Mechanics of High-Flexible Beams Under Live Loads

Abstract

In this paper the mathematical formulation of the equilibrium problem of high-flexible beams in the framework of fully nonlinear structural mechanics is presented. The analysis is based on the recent model proposed by L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. In this model the complete three-dimensional kinematics of the beam is taken into account, both deformations and displacements are considered large and a nonlinear constitutive law in assumed. After having illustrated and discussed the peculiar mechanical aspects of this special class of structures, the criteria and methods of analysis have been addressed. A classification of the structures based on the degree of kinematic constraints has been proposed, distinguishing between isogeometric and hypergeometric structures. External static loads dependent on deformation (live loads) are also considered. The governing equations are derived on the basis of a moment-curvature relationship obtained in L. Lanzoni and A.M. Tarantino: The bending of beams in finite elasticity in J. Elasticity (2019) doi:10.1007/s10659-019-09746-8 2019. The governing equations take the form of a highly nonlinear coupled system of equations in integral form, which is solved through an iterative numerical procedure. Finally, the proposed analysis is applied to some popular structural systems subjected to dead and live loads. The results are compared and discussed.

Appendix: Expressions of Derivatives \(S_{X,X}\), \(S_{Y,Y}\) and \(S_{Z,Z}\)

The derivatives of the functions $S_{J}=2 (\omega _{1}+I_{1}\, \omega _{2} )\lambda _{J}-2 \, \omega _{2}\, \lambda _{J}^{3}+2\, I_{3}\, \omega _{3}\, \frac{1}{\lambda _{J}}$, with $J=X,Y$ and $Z$, contained in the system (20), have the following expressions:

$$\begin{aligned} S_{X,X} &=2 (\omega _{1,X}+I_{1,X}\, \omega _{2}+I_{1}\, \omega _{2,X} )\lambda _{X}-2\, \omega _{2,X}\, \lambda _{X}^{3}+2 (I_{3,X} \, \omega _{3}+I_{3}\, \omega _{3,X} )\frac{1}{\lambda _{X}}, \\ S_{Y,Y} &=2 (\omega _{1,Y}+I_{1,Y}\, \omega _{2}+I_{1}\, \omega _{2,Y} )\lambda _{Y}+2 (\omega _{1}+I_{1}\, \omega _{2} ) \lambda _{Y,Y}-2\, \omega _{2,Y}\, \lambda _{Y}^{3} \\ &\quad {}-6\, \omega _{2}\, \lambda _{Y}^{2}\, \lambda _{Y,Y}+2\, I_{3,Y}\, \omega _{3}\, \frac{1}{\lambda _{Y}}+2\, I_{3}\, \omega _{3,Y}\, \frac{1}{\lambda _{Y}}-2\, I_{3}\, \omega _{3}\, \frac{\lambda _{Y,Y}}{\lambda _{Y}^{2}}, \\ S_{Z,Z} &=2 (\omega _{1,Z}+I_{1,Z}\, \omega _{2}+I_{1}\, \omega _{2,Z} )\lambda _{Z}+2 (\omega _{1}+I_{1}\, \omega _{2} ) \lambda _{Z,Z}-2\, \omega _{2,Z}\, \lambda _{Z}^{3} \\ &\quad {}-6\, \omega _{2}\, \lambda _{Z}^{2}\, \lambda _{Z,Z}+2\, I_{3,Z}\, \omega _{3}\, \frac{1}{\lambda _{Z}}+2\, I_{3}\, \omega _{3,Z}\, \frac{1}{\lambda _{Z}}-2\, I_{3}\, \omega _{3}\, \frac{\lambda _{Z,Z}}{\lambda _{Z}^{2}}, \end{aligned}$$

with $\omega _{i}=\frac{\partial \omega }{\partial I_{i}}$, $I_{i,K}=\frac{\partial I_{i}}{\partial K}$ and $\omega _{i,K}=\frac{\partial }{\partial K} ( \frac{\partial \omega }{\partial I_{i}} )$ for $i=1, 2, 3$, $K=X, Y$ and Z. On the basis of the kinematic model, the following quantities can be computed:

