Discontinuity Stress Formula From Roark's Handbook

[graywolfscvm]

I'm writing a set of calculations that will show the allowable stress for a cylindrical tank with a cone bottom head. Usually we would use ASME code to determine this but the half apex angle of the cone head is greater than 30 degrees which means, according to ASME code, we have to present an analysis proving the discontinuity stress are within limits. I'm using the example from "Roark's Formulas for Stress & Strain", sixth edition, article 12.4, (page 579 - 582). Ok, here is my problem and I apologize in advance if this question is elementary.

I understand most of the example and I see where each equation is comming from. It's just the last part of the example that I can't seem to understand.

On page 581, the example equation shows the summation of the radial deflections for the end of the cone and equating it to the sum of the radial deflection for the cylinder.
0.00944+12.59E-6Q_o+4.677E-6M_o+=1.00796-20.16E-6V_o+6.689E-6M_o

Then the next equation is the same thing but for the meridian rotations.
0.000674+4.677E-6Q_o+3.395E-6M_o=0+6.698E-6V_o-4.452E-6M_o

Finally it shows the equation of the radial forces.
Q_o+5091cos45=V_o

My problem is I can not seem to determine how the forces were solved, (Q_o=-2110, V_o=1490, M_o=2443). I know it's a simple solution but I think I've looked at this too long to see the answer right in front of my face.

If anyone has seen this or has this book and can give me a hint that would be great.

Thank you for taking the time to read my question.

-Steve CVM

 

[israelkk]

This is a set of three linear equations with three unknown Qo, Mo, Vo. You should check your high school math how to solve it.

 

[rhg]

See "A.D. Merkblatt B2" for design of cone shell greater than 30 degrees.

 

[graywolfscvm]

Thank you for your help.

 

[arto]

I did this one out way back when & I have "sin" penned in over "cos" in the 3rd equation {i.e.: Qo+5091sin45=Vo ???[5th ed Roark p500 = an error in Roark ?? I used pen - musta been confident]

Mark's Handbook was solving 3eq's & 3 unknowns using determinants [pg 2-22 in the 8th edition]

 

[JStephen]

In the example, it won't matter, sin 45 = cos 45.

 

[juancito]

There are to good books that deals with this discontinuities situations:
Pressure Vessel Design Handbook by H.Bednar
Mechanical Design of Process Eq`mt. By Farr and Jawad.
Pls have a look on them.