Mechanism of cracking failure in curved stems due to transverse stress under a bending moment

It is essential to understand the failure modes and the failure moments of curved stems since trees could be damaged and may cause severe loss of life and property in urban. In our previous papers, the cracking failure mechanism of hollow erect and curved trunks under bending moment was clarified and we also found that the mechanism of cracking failure of solid trunk is different from the hollow one. While the existing equation to calculate the transverse stress of a solid curved stem under a bending moment is approximate and may cause considerable errors when the initial curvature of stem is small. To solve this problem, a series of novel equations were derived in this study. Among these newly derived equations, 3 of them are especially practical in the assessment of the risk of urban tree for both safety and environment management which are M_c, c_minR and c_criR. The equation of M_c is to calculate the bending moment for a solid curved trunk at which the cracking failure is initiated and this equation is more accurate than the existing equation. The equation of c_minR is to calculate the minimum cR (c: curvature, R: radius of trunk) below which cracking failure will not occur, and the equation of c_criR is to calculate the critical cR which represents equal opportunities for cracking and bending failure of the stem to occur. To exert our model to practice, the equations derived in this paper were applied to literature data (Wood handbook, 1999). From the data of 41 species of softwood and 48 species of hardwood, statistically, hardwoods have larger average values of c_minR and c_criR than softwoods which means that hardwoods are more resistant to cracking failure than softwoods.