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Polska Izba Książki

Copyright © by Wydawnictwo Naukowe PWN SA
Warszawa 2017

eBook based on printed edition from year 2017, (Ist ed.)
Warsaw 2017

ISBN 978-83-01-19206-8

Wydawnictwo Naukowe PWN SA
02-460 Warszawa, ul. Gottlieba Daimlera 2
tel. 22 69 54 321; faks 22 69 54 228
infoline 801 33 33 88
e-mail: pwn@pwn.com.pl; reklama@pwn.pl, www.pwn.pl

Content

From Authors

Introduction

1. Basic principles of material strength

1.1. Introduction

1.2. The division of stresses and strains

1.2.1. Normal stresses

1.2.2. Shear stresses

1.2.3. Reduced stresses

1.2.4. Strains classification

1.3. Strength criteria

1.3.1. Admissible stress for static loads

1.3.2. Stress-strain relations

1.3.3. Static factor of safety

2. Tensile and compressive strength calculations

2.1. Introduction

2.2. Tensile stresses

2.2.1. Gravitational breaking

2.2.2. Hooke’s law

2.3. Compressive stresses

2.4. Deformations

2.4.1 Longitudinal deformations

2.4.2. Lateral deformations

2.5. Calculation of elements under tensile/compressive strength

2.6. Examples of calculations for stretched and compressed elements

3. Tensile stresses in the vessels

3.1. Stresses in cylindrical walls – barrels

3.1.1. Tensile circumferential stresses

3.1.2. Longitudinal tensile stresses

3.2. Stresses and wall thickness of cylindrical vessels

3.3. Stress and wall thickness of spherical vessels

3.4. Calculation of bolts for flange connections in vessels

3.5. Calculation of thin-walled vessel shells

3.6. Combined calculations for thin-walled shells and bolts

4. Surface pressure

5. Shearing and tangential shear stresses

5.1. Introduction

5.2. Shear stresses

5.3. Admissible shear stress

6. Calculation of stresses σr, σśc, σn and τt

6.1. Threaded connections

6.1.1. Calculation of threaded connections

6.2. Welded joints

6.2.1. Calculation of welded joints

6.3. Riveted joints

6.3.1. Calculation of riveted joints

7. Buckling stress

7.1. Introduction

7.2. Slenderness factor λ

7.3. Moments of inertia Ix, Iy, Io, Imin

7.3.1. Steiner theorem

7.3.2. The method of calculating axial moments of inertia for complex cross-sections

7.4. Critical force Fkr

7.5. Ranges of buckling

7.5.1. Elastic buckling

7.5.2. Non-elastic buckling (ductile)

7.6. Effective-length factor α and cases of buckling

7.7. Buckling strength calculations

7.7.1. Calculating admissible loads for components with pre-determined dimensions

7.7.2. Calculation of dimensions for given buckling loads

7.8. Examples of buckling calculations

8. Summary of discussed stresses

8.1. Introduction

8.2. The treatment procedure

8.2.1. Specific example of the calculation process

8.3. Examples of calculations

9. Principles of Mechanics

9.1 Introduction

9.2. Principles of Statics

9.2.1. Units used in mechanics

9.2.2. Newton’s laws

9.2.3. The principles used in statics

9.2.4. Supports, nodes and their reactions

9.2.5. Flat, convergent system of forces

9.2.6. Flat parallel system of forces

9.2.7. Graphical methods of determining reaction forces

9.2.8. Spatial force system

9.2.9. Friction forces

10. Admissible stresses and fatigue

10.1 Admissible stresses in dynamic loads

10.2. Fatigue factors of safety

10.3. Admissible stresses

11. Torsional and bending stresses

11.1. Torsional stress

11.1.1. Admissible stresses

11.1.2. Angle of torsion criterion

11.2. Calculation examples of “pure” twisting

12. Bending stresses

12.1. Bending moment

12.2. Bending resistance modulus

12.3. Admissible bending stresses criterion

12.4. Deflection – criterion of deformation limits

12.5. Analytical method of calculating beams

12.5.1. Example of utilizing the calculation method

12.6. Graphical method of calculating beams

12.7. Other cases of bending stress

12.8. Calculation examples

13. Bending with torsion – reduced moment

13.1. Bending with torsion

13.2. Complex stresses in the calculation of shafts

13.3. Calculation examples

13.3.1. Calculations of agitator shaft of a reactor

13.3.2. Calculations of reduced moments

13.4. Two-plane reduced stresses in shafts

13.4.1. Calculating the shaft being twisted and bent in two planes

13.5. Practical use of shaft calculations

14. Summary of all discussed stresses

14.1. Introduction

14.2. Examples

14.2.1. Project of mounting an advertisement

14.2.2. Project of a motion system

Specifications of unitary symbols applicable in calculation of strength

3 Tensile stresses in the vessels

Cylindrical vessels consist of a cylindrical (boiler drum) part and two bottoms (ends), welded onto it. The ends may be circular flat, dished, spherical or conical. They may also be welded to the flange and then – in a form of screw cap – to the tank. Different types of containers have been presented in figure 3.1.

Fig. 3.1. Types of vessels: a) vertical, cylindrical, closed, b) cylindrical, with flat lid and conical bottom, c) cylindrical, horizontal, with two spherical lids, d) spherical with hatch.

