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Dan Abrams + Magenes Course on Masonry
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Masonry Structures, lesson 8 slide 1
Seismic design and assessment ofMasonry Structures
Seismic design and assessment ofMasonry Structures
Lesson 9October 2004
Masonry Structures, lesson 8 slide 2
Vertical structures: degree of coupling
(a) (b)
(c)
Role of coupling provided by floors and/or spandrel beams
deflected shape and crack pattern
shears and momentsdeflected shape and
crack patternshears and moments
deflected shape and crack pattern
shears and moments
Masonry Structures, lesson 8 slide 3
Vertical structures: degree of coupling
Role of coupling provided by floors and/or spandrel beams
deflected shape and crack pattern
shears and moments
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.
Masonry Structures, lesson 8 slide 4
Vertical structures: degree of coupling
Role of coupling provided by floors and/or spandrel beams
deflected shape and crack pattern
shears and moments
Piers weaker than spandrels: e.g. unreinforced masonry walls w. r.c. slabs and masonry spandrels/deep beams.
Masonry Structures, lesson 8 slide 5
Vertical structures: degree of coupling
Role of coupling provided by floors and/or spandrel beams
deflected shape and crack pattern
shears and moments
Spandrels weaker than piers: e.g. reinforced masonry walls w. r.c. slabs and masonry spandrels/deep beams (similar to coupled r.c. walls)
Masonry Structures, lesson 8 slide 6
Vertical structures: modelingSome possible modelling approaches for multistorey masonry walls
a) cantilever model b) equivalent frame c) equivalent frame with rigid offsets
d) 2-D or 3-D finite element modelling
Masonry Structures, lesson 8 slide 7
Vertical structures: modelingCantilever 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)
Masonry Structures, lesson 8 slide 8
Vertical structures: modelingEquivalent 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 .
Masonry Structures, lesson 8 slide 9
Vertical structures: modelingEquivalent frame, with strong spandrels:
Frame analogy can be simplified for urm buildings with rigid andstrong 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.
Masonry Structures, lesson 8 slide 10
Equivalent frame with rigid offsets: example
H1
Heff
H2
offsetrigid
offsetrigid
deformable length
i
i'
j
j'
height y interstore free H
)/h'h'HD(h'H 31
eff
=
−+=
spandrel beam
pier
joint
F1
F2
Masonry Structures, lesson 8 slide 11
Equivalent frame with rigid offsets: example
Masonry Structures, lesson 8 slide 12
Equivalent frame with rigid offsets: 3-d modeling
nodo
nodo
braccio rigido
cerniera
braccio rigido Pier element
Spandrel element
R.c. beam elem.
Rigid offset
FRONT VIEW
PLAN
jointrigid offset
joint
hinge
rigid offset
Masonry Structures, lesson 8 slide 13
Equivalent frame with rigid offsets: 3-d modeling
17,7
14,3
0,75 0,3
0,6
letto
letto
pranzocucina
bagno
piano rialzato
Four-storey urm existing building
plan
Masonry Structures, lesson 8 slide 14
Example of linear elastic frame model with commercial software
Structural model - Plan
Masonry Structures, lesson 8 slide 15
Example of linear elastic frame model with commercial software
Structural model – 3D view
Masonry Structures, lesson 8 slide 16
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 notnecessarily 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
Masonry Structures, lesson 8 slide 17
Seismic resistance verification of masonry buildings
As will be seen in next lessons, seismic resistanceverification of masonry buildings can in principle becarried out using different methods of analysis:
- linear static
- linear dynamic (modal analysis)
- nonlinear static
- nonlinear dynamic
Masonry Structures, lesson 8 slide 18
Seismic resistance verification of masonry buildings
In most cases, for masonry structures there is no need forsophisticated 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 ofgravity loads.
Masonry Structures, lesson 8 slide 19
Linear elastic analysis, equivalent static procedure
ii. Using appropriate mathematical models, the stiffness of individual walls in each storey is calculated. The stiffness matrixof 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.
∑=
jjj
iidbi sW
sWFF ,
Masonry Structures, lesson 8 slide 20
vi. The storey shear is distributed among the walls according to the structural model adopted and the design values of action effectsare 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.
Linear elastic analysis, equivalent static procedure
Masonry Structures, lesson 8 slide 21
•Masonry buildings were among the first structures in which the need for nonlinear analysis methods was felt in practical design/assessmentprocedures.
•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 analysis, equivalent static procedure (a.k.a. pushover)
Masonry Structures, lesson 8 slide 22
Nonlinear behaviour of a masonry wall (pier)
V
δδ e δu
Vu
Vmax
0,75Vu 0,8V u
cyclic envelope
Kel
Possible bi-linear idealization
Masonry Structures, lesson 8 slide 23
Nonlinear behaviour of a masonry wall (pier)
V
δδ e δu
Vu
Vmax
0,75Vu 0,8V u
cyclic envelope
Kel
Ultimate deflection capacity for masonry piers
Earlier proposals based on ductility
(δu = µu δu ) without reference to failure mode.
e.g. : µu= 2.0-3.0 for urm
µu= 3.0-4.0 for confined masonry
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
θu = 0.8-1.2 % for urm failing in flexure/rockingh
Masonry Structures, lesson 8 slide 24
Storey mechanism idealization
Masonry Structures, lesson 8 slide 25
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:
uixixxiix
uixixeixix
uixxiuixix
eixixelasticxixiixxiix
uuKV
uuuu
VKVV
uuKKuKV
,
,,,
,
,,
if 0 ;0
; if ;
; if ;
>==
≤<==
≤==
Stiffness of wall i :
Center of rigidity:
θθθθ =−⋅+=−⋅−= iRiRyiyRiRxix xxuuyyuu ; )( ; )(
∑∑
∑∑ ⋅
=
⋅
=
ixi
iixi
R
iyi
iiyi
R K
yKy
K
xKx ;
etc.Iterations must be carried out until equilibrium is satisfied at each displacement increment
Masonry Structures, lesson 8 slide 26
Example of nonlinear storey envelope:
Forza alla base-Spostamento
0
200
400
600
800
1000
1200
0 0.01 0.02
Spostamento [m]
Forz
a [K
N]
T ET TO
1° PIANO
When a relatively large number of walls is present, as in most buildings, the storey envelope has a smooth transition from elastic to ultimate.
In general, internal forces distribution at ultimate is governed by strength of walls, not by elastic stiffness, even when a limited inelastic deformation capacity of piers is assumed.
Interstorey displacement at centre of mass (m)
Interstorey shear(kN)