**2. Basic design provisions**

#### **2.1 Initial provisions**

Initially, the diameter *d* and wall thickness *δ* of the tube should be assigned for CFST. Taking into account the research results [7] for columns of circular cross-section, it is recommended to use the following restrictions:

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

$$20\sqrt{\frac{235}{f\_y}} \le \frac{d}{\delta} \le 150\frac{235}{f\_y},\tag{1}$$

where *fy* is a yield stress of the steel shell, MPa.

For monolithic columns, the possibility of loss of stability of the tube wall at the stage of installation of the supporting structures of the frame should be taken into account. The steel tube can be used as a supporting structure for several overlying floors even before it is filled with concrete, which significantly speeds up the process of constructing a building. In this case, local buckling is impossible when

$$\frac{d}{\delta} \le 85 \sqrt{\frac{235}{f\_y}}.\tag{2}$$

If condition (2) is not met, it is necessary to check the stability of the tube walls under the action of corresponding loads. For this purpose, for example, the recommendations of European norm procedure (EN 1993-1-1 Steel Design) can be used.

For a short centrally loaded CFST column, the cross-sectional strength is usually determined. Most researchers use a fairly simple formula for this

$$N = f\_{cc}A + \sigma\_{px}A\_p,\tag{3}$$

where *fcc* is strength of volumetrically loaded concrete core;

*σpz* is axial direction compression in the steel shell in CFST limit state;

*A* and *Ap* are cross-section areas of the concrete core and the steel shell.

Thus, in order to calculate the CFST strength, it is necessary to know the values of the strength of the volumetrically loaded concrete core and the compression in the steel shell. Various approaches and relationships for determining *f cc* and *σpz* are recommended. They are reviewed below.

#### **2.2 Known approaches for determining the strength of a concrete core**

Compression strength is a very important mechanical attribute of CFST concrete core. In the limiting state centrally loaded circular section column, concrete is in the conditions of three-axis compression by axial direction strain *σ<sup>с</sup><sup>z</sup>* and transverse strain *σ<sup>с</sup>r*.

A quite simple relationship, being in fact the Mohr-Coulomb strength condition, is most often used in calculations for such conditions

$$f\_{\mathcal{cc}} = f\_{\mathcal{c}} + k \sigma\_{\mathcal{cr}},\tag{4}$$

where *fc* is concrete unconfined compression strength;

*k* is coefficient of lateral pressure.

Considering experiments, the value of the *k* coefficient is usually taken as constant in this formula: *k* = 4,1 or *k* = 4,0.

Though the Eq. (4) was recommended by American researches F. Richard, A. Brandtzæg and R. Brown as far back as in 1929, it is currently used by many researches, including for designing columns with different types of confinement reinforcement. The relationships to determine the volumetrically loaded concrete recommended by regulations in many countries have been obtained based on this very formula. However, the gained new experimental materials evidence that the Eq. (4) does not always allow to get a valid result.

This is caused by many reasons. One of them is inaccuracies in determination of lateral strain *σсr*. The second reason is ignoring the scale factor. Since CFST frequently have significant cross-sectional dimensions (630 … 1000 mm and more for high buildings), this factor shall be considered. The research [4] devoted to a review of a government program of concrete-filled tubes research carried out in the end of the 20th century in Japan introduces the following relationship

$$f\_{\alpha} = \chi\_c f\_{\varepsilon} + k \sigma\_{cr},\tag{5}$$

in which *γ<sup>с</sup>* scale is factor coefficient determined by the formula

$$
\gamma\_c = \mathbf{1}, \mathbf{67} \cdot d\_c^{-0.112} \ge \mathbf{0}, \mathbf{85}, \tag{6}
$$

where *d<sup>с</sup>* is concrete core diameter in mm.

A similar dependence was proposed in [15].

Regarding such approach as conceptually correct, it is worth mentioning a quite limited range of CFST cross section diameters, where usage of relationships (6) allows to obtain a result acceptable for practical purposes. According to this formula, first, *γ<sup>с</sup>* ≈ 1 when the concrete diameter is 100 mm, and а *γ<sup>с</sup>* ≈0.95 when *d<sup>с</sup>* = 150 mm. In most countries, square-sided cube test pieces or cylinders with a cross-sectional diameter of 150 mm are considered reference concrete. In this case, the regulations provide that *γ<sup>с</sup>* ¼ 1.05 when *d<sup>с</sup>* = 100 mm. Secondly, one has to take *γ<sup>с</sup>* ¼ 0.85 already when the cross-section diameter exceeds 300 mm, which does not correspond to experimental data of researches of large scaled samples with cross section diameters between 630 and 1020 mm.

Considering the results of the research [16], the coefficient *γ<sup>с</sup>* is recommended to be determined by the formula

$$\gamma\_c = 0, \text{75} + 0, \text{25} \left( \frac{d\_0}{d\_c} \right)^{0,5}, \tag{7}$$

where *dо* is reference cylinder diameter taken equal to 150 mm.

