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The code stress category is based on the load category. A sustained load creates sustained stress, and so forth. Stress calculation procedures are all the same regardless of the load or stress category.

The first step of stress calculation is to apply the corresponding category of loads to the piping sys-tem to find out the responding internal forces and moments at each component. This is the so-called structural analysis involving finite element or other appropriate methods. The routine piping stress analysis involves only straight beam and curved beam elements. The structural analysis deals only with the centerline of the piping system. It calculates the total forces and moments acting at the entire piping cross-section of each centerline point. These forces and moments are then used to calculate the stresses.

Calculations of the forces and moments at each component follow the basic principle of structural mechanics, which is unvarying and straightforward. However, the calculation of code stresses differs fairly appreciably among the codes. Lately, there have been significant attempts [12] in ASME to keep the requirements similar among all codes, especially between B31.1 and B31.3. However, some dif-ferences will no doubt remain as they are. These difdif-ferences are the driving force separating the code into different sections in the first place.

The standard stress formulas given by the codes, such as B31.1, include only the moments. In other words, the forces, except pressure force, are ignored in stress calculations. This is because the stresses due to forces are generally very small. In cases where stresses due to forces are significant, they will be reflected in the unusually huge loads generated at the anchors and restraints. Once these huge loads

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at the anchors and restraints have been reduced to an acceptable level, pipe stresses due to forces will be reduced to an insignificant level.

Figure 4.10 shows the moments acting at a bend and at a branch connection, which consists of three straight segments of pipe. Because of the different stress intensification factors involved at different component types and orientations, the moments calculated from the structural analysis have to be re-oriented in accordance with the component plane before being used to calculate the stresses. The legs of a bend or a branch connection form the plane of the component. The moment causing the leg to bend in the plane is called the in-plane bending moment, Mi, and the moment causing the leg to bend in the direction perpendicular to the plane is called the out-plane bending moment, Mo. The moment twisting the leg is called the torsion moment, Mt.

The stresses are calculated at the nodes located at the ends of each element. Each node at the ele-ment is independently calculated using the forces and moele-ments occurring at that node point. This ap-proach sounds reasonable for non-branched elements. For a branch connection, however, the forces and moments from all connecting legs are expected to act together at the common branch point, which is the junction of the legs. This naturally raises the question of the validity of calculating the stress on each leg independently. However, this appears to be not much of a concern in non-nuclear piping. Class 1 nuclear piping, to be discussed later in this chapter, considers the forces and moments from all three legs together at a branch connection.

The stresses due to these moments are calculated as follows (see Section 2.5.2 for stress formulas and Table 3.1 for stress intensification factors):

•  In-plane bending stress varies around the cross-section of the pipe with the maximum stress at the extreme fiber equal to

S_bi M_i

Zi_i (4.16a)

•  Out-plane bending stress also varies around the cross-section of the pipe with the maximum stress at the extreme fiber equal to

Fig. 4.10

Torsion momenT, and in-pLane and ouT-pLane Bending momenTs

S_bo Mo

Zi_o

(4.16b)

•  Combined bending stress is the combination of in-plane and out-plane bending stresses. Because the maximum fiber stresses of the in-plane and out-plane bending moments are located 90 deg.

apart, they can be combined by the square root of sum of squares (SRSS) method. That is,

S_b (S_bi)² (S_bo)² (i_iM_i)² (ioM_o)²

Z (4.16c)

The combined bending stress is in the longitudinal direction of the pipe. It has the maximum tensile stress at one point and varies gradually through the cross-section. Eventually, it changes to compression and reaches the maximum compression at the diametrically opposite location.

When combining with other stresses, we have to keep in mind that the bending stress always has equal magnitude of tension and compression in a cross-section.

•  Torsion stress is a shear stress uniformly distributed around the circumference of the cross-section, but varies linearly in the diametrical direction. The maximum shear stress at the outer surface is

W St Mt

Zpi_t Mt

2Zi_t (4.16d)

The polar (torsion) section modulus, Zp, is twice the bending section modulus, Z. The torsion stress intensification factor, it, is currently considered as unity for most of B31 codes.

Stresses given by Eqs. (4.16c) and (4.16d) can be evaluated separately, or in combination, de-pending on the allowable stress criterion given by the design specification or code.

•  Stress intensity and expansion stress. Thermal flexibility and most modern design analyses are based on the combined effect of all multi-dimensional stresses involved. ASME B&PV and B31 codes adopt the maximum shear failure theory in evaluating the combined effect (see Section 2.6.3) The combined quantity used is called the stress intensity, which is twice the maximum shear stress.

