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 Engineering Formulae
  Strength of Materials
 

 

 

Stress

stress

where,       σ=normal stress, or tensile stress, pa

                  P=force applied, N 

                  A=cross-sectional area of the bar, m2

                  =shearing stress, Pa

                  As=total area in shear, m2

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Strain

strain

where,

                =tensile or compressive strain, m/m

                =total elongation in a bar, m

                =original length of the bar, m

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Hooke's Law

Stress is proportional to strain

hooke's law

where,

E=proportionality constant called the elastic modulus or modulus of elasticity or Young’s modulus, Pa

 

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Piosson's Ratio

piosson's ratio

where,

                v=Poisson’s ratio

                =lateral strain

                =axial strain

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Unit Volume Change

unit volume change

where,

                =change in volume

                  =original volume

                   =strain

                   =Poisson’s ratio

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Elongation due to its weight

elongation due to its weight

where,

=total elongation in a material which hangs vertically under its own weight              

W=weight of the material

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Thin Rings

thin rings

where,

                =Circumferential or hoop Stress

                S=Circumferential or hoop tension

                A=Cross-sectional area

                =Circumferential strain

                E=Young’s modulus

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Strain Energy

strain energy

where,

                U=total energy stored in the bar or strain energy

                P=tensile load

   =total elongation in the bar

    L=original length of the bar

    A=cross-sectional area of the bar

    E=Young’s modulus

                U=strain energy per unit volume  

                 strain energy

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Thin Walled Pressure vessels

thin walled pressure vessels

where,

                =normal or circumferential or hoop stress in cylindrical vessel, Pa

                =normal or circumferential or hoop stress in spherical vessel,  Pa  and longitudinal stress around the circumference

                  P=internal pressure of cylinder, Pa

                  r=internal radius, m

                  t=thickness of wall, m

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Mohr's Circle for Biaxial Stress

mohr's circle for biaxial stress

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Pure Shear

pure shear

where,

                =Shearing Stress, Pa

                =Shearing Strain or angular deformation

                G=Shear modulus, Pa

                E=Young’s modulus, Pa

                V=Poisson’s ratio

 

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Torsion formula for Thin walled tubes

torsion formula for thin walled tubes

where,

               =maximum shearing stress, Pa

               =Shearing stress at any point a distance x from the centre of a                           section

                r=radius of the section, m

                d=diameter of a solid circular shaft, m

               =polar moment of inertia of a cross-sectional area, m4

                T=resisting torque, N-m

                N= rpm of shaft

                P=power, kW

               torsion formula for thin walled tubes=angle of twist, radian

                L=length of shaft, m

                G=shear modulus, Pa

                do=outer diameter of hollow shaft, m

                di=inner diameter of hollow shaft, m

 

and

               torsion formula for thin walled tubes

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Torsion formula for Circular Shafts

torsion formula for circular shafts

where,

                =Ip, polar moment of inertia for thin-walled tubes

                r=mean radius

                t=wall thickness

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Flexure Formula

flexure formulae

where,    

                =Stress on any point of cross-section at distance y from the                        neutral axis

               =stress at outer fibre of the beam

               c=distance measured from the neutral axis to the most remote fibre of the beam

                I=moment of inertia of the cross-sectional area about the centroidal axis

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Shear Stress In Bending

shear stress in bending

where,    

                F=Shear force

                Q=statistical moment about the neutral axis of the cross-section

                b=width

                I=moment of inertia of the cross-sectional area about the Centroidal axis.

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Thin-Walled Hollow Members (Tubes)

thin-walled hollow members

where,      =shearing stress at any point of a blue

                  t=thickness of tube

                  q=shear flow

                  T=applied torque

                  R=distance between a reference point and segment ds

                  Π=angle of twist of a hollow tube

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Stress Concentration

stress concentration

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Curved Beam in Pure Bending

curved beam in pure bending

where,     =normal stress

                 M=bending moment

                 dA=cross-sectional area of an element

                 r=distance of curved surface from the centre of curvature

                 A=cross-sectional area of beam

                 R=distance of neutral axis from the centre of curvature

                 R1=distance of centroidal axis from the centre of curvature

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Bending of a Beam

(a) Bending of a Beam Supported at Both Ends

bending of a beam

(b) Bending of a Beam Fixed at one end

bending of a beam

where,     d= bending displacement, m

                F=force applied, N

                I=length of the beam, m

                a=width of beam, m

                b=thickness of beam, m

                Y=Young’s modulus, N/m2

 

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