Lesson9

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

Lesson 9
October 2004

Masonry Structures, lesson 8 slide 1

Vertical structures: degree of coupling
Role of coupling provided by floors and/or spandrel beams

deflected shape and shears and moments
deflected shape shears and moments crack pattern
and crack pattern
(b)
(a)

crack pattern

(c)



Cantilever walls with floor
slabs: e.g. reinforced masonry
walls heavily reinforced, where
out-of-plane stiffness/strength
of floor slabs/ring beams is
negligible compared to
cantilever walls.
crack pattern



Piers weaker than
spandrels: e.g. unreinforced
masonry walls w. r.c. slabs
and masonry spandrels/deep
beams.

deflected shape and crack shears and moments
pattern



Spandrels weaker than
piers: e.g. reinforced
masonry walls w. r.c. slabs
and masonry spandrels/deep
beams (similar to coupled
r.c. walls)

deflected shape and crack shears and moments
pattern


Vertical structures: modeling
Some possible modelling approaches for multistorey masonry walls

a) cantilever model b) equivalent frame c) equivalent frame d) 2-D or 3-D finite
with rigid offsets element modelling


Cantilever model:

Is the most conservative type
of modeling. Traditionally
used for analysis under wind
loads.
For seismic loading and
elastic analysis in the great
majority of the cases will
give very penalizing results
for the designer, especially
for unreinforced masonry
(sketch on board)


Equivalent frame, with or without rigid offsets:

Can be applied both 2-D and 3-D
modeling. Tends to give a more
realistic picture of the response. It is
more complex because it requires the
definition of the stiffness/strength
characteristics of horizontal coupling
elements (ring beam, spandrels…).
The use of rigid offsets can be
appropriate to limit the
deformability of horizontal elements.
Horizontal elements are structural,
their strength should be verified .


Equivalent frame, with strong spandrels:

Frame analogy can be simplified for urm buildings with rigid and
strong spandrels and r.c. floor slabs or ring beams. Flexural
capacity of the walls section is so low that piers may be considered
symmetrically fixed at top and bottom.


Equivalent frame with rigid offsets: example

spandrel beam rigid
F2 offset i
H1
pier i'

F1
deformable
joint length Heff

rigid
j'
offset H2

j

H eff = h'+ 1 D( H − h' )/h'
3

H = free interstore y height


Equivalent frame with rigid offsets: example


Equivalent frame with rigid offsets: 3-d modeling

nodo
joint
braccio
rigid
rigido braccio rigido
joint
nodo rigid offset Pier element
offset
Spandrel element
cerniera
hinge R.c. beam elem.

Rigid offset

FRONT VIEW

PLAN


Equivalent frame with rigid offsets: 3-d modeling

Four-storey urm existing
building

piano rialzato
plan
0,6

0,75 letto 0,3

letto

bagno
14,3

pranzo
cucina

17,7


Example of linear elastic frame model with commercial software

Structural model - Plan


Example of linear elastic frame model with commercial software

Structural model – 3D view


Refined 2-d or 3-d finite element modeling

Refined finite element modeling could be needed:
-In linear elastic analysis, when geometry is rather complicated and no
equivalent frame idealization is possible; its use in terms of stress
evaluation is questionable, since local elastic stresses are not
necessarily related to safety w. respect to collapse of the structure.
-In practice in linear elastic models the integration of the stresses to
obtain forces and moments is often needed to perform safety checks
according to design codes.
- In nonlinear analysis for important structures (e.g. monuments)
provided suitable constitutive models are used
Full nonlinear 3-d f.e.m. modeling of whole buildings is still far from
being a usable tool in real practice


Seismic resistance verification of masonry buildings

As will be seen in next lessons, seismic resistance
verification of masonry buildings can in principle be
carried out using different methods of analysis:

- linear static

- linear dynamic (modal analysis)
- nonlinear static
- nonlinear dynamic


Seismic resistance verification of masonry buildings

In most cases, for masonry structures there is no need for
sophisticated dynamic analyses for seismic resistance verification.
An equivalent static analysis (linear on non linear) can often be be
adequate. In this lesson, attention will be focused on static analysis.
The calculation procedure depends on whether linear or non linear
methods are used for assessing the seismic action effects.
The typical procedure for linear analysis and seismic resistance
verification consists of a series of calculation and steps that are in
general common to all design/assessment codes.

i. The weight of the building, concentrated at floor levels, is
determined by taking into account the suitable combination of
gravity loads.


