Technology & Engineering
Hydrostatic Stress
Hydrostatic stress refers to the pressure exerted by a fluid at rest. It is uniform in all directions and increases with depth in the fluid. This type of stress is important in engineering and technology, particularly in the design and analysis of structures that are subjected to fluid pressure, such as dams, pipelines, and hydraulic systems.
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11 Key excerpts on "Hydrostatic Stress"
- Melvyn Kay(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
27 Chapter 2 Water standing still Hydrostatics 2.1 INTRODUCTION Hydrostatics is the study of water when it is not moving; it is standing still. It is important to civil engineers who are designing water storage tanks and dams. They want to work out the forces that water creates in order to build reservoirs and dams that can resist them. Naval architects designing submarines want to understand and resist the pressures created when they go deep under the sea. The answers come from understanding hydrostatics. The science is simple both in concept and in practice. Indeed, the theory is well established and little has changed since Archimedes (287–212 bc) worked it out over 2000 years ago. 2.2 PRESSURE The term pressure is used to describe the force that water exerts on each square metre of some object submerged in water, that is, force per unit area. It may be the bottom of a tank, the side of a dam, a ship or a submerged submarine. It is calculated as follows: pressure force area = . Introducing the units of measurement . pressure (kN/m ) force (kN) area (m ) 2 2 = Force is in kilo-Newtons (kN), area is in square metres (m 2 ) and so pres- sure is measured in kN/m 2 . Sometimes pressure is measured in Pascals (Pa) in recognition of Blaise Pascal (1620–1662) who clarified much of modern 28 Practical Hydraulics and Water Resources Engineering day thinking about pressure and barometers for measuring atmospheric pressure. 1 Pa = 1 N/m 2 . One Pascal is a very small quantity and so kilo-Pascals are often used so that 1 kPa = 1 kN/m 2 . Although it is in order to use Pascals, kN/m 2 is the measure of pressure that engineers tend to use and so this is used throughout this text (see example of calculating pressure in Box 2.1). 2.3 FORCE AND PRESSURE ARE DIFFERENT Force and pressure are terms that are often confused.
- Melvyn Kay(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
Chapter 2Water standing still
Hydrostatics2.1 INTRODUCTION
Hydrostatics is the study of water when it is not moving; it is standing still. It is important to civil engineers who are designing water storage tanks and dams. They want to work out the forces that water creates in order to build reservoirs and dams that can resist them. Naval architects designing submarines want to understand and resist the pressures created when they go deep under the sea. The answers come from understanding hydrostatics. The science is simple both in concept and in practice. Indeed, the theory is well established and little has changed since Archimedes (287–212 BC ) worked it out over 2000 years ago.2.2 PRESSURE
The term pressure is used to describe the force that water exerts on each square metre of some object submerged in water, that is, force per unit area. It may be the bottom of a tank, the side of a dam, a ship or a submerged submarine. It is calculated as follows:pressure =Introducing the units of measurementforce area.pressure(=kN/m 2).force( kN )area(m 2)Force is in kilo-Newtons (kN), area is in square metres (m2 ) and so pressure is measured in kN/m2 . Sometimes pressure is measured in Pascals (Pa) in recognition of Blaise Pascal (1620–1662) who clarified much of modern day thinking about pressure and barometers for measuring atmospheric pressure.1 Pa = 1 N/m2 .One Pascal is a very small quantity and so kilo-Pascals are often used so that1 kPa = 1 kN/m2 .Although it is in order to use Pascals, kN/m2 is the measure of pressure that engineers tend to use and so this is used throughout this text (see example of calculating pressure in Box 2.1 ).2.3 FORCE AND PRESSURE ARE DIFFERENT
Force and pressure are terms that are often confused. The difference between them is best illustrated by an example:
eBook - PDFSoil Mechanics
A One-Dimensional Introduction
- David Muir Wood(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
Stationary water cannot transmit shear stresses. We can then imagine that these little measuring plates are in fact just measuring planes drawn in the fluid but having no material presence: the resulting stress on each plane is, as before, the same. Such a “hydrostatic” stress in a fluid is more usually described as a fluid pressure and, since we are not usually concerned to try to pull fluids apart, pressure is helpfully deemed to be positive in compression. Another way of visualising this intuitive property is to imagine standpipes with various geometries inserted in the pool at the same depth as shown in Fig. 2.15 . The water in each standpipe will rise to the level of the water surface. The pressure at the mouth of each standpipe is found directly from the height z of the free surface above the opening. We know that objects float in water. Archimedes’ principle says that an object wholly or partially immersed in a fluid (water will be a particular fluid) is buoyed up by a force equal to the weight of the displaced fluid. We can understand this result by applying our recently acquired knowledge of pressure in fluids. Let us consider a cuboidal object wholly submerged in the pool of water with the faces of the cuboid aligned with vertical and horizontal axes, as shown in Fig. 2.16. The lengths of the sides of the object are x , y and z and the top surface of the object is at a depth z below the water surface. There will be forces F x 1 , F x 2 , F y 1 , F y 2 , F z 1 , F z 2 acting on the faces of the object, as shown. We can calculate the magnitude of each of these forces by integrating the water pressure over each face. Because of the chosen 2.7 Water in the ground: Introduction to hydrostatics 25 A B C D E z Figure 2.15. Standpipes A, B, C, D, E in pool of water with opening at depth z : the water rises to exactly the same level in each of the standpipes.
