NON-LINEAR FINITE ELEMENT METHOD The tensile behavior of concrete includes a crack model and a stress-strain relationship

Cracking in concrete can be modelled as smeared, which makes the concrete continuous, and the stiffness matrix is only updated after cracking [25]. A smeared approach recom-mended by the Guidelines for NFLEA, is the total strain-based crack model, where the cracks can be considered fixed or rotating [6]. In the rotating crack model, the cracks co-rotate with the principal stress directions. Due to the rough nature of concrete, some stresses can be transferred over the cracks [39]. For the total strain-based rotating crack model, there are a variety of tensile curves to model the tensile behavior. One of these is the exponential curve [25]. As Figure 2.7 shows, this stress-strain relationship has a softening effect after the maximum tensile strength, f_t, is reached.

Figure 2.7: Exponential tensile stress-strain model [6]

The material parameters included in the exponential softening model are the tensile strength of concrete,f_t, the fracture energy, G_F, and Young’s modulus, E_c, for the elastic region [19]. h_eq is an equivalent element length dependent on the mesh discretization. By the Guidelines for NLFEA [6], G_F and f_t can be calculated by

GF = 73 f_cm^0.18 and ft= 0.3 f_ck^2/3 (2.11) where f_cm is the mean concrete compressive strength and f_ck is the characteristic com-pressive strength in MPa. GF is in Nmm/mm², andft is in MPa.

The concrete behavior in compression is complex. The Guidelines for NLFEA [6] suggests a parabolic stress-strain relationship in compression, as illustrated in Figure 2.8.

Figure 2.8: Parabolic compressive stress-strain model. Modified to our notations [25]

The parabolic model is dependent upon the compressive fracture energy, G_C, the maxi-mum compressive strength, f_c, the Young’s modulus, E_c, the strains, , and the element size, h [16, 25]. G_C is in Nmm/mm² and can be calculated from

G_C = 250 G_F (2.12)

Instead of a sudden decrease to zero compressive strength, the model contains a softening effect after the maximum compressive strength, f_c, is reached. The strain at maximum compressive stress,_c,par, is defined in TNO [25] as

c,par =−5 3

fc

E_c (2.13)

The maximum compressive strength of concrete depends on the multi-directional stress state. The strength increases with confined compressive stresses in the lateral directions, and decreases with tensile stresses. This can be modelled by a 1993 Selby and Vecchio model [39]. In general, the concrete has reduced compressive strength when lateral crack-ing occurs. One of the models for lateral crackcrack-ing is model B, suggested by Vecchio and Collins [42] and is illustrated in Figure 2.9. Here, the compressive strength of the concrete, f_c, is reduced by a reduction factor, β_σcr [25].

2.2. NON-LINEAR FINITE ELEMENT METHOD

Figure 2.9: Model B suggested by Vecchio and Collins [25]

Poisson’s ratio, ν_c, dictates how stresses and strains in different directions are affected by each other. For a cracked and discontinuous volume, this relationship changes, and ν_c changes with cracking. This can be modeled with a damage-based approach, where ν_c is gradually reduced to zero as cracking occurs [25]. The Guidelines for NLFEA [6]

recommends the following relation to approximate E_c: E_c=E_c0 f_cm

10

!1/3

(2.14) where E_c0 = 21500 MPa, and f_cm is in MPa.

For this report, a parabolic stress-strain model was used in compression, and an exponen-tial stress-strain model was used in tension. The total strain-based rotating crack model was used with automatic crack bandwidth. The Vecchio and Collins 1993 model was used for reduction due to lateral cracking, with a minimum reduction factor β_σ^min = 0.6 as used by Belletti et al. [5], although the Guidelines for NLFEA recommend β_σ^min = 0.4.

Poisson’s ratio was modeled with a damage based reduction, and the stress confinement effect used a Selby and Vecchio model. Input material parameters were calculated based on relations from the Guidelines for NLFEA [6] and are presented in Table 2.2. f_cm is adapted to f_c,situ, and f_ck is adapted to the expression presented in (2.4).

