Numerical analysis of failure mechanisms of historical constructions

Transcription

1 Numerical analysis of failure mechanisms of historical constructions J. M. Sieczkowski*, IP. Szotomicki* ' Wroclaw University of Technology, Poland. Abstract This paper presents numerical analysis of complex masonry structures using FEM. The results of computations are very important because the load-carrying ability and deferability of historical masonry structures is still not sufficiently known. Masonry is incapable of carrying tensile forces, and will crack in response to displacements imposed by the external environment. The main objective was prediction of failure mechanisms of complex masonry structures applying an interface and damage model 1 Introduction Nowadays there is a general concern for preservation of historical constructions. This will invariably require to pay greater attention to the reliable assessment of the structural conditions of the monuments and to the correct design of intervention measures. The first step to analysis of historical constructions is the accurate prediction of the actual safety level of these structures. This include the knowledge of both the mechanism of deterioration of the material and of the structural components and the evaluation of degradation with time. Structural engineering is the most important discipline which should cooperate when an historical construction may show structural damages and may need a strengthening intervention. Numerical methods are particularly efficient means for the structural diagnosis, but shall be supported by experimentally found input data. The good tools in numerical analysis of failure mechanics of historical masonry structures are composite interface model and damage model proposed by Simo and Oliver. An important objective of this

2 384 Damage and Fracture Mechanics VI analysis is obtain robust numerical tools, capable of predicting the behaviour of the structure from the linear elastic stage, through cracking and degradation until complete loss of strength. Only then, it is possible to control the serviceability limit state, to fully understand the failure mechanism and asses the safety of the structure. 2 Application of FEM The main limitation of most finite element formulations, when applied to masonry structures, lies in the hypothesis of the continuum, which makes it difficult to simulate the particular kinematics of the masonry modes of failure. Elastic analyses based on the finite element method are only suitable for characterising working conditions subject to states of not very large stresses in which compressive stresses are largely predominant. However, the adoption of suitable constitutive equations which account for the main phenomena related to failure of the materials such as cracking under tension and crushing under compression and makes it possible to reproduce more advanced stages of the response and even to simulate very accuracy failure mechanisms. The finite element method combined with suitable constitutive equations has been successfully used to study complex historical masonry structures. 2.1 Composite interface model Micromodelling consists of simulating the global behaviour of a composite material by means of detailed geometrical discretization, based in the finite element method, and providing non-linear constitutive laws for each component. In the micromodels bricks, bed and vertical mortar joints are simulated separately. Micro-model can include all the basic types of failure mechanisms that characterise masonry. In the following main attention is given to analysis, in which interface elements are used as potential crack, slip or crushing planes. A composite interface model proposed by Lourenco [1] consists of: a) tensile strength function (tension cut off) /l(cr,^) = cr-^(ari) (1) where the yield function can be written: <?!=/; exp(--^-fi) (2) Gf In the above expression, f \ is the tensile strength of the brick/mortar interface and Gf is the energy of failure in mode I (under tension). The associated flow rule and strain softening hypotheses were considered. Assuming that only normal plastic displacement has an influence on the softening of masonry leads to:

3 Damage and Fracture Mechanics VI 385 Figure 1: A diagram of displacement for barrel vault NLIn STRESS StaprlS =18 Figure 2: A diagram of plastic zones for barrel vault

4 386 Damage and Fracture Mechanics VI where: ^ is a plastic multiplier rate. b) function representing the Coulombfrictionmodel, 4 (3) where: /2 (cr, K 2 ) = T\ + otancp(k2 ) - C(KI ) (4) c = CQ exp(- -~^)jancp = tancp^ + (tan<p, - tan<p^ ) (5) / ^0 In the above expression CQ is a coefficient of cohesion, tancp^ is a residual friction angle and G^ is the energy of failure in mode II (under shearing). Assuming that only tangent plastic displacement has an influence on the softening of masonry leads to: c) function representing the elliptic Cap model. where: (?, C^, - a set material parameters, <r - yield value of stress. Using matrix notation, eq. (7) can be rewritten as: (7) /3((7,^) = l(7^(7-(^(^)^ (8) where the projection matrix P equals diag{2,2,c^}. An associated flow rule and a strain hardening/softening hypothesis are considered. The presented method of micro-modelling [2], which employs the composite interface model, supplies interesting results concerning the behaviour of masonry structures. A good example here is a computer analysis of a barrel vault (span - 4 m, height - 2,6 m), which was considered as loaded with its dead-weight. The results are presented in the form of displacements and a drawing of plastic zones (figs. 1,2). 2.2 Damage model Recent investigations support the idea that non linear behaviour of masonry can be modelled using concepts of damage theory only provided an adequate damage function which is defined for taking into account the different response of masonry under tension and compression states. Cracking can, therefore, be interpreted as a local damage effect, defined by the evolution of known material (9)

5 Damage and Fracture Mechanics VI 387 parameters and by one or several functions which control the onset and evolution of damage. The damage function g(t,r) defines the limit of the region of undamaged response and is written at time t as 'g(r,r)=fr-,r<0 (10) where the undamaged complementary energy norm is defined as (11) For Simo's damage model y = 1, whilst for Oliver's damage model it is given as (12) c (13) where cr^, CT^ are the initial damage strengths in tension and compression. a' (i= 1,2,3) are principal undamaged stresses and M = M^O (14) [0 otherwise tr in the damage function (10) is the current damage strength measured with an energy norm and can be given as 4 = max{0,.,max/r} (15) where 0,. denotes the initial damage threshold of the material. The damage flow rule defines damage softening and is given by where ji > 0 is the damage consistency parameter and defines damage loading/unloading conditions according to the Kuhn-Tucker relations In addition, to simplify the calculations in damage analysis, the damage multiplier p, is defined so that With the consistency of the damage condition (10) and definition i = f = p (18) (j\.f* defines the damage rate with respect to the undamaged elastic #T complementary norm. If the damage potential function G is assumed to be independent of d substitution of (18) into (16) will lead to d = G (19)

6 388 Damage and Fracture Mechanics VI with the undamaged condition being enforced so that Simo suggested for masonry that <4),,=o, =0 (20) where A and B are characteristic material parameters and 0,. denotes the initial damage threshold. Alternatively Oliver proposed the damage accumulation function G with a form as where A is a characteristic material parameter and 0^. is the initial damage threshold for Oliver's damage model. A good example application of damage model is numerical analysis of dome which was considered as loaded with its dead-weight The results are presented in the form of principal stresses for tension and compression. 3,4). (21) (22) Figure 3: A diagram of principal tension for cracked dome. MPa.

7 Damage and Fracture Mechanics VI 389 Figure 4: A diagram of principal compression for cracked dome. MPa. 3 Conclusions The analysis of historical constructions is a difficult task due to the large numbers of unknown influence factors. Sophisticated techniques of analysis are essential to understand the behaviour of this type of constructions and to control the importance of the different influence factors. Two numerical approaches were satisfactorily used to carry out a non-linear analysis of the structure of a gothic vault and dome - taking into account the more prominent features of the mechanical response of the structural material. Both approaches include suitable constitutive equations to model the essential characteristics of the behaviour of the material, in particular the inability of masonry to carry tension. References fl] Lourenco, P.B., Analysis of masonry structures with interface elements: Theory and applications, Delft University of Technology, Report no , Delft, Netherlands [2] Sieczkowski, J.M., Szotomicki, J.P., Application of FEM for analysis of masonry structures, Civil and environmental engineering conference, p. 1-8, Bangkok, Thailand 1999.