2-d seismic nonlinear analyses

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reliable, a structural analysis performed on every wall as separated by the remaining structure andunder the action of vertical and seismic loads directly applied is more realistic and, at least, must

accomplish a global analysis.

For this reason, here the main façade of the palace is taken into consideration (see Figure 1). A

detailed comparison between three different 2-D non linear models is discussed. The first model

relies on a equivalent frame approach, in agreement with the OPCM 3274 [2] requirements. Thesecond model consists in a 2D elasto-plastic with damaging FE analysis [3] conducted by means

of the ADINA FE commercial code. Finally, in the third model an analysis at collapse is

conducted by means of a homogenised limit analysis model [4] [5]. As authors experienced, all the

models give both similar failure mechanisms and total ultimate shear at the base.

2. DESCRIPTION OF THE MODELS

For the analysis, three different models are investigated. The first one is based on an

“equivalent frame” approach in accomplishment with the new Italian Seismic Code [2]. A FEmodel is built considering each wall pier as a beam element in which the in-plane behaviour is

modelled as elastic-perfectly plastic. In particular, the OPCM takes into consideration two main

failure conditions: “in-plane rocking” failure and “shear failure with diagonal cracking”. The

evaluation of the rocking strength is based on a simple equilibrium statement which neglects the

tensile resistance of the bed joints, whereas the shear strength is evaluated according to the

Turnsek Kacovic formulation [6]. The type of failure occurring depends on the first limit value

which is reached, either ultimate bending moment or ultimate shear.The horizontal masonry element of a wall, called spandrel wall, can be modelled in the same

way of the piers; such beams are usually placed at the floor level. Rigid offsets can be introduced

at the intersection between piers and spandrel walls in order to take into account the higher

stiffness of the masonry.

The non-linear analysis has been stopped when one pier has reached its ultimate displacement.

The displacement of a control point (point A in Figure 1) versus the corresponding total shear isdiagrammed in the so-called capacity curve. The results are obtained by using the commercial

software Aedes Pce 5.0. The equivalent frame model is shown in Figure 2.

Figure 2: Equivalent frame model.

The Young modulus E , the cohesiont

0 and the compressive strength are respectively 1650,0.12 and 2.2, all values in MPa. The loads considered in the analysis are the vertical actions (self-

weight and live load) and the seismic action. The seismic action is introduced by the application of

a set of equivalent horizontal loads distributed linearly along the height.

In the second analysis, a two-dimensional homogeneous plane stress model is developed in the

software ADINA context. The numerical model chosen is the model described in [3] which

basically employs two features, a strain softening for the weakening of the material under

compression, failure envelopes which involve cracking in tension and crushing in compression.

The adopted mesh is represented in Figure 3 where eight-nodes isoparametric elements are

involved. The mechanical properties used in the analysis are the following: initial E equal to 1650



MPa, crushing stress limit f c equal to 2.2 MPa, crushing strain ec equal to 0.002, ultimate stress f u equal to 1.8 MPa, ultimate strain eu equal to 0.0035, tensile limit f t equal to 0.1 MPa, Poisson ratio

equal to 0.2.

Figure 3: Adina discretisation.

The last analysis is based on an upper bound limit analysis approach combined with

homogenisation. The anisotropic homogenized failure surface for masonry is provided making use

of an admissible and equilibrated model originally presented in [4]. For the example at hand,

bricks are assumed of dimensions 55x250x120 mm and infinitely resistant, whereas joints are

assumed 10 mm thick with a Mohr-Coulomb failure criterion (cohesion 1.0=c MPa and friction

angle °=Φ 20 ) with compressive cut-off 2.2=cf MPa. The structural kinematic model is based

on the formulation by Sloan and Kleeman [7], consisting in a triangular discretization of the 2D

domain, Figure 4, and in the introduction of discontinuities of the velocity field along the edges of

adjacent triangles. For each element, two velocity unknowns per node are introduced, i.e. the

velocity field is assumed linear inside each element, with constant strain rate field. Jumps of

velocities at interfaces are supposed to vary linearly. It can be shown that the problem can be set asa linear programming in which the internal dissipated power is minimized. The final collapse

multiplier furnishes the maximum percentage of the vertical load beared by the structure as

seismic action.

Figure 4: Homogenised limit analysis discretisation

3. NUMERICAL RESULTS

In Figure 5 the force-displacement curve, i.e. the total

shear force versus the control point displacement, is presented

for the described models. The correspondent failure

mechanisms are presented in Figures 6, 7, 8. The results of the

models C and L present common features in terms both of

ultimate strength and of failure mechanism. Such a collapsemechanism is determined by the rocking of the first three piers

with the consequent failure of the spandrel walls. The ultimate

shear values are: 6000 kN for the model C, 5640 kN for the

model L. On the other hand, the results related to the model F

show that the ultimate strength is 20% lower (equal to 4500

kN) and that the behaviour results to be less fragile. The curve

Figure 5: Capacity curves

F = equivalent frame model

C = continuous model

L = limit analysis.



is stopped when the first pier reaches its ultimate displacement; such a limit value occurs in thefirst left pier. Furthermore, the value of the ultimate strength depends strongly on the size of the

rigid offset assumed. In the present analysis the rigid zone is spanned on the entire intersection

pier-spandrel as shown in Figure 2 by a thicker line.

Figure 6: Plastic hinges for the model F; blu=bending, red=shear.

Figure 7: Cracks pattern for continuous model C. Red arrow means cracked zone.

Figure 8: Model L: total plastic dissipation, contour plot.

References

[1] Ramos L, Lourenço PB, Modeling and vulnerability of historical city centers in seismic areas:

a case study in Lisbon. Engineering Structures, 26, 2004, 1295-1310.

[2] O.P.C.M. 3274, 20/03/2003, Primi elementi in materia di criteri generali per la classificazione

sismica del territorio nazionale e di normative tecniche per le costruzioni in zona sismica [in

Italian] & O.P.C.M. 3431/05 09/05/2005, Ulteriori modifiche ed integrazioni all'OPCM

3274/03 [in Italian].

[3] Bathe KJ, Walczak J, Welch A, Mistry N, Nonlinear analysis of concrete structures, Computer

& Structures 32(3-4), 1989, 563–590.

[4] Milani G, Lourenço PB, Tralli A. Homogenised limit analysis of masonry walls. Part I: failuresurfaces, Computers and Structures, 84, 2006, 166-180.

[5] Milani G, Lourenço PB, Tralli A. Homogenised limit analysis of masonry walls. Part II:

structural examples, Computers and Structures, 84, 2006, 181-195.

[6] Turnšek V, Cacovic F, Some experimental results on the strength of brick masonry walls,

Proc. Of the 2nd

Int. Brick Masonry Conference, Stoke on Trent, 1971, 149-156.

[7] Sloan SW, Kleeman PW, Upper bound limit analysis using discontinuous velocity fields,

Computer Methods in Applied Mechanics and Engineering 127(1-4), 1995, 293–314.

2-d seismic nonlinear analyses

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