$$\begin{aligned} &\lambda _{Y,Y}=\frac{\partial \lambda _{Y}}{\partial Y}=- \frac{1}{r(s)}\, e^{-\frac{Y}{r(s)}}, \\ &\lambda _{Z,Z}=\frac{\partial \lambda _{Z}}{\partial Z}=r(s) \bigl[1-e^{- \frac{Y}{r(s)}} \cos \beta (s) \bigr]\theta ^{(2)}(s),\text{ with } \frac{1}{R(s)}=\theta ^{(1)}(s), \\ &I_{1}=2\, e^{-\frac{2Y}{r(s)}}+ \biggl[1+\frac{r(s)}{R(s)}\, \bigl(1-e^{- \frac{Y}{r(s)}}\cos \beta (s) \bigr) \biggr]^{2}, \\ &I_{3}=e^{-\frac{4Y}{r(s)}} \biggl[1+\frac{r(s)}{R(s)}\, \bigl(1-e^{- \frac{Y}{r(s)}}\cos \beta (s) \bigr) \biggr]^{2}, \\ &I_{1,X}=2\, \lambda _{Z}\, \frac{1}{R(s)}\, e^{-\frac{Y}{r(s)}}\, \sin \beta (s), \\ &I_{1,Y}=-\frac{4}{r(s)}\, e^{-\frac{2Y}{r(s)}}+2\, \lambda _{Z}\, \frac{1}{R(s)}\, e^{-\frac{Y}{r(s)}}\, \cos \beta (s), \\ &I_{1,Z}=2\, \lambda _{Z}\, r(s)\, \bigl(1-e^{-\frac{Y}{r(s)}} \, \cos \beta (s) \bigr)\theta ^{(2)}(s), \\ &I_{3,X}=2\, e^{-\frac{5Y}{r(s)}}\, \lambda _{Z}\, \frac{1}{R(s)}\, \sin \beta (s), \\ &I_{3,Y}=-\frac{4}{r(s)}\, e^{-\frac{4Y}{r(s)}}\, \lambda _{Z}^{2}+ \, 2\, e^{-\frac{5Y}{r(s)}}\, \lambda _{Z}\, \frac{1}{R(s)}\, \cos \beta (s), \\ &I_{3,Z}=2\, e^{-\frac{4Y}{r(s)}}\, \lambda _{Z}\, r(s)\, \bigl(1-e^{- \frac{Y}{r(s)}}\, \cos \beta (s) \bigr)\theta ^{(2)}(s). \end{aligned}$$

For a Mooney-Rivlin material the following derivatives, contained in the previous equations, are calculated:

$$\begin{aligned} &\omega _{1}=a,\quad \omega _{1,X}=\omega _{1,Y}=\omega _{1,Z}=0, \\ &\omega _{2}=b,\quad \omega _{2,X}=\omega _{2,Y}=\omega _{2,Z}=0, \\ &\omega _{3}=c-\frac{d}{2I_{3}}, \\ &\omega _{3,X}=\frac{d}{R(s)}\, \frac{1}{\lambda _{Z}^{3}}\, e^{ \frac{3Y}{r(s)}}\, \sin \beta (s), \\ &\omega _{3,Y}=-\frac{d}{2}\, e^{\frac{4Y}{r(s)}} \biggl( \frac{4}{r(s)}\, \frac{1}{\lambda _{Z}^{2}}-\frac{2}{R(s)}\, \frac{1}{\lambda _{Z}^{3}}\, e^{-\frac{Y}{r(s)}}\cos \beta (s) \biggr), \\ &\omega _{3,Z}=d\, e^{\frac{4Y}{r(s)}}\, \frac{1}{\lambda _{Z}^{3}} \, r(s) \, \bigl(1-e^{-\frac{Y}{r(s)}}\cos \beta (s) \bigr)\theta ^{(2)}(s). \end{aligned}$$