How can we calculate the minimum bend radius(the radius which it starts to buckle) of a annular cylindrical tube? Intuitively I guess that for two tubes having same wall thickness but different outer diameters, the larger diameter one will have larger bend radius. Is there an analytical way to confirm this?

• This would also be a function of the material you intend to use. Different materials make different sections behave differently. Jun 6, 2016 at 11:49
• @Han-KwangNienhuys Thanks for the link. When we order PVC tubing the manufacturers have a minimum bending radius specification. Is it that they come up with that number experimentally ? Jun 6, 2016 at 14:23
• Just trying it out and writing the number in the data sheet sounds more practical than calculating it. Jun 6, 2016 at 17:01

There is an analytical solution to the buckling of a tube under flexure which was first proposed by Brazier in 1927. There is also a more detailed and helpful discussion of Brazier's solution in Calladine's book 'Theory shell structures'. (Apologies for the pay-walled links ...)

I won't repeat the analysis here but there are two main points which are are considered in the analysis: Firstly, consideration of the Brazier effect which is the flattening (ovalisation) of a tube when it is put under bending. You can see this effect by experimenting with a straw. Secondly, the local buckling condition of a cylindrical shell segment.

The results of this analysis are summarized by Calladine2 in the following plot:

This is a dimensionless plot with the following dimensionless parameters: $s_{cr}$ is the critical stress for local buckling, $s$ is the extreme fibre stress, $m$ is the bending moment, $\zeta$ is a flattening parameter, and $\frac{1}{c}$ is the radius the tube is bent to.

The radius at which buckling occurs is the the radius corresponding to the intersection of the $s_{cr}$ and $s$ curves as shown on the plot. You can calculate the actual (dimensioned) radius of curvature by reading $c$ from the plot and unraveling the dimensionless parameters. In which case you get the minimum bend radius R:

$$R = \frac{3^\frac{1}{2}r^2}{2 c t}$$

where $t$ is the tube thickness, and $r$ is the radius of the tube. Therefore the minimum bend radius increases with the square of the tube radius. There will be lots of other practical aspects which will influence the bending of a tube in reality which are not considered in this analysis.

There is also a practical element in that there are quite a few different methods for bending tube ranging from a completely unsupported bend through simple jigs, draw benders (which provide external support) to serious industrial NC mandrel benders which support the tube from the inside during bending and can give sophisticated control of strain rate and other parameters.

There is also a difference between creating a bend at a point and rolling a continuous radius.

In practice the behaviour can vary significantly even from one batch of stock to the next so any calculations need to be treated with some degree of caution.