Table 2.2: Concrete properties

Property Value

Initial Poisson’s ratio ν_c= 0.15

Tensile strength f_ct,situ = 0.3(f_c,situ−6)^2/3 Fracture energy G_F,situ = 73f_c,situ^0.18 Compressive fracture energy G_C,situ = 250G_F,situ Young’s modulus E_c,situ=E_c0^fc,situ₁₀ ^1/3

Reinforcement steel

The material model recommended by the Guidelines for NLFEA [6] for reinforcement is rather simple compared to that of concrete. An isotropic bi-linear elasto-plastic stress-strain relationship, as illustrated in Figure 2.10, is sufficient.

Figure 2.10: Bilinear stress-strain relationship for reinforcement [6]

This is a form of von Mises plasticity, where Hooke’s law is valid for the linear part, as _sy = f_sy

E_s (2.15)

where sy is the yield strain, Es is Young’s modulus, and fsy is the yield strength. The ultimate strain, _su, and ultimate strength, f_su, are needed to define the elasto-plastic part. The following relationship is used:

_su = f_su−f_sy

E_har +_sy (2.16)

whereEhar is the hardening modulus.

2.2. NON-LINEAR FINITE ELEMENT METHOD There are several ways to create a complete stress-strain relation for reinforcement. For a class C reinforcement, the Guidelines for NLFEA [6] provide recommendations for the ultimate and yield stress relation, ^{f su}

f sy k, as well as a characteristic ultimate strain limit, _uk:

1.15≤(f_su

f_sy)_k ≤1.35 and _uk ≥7% (2.17)

The steel parameters used in this report are summarized in Table 2.3. Note that Ehar

is not a fixed parameter, but depends upon f_sy, _sy and f_su. Perfect bonding between concrete and reinforcement was assumed.

Table 2.3: Reinforcement properties

Property Value

Young’s modulus E_s = 200 000 MPa

Ultimate strain _su = 7.5%

Yield strain _sy = ^f_E^sy

s

Ultimate stress f_su = 1.25 f_sy

Platen

The platens were modeled as steel plates with a linear elastic behavior and material properties as in Table 2.4.

Table 2.4: Platen properties

Property Value

Young’s modulus Ep = 200 000 MPa

Poisson’s ratio ν_p = 0.3

The properties for the interface elements are given in Table 2.5. These stiffnesses are collected from a benchmark experiment similar to the beam under consideration, as used by Belletti et al. [5].

Table 2.5: Interface properties

Property Value

Normal stiffness K_n = 36 300 N/mm³

Shear stiffness K_s = 3.63·10⁻⁸ N/mm³

2.3 Reliability

In a reliability assessment, the probability of failure within a lifetime of a structure is sought. Calculating failure probabilities exactly is difficult, hence many approximate methods have been developed. This is a topic highly relevant for design codes as they carry the responsibility of ensuring certain safety levels. This section presents general concepts of reliability, as well as reliability methods to estimate failure probabilities.

2.3.1 General concepts

A structure or a technical system needs to meet certain requirements in regards to safety and serviceability. These can be assessed with ultimate limit state or service limit state.

Mathematically, this can be expressed as a limit state condition on the form

G(x)>0 (2.18)

where xis the vector of random variables xi. For a structural problem,xi can represent properties like dimensions of components, stiffness values, loads, etc. A structure failing to meet the limit state requirements can be expressed asG(x)<0. The equalityG(x) = 0 is defined as the limit state function (LSF). A LSF separates the failure domain from the safe domain, and is used to express a structure’s probability of failure:

p_f =P(G(x)<0) =

Z

D

f_x(x₁, ..., x_n)dx₁...dx_n (2.19) wheref_x(x₁, ..., x_n) is the joint PDF for all random variables, andD is the failure domain defined by G(x) <0. Computing this integral is in many ways what reliability theory is about.

2.3. RELIABILITY

In document Reliability Analysis of Reinforced Concrete using Non-Linear Finite Element Analysis (sider 36-42)