**Figure 6.**

*Generalized calculation diagram of steel, operating under conditions of complex stress state.*


#### **Table 1.**

*Coordinates of characteristic points of the generalized steel deformation diagram, constructed in the axes σpi* � *εpi.*

calculating the strength of CFST columns. Formulas for calculating these coefficients are given in work [16].

The solution of the Eqs. (37) and (38), taking into account the joint deformation of concrete and steel tube, allows obtaining the formula for calculating the lateral pressure

$$
\sigma\_{cr} = \frac{\left(\upsilon\_p - \upsilon\_{xr}\frac{d\_c}{d\_c + \delta}\frac{\nu\_{ci}}{\mu\_{ci}}\right)\varepsilon\_{cx}}{K\_p + K\_c},
\tag{39}
$$

in which *Kp* and *Kc* are the parameters defining condition of steel shell and concrete core.

$$K\_p = \frac{\mathbf{0}, \mathbf{5}\nu\_p}{\nu\_p E\_p} \left[ \nu\_p \left( \frac{d}{\delta} - \mathbf{1} \right) - \left( \frac{d}{\delta} + \mathbf{1} \right) \right];\tag{40}$$

$$K\_{\varepsilon} = \frac{\beta\_r}{\nu\_{ci} E\_c} \left( \frac{2 \nu\_{cr}^2 \nu\_{cz}}{\nu\_{ci}} + \nu\_{rr} - \frac{\nu\_{ci}}{\nu\_{cr}} \right). \tag{41}$$

When the strain *ε<sup>с</sup><sup>z</sup>* and lateral pressure *σcr* are known, all other components of the stress–strain state of CFST column can be calculated. The strains are incrementally increased until the stress *σ<sup>с</sup><sup>z</sup>* reaches the strength of volumetrically compressed concrete *f* ð Þ *n cc* (**Figure 4**), previously calculated using formulas (15) and (16). The calculation is performed on a computer.

After that we compare the last value of strain *εс<sup>z</sup>* with the strain calculated on a formula (34) *ε* ð Þ *n cc*<sup>1</sup> in top of the deformation chart of concrete. In the existing incoherence *εcz* � *ε* ð Þ *n cc*1 <sup>&</sup>gt; <sup>Δ</sup>*<sup>ε</sup>* (Δ*ε*� the accuracy of calculations set by the estimator) we specify value of an exponent *m* in a Eq. (34) and repeat all calculations.

Upon termination of calculations we receive arrays of numerical data for deformation charting of concrete core f g *<sup>ε</sup>cz* � f g *<sup>σ</sup>cz* and steel shell *<sup>ε</sup>pz* � *<sup>σ</sup>pz* .

### **3.3 The second stage of calculation**

At the second stage, the bearing capacity of the eccentrically loaded CFST element is calculated. The design scheme of the normal section of element is shown in **Figure 7**.

In the calculation process, the deformation of the most compressed fiber of the concrete core *ε<sup>c</sup>* max is increased step-by-step. At each step, using the Bernoulli hypothesis, the diagram of strain fiber of the steel tube *ε<sup>p</sup>* min . The search for this value is carried out with a gradual shortening strain decrease (starting from *ε<sup>c</sup>* max ) or the build-up of the elongation strain *ε<sup>p</sup>* min (starting from zero).

The normal section of the calculated element is conditionally divided into small sections with areas of concrete *A<sup>с</sup><sup>i</sup>* and steel shell *Apk*. In the presence of longitudinal reinforcement, the cross-sectional area of each bar is designated as *Asj*.

The origin of coordinates is aligned with the geometric center of the element's cross section. If the Bernoulli hypothesis is observed, there is a strain in the center of each section of concrete and steel tube. With known strains, the corresponding stresses are determined according to the results of the first stage of the calculation. The stresses are assumed to be evenly distributed within each section of concrete

#### **Figure 7.**

*Design model of the normal section of the CFST element deformations of the normal cross section is designed, corresponding to the equilibrium condition of the calculated element. In order to develop such a diagram it is required to find the corresponding value of the strain of the least compressed (stretched).*

and steel tube. After each step of strain *ε<sup>c</sup>* max increasing, it is necessary to ensure that the equilibrium conditions are met:

$$N\_x = \sum\_i \sigma\_{xi} A\_{ci} + \sum\_k \sigma\_{puk} A\_{pk} + \sum\_j \sigma\_{sj} A\_{sj};\tag{42}$$

$$N\_{z}e\_{0} = \sum\_{i} \sigma\_{xi} A\_{ci} Z\_{mi} + \sum\_{k} \sigma\_{pzk} A\_{pk} Z\_{pnk} + \sum \sigma\_{j} A\_{j} Z\_{mj},\tag{43}$$

in which *Zcni*, *σczi* � the coordinate of the gravity center of the *i*-th section of concrete and the stress of the axial direction at the level of its gravity center; *Zpnk*, *σpzk* � the coordinate of the gravity center of the steel shell *k*-th section and the stress of the axial direction at the level of its gravity center; *Zsnj*, *σsj* � the coordinate of the gravity center of the of the longitudinal reinforcement *j*-th bar and the stress in it.

