How to calculate the equivalent stiffness of concentric, composite tubes in bending?

[yamoffathoo]

I would like to determine the effect on natural frequencies in bending by adding incremental layers of a composite wrap to the entire length of a stainless steel tube.
For this application, the tube is simply supported at one end and anchored at the other with uniformly distributed mass of just the composite assembly.

The stainless steel tube has the following properties:
Inner Radius = 0.185”
Outer Radius = 0.25”
Young’s Modulus = 30,000,000 psi
Poisson’s Ratio = .27
Density = .289 lbm/in^3

The composite wrap has the following properties:
Inner Radius = 0.25” (same as outer radius of tube)
Young’s Modulus (Circumferential and Axial) = 5,662,000 psi
Poisson’s Ratio = 0.091
Density = 0.0683 lbm/in^3
a single wrap thickness is 0.051"

Roark & Young 5th ed shows how to do it with beams by developing the equivalent width technique within the proportional limits for all materials, but I haven't found a method for hollow circular sections.
FEA is an option but I was hoping to apply TkSolver and its iterative equation solution capability.

 

[ANE91]

This is commonly investigated for tower-type structures and chimneys, where the structure narrows with height, but there is typically no composite component to the evaluation.

Why not simply transform the wrap into steel and run the usual dynamic analysis?

 

[SWComposites]

as a first order approximation, just scale the composite thickness down by the ratio of moduli, and add that adjusted thickness to the steel thickness.

 

[HTURKAK]

I would like to determine the effect on natural frequencies in bending

In this case , three parameters ( E, I , m ) of the combined section are required. The bending stiffness ( E*I ) of the combined section will be superposition of two cylinders .
EI of total section= EI of steel tube + EI of wrap
and m= m1+m2 .

If you were looking for the stresses ( due to bending , torsion ..) this is another story.

 

[yamoffathoo]

Let me know if you agree with my equations and whether the incremental wrap produced the result expected, thanks.

 

[yamoffathoo]

My apologies - trimmed the Rules by accident.

 

[GregLocock]

Let's put our BoFP hats on and try and do it by hand. EI_both=E_steel*I_steel+ E_wrap*sum(I_wrap_n) where n is the number of wraps

m_both=M_steel+sum(m_wrap_n)

f is proportional to sqrt(EI/m)

So your f curve looks right, thin layers of wrap are acting like mass loading, due to the low value of E_wrap/E_steel, but as sum(I_wrap_n) increases it starts to dominate and the higher EI_wrap/m_wrap dominates. Eventually it should trend toward the value for a pure composite tube.

 

[yamoffathoo]

Thanks Greg Locock,
This 40" length of tube is responding to a process pulsation that caused the tube-to-nozzle weld (at the anchored end) to fail in fatigue after 2 years of continuous operation that subsequently caused loss of production to make the repair.
The 40" span is not close to walls or floors where intermediate supports could be added, so the thought was to add insulation or a composite wrap to shift the natural frequency.
If my calculations are correct, Carbon Seal wrap would require too much mass to significantly move the natural frequency.
Is there a way to determine the dampening effect of the wrap?

 

[GregLocock]

There may be some brave soul who thinks they can calculate it, I'd be more inclined (being Mr Empirical) to bash a wrapped tube with a stick. My guess is that compared with a steel tube at say 0.01 the impedance mismatch at the interface, plus the naturally higher damping of the composite (go and hit a glass fiber tube for comparison) will take you up to 0.05. You could of course add either a harmonic damper or some constrained layer damping (a tube filled with bitumastic goo or sand or lead pellets)

 

[yamoffathoo]

Thanks Greg,

I received an E.Mail notification with your response which included the following statement at the end:

"You don't have to wrap the whole tube evenly, the maximum bending and least impact from the mass loading is at the root."

This statement does not appear in the text above...did you reconsider and remove it later and if not, (because there is a glitch in the software), how far down the tube from the nozzle would you recommend wrapping the tube?

I should mention that this tube is a drain line on a hydraulic system containing a nasty phosphate ester that operates at 1,450 psig, so the wrap would have a "tell-tale" tube embedded in the wrap at the nozzle/tube weld leading to a container that can be monitored daily.

In addition, the 40" span continues after the support for another 48" to two, closely spaced, closed gate valves supported off a wall, which raises the possibility of an acoustic resonance happening in this dead ended branch.

Would you consider partially opening the valve closest to the nozzle to disrupt a standing wave and if so, by how much?

 

[GregLocock]

How odd. Yes I definitely wrote that I must have overwritten it when adding the untuned dampers bit.

I don't have a number, I'd work it out. The optimum shape for a cantilever is frequently studied. However you need to consider the entire system, not just one part of it.