eBook - PDFMechanics of Deformable Bodies
Lectures on Theoretical Physics
- Arnold Sommerfeld(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
This equation actually amounts to a definition of the fluid in the state of rest—in contrast to the state of solidification: glass which has no crystalline structure must be characterized physically as a fluid in the solid state for which Eq. (1) is, of course, incorrect. A fundamental conclusion from (1) is that the hydrostatic pressure is associated with a spherical tensor quadric. Any diameter of the quadric may be taken as a principal axis, because (1) must be true in any coordinate system. The complete symmetry of the tensor of hydrostatic pressure can be expressed as follows. At a given point the pressure acts on any surface element in the direction of its normal i, its magnitude being the same for all directions i; p xl = p 22 = P33 for any three orthogonal directions. Pascal 4 , it appears, was the first to perceive this law. W e can summarize this result in the statement that the hydrostatic pressure is a scalar quantity. It will be denoted by p, as is usually done with omission of the subscripts that are now abundant. Note that there is no preferential loading of the horizontal cross sections if a vertical force such as gravity acts on the fluid; the same pressure p is transferred 4 Blaise Pascal (1624-1662), the great geometer (Pascal's theorem) and mathemati-cian (Pascal's triangle, foundation of the calculus of probabilities), a great writer and theologian too; he studied the laws of air pressure and at his instigation the first baro-metrical determination of an altitude was carried out. 4 2 MECHANICS OF DEFORMABLE BODIES IIL6] through any cross-section (also a vertical one). The unidirectional char-acter of the force does not interfere with the symmetry of the pressure. In order to calculate the pressure if magnitude and direction of the external force are given, let F denote the external force per unit of fluid volume, so that the dimension of F is dyne/cm 3 . Then the volume element A T is acted upon by the force FAr.
eBook - ePub- Ron Darby, Raj P. Chhabra(Authors)
- 2016(Publication Date)
- CRC Press(Publisher)
4 Fluid Statics“Archimedes’ finding that the crown was of gold was a discovery; but he invented the method of determining the density of solids. Indeed, discoverers must generally be inventors; though inventors are not necessarily discoverers.”—William Ramsay, 1852–1916, ChemistI. STRESS AND PRESSUREThe forces that exist within a fluid at any point may arise from various sources. These include gravity, or the “weight” of the fluid, an imposed static pressure from an external source, an external driving force such as a pump or compressor, and the internal resistance to relative motion between fluid elements, or inertial effects resulting from the local velocity and the mass flow rate of the fluid (e.g., the transport or rate of change of momentum).Any or all of these forces may result in local stresses within the fluid. “Stress” can be thought of as a (local) “concentration of force,” or the local force per unit area that acts upon an infinitesimal volume of the fluid. Now both force and area are vectors, the direction of the area being defined by the normal vector that points outward relative to the fluid volume bounded by the surface. Thus, each stress component has a magnitude and two directions associated with it, that is, the direction of the force that acts on the fluid element, Fj , and the orientation of the surface of the element upon which the force acts, Ai. These are the characteristics of a “second-order tensor” or “dyad.” If the direction in which the local force acts is designated by subscript j (e.g., j = x, y, or z in Cartesian coordinates) and the orientation (i.e., the normal to the surface) of the local area element upon which it acts is designated by the subscript i, then the corresponding stress component (σij
eBook - ePub- William S. Janna(Author)
- 2020(Publication Date)
- CRC Press(Publisher)
2 Fluid StaticsIn this chapter, we study the forces present in fluids at rest. Knowledge of force variations—or, more appropriately, pressure variations—in a static fluid is important to the engineer. Specific examples include water retained by a dam or bounded by a levee, gasoline in a tank truck, and accelerating fluid containers. In addition, fluid statics deals with the stability of floating bodies and submerged bodies and has applications in ship hull design and in determining load distributions for flat-bottomed barges. Thus, fluid statics concerns the forces that are present in fluids at rest, with applications to various practical problems.After completing this chapter, you should be able to:- Discuss pressure and pressure measurement;
- Apply the hydrostatic equation to manometers;
- Develop equations for calculating forces on submerged surfaces;
- Examine problems involving stability of partially or wholly submerged objects.