When both equilibrium conditions are met, the value of the compressive force *Nz* corresponding to the given strain *ε<sup>c</sup>* max is fixed. Next, the strain of the most compressed fiber of the concrete core increases and all calculations are repeated. The limiting values of this strain *εccu* can be accepted according to the recommendations [21].

The problem of determining the strength reduces to finding the value of the strain of the most compressed fiber *ε<sup>c</sup>* max ≤*εcc*1, corresponding to the maximum value of the compressive longitudinal force *Nu*. The calculation results show that under certain design parameters of CFST columns, the strain *ε<sup>c</sup>* max does not reach *εcc*1. Then the stress in the concrete *σ<sup>с</sup><sup>z</sup>* cannot achieve its strength at triaxial compression. This design situation occurs when using low strength concrete and a strong steel shell with a small ratio *d* /*δ.* Therefore, the criterion for the loss of the strength of the column is the achievement of the maximum value of compressive force in the process of increasing the strain of the most compressed fiber.

The proposed method makes it possible to limit the axial strains of the columns. It is known from experiments that the strain of compressed CFST elements can reach 5 ÷ 10% [16]. With such strains, the operation of the columns of the buildings becomes impossible. Thus, excessive strain can determine the ultimate limit state of the CFST column. The maximum permissible values of these strains can be set by a structural engineer, depending on a specific design situation for a designed building or a structure.

#### **3.4 Calculation of flexible elements**

Due to the complex nature of load resistance of CFST columns, in design practice, as a rule, the simplified methods of calculation of their bearing capacity are used. At that, flexibility is usually taken into account by the coefficient of longitudinal bending, determined according to empirical relationships. In the monograph we consider the deformation calculation of CFST column bearing capacity.

A rod of a circular cross-section with a constant length, loaded by a compressive force N applied to the ends with the same initial eccentricity *e*<sup>0</sup> (no less accidental than *ea*) and hinged at its ends is regarded as a basic case. The deformation scheme of such a rod is shown on **Figure 8**.

According to the known positions of structural mechanics, if we apply force N along the axis that coincides with the physical gravity center of an elastic rod crosssection, the rod will remain a rectilinear one until the force reaches the value of the critical load Nu corresponding to the moment of stability loss. Only after that the middle part of the rod will receive the corresponding deflection *fcr*.

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

**Figure 8.** *The scheme of a compressed rod deformation.*

A bending moment *M* will appear in any section along the length of the bar from the compressive force *N*. The moment *M* is calculated by the formula

$$M = N(e\_0 + \jmath),\tag{44}$$

where *y* is the horizontal displacement value of the cross-section in question.

With the increase of the bending moment, the strength of a compressed rod normal section decreases, which must be taken into account during the calculation. On the other hand, the axial load increase to a critical value in the columns of great flexibility can lead to a very significant increase of transverse deformations - the loss of stability of the second kind. With a certain transverse deflection, the compressive load reaches a maximum value, after which its decrease is observed with a further deflection increase (**Figure 9**). At the same time, the strength properties of materials from which the column is made will not be implemented fully.

**Figure 9.** *The dependence of compressive force on deflection*

The main assumptions that are directly relevant to this study are the following ones:


The flexibility of the column is determined for the reduced cross-section. For the base case under consideration, this flexibility can be approximated by the following formula:in which *l* is the estimated length of the rod; *(EI)eff*, *(EA)eff* are effective stiffness of the most loaded reduced section for bending and compression.

$$
\lambda\_{\sharp \overline{\mathcal{U}}} = l \cdot \sqrt{\frac{(E \mathcal{A})\_{\sharp \overline{\mathcal{U}}}}{(E \mathcal{I})\_{\sharp \overline{\mathcal{U}}}}}.\tag{45}
$$

It is recommended to calculate the stiffness ð Þ *EI eff* and ð Þ *EA eff* in the first approximation by the following formula:

$$(EI)\_{eff} = 0, \mathfrak{S}E\_c I\_c + 0, \mathfrak{S}E\_p I\_p + E\_s I\_s;\tag{46}$$

$$(EA)\_{\mathcal{eff}} = \mathbf{0}, \mathbf{5}E\_c A\_c + \mathbf{0}, \mathbf{5}E\_p A\_p + E\_s A\_s,\tag{47}$$

where *Iс*,*Ip*,*Is* are the moments of inertia of a concrete core, a steel tube and a longitudinal reinforcement; *Eс*, *Ep*, *Es* are the moduli of elasticity of concrete, steel case and longitudinal reinforcement.

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

#### **Figure 10.**

*The design scheme of a flexible pipe-concrete column: a - the decomposition of the compressed rod along the length; b - distribution diagrams of concrete relative deformations in Section 2 and 3.*

Flexibility can have a significant effect on the load capacity of compressed elements when the condition *λeff* >*λ*<sup>0</sup> is performed, in which the threshold value of flexibility is calculated by the following formula

$$
\lambda\_0 = \pi \sqrt{\mathbf{0}, \mathbf{0} \mathbf{1} (\mathbf{E} \mathbf{A})\_{\mathrm{eff}} / N\_{\mathrm{u0}}},\tag{48}
$$

where *Nu0* is the strength of a short, centrally compressed CFST element.

