Maximum Bending Stress and Pressure Vessel Design

If I am picturing your case as you describe - can you not just consider the upper drum as a uniform load across its length?

.......... and be sure to include ligament efficiency as per UG-53 (assuming you are designing to ASME) - and watch for buckling.

............ wait a minute.

Could you describe a little more clearly what you are trying to do. Are you "hanging" the lower drum by the tubes in between?

i'm designing based on IBR code.it's the bi drum boiler.in this the upper drum is hanged by the u rod support.the lower drum is supported by the boiler bank tubes which are connected between upper and lower drum.as per regulation 273 if i want to find out the maximum bending moment acting at the lower drum.how could i calculate this.?in the case of upper drum there is u rod supporting point with that reference i can found out the maximum moment acting at upper drum.how could i found out the maximum bending moment in lower drum in this case.

As per IBR reg 273

The resultant bending moment M at any section shall be the algebraic sum of the bending moments due to the eccentricity of the end pressure and that due to the externally applied loads.

where,

M = Resultant bending moment due to eccentricity in pound-inches.
P = Design pressure in pounds per square inch.
D = Internal diameter of drum or header in inches.
e = Eccentricity of the nett cross section i.e., the distance from the neutral axis of the nett section to the drum or header axis in inches.

The bending moment (M_w) due to externally applied loads shall be calculated by treating the drum or header as a beam carrying the externally applied loads, including the self-weight of the drum or header and its contents under working conditions.

in calculating the longitudinal stress due to bending in a drum supported at or near its ends and connected to a lower drum by a bank of tubes (so arranged as to form substantial struts between the drums) the value of the moment of inertia I_a used in the formula in sub-regulation (c) shall be:

Moment of inertia of upper drum (I_a) plus a proportion (S) of the moment of inertia of lower drum (I_c).

where,

a = The angle in degrees between the vertical and the line joining the centres of the upper and lower drums. Where a is equal to or greater than 15½°; S shall be taken as O. In no case shall the actual value of I_a used in Equation 72d be taken as more than 1.33 times the moment of inertia of the upper drum (I_b),

I am confused - it seems pretty straightforward to me.

thank you for your reply.in my case the angle "a" is less than 15 1/2°. sorry to say i couldn't understand how to calculate the bending moment due to load for lower drum in my case,because it is hanged by boiler bank tubes in reg.273 it is given as (in no case shall the actual value of Ia used in Equation 72d be taken as more than 1.33 times the moment of inertia of the upper drum (Ib) is it mean that lower moment of inertia can be calculated as any value less than 1.33 times the moment of inertia of upper drum.

I interpret it as this:

Lower Drum: Calculate bending moments as per beam theory: The bending moment (M_w) due to externally applied loads shall be calculated by treating the drum or header as a beam carrying the externally applied loads, including the self-weight of the drum or header and its contents under working conditions.

Do that irrespective of anything to do with the upper drum.

Then, if by your situation, equation 72(e):

Moment of inertia of upper drum (I_a) plus a proportion (S) of the moment of inertia of lower drum (I_c).

works out to be more than 1.33 the actual moment of inertia of the upper drum.

You have to use that value (i.e. 1.33X) and nothing more (even if the above equation works out to more)

thank you i understood well to find the moment of inertia of lower drum in this case we can realize the lower drum is supported by each tube attached to it,in this case is it possible to consider the bending stress to be negligible. in reg.273 (72b) how could i find out the mw value(bending moment due to load) for load.shall i consider the extremity tube for finding out the bending moment,because the lower drum is supported by more no of bank tubes attached to it.

Yes - the bending moment of the lower drum will be negligible because you probably [I don't know enough about any other specific loadings in your case] have a uniform load equal and opposite a uniform load (the tubes supporting versus the vessel weight/unit length) - depending on the length of the drum that the upper tubes are installed along; if the tubes are over the majority of the length you will have some bending at the ends.

I am not sure about the second part of your question - but from 273b

The bending moment (M_w) due to externally applied loads shall be calculated by treating the drum or header as a beam carrying the externally applied loads, including the self-weight of the drum or header and its contents under working conditions.

So treat the upper drum as a beam: a uniform load from the lower drum + weight of upper drum and tubes + weight of contents = "whatever support system you have"

yes-i understood hence it is advisable to check the tension stress induced in the tube which have to support the lower drum. regarding the explanation for finding out the bending moment for upper drum i accept it.but it is not clear to find out the bending moment due to load for lower drum.