**3. Deformation calculation of strength**

#### **3.1 General provisions**

In a number of earlier published works it is shown that the most reliable calculations of the bearing capacity of CFST columns, taking into account their design features, can be carried out on the basis of nonlinear deformation model. The calculation sequence of similar designs for deformation model is in detail stated in [16].

The calculations are based on the assumptions specified in the EN 1992-1-1 standard. They are listed in the introduction. While processing the experimental data the values of random eccentricity are taken three times less than the values recommended by standards for design purposes. Thus, the centering of the samples along the physical axis is taken into account.

The calculation is based on the relationships between stresses and strains for the concrete core *σ<sup>с</sup><sup>z</sup>* � *ε<sup>с</sup>z*, steel tube *σpz* � *εpz* and reinforcement (if any) *σ<sup>s</sup>* � *εs*. The concrete core and steel tube operate under conditions of volumetric stress state, which can change quantitatively and qualitatively with increasing load (**Figure 3**). The accuracy of calculations largely depends on the reliability of the adopted diagrams. At that, the diagrams contained in the regulatory documents are not suitable to evaluate the strength resistance of a concrete core and a steel shell. Therefore, reliable diagrams *σ<sup>с</sup><sup>z</sup>* � *ε<sup>с</sup><sup>z</sup>* and *σpz* � *εpz* need to be constructed initially. The form of a diagram set is recommended to be multipoint one. It is shown in [17] that such a method is the most universal one. At the second stage the strength of the compressed CFST is calculated.

#### **3.2 The first stage of calculation**

At the first stage, the deformation diagrams of the concrete core and the steel tube are constructed for the axial direction of the element. For this purpose, the load resistance of a short centrally compressed CFST element is considered. Load is imposed quickly. The concrete core is considered as a transversely isotropic body. The steel tube is considered to be an isotropic body. In the tube the stresses arise in the axial, circumferential and radial directions – *σpz*, *σp<sup>τ</sup>*, *σpr*. The stress signs of the concrete core and steel shell depend on the load level. At low load levels, the value of the coefficient of transverse strains of steel exceeds the value of the coefficient of transverse strains of concrete. For these levels, there is no triaxial compression of concrete (**Figure 3b**). When the value of the coefficient of transverse strains of concrete exceeds the value of the coefficient of transverse strains of steel, the volumetric compression of concrete takes place (**Figure 3c**).

*Bearing Capacity of Concrete Filled Steel Tube Columns DOI: http://dx.doi.org/10.5772/intechopen.99650*

**Figure 3.**

*Tension of steel tube and concrete core of the central compressed CFST column: a – scheme of loading; b – at low loading levels; c – at high loading levels.*

*Branch of concrete deformation charts at step-by-step strengthening of axial deformations: 1 - uniaxial compression, 2,3 - volume compression at the intermediate stages of deformation; 4 - volume compression in a limit state.*

Curvilinear deformation diagrams are accepted for the concrete core. The coordinates of vertex of each diagram depend on the lateral pressure on the concrete from the steel tube. It is assumed that with an increase of the compressive force *N*, the lateral pressure on the concrete *σcr* goes up from zero to a certain limiting value. Therefore, the calculation requires the use of many such diagrams (**Figure 4**).

The coordinates of vertex of each diagram determine the strength of the concrete core (uniaxially compressed *fc* or volumetrically compressed *f cc*) and its strain (*ε<sup>с</sup>*<sup>1</sup> or *εcc*<sup>1</sup> respectively).

There are many proposals for determining the strain *εcc*<sup>1</sup> in the literature. The main disadvantage of the above formulas is that they are all obtained from the results of the corresponding experiments. It greatly limits the scope of their application.

Let's show how one can get the corresponding formula based on the phenomenological approach.

**Figure 5** shows the stress–strain diagram of compressed concrete, corresponding to the maximum reached stress and compare it with the uniaxial compressed concrete diagram. It follows from the above that the initial modulus of elasticity *E<sup>с</sup>* for both diagrams is the same.

The strain *εcc*<sup>1</sup> at the vertex of the diagram *σ<sup>с</sup><sup>z</sup>* � *ε<sup>с</sup><sup>z</sup>* is made up of elastic *εel* and plastic *ε*pl components

$$
\varepsilon\_{\rm cc1} = \varepsilon\_{\rm el} + \varepsilon\_{\rm pl}. \tag{31}
$$

Elastic strain *εel* is associated with the elastic part of the strain of uniaxially compressed concrete *ε*<sup>0</sup> el by the following relationship:

$$
\varepsilon\_{\rm el} = \varepsilon\_{\rm el}^{\prime} \frac{f\_{\rm cc}}{\gamma\_{\rm c} f\_{\rm c}}.\tag{32}
$$

Plastic strain *ε*pl is associated with the plastic part of the strain of uniaxially compressed concrete *ε*<sup>0</sup> pl by a similar relationship:

$$
\varepsilon\_{\rm pl} = \varepsilon\_{\rm pl}^{\prime} \left( \frac{f\_{\rm cc}}{\chi\_{\rm c} f\_{\rm c}} \right)^{m}, \tag{33}
$$

where *m* is the exponent, *m* > 1.

The parameter *m* takes into account the fact that the increase in strains of volumetrically compressed concrete is more intense than the increase in its strength.

