⚙️SOM:

🤔 KERN of Column section -

The kern of a section is the region in which a compressive point load may be applied without producing any tensile stress on the cross section.

● Shape of KERN in eccentric loading -

Column section        Kern shape
Rectangular                 Rhombus
Square                         Square
Circular                        Circle
I section                       Rhombus

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⚙️SOM:

SHEAR CENTRE-

● Shear centre -  a point from which a concentrated load passes but  there will be only bending and no twisting.

● It is also called as centre of of flexure.

● Resultant of shear passes through shear centre.

● Shear centre always lies on axis of symmetry if exists.

● Shear centre (e) for various sections
Channel                   =>  e = b²h²t / 4I
Semicircular           =>  e = 4R / 3.14
Open circular slit    =>  e = 2R

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🤔Pure bending:
 ( Theory of simple bending) :

🔆It is a condition of stress where a bending moment is applied to a beam without the simultaneous presence of axial, shear, or torsional forces.

🔆Pure bending occurs only under a constant bending moment (M) since the shear force (V), has to be equal to zero.

🔆In reality, a state of pure bending does not practically exist, because such a state needs an absolutely weightless member.

🔆The state of pure bending is an approximation made to derive formulas.

⭐️⭐️⭐️
Assumptions:

✴️The material of the beam is homogeneous and isotropic.

✴️The value of Young's Modulus of Elasticity is same in tension and compression.

✴️The transverse sections which were plane before bending, remain plane after bending also.

✴️The beam is initially straight and all longitudinal filaments bend into circular arcs with a common centre of curvature.

✴️The radius of curvature is large as compared to the dimensions of the cross-section.

✴️Each layer of the beam is free to expand or contract, independently of the layer, above or below it.

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🤖pure shear

🔰It is three-dimensional 
homogeneous flattening of a body.

🔰It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly.

🔰For soft materials, such as rubber, a strain state of pure shear is often used for characterizing 
hyperelastic & fracture mechanical
 behaviour.

🔰Pure shear is differentiated from simple shear in that pure shear involves no rigid body rotation.

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🔵 SOM

RATIOS OF SHEAR STRESS -

                                 Ratio
Shape             (τ)max / ( τ )avg

Rectangle           1.5
Circle                  1.33
Triangle              1.5
Diamond             1.125

                               Ratio
Shape            (τ)neutral axis / ( τ ) avg

Rectangle           1.5
Circle                  1.33
Triangle              1.33
Diamond             1

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⚙️ SOM

🫥TYPES OF ELASTIC CONSTANTS -

● Young’s modulus or Modulus of elasticity (E) = Normal stress/ Normal strain

● Shear modulus or Modulus of Rigidity (G) = Shear stress/ Shear strain

● Bulk modulus (K) = Direct stress/ Volumetric strain

RELATIONSHIP BETWEEN E, G, and K -

◇ E = 2G(1 + 1/m)  or     E = 2G(1+µ)
◇ E = 3K (1-2/m)   or     E = 3K (1-2µ)
◇ E = 9KG/ (G + 3K)

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🔵SOM

Number of total elastic constants (TEC) and independent elastic constants (IEC) based on material -

MATERIAL      TEC       IEC

Isotropic           4           2
Orthotropic      12         9
Anisotropic       ∞          21

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⚙️SOM

🤔Basics of SFD and BMD -

Shear force - It is  algebraic sum of forces acting on beam at either left or right of considered section

Moment of force - It can be formulated as the product of force and the length of the line crossing over the point of support.

Couple - It is a moment that is generated of two forces having the same magnitude, acting in the opposite direction equidistant from the reaction point.

Torque - It is a moment having a tendency to to twist a body around its axis of rotation.

The point of Inflection - It can be defined as the point in the bending moment diagram where the bending moment is becomes ‘ 0 ’.

Point of contraflexure - Point at which the bending moment curve of the beam  changes its sign.

Shear Span - Span having constant shear force value is termed as shear span.

Pure bending - when shear force is zero and value of bending moment is constant then it is called as pure bending.

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🔥SOM Very IMP One liners:

🔴1st derivative of deflection(∆)gives slope(θ)

🔴2nd derivative of deflection(∆) gives moment(M)

🔴 3rd derivative of deflection(∆) gives shear force(S)

🔴 4th derivative of deflection(∆) gives intensity of loading(W)

🔴Slope of bending moment diag. gives shear force

🔴Negative slope of shear force diag. gives loading intensity

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