The tensile stresses occurring in the tanks can be divided, depending on where they occur, into stress: in the walls, in flange bolts, in tank’s fixture (suction and discharge piping, measuring connectors, valves, liquid level indicators, glass sighting tubes, lifting eye bolts etc.).

The theoretical basis concerning both tensile σr and admissible kr stress together with described by the Hooke’s law elongations ∆l, presented in the previous chapter, are still valid, but only for linear loads and longitudinal elements, such as rods, bolts, or pipes.

Walls of the vessels, which are classified as flat and spatial elements, require individual theoretical introduction concerning stress and strength of thin-walled coatings.

A vessel is considered thin-walled if the actual thickness of the walls grz is less than the thickness limit ggr, which is linked to the diameter of the vessel D through the following relation:

(3.1)

3.1. Stresses in cylindrical walls – barrels

Cylindrical vessel consist of a cylindrical part called a barrel and two bottoms. If there is pressure p within the vessel, then it acts both on the surface of the barrel, as well as bottom surfaces. In case of barrel coating, shown in Fig. 3.2, there are two types of stress, circumferential σro and longitudinal σrw. They stretch the coating as shown in an example of its rectangular element (Fig. 3.2).

Fig. 3.2. The tensile stresses in the cylindrical axisymmetric shell

In order to analyze the value of these stresses, let us imagine a vessel being cut by two planes perpendicular to each other: one perpendicular to the base – along the symmetry axis (Fig. 3.3) and second parallel (Fig. 3.4).

Fig. 3.3. Circumferential stress.

Fig. 3.4. Longitudinal breaking tensions

3.1.1. Tensile circumferential stresses

According to the definition, stress σro is the ratio of force F to the cross-sectional area A of the vessel, where the force is a product of pressure p and area of projected vessel surface parallel to the cross-section of that surface Ap.

(3.2)

thus, for D ≈ Dw and admissible tension stress kr: stresses σro and wall coating thickness g:

(3.3)

3.1.2. Longitudinal tensile stresses

Similar reasoning carried out for an imaginary longitudinal cross-section (Fig. 3.4) returns the dependence of longitudinal stress in cylindrical vessel walls.

After the imaginary cutting of the vessel with a plane, parallel to its base, through the action of a force F, the longitudinal tensile stress σrw, arises at a plane A.

(3.4)

For thin-walled vessels (3.1), where D >> g, Dz = Dw + 2g ≈ Dw = D, circular ring area A equals approximately:

(3.5)

and then, after substituting (3.5) to (3.4) and shortening, we obtain:

(3.6)

3.2. Stresses and wall thickness of cylindrical vessels

A comparison of equations defining circumferential (3.3) and longitudinal (3.6) stresses shows that the former is, for identical conditions (p, D, kr), twice as high σro = 2σrw, therefore the wall thickness of the cylindrical vessel with prescribed pressure p and diameter D is equal to the thickness calculated from the circumferential stress gcyl = gro and hence – should be established from equation (3.3).

Exceptions include cases where the vessel contains a layer of liquid of height H and density γ, with hydrostatic pressure ph = H ∙ γ, which is comparable to or greater than the gas pressure p inside the vessel. In such cases, longitudinal stresses σrw might have equal, or even greater value than the circumferential σro. In this scenario, the thickness must be calculated from the higher one of these values, through one of the following relationships:

(3.7)

3.3. Stress and wall thickness of spherical vessels

The stress occurring in a tank wall or in the bottoms of spherical vessel, results from the symmetry condition for stresses in spherical coatings, where there is only one type of stress – the longitudinal tension stress σrw.

The longitudinal stresses occurring in the cross-section of the vessel and their directions have been illustrated in Fig. 3.5 and relations (3.4) – (3.6), which show that wall thickness of a spherical vessel or bottom gd can be written as:

(3.8)

Fig. 3.5. Tensile stresses in the spherical shell

3.4. Calculation of bolts for flange connections in vessels

Example 3.1.

Calculate how many studs nsr with a diameter d = 36 mm (Figure 3.6) must be used to connect the cover with the vessels drum of diameter D = 2.2 m, if the pressure inside the vessel is p = 16 at, and kr = 130 MPa.

Fig. 3.6. Flange joint with a stud

SOLUTION:

Data: D = 2.2 m

p = 16 at ≈ 1.6 MPa

d = 36 mm = 3.6 ∙10-2 m

kr = 130 MPa

Searches: nsr

An upward force F acts on the cover of the vessel – it is derived from pressure, and equal to the product of the cross-section of the vessel and pressure. This force in studs cross-section, with a total area Asr generates tensile stress:

For technical reasons, the nearest value which is a multiple of 3 or 4 is chosen: nsr = 48.

Example. 3.2.

Fig. 3.7. The joint flange with hexagonal bolt screwed into the flange

Calculate how many steel bolts nsr, with kr = 120 MPa, and diameter d = 16 mm (M16) must twist the tank cap, of a diameter D = 1.4 m, wherein the pressure inside is p = 20 at.

SOLUTION:

Data: D = 1.4 m

p = 20 at = 2 MPa

kr = 120 MPa

d = 16 mm = 0.016 m

Searches: nsr

The number of bolts in the cover nsr can be calculated from tensile stress σr, which on their surface Asr causes a force F, generated by a pressure pmaks, to act on the vessel lid Ap.