Thus, the total deformation of the volume-compressed concrete at the maximum stress is determined by the formula

$$
\varepsilon\_{c\varepsilon1} = \varepsilon\_{c1} a\_c^m \left[ 1 - \frac{\chi\_c f\_c}{\varepsilon\_{c1} E\_c} \left( 1 - a\_c^{(1-m)} \right) \right]. \tag{34}
$$

The performed statistical analysis showed that the best match with the results of the experiments corresponds to a value of *m,* calculated by the formula

$$m = 1.\overline{\mathcal{I}} + \frac{3.5}{\sqrt{\chi\_c \overline{f\_c}}},\tag{35}$$

where *fc* is in MPa.

According to the recommendations of [21] the ultimate strain of a volumecompressed concrete is determined by the formula

$$
\varepsilon\_{\rm cc2} = \varepsilon\_{\rm c2} \frac{\varepsilon\_{\rm cc1}}{\varepsilon\_{\rm c1}},
\tag{36}
$$

where *ε*c2 is the ultimate strain for uniaxial compressed concrete.

**Figure 5.** *The graphs of deformation for uniaxial compressed (1) and volume-compressed (2,3) concrete.*

*Bearing Capacity of Concrete Filled Steel Tube Columns DOI: http://dx.doi.org/10.5772/intechopen.99650*

When coordinates of parametric points of the deformation charts of volumetrically compressed concrete are known, it is possible to calculate the bearing capacity of CFST columns based on the deformation model analysis.

To construct the diagrams *σс<sup>z</sup>* � *εс<sup>z</sup>* and *σpz* � *εpz*, a step-by-step increase in the strains of the concrete core and steel tube is carried out while ensuring the condition *εс<sup>z</sup>* ¼ *εpz*. All components of the stress–strain state of concrete and steel are calculated at each *j*-th step.

The analytical relationship between strains and stresses for any point of the concrete core is written in the form of a system of equations:

$$
\begin{Bmatrix} \varepsilon\_{cx} \\ \varepsilon\_{cr} \end{Bmatrix} = \frac{1}{E\_c} \times \begin{bmatrix} \nu\_{cx}^{-1} & -2\nu\_{cr}\nu\_{ci}^{-1} \\ -\nu\_{cr}\nu\_{ci}^{-1} & \left(\nu\_{cr}^{-1} - \nu\_{rr}\nu\_{ci}^{-1}\right) \end{bmatrix} \times \begin{Bmatrix} \sigma\_{cx} \\ \sigma\_{cr} \end{Bmatrix}.\tag{37}$$

The elastic–plastic properties of concrete are taken into account by the coefficients of elasticity *νck* (*k = z, r, i*) and variable coefficients of transverse strains *υzr*, *υrr*. The subscripts *z* and *r* are used for axial and transverse directions, and the subscript *i* is used for the coefficient of elasticity depending on the intensity of stress and intensity of strain.

The values of the intensity of stresses and strains are calculated using the wellknown formulas of solid mechanics. Using the coefficients of elasticity *ν<sup>с</sup><sup>i</sup>* and transverse strains *υzr*, *υrr*, the strains along one direction (axial or transverse) depending on the stresses of the other direction are calculated in a matrix of system pliability (37).

The stress state of a steel tube obeys the hypothesis of a uniform curve [22]. In accordance with this hypothesis, the dependence *σpi*-*εpi*, obtained under uniaxial tension, is accepted for complex stress states. Here *σpi* is the intensity of stresses, and *εpi* is the intensity of strains.

The initial diagram *"σ<sup>p</sup>* � *εp"* is recommended to be tri-linear (rules of Russia - CR 266.1325800.2016). However, when modelling steel sections under a complex stress state, it is advisable to use a deformation diagram calculated using the generalized parameters *σpi* ¼ *σpi= f <sup>y</sup>* and *εpi* ¼ *εpiEp= f <sup>y</sup>* (**Figure 6**). The coordinate values of the characteristic points of the generalized diagram can be taken from **Table 1**.

Communication between strains and stresses for any point of an external steel shell in elastic and elasto-plastic stages can be presented the following equations system:

$$\begin{Bmatrix} \varepsilon\_{px} \\ \varepsilon\_{pr} \\ \varepsilon\_{pr} \end{Bmatrix} = \frac{\mathbf{1}}{\nu\_p E\_p} \times \begin{bmatrix} \mathbf{1} & -\nu\_p & -\nu\_p \\ -\nu\_p & \mathbf{1} & -\nu\_p \\ -\nu\_p & -\nu\_p & \mathbf{1} \end{bmatrix} \times \begin{Bmatrix} \sigma\_{px} \\ \sigma\_{pr} \\ \sigma\_{pr} \end{Bmatrix} . \tag{38}$$

Here *σpz*, *σp<sup>τ</sup>*, *σpr* are normal (main) stresses in a tube in the axial, circumferential and radial directions; *εpz*, *εp<sup>τ</sup>*, *εpr* are strains of a steel tube in the corresponding directions; *Ep* is the initial module of tubes elasticity; *ν<sup>p</sup>* is coefficient of steel elasticity; *υ<sup>p</sup>* is coefficient of tubes cross strain.

The stresses and strains acting on the principal planes are used in Eqs. (37) and (38). Experiments show [16] that in the stage of yield Chernov-Luders lines appear on the surface of the steel tube. These lines are angled 45° to the longitudinal axis of the CFST. Therefore, shear stresses and shear strains are equal to zero here.

The stress–strain states of the concrete core and steel tube largely depend on the values of the coefficients of transverse strain and the coefficients of elasticity of the materials. Therefore, their reliable determination is very important when