The compressive stress in the longitudinal reinforcement *σ<sup>s</sup>* is determined from the condition of its joint deformation with the concrete core. This takes into account the limitation *σ<sup>s</sup>* ≤ *fy,s*.

The calculation is based on the step-iteration method. During the second stage, an eccentrically loaded compressed element is divided along its length into n equal segments, at that *n* ≥*6* (**Figure 10**). Normal sections at the end of each segment are divided into small sections conventionally with the areas of concrete *Aci* and steel *Apk* tube.

The area of one rod of longitudinal reinforcement is *Asj*. Then the calculation process is performed in the following sequence. First, only one normal cross-section of a rod is considered, in which the maximum bending moment arises. This cross section is located in the middle of the column height for the articulated column loaded by a compressive force N with the initial eccentricity *e*0. The strain of the most compressed fiber of the concrete core *εcz* max is increased stepwise in this section.

At each step, the relative deformation of the least compressed (stretched) fiber *εcz* min is determined, corresponding to the conditions of equilibrium cross section. The equilibrium conditions are written in the form of the following equation system:

$$N = (EA)\_{\mathfrak{gl}} e\_0;\tag{49}$$

$$N(e\_0 + f) = (EI)\_{eff} \frac{1}{r},\tag{50}$$

where N is the longitudinal compressive force corresponding to the accepted deformation diagram; *ε*<sup>0</sup> is a fiber relative deformation located at the gravity center of calculated section; *f* is the deflection at the point of maximum bending moment; 1 *<sup>r</sup>* is the curvature of the longitudinal axis in the considered cross-section, determined by the following formula

$$\frac{\mathbf{1}}{r} = \frac{\varepsilon\_{\text{cx\,max}} - \varepsilon\_{\text{cx\,min}}}{d - 2\delta}. \tag{51}$$

Cross-section stiffnesses ð Þ *EA eff* and ð Þ *EI eff* are found taking into account the corresponding elastic coefficients of concrete and steel [7].

The effect of longitudinal bending is taken into account via the eccentricity of the longitudinal force increase by the amount of rod deflection *f* in the calculated section. In the first approximation, the deflection value is determined depending on the curvature of the calculated normal section. Taking into account the dependence (51), we can write the following formula

$$f = \frac{l^2}{\pi^2} \frac{\varepsilon\_{cz \text{ max}} - \varepsilon\_{cz \text{ min}}}{d - 2\delta},\tag{52}$$

where *l* is the estimated length of the considered rod.

An improved deflection value *f* should be found at each calculation step for a more reliable calculation of a compressed rod longitudinal bending. This can only be done by adjusting the stiffness along a rod length.

The numerical solution of the problem of calculating the deflection [16] with the number of partitions n = 6 allows us to obtain the following formula

$$f = \frac{l\_0^2}{266} \left( \frac{\mathbf{1}}{r\_0} + 6\frac{\mathbf{1}}{r\_1} + 12\frac{\mathbf{1}}{r\_2} + 8\frac{\mathbf{1}}{r\_{\text{max}}} \right),\tag{53}$$

where <sup>1</sup> *r*0 is the curvature of the element on the upper (lower) supports; <sup>1</sup> *ri* is the curvature of the element in the i-th section; <sup>1</sup> *<sup>r</sup>*max is the curvature in the middle of the height.

The problem under consideration is solved as follows. The deviations y of the longitudinal axis of the compressed rod from the vertical are calculated in the sections at the boundaries of each segment into which an element is divided with the deflection found in the first approximation according to the formula

$$y = f \sin\left(\pi z / l\right). \tag{54}$$

Then the distribution of the relative deformations is established for these crosssections, using the Eqs. (49) and (50) and by the replacement of *f* into *y*. Moreover, during the determination of *εcz* max and *εcz* min for each section, it is necessary to satisfy two conditions:


Let's note that the stiffness characteristics ð Þ *EA eff* and ð Þ *EI eff* depend on the parameters of the strain diagram. Therefore, they will be different for each section. *Bearing Capacity of Concrete Filled Steel Tube Columns DOI: http://dx.doi.org/10.5772/intechopen.99650*

After the determination of *εcz* max and *εcz* min according to the Eq. (51), the curvatures in the support and intermediate sections of the rod are found, and by the Eq. (53) the deflection f is specified. The process of deflection refinement can be repeated until a predetermined calculation accuracy is achieved.

They record the value of the compressive longitudinal force *N* for the assumed value of the relative strain of the most compressed fiber of the concrete core *εcz* max of the average cross-sectional rod and the refined deflection *f*. Then the strain of the most compressed fiber of the concrete core *εcz* max is increased and the whole procedure of calculations is repeated. Thus, the dependence "*N f* " is developed (see **Figure 9**). The maximum value of the longitudinal force *Nu*, perceived by the rod, is taken as the bearing capacity.
