lesson9

Masonry Structures, lesson 8 slide 1

Seismic design and assessment ofMasonry Structures

Seismic design and assessment ofMasonry Structures

Lesson 9October 2004


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


shears and moments





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.





shears and moments

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





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)


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


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)


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 .


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.


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


Equivalent frame with rigid offsets: example


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


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


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 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


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


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.


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 ,


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 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)


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


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


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:

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


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)

lesson9

Documents

vertical structures

example masonry structures

board masonry structures

masonry structures t

important structures

d viewmasonry structures

pattern cmasonry structures

reinforced masonry walls