New analytical equations are developed to calculate stresses in an elbow submitted to an in-plane bending moment. If a detailed stress analysis is asked (for example, for the elastic stress intensity factor calculation $KI),$ available code equations are not accurate enough because their determination is based on an axi-symmetric analysis, where the elbow is compared to a torus, and therefore the end effects are not taken into account. This work relies on elastic finite element calculations using shell elements. Longitudinal and circumferential stresses are calculated in the median section and in the junction section between the elbow and the straight pipe. Elbows considered here have a bend angle $αc=45,$ 90, and 180 deg and show a ratio $rm/t$ between 5 and 20 and a parameter λ up to 1. Analytical expressions are proposed to fit the membrane and bending components of each stress. Coefficients of these equations are tabulated and expressed only as a function of the ratio $rm/t$ and the parameter λ for the median section and the junction section. An interpolation methodology is also given for intermediate elbow angles: these analytical relations give a simple solution to have an accurate estimate of the elastic stress distribution in the median section and the junction section in a elbow submitted to an in-plane bending moment, for a wide range of elbow angles, ratios $rm/t$ and parameters λ.

1.
ASME, ‘NB 3685: Class 1 piping’ Section III, Division 1. 1992.
2.
RCC-M, ‘B 3680: tuyauterie de niveau 1’ Tome 1, Vol. B. 1988.
3.
RCC-MR, ‘RC 3680: mate´riels de niveau 2’ Tome 1, Vol. C. 1993.
4.
Marie, S., and Ne´de´lec, M., 2000, “Simplified Method to Determine J for Cracked Elbows Submitted to Internal Pressure and In-Plane Bending Moment,” Proc., ASME PVP Conference, Seattle, WA.
5.
Von
Karman
, 1911, “U¨ber die forma¨nderung du¨nnwandieger Rohe, insbersondere federnder ausgliechrohre,” Zeitschrift des vereines deutscher ingenieure, 55, pp. 1889–1895.
6.
Vigness
,
I.
,
1943
, “
Elasticproperties of Curved Tubes
,”
Trans. ASME
,
65
, pp.
105
120
.
7.
Rodabaugh
,
E. C.
, and
George
,
H. H.
,
1957
, “
Effect of Internal Pressure on Flexibility and Stress-Intensification Factors of Curved Pipe or Welding Elbows
,”
Trans. ASME
,
79
, pp.
939
948
.
8.
Gross
,
N.
,
1953
, “
,”
Proc., Inst. Mech. Eng.
,
1B
, pp.
465
479
.
9.
Dodge, W. G., and Moore, S. E., 1972, “Stress Indices and Flexibility Factors for Moment Loadings on Elbows and Curved Pipe,” WRC Bulletin No. 179.
10.
Rodabaugh, E. C., Iskander, S. K., and Moore, S. E., 1978, “End Effects on Elbows Subjected to Moment Loading,” ORNL/Sub/2913-7 Report, Mar.
11.
ASME, ‘Code Case N-319’ Section III, Division 1. 1992.