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Explanation

• The flow rate through a pipe is usually measured by providing a coaxial area contraction within the pipe and by recording the pressure drop across the contraction, which is the application of Bernoulli’s equation.
• Bernoulli's principle can be derived from the principle of conservation of energy.
• Bernoulli's theorem states that, in a steady flow, the sum of all forms of energy in a fluid along a streamline is the same at all points on that streamline
• Bernoulli equation represented in head form (the total energy per unit weight):

   \(\frac{P}{\rho g} + \frac{{{v^2}}}{{2g}} + Z = Constant\)

• Three flow meters primarily operate on this principle:

1. Pitot tube
2. Venturimeter
3. Orifice meter

Concept:

• Piezometer is the simplest form of manometer used for measuring gauge pressure
• One end of this manometer is connected to the point where the pressure is to be measured and the other end is open to the atmosphere
• As one end is open to the atmosphere so it can’t be used for measuring the gas pressure
• Piezometer is used to measure the static pressure (Gauge) in the pipe.
• It measures the pressure in the form of the height (head)of the liquid column in the piezometer.
• Gas pressure cannot be measured by means of piezometers because a gas forms no free atmosphere surface and it can’t be used when large pressures in the lighter liquid are to be measured. So gauge pressure is used for measuring low pressure.

Explanation:

Specific weight:

The specific weight (also known as the unit weight) is the weight per unit volume of a material. Weight is a force. The SI unit for specific weight is N/m3. 

\(\gamma =\rho g\)

A commonly used value is the specific weight of water on Earth at 4°C which is 9.81 kN/m3.

Mass density: 

• Mass density represents the mass (or the number of particles) per unit volume of a substance, material, or object. Mass density is also known as density.
• Its SI unit is kg/m3

Mass Density = \(\frac{Mass}{Volume}\)

Specific gravity or Relative density:

• The ratio of density or specific weight of a subject matter to that of a standard matter is known as specific gravity relative density.

\({\rm{S}} = \frac{{{{\rm{\rho }}_{{\rm{subject}}}}}}{{{{\rm{\rho }}_{{\rm{standard}}}}}}{\rm{or}}\frac{{{{\rm{\gamma }}_{{\rm{subject}}}}}}{{{{\rm{\gamma }}_{{\rm{standard}}}}}}\)

Concept:

• The Unwin’s formula is used to determine the size of the rivet.
• Unwin's formula gives a relation between the hole diameter of the rivet and the thickness of the connected plates.
• The 'd' in the formula represents the nominal diameter of the rivet in mm and 't' represents the thickness of the plate

According to Unwin’s formula,

The nominal diameter (ɸ) of the rivet is given by:

ɸ = 6.04√t        {t = thickness of plate in “mm”}

ɸ = 1.91√t        {t = thickness of plate in “cm”}

Explanation:

Twisting moment impart an angular displacement of one end cross-section with respect to the other end and it will setup shear stresses on any cross section of the bar perpendicular to its axis.

Torsion Equation of a shaft is given by,

\(\frac{T}{J} = \frac{τ }{r} = \frac{{Gθ }}{L}\)

Polar section modulus of shaft is given by,

\({Z_p} = \frac{J}{r} = \frac{{\frac{{\pi {D^4}}}{{32}}}}{{\frac{D}{2}}} = \frac{{\pi {D^3}}}{{16}}\)

Where, T = Torque, J = Polar moment of inertia, τ = Shear stress, r = Radius of shaft, G = Shear modulus, θ = Angle of twist and L = Length of shaft

Shear stress distribution across the section of the circular shaft is shown below.

Using Torsinal formula

\(\frac{T}{J} = \frac{τ }{r} = \frac{{G\theta }}{L}\)

Forgiven T and J; τ ∝ r i.e.

The shear stress distribution is linear.

Shear stress at centre, τcentre = 0

\(S{F_{xx}} = - Wx\)

\(S{F_{x = 0}} = 0;S{F_{x = L}} = wL\)

Shear force at free end is zero and at fixed end it will be wL.

