Graphical Method of Analysis MCQ Quiz in বাংলা - Objective Question with Answer for Graphical Method of Analysis - বিনামূল্যে ডাউনলোড করুন [PDF]

Concept:

• Mohr's circle is a geometric representation of the two-dimensional stress state and is very useful to perform quick and efficient estimations.
• It can help to readily find stresses on any specified plane by using a diagram instead of complicated computation.
• Each point on the mohr circle represents a point.
• the x-axis represents the direct stress and the y-axis represents the shear stress.
• since in any loading condition, the principal plane (plane on which shear stress is zero) will be there hence the centerline of the mohr circle will always be x-axis.

so option 1 & 2 can never be the case.

in given stress condition direct stress is zero and only shear stress are there this is the condition of plane shear stress.

and in plane shear stress condition principal plane, σ₁ = -σ₂ 

only option 3 satisfied this condition.

Stress on x face, A(200, 0) and Stress on y face, B(0, 0)

Maximum shear stress = Radius of Mohr circle

τ_max= σ/2 = 100 MPa

Concept:

Stress amplitude (σ_v) \(= \frac{{{σ _{max}}~ -~ {σ _{min}}}}{2}\)

Given:

σ_max = 400 MPa, σ_min = 200 MPa

σ_v \(= \frac{{400~ -~ 200}}{2} = 100\ MPa\)

Concept:

Mohr's circle is shown below:

• On the X-axis principal stress is drawn and on the Y-axis shear stress is drawn.
• The center of Mohr's circle lies on X-axis, therefore, the Y-coordinate of the center will be zero always and the x-coordinate is the average of principal stress.
• ∴ Coordinates of the center are \(\left[ {\frac{{\left( {{\sigma _{xx}} + {\sigma _{yy}}} \right)}}{2},\;0} \right]\).

Concept:

Maximum and minimum values of normal stresses occur on planes of zero shearing stress. The maximum and minimum normal stresses are called the principal stresses, and the planes on which they act are called the principal plane.

\({\sigma _{max,\;min}} = \frac{{{\sigma _{xx}}\;+\;{\sigma _{yy}}}}{2} \pm \sqrt {{{\left( {\frac{{{\sigma _{xx}}\;-\;{\sigma _{yy}}}}{2}} \right)}^2} +\;\tau _{xy}^2} \)

But the maximum shear stress planes may or may not contain normal stresses as the case may be.

\({\tau _{max}} = \frac{{{\sigma _{max}}\;-\;{\sigma _{min}}}}{2} = \sqrt {{{\left( {\frac{{{\sigma _{xx}}\;-\;{\sigma _{yy}}}}{2}} \right)}^2} +\;\tau _{xy}^2} \)

which is equal to the radius of the Mohr's Circle.

Calculation:

Given:

\({\sigma _{xx}} = 40~MPa, ~{\sigma _{yy}} = 100~MPa,{\tau _{xy}} = 40~MPa\)

Radius of Mohr's circle:

 \(\tau_{max}= \sqrt {{{\left({\frac{{{\sigma _{xx}} - {\sigma _{yy}}}}{2}} \right)}^2} + \tau _{xy}^2} \)

\(\therefore\sqrt{\left ( \frac{40\;-\;100}{2} \right )^2\;+\;40^2}\Rightarrow50\;MPa\)

Concept:

Centre \(H = \frac{{{\sigma _1} + {\sigma _2}}}{2}\)

Radius = \(\frac{{{\sigma _1}\; - \;{\sigma _2}}}{2}\)

Calculation:

Given, σ₁ = OA = + 250 MPa, σ₂ = OB = - 100 MPa

\(H = \frac{{250\; + \;\left( { - 150} \right)}}{2} = 50\) MPa

Radius = \(\frac{{{\sigma _1}\; - \;{\sigma _2}}}{2}\)

\(R = \frac{{250 - \left( { - 150} \right)}}{2} = 200\;MPa\)

Radius of the Mohr circle HC = HE = 200 MPa

Normal stress on the plane HC = OF = 200 MPa

OH = 50 Mpa. ∴ HF = OF - OH = 200 - 50 = 150 MPa

From the triangle HFC, \(FC = \sqrt {H{C^2} - H{F^2}}\) = \(\sqrt {{{200}^2} - {{150}^2}}\) = ​ ​50√7 MPa

The magnitudes of the shearing stress on a plane on which the normal stress is 200 MPa, FC = 50√7 MPa

The plane at right angle to the plane HC will be shown by 2 × 90° = 180 it is shown by the plane HE. 

From the similiar triangle HCF and HEG 

\(\frac{{HF}}{{HC}} = \frac{{HG}}{{HE}}\)

As HC = HE = R

So, HF = HG

150 = HO + OG

150 = 50 + OG

OG = 100 MPa (Compressive)

Tensile and the normal stress on a plane at right angle to the plane HC = OG = 100 MPa (Compressive)

Concept:

Maximum and minimum values of normal stresses occur on planes of zero shearing stress. The maximum and minimum normal stresses are called the principal stresses, and the planes on which they act are called the principal plane.

