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Chapter Objectives
Determine stresses developed in thin-walled pressure vessels
Determine stresses developed in a member’s cross section when axial load, torsion, bending and shear occur simultaneously.
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In-class Activities
Thin-Walled Pressure Vessels
Review of Stress Analyses
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3. THIN-WALLED PRESSURE VESSELS
Assumptions:
Inner-radius-to-wall-thickness ratio ≥ 10
Stress distribution in thin wall is uniform or constant
Cylindrical vessels:
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4. THIN-WALLED PRESSURE VESSELS (cont)
Spherical vessels:
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EXAMPLE 1
A cylindrical pressure vessel has an inner diameter of 1.2 m and a thickness of 12 mm. Determine the maximum internal pressure it can sustain so that neither its circumferential nor its longitudinal stress component exceeds 140 MPa. Under the same conditions, what is the maximum internal pressure that a similar-size spherical vessel can sustain?
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EXAMPLE 1 (cont)
Solutions
The maximum stress occurs in the circumferential direction.
The stress in the longitudinal direction will be
The maximum stress in the radial direction occurs on the material at the inner wall of the vessel and is
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EXAMPLE 1 (cont)
Solutions
The maximum stress occurs in any two perpendicular directions on an element of the vessel is
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8. REVIEW OF STRESS ANALYSES
Normal force P leads to:
Shear force V leads to:
Bending moment M leads to:
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9. REVIEW OF STRESS ANALYSES (cont)
Torsional moment T leads to:
Resultant stresses by superposition:
Once the normal and shear stress components for each loading have been calculated, use the principal of superposition to determine the resultant normal and shear stress components.
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EXAMPLE 2
A force of 15 kN is applied to the edge of the member shown in Fig. 8–3a. Neglect the weight of the member and determine the state of stress at points B and C.
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EXAMPLE 2 (cont)
Solutions
For equilibrium at the section there must be an axial force of N acting through the centroid and a bending moment of N•mm about the centroidal or principal axis.
Stress components
Normal Force
Bending moments
The maximum stress is
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EXAMPLE 2 (cont)
Solutions
Super Position
The location of the line of zero stress can be determined by proportional triangles
Elements of material at B and C are subjected only to normal or uniaxial stress.
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EXAMPLE 3
The tank in Fig. 8–4a has an inner radius of 600 mm and a thickness of 12 mm. It is filled to the top with water having a specific weight of γw = 10 kNm3. If it is made of steel having a specific weight of γst = 78 kNm3, determine the state of stress at point A. The tank is open at the top.
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EXAMPLE 3 (cont)
Solutions
The weight of the tank is
The pressure on the tank at level A is
For circumferential and longitudinal stress, we have
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EXAMPLE 4
The member shown in Fig. 8–5a has a rectangular cross section. Determine the state of stress that the loading produces at point C.
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EXAMPLE 4 (cont)
Solutions
The resultant internal loadings at the section consist of a normal force, a shear force, and a bending moment.
Solving,
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EXAMPLE 4 (cont)
Solutions
The uniform normal-stress distribution acting over the cross section is produced by the normal force.
At Point C,
In Fig. 8–5e, the shear stress is zero.
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EXAMPLE 4 (cont)
Solutions
Point C is located at y = c = 0.125m from the neutral axis, so the normal stress at C, Fig. 8–5f, is
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EXAMPLE 4 (cont)
Solutions
The shear stress is zero.
Adding the normal stresses determined above gives a compressive stress at C having a value of
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EXAMPLE 5
The rectangular block of negligible weight in Fig. 8–6a is subjected to a vertical force of 40 kN, which is applied to its corner. Determine the largest normal stress acting on a section through ABCD.
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EXAMPLE 5 (cont)
Solutions
For uniform normal-stress distribution the stress is
For 8 kNm, the maximum stress is
For 16 kNm, the maximum stress is
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EXAMPLE 5 (cont)
Solutions
By inspection the normal stress at point C is the largest since each loading creates a compressive stress there
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