2.1 PRESSURE AND PRESSURE MEASUREMENT
Because our interest is in fluids at rest, let us determine the pressure at a point in a fluid at rest. Consider a wedge-shaped particle exposed on all sides to a fluid as illustrated in Figure 2.1a . Figure 2.1b is a free-body diagram of the particle cross section. The dimensions Δx , Δy , and Δz are small and tend to zero as the particle shrinks to a point. The only forces considered to be acting on the particle are due to pressure and gravity. On either of the three surfaces, the pressure force is F = pA . By applying Newton’s second law in the x - and z -directions, we get, respectively,FIGURE 2.1 A wedge-shaped particle.∑F x=p xΔ z Δ y −p sΔ s Δ y sin θ =ρ 2Δ x Δ y Δ za x= 0
where∑F z=p zΔ x Δ y −p sΔ s Δ y cos θ − ρ gΔ x Δ y Δ z2=ρ 2Δ x Δ y Δ za z= 0p x , p z , and p s are average pressures acting on the three corresponding facesa x and a z are the accelerationsρ is the particle densityThe net force equals zero in a static fluid. After simplification, with a x = az
- James W. Murdock(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
2.1 INTRODUCTION This chapter is concerned with establishing the basic relations of fluid statics. Included are the basic equation of fluid statics, pressure-height relations for incompressible fluids and for ideal gases, pressure-sensing devices, liquid forces on plane and curved surfaces, stress in pipes due to internal pressure, acceleration of fluid masses, and finally buoyancy and flotation. The following sections of this chapter may be of special interest to designers: Section 2.5, Pressure-Sensing Devices; Section 2.7, which de scribes the 1976 U.S. Standard Atmosphere; and Section 2.10, which includes ANSI/ASME Code equations for pipe stress. This chapter may be used as a text for tutorial or for refresher purposes. Each concept is explained and derived mathematically as needed. As in Chapter 1, the minimum level of mathematics is used for derivations con sistent with academic integrity and clarity of concept. There are 16 tutorial type examples of fully solved problems. 2.2 FLUID STATICS Fluid statics is that branch of fluid mechanics that deals with fluids that are at rest with respect to the surfaces that bound them. The entire fluid mass may be in motion, but all fluid particles are at rest with each other. 46 Fluid Statics 47 There are two kinds of forces to be considered: (1) su rface f o r c e s , forces due to direct contact with other fluid particles or solid walls (forces due to pressure and tangential, that is, shear stress), and (2) b o d y f o r c e s , forces acting on the fluid particles at a distance (e.g., gravity, magnetic field, etc.). Since there is no motion of a fluid layer relative to another fluid layer, the shear stress everywhere in the fluid must be zero and the only surface force that can act on a fluid particle is normal pressure force. Because the entire fluid mass may be accelerated, body forces other than gravity may act in any direction on a fluid particle.
- John I. Hochstein, Andrew L. Gerhart(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
A general class of problems involving fluid motion in which there are no shearing stresses occurs when a mass of fluid undergoes rigid-body motion. For example, if a con- tainer of fluid accelerates along a straight path, the fluid will move as a rigid mass (after the initial sloshing motion has died out) with each particle having the same acceleration. Since there is no deformation, there will be no shearing stresses and, therefore, Eq. 2.2 applies. Similarly, if a fluid is contained in a tank that rotates about a fixed axis as shown by the figure in the margin, the fluid will simply rotate with the tank as a rigid body, and again Eq. 2.2 can be applied to obtain the pressure distribution throughout the moving fluid and the free surface shape. ω Photograph courtesy of Geno Pawlak. Chapter Summary and Study Guide Chapter Summary In this chapter, the pressure variation in a fluid at rest is considered, along with some important consequences of this type of pressure var- iation. It is shown that for incompressible fluids at rest the pressure varies linearly with depth. This type of variation is commonly referred to as a hydrostatic pressure distribution. For compressible fluids at rest, the pressure distribution will not generally be hydrostatic, but Eq. 2.4 remains valid and can be used to determine the pressure dis- tribution if additional information about the variation of the specific weight is specified. The distinction between absolute and gage pres- sure is discussed along with a consideration of barometers for the measurement of atmospheric pressure. Pressure-measuring devices called manometers, which utilize static liquid columns, are analyzed in detail. A brief discussion of me- chanical and electronic pressure gages is also presented. Equations for determining the magnitude and location of the resultant fluid force acting on a plane surface in contact with a static fluid are developed.
