lesson14
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Dan Abrams + Magenes Course on MasonryTRANSCRIPT
Masonry Structures, lesson 14 slide 1
Seismic design and assessment ofMasonry Structures
Seismic design and assessment ofMasonry Structures
Lesson 14October 2004
Masonry Structures, lesson 14 slide 2
The reality of most historical centers: what is “the building”?
(Carocci et al., 1993)
Masonry Structures, lesson 14 slide 3
(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
Masonry Structures, lesson 14 slide 4
Damage in most vulnerable buildings:
Tyipically, partial overturning mechanisms of façade walls or corner walls
Masonry Structures, lesson 14 slide 5
Typical distribution of damage in historical
centers.
Masonry Structures, lesson 14 slide 6
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
Masonry Structures, lesson 14 slide 7
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.
Masonry Structures, lesson 14 slide 8
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
Masonry Structures, lesson 14 slide 9
Catalog of damage mechanisms due to earthquakes:
OUTOUT--OFOF--PLANE INSTABILITY OF PLANE INSTABILITY OF DOUBLEDOUBLE--LEAF WALLSLEAF WALLS
Masonry Structures, lesson 14 slide 10
Catalog of damage mechanisms due to earthquakes:
GLOBAL OVERTURNING OF FAGLOBAL OVERTURNING OF FAÇÇADESADES
Masonry Structures, lesson 14 slide 11
Catalog of damage mechanisms due to earthquakes:
GLOBAL OVERTURNING OF FAGLOBAL OVERTURNING OF FAÇÇADESADES
Masonry Structures, lesson 14 slide 12
Catalog of damage mechanisms due to earthquakes:OVERTURNING OF FAOVERTURNING OF FAÇÇADES ADES CARRYING OVER CORNER “WEDGES”CARRYING OVER CORNER “WEDGES”
Masonry Structures, lesson 14 slide 13
Catalog of damage mechanisms due to earthquakes:
PARTIAL OVERTURNING OF FAPARTIAL OVERTURNING OF FAÇÇADESADES
Masonry Structures, lesson 14 slide 14
Catalog of damage mechanisms due to earthquakes:OVERTURNING OF FAOVERTURNING OF FAÇÇADESADES
Masonry Structures, lesson 14 slide 15
Catalog of damage mechanisms due to earthquakes:PARTIAL OVERTURNING OF PARTIAL OVERTURNING OF FAFAÇÇADES: EFFECT OF OPENINGSADES: EFFECT OF OPENINGS
Masonry Structures, lesson 14 slide 16
Catalog of damage mechanisms due to earthquakes:PARTIAL OVERTURNING OF PARTIAL OVERTURNING OF FAFAÇÇADES: EFFECT OF OPENINGSADES: EFFECT OF OPENINGS
Masonry Structures, lesson 14 slide 17
Catalog of damage mechanisms due to earthquakes:
PARTIAL OVERTURNING OF PARTIAL OVERTURNING OF FAFAÇÇADES: EFFECT OF OPENINGSADES: EFFECT OF OPENINGS
Masonry Structures, lesson 14 slide 18
Catalog of damage mechanisms due to earthquakes:
DAMAGE DUE TO THRUST FROM DAMAGE DUE TO THRUST FROM ROOF STRUCTUREROOF STRUCTURE
Masonry Structures, lesson 14 slide 19
Catalog of damage mechanisms due to earthquakes:
DAMAGE DUE TO THRUST FROM DAMAGE DUE TO THRUST FROM ROOF STRUCTUREROOF STRUCTURE
Masonry Structures, lesson 14 slide 20
Catalog of damage mechanisms due to earthquakes:
DAMAGE DUE TO THRUST FROM DAMAGE DUE TO THRUST FROM ROOF STRUCTUREROOF STRUCTURE
Masonry Structures, lesson 14 slide 21
Catalog of damage mechanisms due to earthquakes:
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
Masonry Structures, lesson 14 slide 22
Catalog of damage mechanisms due to earthquakes:
LOCAL DAMAGE IN WALL LOCAL DAMAGE IN WALL ––JOISTS JOISTS CONNECTIONS CONNECTIONS
Masonry Structures, lesson 14 slide 23
Catalog of damage mechanisms due to earthquakes:LOCAL DAMAGE DUE TO LOCAL DAMAGE DUE TO POUNDING OF ADJACENT POUNDING OF ADJACENT BUILDINGSBUILDINGS
Masonry Structures, lesson 14 slide 24
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.
Masonry Structures, lesson 14 slide 25
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
Masonry Structures, lesson 14 slide 26
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
Masonry Structures, lesson 14 slide 27
By putting: Hx
xhHx
h )1( and 112
−==
1)1)(/(2
−++
=x
xxPSxHBλ
SSPx +
+= 21
By requiring that dλ/dx = 0 we obtain:
then :
Masonry Structures, lesson 14 slide 28
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.
Masonry Structures, lesson 14 slide 29
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
Masonry Structures, lesson 14 slide 30
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.
Masonry Structures, lesson 14 slide 31
Friction forces
Masonry Structures, lesson 14 slide 32
Friction forces
(De Felice and Giannini, 2001)friction along toothed crack
Masonry Structures, lesson 14 slide 33
Forces coming from tie rods
α0P1
P1
θ1
Masonry Structures, lesson 14 slide 34
Forces coming from tie rods
The effect of tie rods can be introduced as an external force whose value depends on displacement
α0P1
P1
θ1
F1
Masonry Structures, lesson 14 slide 35
Forces coming from tie rods
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
Masonry Structures, lesson 14 slide 36
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
Masonry Structures, lesson 14 slide 37
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)
Masonry Structures, lesson 14 slide 38
bh
l
L
hs
T
Qr
L1
L2
φ
Qr
Qf
1
2
i
n
L1 L2
α
(Restrepo-Velez, 2004)
Masonry Structures, lesson 14 slide 39
(D’Ayala and Speranza, 2002)
Masonry Structures, lesson 14 slide 40
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.
Masonry Structures, lesson 14 slide 41
Use of rigid-body analysis for seismic assessment
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).
Masonry Structures, lesson 14 slide 42
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 ∑∑∑∑
==
+
+==
Masonry Structures, lesson 14 slide 43
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 ∑∑∑∑
==
+
+==
Masonry Structures, lesson 14 slide 44
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
Masonry Structures, lesson 14 slide 45
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.
Masonry Structures, lesson 14 slide 46
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
Masonry Structures, lesson 14 slide 47
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
Masonry Structures, lesson 14 slide 48
4’’c. Define effective secant period at 0.4 du* on capacity curve as:
*
** 2
s
ss a
dT π=
Masonry Structures, lesson 14 slide 49
Use of rigid-body analysis for seismic assessment
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.