This formula does not need any limitations in a quite wide range of *d<sup>с</sup>* = 100 to 3000 mm, which is convenient for practical calculations.

Another reason of the results obtained by the Eq. (4) not always corresponding to experimental data is the value of the coefficient of lateral pressure *k* = 4.1 taken as constant here. The research [17] shows theoretically that this value is variable. Our researches [16] found that the lateral pressure *σ<sup>с</sup><sup>r</sup>* reaches sometimes a value of 10÷15 MPa and more for CFST concrete cores before concrete destruction. Meanwhile, the values of the coefficient of lateral pressure can be within a range *k* =2,5÷7. Therefore, it is obvious that even insignificant inaccuracies in determination of *k* frequently lead to significant errors in determination of concrete core strength *f cc* and load-carrying ability of a designed element.

Some of researches recommend considering this point. For example, in the research [18] it was correctly mentioned that, other factors being equal, the value of the coefficient of lateral pressure decreases while this pressure increases. A formula is recommended for its determination

$$k = \mathfrak{G}, \Im(\sigma\_{cr})^{-0, \Im}. \tag{8}$$

However, recently a formula of J. Mander has been used more frequently than others [19].

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

$$\frac{f\_{cc}}{f\_c} = 2,254\sqrt{1+7,94}\frac{\sigma\_{cr}}{f\_c} - 2\frac{\sigma\_{cr}}{f\_c} - 1,254.\tag{9}$$

This formula was received based on the results of statistical processing of a large amount of experimental data and is usable for not only medium- but also high-strength concrete with *fc* of up to 120 MPa.

However, two main disadvantages of the Eq. (9) should be mentioned. First, lateral pressure *σс<sup>r</sup>* shall be known in CFST limit state to use it. As previously noted, this pressure is unknown when the load-carrying ability of such columns is calculated. Experiments with 180 samples of concrete-filled tubular elements [16] showed that *σс<sup>r</sup>* depends on geometry and design parameters of a designed column and may vary in wide limits. In addition, the relationship (9) is correct only for normal concrete. E.g. it is well known that fine grain concrete resists volumetric compression somewhat worse [17]. That is why other relationships shall be obtained for other concrete types, which causes certain inconveniences in calculations.

Processing of a number of experimental data evidences the existence of a stable relationship between *σ<sup>с</sup><sup>r</sup>* and a constructive coefficient of concrete-filled tubes *ξ* determined with use the formula

$$\xi = \frac{f\_{\mathcal{J}} A\_p}{f\_c A}. \tag{10}$$

The appropriate formulas are used in Chinese Technical Code for CFST structures (GB50936–2014).

#### **2.3 State of stress in steel tube**

Two methods to assess state of stress in a steel shell are known. The first one hypothesizes that a steel tube acts only transversely in limit state. In this case, the axial direction compression in the steel shell *σpz* is equal to zero. Then hoop stress determining the value of the lateral pressure in concrete reaches the yield stress of steel *σp<sup>τ</sup>* ¼ *f <sup>y</sup>*. However, in general, it does not correspond to the real state of stress in a steel shell. Most researchers believe that the value of stress *σpz* depends on geometry and design parameters of CFST.

In the limiting state, the stress intensity in the steel shell reaches the yield point. During the central compression of a short CFST element, the steel shell experiences a compression-tension-compression stress state. Radial compressive stresses in the wall of steel tubes with *d/δ* ≥ 40 are small and they are usually neglected. Then the plane stress state "compression-tension" is considered for the tube. For this case, the Hencky-Mises yield criterion is written as follows:

$$
\sigma\_{p\underline{\pi}}^2 + \sigma\_{p\underline{\pi}}^2 - \sigma\_{p\underline{\pi}}\sigma\_{p\underline{\pi}} = f\_y^2,\tag{11}
$$

where *σp<sup>τ</sup>* is the steel tube hoop stress in CFST limit state. Then the stress *σpz* can be calculated using the formula

$$
\sigma\_{p\underline{r}} = \sqrt{f\_y^2 - \mathbf{0}, \mathbf{75}\sigma\_{p\underline{r}}^2} - \mathbf{0}, \mathbf{5}|\sigma\_{p\underline{r}}|.\tag{12}
$$

Let us mention that the Eq. (12) is correct for thin-shell tubes when d/δ ≥ 40. These very tubes are generally used as steel shells for CFST.

The hoop stresses averaged by thickness in the steel shell for thin-shell tubes can be expressed through the lateral pressure by the following relationship with accuracy sufficient for practical calculations

$$
\sigma\_{p\tau} = -2\sigma\_{cr}\frac{A}{A\_p}.\tag{13}
$$

Consequently, the axial direction compression in the steel shell depend on its yield stress *f <sup>y</sup>*, the value of the lateral pressure from the concrete core *σcr*, and ratio of the column reinforcement.

