Stress analysis of the flange ring under preloading conditions involves a detailed examination of deformation coordination between the flange and the shell. The total radial displacement of the flange ring is calculated as: w0f = wp0sf + wM0sf + wM0Hf. The external torque M0H is given by M0H = p0b(D0b - D0g)/2 = p0bS0, based on assumptions (2) and (5), where wM0sf = 0, wM0Hf = 0, and wp0sf = 0.
The axial stress at the edge of the flange ring is equal to the axial stress of the shell at the same location. According to assumption (4), the bending stress in the shell due to the edge moment M0s is S0z = 12M0s / xD³. Since the flange ring contacts the inner edge of the shell at x = -D/2, it experiences maximum compressive bending stress, while at x = D/2, it experiences maximum tensile bending stress. At x = D/2, the axial stress reaches its peak value: (S0z)max = 6M0s / D².
The radial stress in the flange ring is composed of the bending stress from the moment M0r and the radial stress caused by the edge force p0s. It can be expressed as S0r = 12M0rzt³f - p0sf, with M0r = M0s - p0st³f / 2. The maximum radial tensile stress occurs at the inner diameter of the flange ring, z = tf/2 (point a): 6M0rt²f - p0stf. Conversely, at z = -tf/2 (point b), the maximum radial compressive stress is 6M0rt²f + p0stf. Therefore, the maximum radial stress at point a is (S0r)max = 6M0rt²f - p0stf.
The circumferential stress of the flange ring is the sum of the bending stress caused by the corner H0f and the membrane stress from the pressure -p0stf. It can be written as S0t = SH0ft + Sp0st = EfH0fz/r - p0stf(D0 + Di)/2(D0 - Di). When r = Di/2 and z = tf/2 (point a), the maximum circumferential tensile stress is EftfDiH0f + p0stfln(D0/Di). At r = Di/2 and z = -tf/2 (point b), the maximum circumferential compressive stress is EftfDiH0f - p0stfln(D0/Di). Hence, the maximum circumferential stress is (S0t)max = EftfDiH0f + p0stfln(D0/Di).
Under operating conditions, the flange ring deformation coordination includes the total radial displacement of the flange ring: wpf = wpsf + wMpsf + wMpHf + wpcf. The external torque MpH is defined as MpH = ppgsp + p1(Dpb - Di/2) + p2(2Dpb - Di - Dpg/4). Based on assumptions (2) and (4), the radial displacement of the flange ring is wpf = 0.
The total radial displacement of the shell is wpH = wpsH + wMpsH + wpcH, where wpsH = pps²k³D, wMpsH = Mps²k²D, and wpcH = pcDi² / 8EfD₀(2 - L). The total corner of the shell HpH = HpsH + HMpsH + HpcH, with HpsH = -pps²k²D, HMpsH = -MpskD, and HpcH = 0. According to assumption (4), the radial displacements and rotation angles of the flange and shell are equal, i.e., wpf = wpH and Hpf = HpH.
The axial stress at the edge of the flange ring equals the transverse stress of the shell. Under the action of the edge moment Mps and internal pressure pc, the total axial stress of the shell is Spz = 12Mps / xD³ + pc(Di + D) / 4D. At x = D/2, the maximum axial tensile stress is 6Mps / D² - pc(Di + D) / 4D, and at x = -D/2, the maximum compressive stress is 6Mps / D² + pc(Di + D) / 4D. Therefore, the maximum axial stress is (Spz)max = 6Mps / D² - pc(Di + D) / 4D.
The radial stress of the flange ring is the sum of the bending stress from the moment Mpr, the internal pressure strength pc, and the radial stress caused by the edge force pps. It is given by Spr = 12Mprzt³f - ppstf + pc, with Mpr = Mps - ppstf / 2. At the inner diameter, z = tf/2 (point a), the maximum radial tensile stress is 6Mpr / t²f - ppstf + pc, while at z = -tf/2 (point b), the maximum radial compressive stress is 6Mpr / t²f + ppstf - pc. Thus, the maximum radial stress is (S0r)max = 6Mpr / t²f - ppstf + pc.
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