Differences

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--- boiler:metallurgy:hoop_stresses [2020/04/21 22:01] – fornax
+++ boiler:metallurgy:hoop_stresses [2020/05/11 19:27] (current) – fornax
@@ Line 1: / Line 1: @@
-====== Hoop Stresses in boiler tube ======
+====== Tensile Stresses in a boiler tube ======
+These formulas are for a tube or cylinder where **wall thickness is more than 1/20 of diameter**. For a thick wall tube, scroll to the bottom of this page.
+{{ boiler:metallurgy:tube_stresses.png}}
+===== Stress in Axial Direction =====
+Axial direction is one that goes along the tube longitudinally.
+The stress in axial direction at a point in the tube or cylinder wall can be expressed as:
+σ<sub>a</sub> = (p<sub>i</sub> r<sub>i</sub><sup>2</sup> - p<sub>o</sub> r<sub>o</sub><sup>2</sup> )/(r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>)
+where\\
+σ<sub>a</sub> = stress in axial direction (MPa, psi)\\
+p<sub>i</sub> = internal pressure in the tube or cylinder (MPa, psi)\\
+p<sub>o</sub> = external pressure in the tube or cylinder (MPa, psi)\\
+r<sub>i</sub> = internal radius of tube or cylinder (mm, in)\\
+r<sub>o</sub> = external radius of tube or cylinder (mm, in)\\
+===== Stress in Circumferential Direction - Hoop Stress =====
+The stress in circumferential direction - hoop stress - at a point in the tube or cylinder wall can be expressed as:
+σ<sub>c</sub> = [(p<sub>i</sub> r<sub>i</sub><sup>2</sup> - p<sub>o</sub> r<sub>o</sub><sup>2</sup>) / (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>)] - [r<sub>i</sub><sup>2</sup> r<sub>o</sub><sup>2</sup> (p<sub>o</sub> - p<sub>i</sub>) / (r<sup>2</sup> (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>))]
+where\\
+σ<sub>c</sub> = stress in circumferential direction (MPa, psi)\\
+r = radius to point in tube or cylinder wall (mm, in) (r<sub>i</sub> < r < r<sub>o</sub>)\\
+maximum stress when  r = r<sub>i</sub> (inside pipe or cylinder)\\
+===== Stress in Radial Direction =====
+Radial direction is one going through the wall thickness, such as from outside surface to the inside surface.\\
+The stress in radial direction at a point in the tube or cylinder wall can be expressed as:
+σ<sub>r</sub> = [(p<sub>i</sub> r<sub>i</sub><sup>2</sup> - p<sub>o</sub> r<sub>o</sub><sup>2</sup>) / (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>)] + [r<sub>i</sub><sup>2</sup> r<sub>o</sub><sup>2</sup> (p<sub>o</sub> - p<sub>i</sub>) / (r<sup>2</sup> (r<sub>o</sub><sup>2</sup> - r<sub>i</sub><sup>2</sup>))]
+maximum stress when  r = ro (outside pipe or cylinder)\\
+===== Resultant Stress =====
+Combined stress in a single point in the cylinder wall cannot be described by a single vector using vector addition. Instead stress tensors (matrixes) describing the linear connection between two physical vectors quantities can be used.
+Reference: [[https://www.engineeringtoolbox.com/stress-thick-walled-tube-d_949.html|Link]]
+====== Hoop Stresses in a thin walled tube or Cylinder ======
 When a thin-walled tube or cylinder is subjected to internal pressure a hoop and longitudinal stress are produced in the wall.
-For the thin walled equations below the wall thickness is less than 1/20 of tube or cylinder diameter.
+For the thin walled equations below the **wall thickness is less than 1/20 of tube or cylinder diameter**.
 The hoop stress is acting circumferential and perpendicular to the axis and the radius of the cylinder wall. The hoop stress can be calculated as
-σh = p d / (2 t)
+σ<sub>h</sub> = p d / (2 t)
 where
-σh = hoop stress (MPa, psi)\\
+σ<sub>h</sub> = hoop stress (MPa, psi)\\
 p = internal pressure in the tube or cylinder (MPa, psi)\\
 d = internal diameter of tube or cylinder (mm, in)\\
 t = tube or cylinder wall thickness (mm, in)\\