### **2.4 Central compression strength**

The literature review shows that obtaining a reliable formula for determining the strength of volumetric compressed concrete of CFST elements is not an easy task. Most often, empirical formulas, which have significant limitations depending on the conditions of carried out experiment, are used. In case of structural changes or the use of new types of concrete and steel grades, other formulas will be needed. In this case, it is necessary to correctly determine the lateral pressure of a steel tube *σcr* on concrete, which directly affects both the strength of the concrete *f cc* and the stress in the tube *σpz*.

In this regard, it is important to obtain theoretically based, universal formulas for determining *f cc*, *σpz* and the strength of CFST. The solution to this problem is proposed on the basis of the known strength function of volumetric compressed concrete [17]. In the case of uniform lateral pressure, the result of solving this function is Eq. (5) with a variable value *k* depending on the level of lateral pressure *m* ¼ *σ<sup>с</sup>r= f cc* and the type of concrete. For its determination, a formula is recommended

$$k = \frac{1 + a - am}{b + (c - b)m},\tag{14}$$

where *a*, *b* are material coefficients determined based on experiments;

*c* is a parameter determining the nature of strength surface in the area of all-around compression (for a dense concrete core, the strength surface is open, and *c* = 1).

The average values of strength of normal concrete, calculated with a reliability of 50%, correspond to the coefficients *b* = 0.096 and *a* = 0.5*b*.

The analysis of relationship (14) shows that with high levels of sidework (with *m* ! 1Þ, the value of the lateral pressure coefficient is *k* ! 1. In such cases, concrete destruction will be of shear nature, according to Coulomb's law. With the abovementioned coefficients *k* for CFST, volumetrically loaded concrete destruction occurs due to combinations of break and shear, which corresponds to numerous experimental data.

Inserting the Eq. (14) into the Eq. (5) and performing some transformations, we will obtain:

$$f\_{\ll} = a\_{\mathfrak{c}} \chi\_{\mathfrak{c}} f\_{\mathfrak{c}};\tag{15}$$

$$a\_c = 0, 5 + 0, 7\overline{5\sigma} + 0, 25\sqrt{\left(\overline{\sigma - 2}\right)^2 + 16\overline{\sigma}/b},\tag{16}$$

where *σ* is a relative value of the lateral pressure from the steel shell on the concrete core in limit state *σ* ¼ *σcr= γ<sup>c</sup> f <sup>c</sup>* � �.

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

Using the relationship (12) and performing some little manipulations, we can write the Eq. (12) as follows

$$
\sigma\_{pz} = \chi\_c f\_c \left( \sqrt{\xi^2 - 3\overline{\sigma}^2} - \overline{\sigma} \right) \frac{A}{A\_p}. \tag{17}
$$

The formula for *σ* calculation is received from solving the task of determination of the maximum compression force received by a short centrally loaded column. Inserting (15), (16) and (17) into the Eq. (3), we obtain the following equation

$$N = \gamma\_c f\_c A \left[ 1 + \left( \frac{\overline{\sigma} - 2}{4} + \sqrt{\left( \frac{\overline{\sigma} - 2}{4} \right)^2 + \frac{\overline{\sigma}}{b}} - \frac{\overline{\sigma}}{2} + \sqrt{\xi^2 - 3\overline{\sigma}^2} \right) \right]. \tag{18}$$

It is obvious that the total axial force received by concrete and steel with standard cross-section depends only on relative lateral pressure *σ* with fixed values of geometry and design parameters of CFST ( *f <sup>c</sup>*, *f <sup>y</sup>*, *A*, *Ap*). For illustrative purposes, **Figure 1** represents diagrams of changes of relative forces received by concrete *N<sup>с</sup>* and the steel shell *Np* and their sum *N* depending on *σ* value. All forces are determined here in relation to the destructive load.

**Figure 1** shows that the graph of the total force change has a maximum point. The maximum compressive force can be found from the equation *<sup>d</sup> <sup>d</sup><sup>σ</sup>* ð Þ¼ *N*ð Þ *σ* 0*:* After determining the derivative we have the equation

$$\left(\frac{b(\overline{\sigma}-2)+8}{\sqrt{b}\sqrt{b(\overline{\sigma}-2)^2+16\overline{\sigma}}}-\frac{12\overline{\sigma}}{\sqrt{\xi^2-3\overline{\sigma}^2}}-1\right)=0.\tag{19}$$

As a result of solving Eq. (19), the following formula was obtained

$$
\overline{\sigma} = \mathbf{0}, 4\mathbf{8}e^{-(a+b)}\mathcal{J}^{0.8}.\tag{20}
$$

Thus, the necessary formulas to calculate the strength of a short centrally loaded CFST have been received.

#### **Figure 1.**

*Diagrams of changes of relative compressive forces received by concrete (1) and the steel shell (2) and their sum (3) depending on σ value.*