\({\sigma _{max,\;min}} = \frac{{{\sigma _{xx}} + {\sigma _{yy}}}}{2} \pm \sqrt {{{\left( {\frac{{{\sigma _{xx}} - {\sigma _{yy}}}}{2}} \right)}^2} + \tau _{xy}^2} \)

But the maximum shear stress planes may or may not contain normal stresses as the case may be.

\({\tau _{max}} = \frac{{{\sigma _{max}} - {\sigma _{min}}}}{2} = \sqrt {{{\left( {\frac{{{\sigma _{xx}} - {\sigma _{yy}}}}{2}} \right)}^2} + \tau _{xy}^2} \)

which is equal to the radius of the Mohr's Circle.

Calculation:

\({\tau _{max}} = \frac{{{\sigma _{max}} - {\sigma _{min}}}}{2} \)

\({\tau _{max}} = \frac{{{{\rm{\sigma }}_1} - {{\rm{\sigma }}_3}}}{2}\)

Mohr's circle is very useful in determining the relationships between normal and shear stresses acting on an inclined plane at a point in a stressed body. It is helpful in finding maximum and minimum principal stresses, maximum shear stress, etc.

Major Principal stress is given as:

\({\sigma _{p1}} = \frac{{{\sigma _1} + {\sigma _2}}}{2} + \sqrt {{{\left\{ {\frac{{{\sigma _1} - {\sigma _2}}}{2}} \right\}}^2} + {\tau _{xy}}^2} \)

Minor Principal stress is given as:

\({\sigma _{p2}} = \frac{{{\sigma _1} + {\sigma _2}}}{2} - \sqrt {{{\left\{ {\frac{{{\sigma _1} - {\sigma _2}}}{2}} \right\}}^2} + {\tau _{xy}}^2}\)

Maximum Shear stress is given as:

\({\tau _{max}} = \frac{{\left( {{\sigma _{p1}} - {\sigma _{p2}}} \right)}}{2}\)

Explanation:

Mohr Circle:

• It is a two-dimensional graphical representation (σ as x-axis and τ as y-axis) of the state of stress inside a body.
• The abscissa and ordinate of each point on the circle are the magnitudes of the normal stress and shear stress components, respectively.
• In other words, the Mohr circle is the locus of those points which represents the normal and shear stress on various planes passing through a point on a loaded body.

Properties of Mohr Circle:

• The Centre of the Mohr circle always lies on the x-axis (σ-axis) i.e. Mohr circle is always symmetrical about the σ-axis.
• The co-ordinate of centre is \(\left ( \frac{σ_x\;+\;σ_y}{2} \right )\;or\;\left ( \frac{σ_1\;+\;σ_2}{2} \right )\) which represents normal stress on the plane of τmax.
• The radius of the Mohr circle represents maximum shear stress i.e. \(τ_{max}= \sqrt {{{\left({\frac{{{σ _{x}}\;-\;{σ _{y}}}}{2}} \right)}^2} + τ _{xy}^2}\;\Rightarrow\frac{σ_1\;-\;σ_2}{2}\)
• If two-point on the circumference of the Mohr circle subtends an angle 2θ at centre, then the angle between those planes will be θ.

Special case:

1. Hydrostatic loading / Hydrostatic stress:

In the case of hydrostatic fluid, equal and alike normal stress acts on two mutually perpendicular planes without any shear i.e. σx = σy = σ and τxy = 0.

\(Centre =\left ( \frac{σ_x\;+\;σ_y}{2} \right )\;and\;Radius=τ_{max}\Rightarrow\sqrt {{{\left({\frac{{{σ _{x}}\;-\;{σ _{y}}}}{2}} \right)}^2} + τ _{xy}^2}\)

∴ centre is at (σ, 0) and radius = 0, which represents a point on x-axis / σ-axis or normal stress axis.

2. Pure shear:

In pure shear σx and σy = 0, τxy = τ

\(Centre =\left ( \frac{σ_x\;+\;σ_y}{2} \right )\;and\;Radius=τ_{max}\Rightarrow\sqrt {{{\left({\frac{{{σ _{x}}\;-\;{σ _{y}}}}{2}} \right)}^2} + τ _{xy}^2}\)

∴ centre is at origin (0, 0) and radius = τ.

Graphical Method:

\(\begin{array}{l} {\tau _{max}} = R = \frac{1}{2}\left( {AB} \right)\\ = \frac{1}{2}\sqrt {{{\left( {60 - 70} \right)}^2} + {{\left( {5 + 5} \right)}^2}} = \frac{1}{2}\sqrt {200} = 5\sqrt 2 \end{array}\)

Analytical Method:

Radius of Mohr circle

\(\begin{array}{l} = \;\sqrt {{{\left( {\frac{{{\sigma _x} - {\sigma _y}}}{2}} \right)}^2} + {\tau _{x{y^2}}}} \\ = \;\sqrt {{{\left( {\frac{{70 - 60}}{2}} \right)}^2} + {{5}^2}} \; = \;5\sqrt 2 \end{array}\)