts ∆ d (Ts ) =a g S ⎜ 1.9 + 2.4 ⎟
4π ⎝
2
H⎠

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


Lesson14

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