# Principle of Material Strenght

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

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 principal stresses reaches the failure value σe’ as in simple tension value, (unidirectional). Mathematically, Mathematically failure occurs when σ1=σe’ or σ2=σe’ or σ3=σe’

Maximum Shear Stress Theory (= Tresca/Guest Theory)

Assuming the failure will occur in a complex stress system when the maximum shear stress reaches the value of the shear stress at failure in simple tension. • The shear stress at which the material fails under tensile test: •

τe =

σ e'

2

• For system of combined stresses, 3 shear stresses:

τ1 = ±

σ1 − σ 2

2

;

τ2 = ±

σ 2 −σ3

2

;

τ3 = ±

σ 3 − σ1

2

• Failure occurs when:

τ 1 = τ e or

τ 2 = τ e or

τ3 =τe

Maximum Strain Energy Theory

Assuming failure will occur in a complex stress system when the total strain energy per unit volume reaches the value at which failure occurs in simple tension. Suitable for predicting the failure of ductile materials under complex loading. Most design codes uses Maximum Shear Stress Theory and M i Th d Maximum St i E Strain Energy Th Theory.

ELASTIC STABILITY

Under certain loading conditions failure of a structure can occur not through plastic failure but failure, by buckling or wrinkling: Buckling results in a gross and sudden change of shape of the structure, Plastic failure, the structure retains the same basic shape.

This mode of failure will occur when the structure is not elastically stable, lacks of sufficient stiffness, rigidity to withstand the load. The stiffness of a structural material is dependent not on the basic strength of the material but on:

Stress (σ) - Strain (ε)

Stress is defined as: Load F σ= = X - sectional area Ao i l Strain is defined as: Change of l th Δl Ch f length = Original length lo

l2 l1

ε=

Strain due to elongation of sample: g p l −l εe = 2 1 l1 Strain due to compression of sample: d −d εc = 1 2 d1 Poisson’s ratio is defined as: P i ’ ti i d fi d ε ν= c

Force F

d2 d1

εe

Stress (σ) – Strain (ε) Curve...

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