Principle of Material Strenght
CLASSIFICATION OF PRESSURE VESSELS
Divided into two classes d Di id d i t t l depending on th di the ratio of the wall thickness to vessel diameter (t/D): Majority of the vessels used in chemical and allied industries are classified as thinwalled vessels. Thick-walled vessels are used for ↑ ↑P.
PRINCIPAL STRESSES
The state of stress at a point in a structural member under a complex system of loading is described by the g p p magnitude and direction of the principle stresses. Principle stresses = maximum values of the normal stresses at the point; which p act on planes on which the shear stress is zero. In a two-dimensional stress system, the principal stresses at any point are related to the normal stress in the x and y d directions, σx and σy, and the shear d d h h stress, τxy at the point of the following eqn.: 1 2 2 Principal P i i l stresses, σ 1 , σ 2 = (σ y + σ x ) ± (σ y − σ x ) + 4τ xy 2 The maximum shear stress at the point is equal to: 1 Maximum shear stress = (σ 1 − σ 2 ) 2
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MATERIAL STRENGTH
Example of principal stress at vessel wall:
General example of symmetrical vessel at an axis:
PRINCIPAL STRESSES IN PV WALL
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Element abcd a b d c
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• Principal stresses: (1) σ1 = meridional / longitudinal stress, acting along a meridian 0-0 axis. (2) σ2 = circumferential/ tangential/hoop stress, acting along parallel to 0-0 axis. (3) σ3 = radial stress acting stress, normal to 0-0 axis. • For thin wall, σ3 the p ), tensile strength of the material. The failure point in a simple tension is taken as the yield-point stress, σe’.
THEORIES OF FAILURE – combined stresses
For components subjected to combined stresses (normal or shear stresses), failure analysis becomes more complicated.
Bending moment stress B di t t Longitudinal stress Shear stress
Circumferential stress
3 commonly used theories to analyze failure under combined stresses:
Maximum Principal Stress Theory
Assuming failure occurs when one of the
Divided into two classes d Di id d i t t l depending on th di the ratio of the wall thickness to vessel diameter (t/D): Majority of the vessels used in chemical and allied industries are classified as thinwalled vessels. Thick-walled vessels are used for ↑ ↑P.
PRINCIPAL STRESSES
The state of stress at a point in a structural member under a complex system of loading is described by the g p p magnitude and direction of the principle stresses. Principle stresses = maximum values of the normal stresses at the point; which p act on planes on which the shear stress is zero. In a two-dimensional stress system, the principal stresses at any point are related to the normal stress in the x and y d directions, σx and σy, and the shear d d h h stress, τxy at the point of the following eqn.: 1 2 2 Principal P i i l stresses, σ 1 , σ 2 = (σ y + σ x ) ± (σ y − σ x ) + 4τ xy 2 The maximum shear stress at the point is equal to: 1 Maximum shear stress = (σ 1 − σ 2 ) 2
[
]
MATERIAL STRENGTH
Example of principal stress at vessel wall:
General example of symmetrical vessel at an axis:
PRINCIPAL STRESSES IN PV WALL
O
Element abcd a b d c
O
• Principal stresses: (1) σ1 = meridional / longitudinal stress, acting along a meridian 0-0 axis. (2) σ2 = circumferential/ tangential/hoop stress, acting along parallel to 0-0 axis. (3) σ3 = radial stress acting stress, normal to 0-0 axis. • For thin wall, σ3 the p ), tensile strength of the material. The failure point in a simple tension is taken as the yield-point stress, σe’.
THEORIES OF FAILURE – combined stresses
For components subjected to combined stresses (normal or shear stresses), failure analysis becomes more complicated.
Bending moment stress B di t t Longitudinal stress Shear stress
Circumferential stress
3 commonly used theories to analyze failure under combined stresses:
Maximum Principal Stress Theory
Assuming failure occurs when one of the