Explanation:

Young's modulus:

• The mechanical property of a material to withstand the compression or the elongation with respect to its length of linear elastic solids like rods, wires etc is called Young's modulus.
• It is also referred to as the Elastic Modulus or Tensile Modulus
• It gives us information about the tensile elasticity of a material (ability to deform along an axis).

The formula of young modulus is given as,

\(E =\frac {Normal ~stress}{Normal ~strain}= \frac {\sigma}{\epsilon}\)

where E is Young’s modulus in Pa, 𝞂 is the uniaxial stress in Pa,ε is the Normal strain or proportional deformation.

Modulus of rigidity:

• It is also known as shear modulus.
• It is the mechanical property of a material due to which it withstand shear stress and resists torsion.
• It is the ratio of shear stress to the corresponding shear strain within the elastic limit. This is denoted by G and given by.

\(\therefore {G} = \frac{{Shear;stress}}{{Shear;strain}} = \frac{\tau }{\phi }\)

Bulk modulus:

• It is the mechanical property of a material due to which it resists the change in volume due to external pressure or equal stress in all directions.
• The concept of bulk modulus can be used in case of Hydrostatic loading.
• It is defined as the ratio of normal stress to the volumetric strain and denoted by 'K' and given by,

\(K = \frac {Normal ~stress}{Volumetric ~strain}= \frac {\sigma }{\epsilon _v}\)

Poisson's ratio:

• When the body is loaded within its elastic limit, the ratio of lateral or transverse strain and linear or longitudinal strain is constant. This constant is known as Poisson's ratio. It is given by,

Additional Information

Relations between Young's modulus(E), Shear modulus(G), Bulk modulus(K), and Poisson's ratio(μ) 

• E = 2G (1 + μ)

• E = 3K (1 – 2μ)

• \({\bf{E}} = \frac{{9{\bf{KG}}}}{{3{\bf{K}}~ + ;{\bf{G}}}}\)

• \({\bf{\mu }} = \frac{{3{\bf{K}}~ - ~2{\bf{G}}}}{{2{\bf{G}} ~+ ~6{\bf{K}}}}\)

Explanation:

Strain Energy: 

• Strain Energy of the member is defined as the internal work done in deforming the body by the action of externally applied forces.
• Strain energy is given by:
• \(Strain;Energy = ;\frac{1}{2} \times Stress \times Strain \times Volume\)

Resilience:

• Resilience is the total strain energy stored in a given volume of material within the elastic limit.
• On removal of load, this energy is released.
• In other words, it is the area under the load-deflection curve within the elastic limit.

Proof resilience:

• The maximum energy which can be stored in a body up to the elastic limit is called proof resilience.

Modulus of resilience:

• Modulus of resilience is defined as proof resilience per unit volume. It is the area under the stress-strain curve up to the elastic limit.

Explanation:

The main difference between the angle of friction and the angle of repose is that:

The angle of friction is defined as the angle between the normal reaction force and the resultant force of normal reaction force and friction when an object just begins to move,

Whereas the angle of repose is defined as the minimum angle of an inclined plane which causes an object to slide down the plane.

In general, The angle of repose is the maximum angle that a surface can be tilted from the horizontal, such that an object on it is just able to stay on the surface without it sliding down.

Explanation:

Pressure at the top of the wall = 0

Pressure at the bottom of the wall = ρgh

Total Resultant Pressure = \(\frac{{\rho g{H^2}}}{2}\)

For a plane surface immersed in a liquid in such a way that the plane of the surface makes an angle θ with the free surface of the liquid:

A = Total area of an inclined surface

h̅ = Depth of centre of gravity of inclined area from a free surface

h* = Distance of centre of pressure from the free surface of a liquid

\({h^*} = \frac{{{I_G}{{\sin }^2}\theta }}{{A̅ h}} + \bar h\)

For vertical plane surface: θ = 90°

\({h^*} = \frac{{{I_G}}}{{A \bar h}} + \bar h\)

\({h^*} = \frac H2+\frac{bH^3 \times2}{12\times b\times H\times H}\)

\(h^*=\frac H2+\frac H6\)

\(h^*=\frac {2H}3\)

This resultant pressure will act through the centroid of the pressure diagram i.e. at H/3 from the base of the wall. Hence it acts at 2H/3 from the free surface of the wall.