Lumped Damage Mechanics: Reinforced Concrete Frames

Lumped Damage Mechanics: Reinforced Concrete Frames

DOI: 10.4018/978-1-4666-6379-4.ch010

Abstract

As aforementioned, buildings in seismic zones must be designed to behave elastically under service loads or earthquakes of small intensity, and they can enter in the plastic range for events of intermediate intensity. Severe earthquakes are defined as those that are improbable but not impossible to happen during the lifetime of the structure. In these cases, structural damage, even damage that cannot be repaired, is allowed as long as there is no structural collapse. In order to design or certify safe structures, it is necessary to have computational tools that allow for the quantification of structural damage and that are able to describe structural behavior accurately near collapse. The elasto-plastic models present serious limitations in this sense. Damage and fracture mechanics represent a more rational option. The goal of this chapter is to describe how the concepts presented in Chapter 9 can be included in the mathematical models for the analysis of framed structures and its numerical implementation in structural analysis programs.
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10.1 The Lumped Damage Model

Consider a slender RC frame element in the planar case. It is assumed that not only yielding of the reinforcement, but concrete cracks are also localized at the extremes of the element. The lumped plasticity model that is described in section 7.1.1 is modified by assuming that concrete cracks can be lumped at the plastic hinges as well: plastic hinges become inelastic hinges and the “lumped plasticity model” develops into the “lumped damage model”. In this and the following chapters, the terms “inelastic hinge” or “plastic hinge with damage” are used indistinctly. In order to represent the appearance and propagation of these cracks, a new set of internal variables is introduced. The term is denoted “damage array” of the element b. The variables di and dj are damage parameters that take values between zero and one; however, they represent densities of macroscopic cracks, as in fracture mechanics, instead of micro defects (see Figure 1a). The first variable in characterizes the crack density that can be lumped at the inelastic hinge i while the second one is related to cracking lumped at inelastic hinge j (see Figure 1b).

Figure 1.

a) Cracks at the end of a frame member b) Lumped damage model

The hypothesis of equivalence in deformation described in section 9.2.2, Equation (9.2.8) can be written in the case of a frame element as:

(10.1.1) where is, again, the matrix of generalized deformations of the elastic beam-column. is the matrix of plastic deformations; the plastic elongations are neglected as it is usual in RC elements, thus . The last matrix, , contains the damage rotations; i.e. the additional rotations due to concrete cracking.

The deformations of the elastic beam-column can be expressed as a function of the generalized stresses using (Equation 3.3.9):

(10.1.2)

The terms are, again, the flexibility matrix of a slender element, the generalized stresses and the initial deformations such as defined in chapter 3 (Equation 3.3.9, 3.2.2 and Table 3).

Table 3.
Force vs. displacement curve values in the case of imposed damage values
d1m1(KN m)t (m)P (KN)
0.000.00000.00000.00000.00000.0000
0.004.00390.00000.00180.00262.8599
0.0511.21690.00000.00540.00758.0121
0.1015.15480.00000.00770.010710.8249
0.1518.09010.00000.00970.013612.9215
0.2020.44960.00000.01160.016314.6069
0.2522.40660.00000.01360.019016.0047
0.3024.04960.00000.01560.021917.1783
0.3525.42880.00920.02700.037918.1634
0.4026.57340.02050.04060.056918.9810
0.4527.49940.03290.05570.077919.6424
0.5028.21290.04690.07260.101620.1521
0.5528.71150.06300.09200.128820.5082
0.6028.98440.08180.11480.160720.7031
0.6529.01030.10450.14220.199120.7217

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