However, this combined quantity is not called the stress intensity in B31 because it uses only one-half of the theoretical stress intensification factors. Rather, it is often referred to as the expansion stress. This is because in B31 codes, this combined quantity was used first in the evaluation of thermal expansion stress range, SE, but not in anything else.

Stress intensity can be calculated based on the general two-dimensional formula given by Eq.

(2.19). In the B31 evaluation of thermal expansion stress, the cyclic effect of pressure is not included. The pressure effect is implied in the establishment of the allowable stress. Therefore, when evaluating expansion stress we have only the longitudinal bending stress and the torsion shear stress to consider. By setting Sy in Eq. (2.19) as zero, we have

S_E S_i S²_b 4W² (iiM_i)² (ioMo)² Equation (4.17) is used in B31.3 with it set to unity. The above equation can be simplified by conservatively using a uniform stress intensification factor, i, which is the greatest of (ii, io, and it).

Taking this conservative approach, we have a new working formula as follows:

S_E Si i

ZM_R where, M_R M_i² Mo²  M_t² (4.18) Equation (4.18) is used by B31.1, CEN, and nuclear piping codes.

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•  Stress calculation for branch leg. For branch connections, pipe stresses are calculated for each leg of the pipe using the moments, acting at the centerline intersection point, and the section modulus of the leg being evaluated. However, stresses at the branch leg (leg-3) are calculated differently. For the branch leg, the moments used are still the ones that are acting on the leg, but the section modulus used is an effective section modulus. This can be explained with the help of Fig. 4.11.

Figure 4.11 shows the potential locations of failure due to branch moments [13, 14]. When the stress intensification factor is significant, the failure will most likely occur at the header wall right next to the wall-to-wall junction. This is shown as “Failure Location A” in the figure. If the stress intensification is not significant, the pipe will crack at the branch pipe wall next to the junction marked as “Failure Location B.” B31 codes use the equivalent section modulus to cover both potential failure locations.

At location A, in addition to stress intensification, the strength and stress of the pipe depend on the wall thickness of the header, not the wall thickness of the branch. Therefore, it is logical to use a section modulus based on the header thickness. This is the first consideration of the effective section modulus, which is defined as

Ze S r²_bte (4.19)

where

Ze = effective section modulus to be used, instead of the branch section modulus, Zb, in calcu-lating stress at branch leg

rb = mean radius of the branch pipe. For integrally reinforced connections with sufficient length of uniform cross-section, the reinforcement section can be treated as the branch pipe for the connection

te = effective thickness. It is the smaller of Th or (iTb). This Th covers location A and (iTb) cov-ers location B of Fig. 4.11

Th = header thickness, excluding reinforcement Tb = branch thickness, excluding reinforcement

i = applicable stress intensification factor; can be conservatively taken as the smallest of all the factors involved

The inclusion of iTb in the definition of te is supposed to take care of location B. The application of Eq. (4.19) can be very confusing in actual cases. Because each component of the moment has its own stress intensification factor, this leads to the necessity of using a different equivalent sec-tion modulus for each moment component. Theoretically, it requires a different equivalent secsec-tion

Fig. 4.11

sTresses aT Branch pipe

modulus for torsion moment, in-plane bending moment, and out-plane bending moment. Even with the approach of conservatively using a uniform stress intensification factor for all moments, as in B31.1, we still have to consider the different intensification factors for different categories of stresses. For instance, B31.1 uses 0.75i as the stress intensification factor for the sustained and occasional stresses. In this case, 0.75iTb has to be used instead of iTb.

The other likely confusion comes from the corrosion allowance and manufacturing under-toler-ance of the wall thickness. Nominal wall thickness is used in expansion stress calculations, but the net wall thickness subtracting the allowance is generally used in sustained stress calculations.

This will result in a different equivalent section modulus than that given by Eq. (4.19). This confu-sion can be alleviated by the use of equivalent stress intensification factors instead.

•  Equivalent stress intensification factors can be used to simplify the process of evaluating the stresses at the branch leg. In this method, the stress at the branch leg is calculated with the unmodified original branch leg section modulus, but with an equivalent stress intensification factor, which is defined as

i_e,x ix Z_b

Sr_b²t_h ix(ZR) (4.20)

where

ie,x = equivalent stress intensification factor for i_x; ie,x ³ 1.0

ix = stress intensification factor of the applicable moment component (=ii, io, it, i, 0.75i, etc.) Zb = section modulus of the branch pipe

th = applicable header thickness

ZR = ratio of branch pipe section modulus versus the section modulus of the equivalent header ring, as shown in the formula

This equivalent stress intensification factor takes care of location A, and the limitation of the ie,x ³ 1.0 will automatically take care of location B.