Linear elastic analysis, equivalent static procedure

ii. Using appropriate mathematical models, the stiffness of
individual walls in each storey is calculated. The stiffness matrix
of the entire structure is evaluated.
iii. The period of vibration T is calculated when necessary and the
ordinate of the design response spectrum Sd(T) is determined.
iv. Assuming that Sd(T) is normalized w. respect to gravity
acceleration, the design base shear is determined as Fb,d = Sd(T)W
where W is the weight of the seismic masses.
v. The base shear is distributed along the height of the building
according to a specified rule, derived from a predominant first-
mode response, e.g.
Wi si
Fi = Fb ,d
j
∑W j s j


Linear elastic analysis, equivalent static procedure

vi. The storey shear is distributed among the walls according to the
structural model adopted and the design values of action effects
are calculated combining seismic loading and other actions (dead
load, variable loads….)
vii. Finally the design resistance of wall sections is calculated and
compared to the design action effects.


Nonlinear analysis, equivalent static procedure (a.k.a. pushover)

•Masonry buildings were among the first structures in which the need
for nonlinear analysis methods was felt in practical design/assessment
procedures.
•Simplified nonlinear static procedures were developed and adopted in
some national codes in Europe as early as the late Seventies, after the
Friuli 1976 earthquake.
•These procedures were based on the concept of “storey mechanism”,
in which it is assumed that collapse or ultimate limit state of the
structure is due by a shear-type failure of a critical storey.
•The bases of this method are also useful to introduce further
developments in nonlinear modeling and nonlinear static procedures
as defined in most recent codes.


Nonlinear behaviour of a masonry wall (pier)

Possible bi-linear idealization

V cyclic envelope
Vmax
Vu
0,75Vu
0,8V u
K
el

δe δu δ


Nonlinear behaviour of a masonry wall (pier)

Ultimate deflection capacity for masonry piers

Earlier proposals based on ductility V cyclic envelope
Vmax

(δu = µu δu ) without reference to failure mode. Vu
0,75Vu
e.g. : µu= 2.0-3.0 for urm 0,8V u
K
el
µu= 3.0-4.0 for confined masonry
δe δu δ
µu= 4.0-5.0 for reinforced masonry

More recent proposals based on drift (θ= δ/h) limits:
e.g. : θu = 0.4-0.5 % for urm failing in shear
h
θu = 0.8-1.2 % for urm failing in flexure/rocking


Storey mechanism idealization


Storey mechanism idealization (3-d, assuming rigid floor)

The method can be implemented by progressively increasing the displacement of the
center of the seismic force C, and applying the equations developed for the elastic case,
considering a modified stifness for each pier as follows:

u ix = u Rx − θ ⋅ ( y i − y R ) ; u iy = u Ry + θ ⋅ ( xi − x R ) ; θi =θ
Stiffness of wall i :

Vix = K xi uix ; K xi = K xi ,elastic if uix ≤ uix ,e ;
Vix ,u
Vix = Vix ,u ; K xi = if uix ,e < uix ≤ uix ,u ;
uix
Vix = 0; K xi = 0 if uix > uix ,u

Center of rigidity:

∑ K yi ⋅ xi ∑ K xi ⋅ yi
xR = i ; yR = i
∑ K yi ∑ K xi
i i Iterations must be carried out until equilibrium is satisfied
etc. at each displacement increment


Example of nonlinear storey envelope:

When a relatively large number 1200
Forza alla base-Spostamento
T ET T O
of walls is present, as in most 1° PIANO

buildings, the storey envelope 1000

has a smooth transition from 800
Interstorey shear (kN)

elastic to ultimate.
Forza [KN]

600

In general, internal forces
distribution at ultimate is 400

governed by strength of walls, 200

not by elastic stiffness, even
when a limited inelastic 0
0 0.01 0.02

deformation capacity of piers is Spostamento [m]
Interstorey displacement at centre of mass (m)

assumed.


Lesson9

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (18)

Andere mochten auch

Andere mochten auch (17)

Mehr von Teja Ande

Mehr von Teja Ande (8)

Lesson9