eBook - PDF- Minoo H Patel, Joel A Witz(Authors)
- 2013(Publication Date)
- Butterworth-Heinemann(Publisher)
Again, hydrostatic stability calcula-tions for the semisubmersible with articulated columns of Chapter 9 illustrates types of interaction that can take place between the hydrostatics and compliance. The above considerations invariably require that the hydrostatic stability of a comphant vessel be examined from first principles as far as is possible. This is the approach taken in this chapter with the introduction of a new pressure integrafion technique for hydrostatic stability calculations. 3.1 Theoretical background to pressure integration technique for hydrostatic analysis Consider the arbitrary body shown in Figure 3.1(a) floating at the free surface between a 'heavy' and a 'light' fluid such as an air-water interface. 49 50 Hydrostatic analysis Light Fluid Light Fluid Surface 'Surface Heavy luid Figure 3.1(a). Free floating body equilibrium, (b) Floating body equilibrium with attached loads There are two forces acting on the body. The first force is the weight of the body acting vertically downwards. The second force is due to the fluid pressure acting on the body's submerged surface. The incremental force, dF, acting on the body due to fluid pressure is d¥ = pg{d- z)dS (3.2) By integrating over the submerged surface the total force, F, is F = 9g{d-z) n(x) dS (3.3) where η is the unit normal vector acting into the body and is a function of position, X. Similarly, the incremental moment dM is dM = X X dF = X X ρ g (d - z) dS (3.4) Integrating Equation (3.4) over the submerged surface gives Μ = xX pg{d - ζ) n(x) d5 Js Resolving forces and moments due to body weight gives F = Wk (3.5) (3.6) The hght fluid is at a constant pressure equal to the free surface pressure. The pressure is assumed to be constant across the free surface.
- Allan D. Kraus, James R. Welty, Abdul Aziz(Authors)
- 2011(Publication Date)
- CRC Press(Publisher)
12 Fluid Statics Chapter Objectives • To develop the basic equation of fluid statics. • To describe the use of manometers to measure pressure. • To evaluate hydrostatic forces on planar and nonplanar surfaces immersed in a fluid. • To develop the concept of buoyancy. • To examine the stability of floating objects. • To describe hydrostatic forces under conditions of uniform rectilinear acceleration. 12.1 Introduction In Chapter 11, we defined a fluid according to its response to a shearing stress and we commented that a fluid at rest would therefore not be subjected to shear stresses. In this chapter we will consider the behavior of fluids at rest, that is, in a static condition. We should distinguish, at the outset, between inertial and noninertial reference frames. In general, we will consider a static situation to be one that is stationary relative to the earth’s surface; this is what is meant by an inertial reference frame. If a fluid is stationary relative to a coordinate system that has some acceleration of its own, such a reference frame is termed noninertial. An example of a noninertial reference frame would be a fluid inside an aircraft as it executes a maneuver. Our considerations will be, in general, for inertial reference frames and we will note, as exceptions, those occasions where noninertial reference frames are employed. 12.2 Pressure Variation in a Static Field In the absence of shear stresses a fluid will experience only gravitational and pressure forces. A representative fluid element of differential size in a static fluid is shown in Figure 12.1. Its dimensions, as indicated, are dx, dy , and dz . Fluid pressure, P ( x, y, z ), which acts on all six faces of the element, will produce force components in each of the coordinate directions. Because pressure always acts inward, the x -component pressure forces, which result from the pressure exerted on the x -faces are 361
eBook - PDFMechanics of Deformable Bodies
Lectures on Theoretical Physics, Vol. 2
- Arnold Sommerfeld(Author)
- 2016(Publication Date)
- Academic Press(Publisher)
CHAPTER II STATICS OF DEFORMARLE BODIES 5. Concept of Stress; General Classification of Deformable Bodies The term statics will be used throughout this book in the same sense as in the mechanics of a finite number of degrees of freedom, where it was applied according to the following definition: Statics treats of the forces to which matter is subjected regardless of the motion caused by them (cf. Vol. I, 5). One distinguishes between external and internal forces. Once a me-chanical system has been delimited, the internal forces of the system are those which obey Newton's third law (equality of action and reaction). Among the external forces, gravity and centrifugal force 1 are commonly en-countered in mechanics of deformable bodies. Capillary forces occur also; they act between internal and surface molecules or between a solid boundary and the molecules of the surrounding fluid. Examples of internal forces are the reactions (tension and pressure forces, surface forces) that have their origin in the distribution of matter over a limited space. They play the most important part 2 in the mechanics of deformable bodies. In solids the reactions measured per unit of area are known as stresses; in the case of fluids we usually speak of pressures (negative stresses); both have the dimension of a force per unit area, dyne/cm 2 in the CGS system. To analyze the stress concept in the case of an elastic solid, we imagine a plane surface cutting through the body and consider the interaction between the two parts of the body across this surface. In Fig. 6 such a surface of separation has been drawn normal to the x-axis, and the two parts of the body have been disjoined for the sake of clarity in the drawing. On the part to the left a positive x-surface has been marked by cross hatching.
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