# Description

B31.3 and B31J output different force and moment in tee section. What is the factor to produce such a result?

# Answer

It introduces the flexibility factor for calculation of force and moment in B31j.

## A. How to calculate force and moment in B31.3

The reaction force and bending moment at fixed point are given below equation in case free point is moved 150mm.

F=3EIU/L^3 = 32953 (N)

M= FL = 32953(Nm)

where, E : modulus of longitudinal elasticity (0.20271E+06(N/mm2))

I : moment of inertia of area (361247(mm4))

U : movement of free edge (150(mm))

L : Length of branch (1000(mm))

## B. How to calculate force and moment in B31J

In B31j, to account for the flexibility factor at outer surface of header, a rotating spring is inserted. The spring has In-plane, Out-Plane, Torsion component but it is used only torsion in this article.

A sample model has a below flexibility data;

----------------------------------------------------------------------------------------------------------------

T E E F L E X I B I L I T I E S D A T A L I S T I N G

Flexibility fac. k Flange Factor (c) Rotational Stiffness

Point Point In-pl Out-pl Torsn In-pl Out-pl Torsn In-pl Out-pl Torsn

Name Seg-Type (none) (none) (N.m/deg )

----- -------- ------ ------ ------ ------ ------ ------ -------- -------- --------

A01 A-Header Rigid Rigid Rigid 1.000 1.000 1.000 .136E+11 .136E+11 .136E+11

B-Branch 1.020 3.906 Rigid 1.000 1.000 1.000 .229E+05 .597E+04 .136E+11

Note: The rotational stiffness values displayed are calculated for branch and

run independently. Program uses "simultaneous use" stiffness values in the

analysis by combining branch and run stiffness values.

Header : Kr = 0.136E+11(N.m/deg)

Branch : Kb = 0.229E+05(N.m/deg)

Branch stiffness is given as below equation;

K = 1 / (1/Kb - 1/2Kr) = 0.229E+05(N.m/deg) = 0.131E+10(N.mm/rad)

Because B31J treats the length from tee cross point to outer surface of header as rigid body, the deformable branch length is given as below equation;

H = L - Do/2 = 771.4(mm)

Where, Do : Header outer diameter (457.2(mm))

The 150 (mm) displacement at the end of the branch pipe is a composite of the component due to rotation from flexible factor and the component due to bending deformation of the member with length H.

The force F and bending moment M at header surface when free edge is moved 150mm is given by below equation;

F = U / (H^2/K + H^3/3EI) = 58985(N)

M = FH = 45501(N.m)

(note : If the diameter ratio (d/D) of the header to the branch is 0.5 or less, the AutoPIPE report will output the forces and moments at the location of the outer surface of the header. If the ratio is greater than 0.5, the forces and moments at the intersection of the tees are output.)

The moment at tee point is given by below equation;

M = FL = 58985(N.m)

# Example

To check the theory, we can create the simple model like below;

This model is cantilever including tee section and given imposed displacement at free edge. To avoid gravity term, the imposed displacement is given to T1 case.

The result is below;

You can check the force and moment value indicate similar value as theoretical one at tee section of branch side.

# See Also

**Results Post Processing in AutoPIPE**