### Stability coefficient

Previous studies on the behavior of original bamboo columns have focused on short columns [33, 39, 40]. However, columns used in practical engineering generally have a larger slenderness ratio, and research relating to the performance of slender bamboo columns is limited. For slender columns, the increased slenderness ratio inevitably leads to lateral deflection and consequent secondary moment, which produce an adverse effect on the columns. The bamboo culm diameter is gradually decreased from basal to top, named taper shape, which results in a gradual change for the second moment of area along the length. The imperfection may reduce the buckling resisting capacity of bamboo columns. Due to various factors, the original bamboo columns may have initial eccentricity, which may produce an additional bending moment and corresponding lateral deflection under axial loading. Besides, bamboo is oval in shape rather than perfect hollow cylinder, which may result in Brazier effect when the specimens under bending moment. Due to the Brazier effect, the longitudinal tension and compression, resisting the applied bending moment, also tend to flatten or ovalize the cross section. As the curvature increases, the flexural stiffness decreases, so as to reduce bearing capacity.

The effect of the additional bending moment on the short column is smaller than that on the long column, which can generally be ignored. For long columns, the additional bending moment and lateral deflection may increase with increasing axial load, eventually leading to instability failure. Therefore, slender columns should be designed considering the second-order effects produced by the inevitable initial eccentricity under axial loading. According to “CECS 434-2016 Technical specification for round bamboo-structural building [41]”, the stability coefficient, shown in Eq. (9), is used to study the reduction degree of the bearing capacity for the bamboo column.

$$ \varphi \,\, = \;\,\frac{{P_{l} }}{{P_{s} }}, $$

(9)

where *φ* represents the stability coefficient and *P*_{l} and *P*_{s} represent the ultimate bearing capacity of the long column and short column, respectively.

For the original bamboo short column, *P*_{s} should meet the following requirements:

$$ P_{s} = f \cdot A, $$

(10)

where *f* represents the compressive strength of the original bamboo and *A* represents the sectional area of the original bamboo.

Figure 13 shows the relationship between the stability coefficient and the slenderness ratio of the test results. The stability coefficient decreases with increasing slenderness ratio. For the tests conducted with the same slenderness ratio, the stability coefficient of the original bamboo specimen with a diameter–thickness ratio of 5.7 was larger than those with ratios of 5.5 and 4.8, which means that increasing the diameter–thickness ratio is useful for enhancing the stability coefficient. In addition, when the diameter–thickness ratios are the same, the stability coefficient decreases quickly with increasing slenderness ratio. Therefore, decreasing the slenderness ratio and increasing the diameter–thickness ratio are two effective methods to enhance the stability of bamboo columns and mitigate local buckling to improve the load capacity of original bamboo columns under axial compression. The regression method is analyzed to calculate the stability coefficient of original bamboo specimens with the same diameter–thickness ratio, as follows:

$$ \varphi = ({1 - 0}{\text{.000081}}\lambda^{2} { - 0}{\text{.004653}}\lambda ) \cdot ({1 + 0}{\text{.024140}}\xi^{2} { - 0}{\text{.150806}}\xi ) \, (\overline{\xi } = 4.8), $$

(11)

$$ \varphi = ({1 + 0}{\text{.000034}}\lambda^{2} { - 0}{\text{.011556}}\lambda ) \cdot ({1 + 0}{\text{.029165}}\xi^{2} { - 0}{\text{.163920}}\xi ) \, (\overline{\xi } = 5.5), $$

(12)

$$ \varphi = ({1 - 0}{\text{.001148}}\lambda^{2} { + 0}{\text{.049163}}\lambda ) \cdot ({1 + 0}{\text{.019136}}\xi^{2} { - 0}{\text{.172517}}\xi ) \, (\overline{\xi } = 5.7). $$

(13)

Moreover, the stability coefficient of original bamboo specimens with different diameter–thickness ratios can be calculated by a similar approach considering the diameter–thickness ratio, as follows:

$$ \varphi = ({1 - 0}{\text{.000040}}\lambda^{2} { - 0}{\text{.007868}}\lambda ) \cdot ({1 + 0}{\text{.019662}}\xi^{2} { - 0}{\text{.111979}}\xi ). $$

(14)

### Verification of the proposed model

Appendix B lists the calculated stability coefficients based on Eq. (9) and Eq. (10), and the average axial compressive strength of three short columns (*H* = 200 mm and *D*_{t} = 100 mm) is used as the compressive strength of the original bamboo materials. The calculated stability coefficient using Eq. (9) was compared with the experimental results, as shown in Fig. 14. The average ratio value (AV) of the calculated value to the experimental value is 1.01, the standard deviation (SD) is 0.08, and the mean absolute error (AAE) of the calculated value is 0.06. The regression test points are evenly distributed on both sides of the theoretical line, and the calculated values, calculated by Eq. (14), are in good agreement with the experimental values. The calculated results demonstrate that the proposed equation exhibited good accuracy in predicting the stability coefficient of the original bamboo columns.

The proposed model of the stability coefficient can be considered to calculate the axial load-bearing capacity of the original bamboo columns considering the influence of the slenderness ratio and diameter–thickness ratio as follows:

$$ P = \varphi \cdot f \cdot A, $$

(15)

where *P* represents the bearing capacity of the original bamboo; *φ* represents the stability coefficient; *f* represents the compressive strength; and *A* represents the sectional area.

To verify the validity of the proposed model, the experimental results of the ultimate bearing capacity are compared with those calculated by Eq. (15). As seen from Fig. 15, the proposed model is applicable and effective, and the proposed formula is valid for the rapid prediction of ultimate bearing capacity.