True strain (?_T) = ln (1 + ?)

Calculation:

Given:

Engineering strain (?) = $(\varepsilon ) = 0.100{\rm{\% }} = \frac{{0.1}}{{100}} = 0.001$

True strain (?_T) = ln (1 + ?)

?_T = ln (1 + 0.001)

?_T = 0.099%

In the case of 3-D analysis, maximum shear stress can be calculated as:

\(\left[ {\frac{{{\sigma _1} - {\sigma _2}}}{2},\frac{{{\sigma _2} - {\sigma _3}}}{2},\frac{{{\sigma _1} - {\sigma _3}}}{2}} \right]\)

And In the case of plane stress analysis, maximum shear stress can be calculated by,

\({\tau _{{\bf{max}}}} = \frac{{{\sigma _1} - {\sigma _2}}}{2}\)

where

σ₁ and σ₂ are principal stresses in a plane.

Calculation:

Given:

σ1 = 100 MPa

σ₂ = 40 MPa

Now, In-plane stress analysis, we have

\(\begin{array}{l} {\tau _{{\bf{max}}}} = \frac{{{\sigma _1} - {\sigma _2}}}{2}\\ \\ {\tau _{{\bf{max}}}} = \frac{{100 - 40}}{2} \end{array}\)

= 30 MPa

Creep is the plastic permanent deformation of a structure under constant load for a very long period of time.

It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increases as they near their melting point.

The creep strains occur due to dead load only.

Factors affecting creep

1. Type of loading
2. Magnitude of load
3. Time
4. Temperature

If the temperature is more, creep is more, and if the temperature reaches half of melting point temperature the creep is intolerable. Such a temperature is defined as homologous temperature.

• For most sections used for beams, the maximum shear stress will occur at the neutral axis of the section in an inverted T-section when subjected to a shear force.
• Most applications require that the maximum shearing stress be determined to evaluate the acceptability of the stress relative to some criterion of design.

Poisson's ratio is the ratio of transverse strain to longitudinal strain.

For perfectly isotropic elastic material, Poisson’s ratio is 0.25 but for most of the materials.

The value of Poisson's ratio lies in the range of 0 to 0.5.

Poisson's ratio for various materials are:

• Cork: 0.0
• Aluminum: 0.31
• Cast iron: 0.21 – 0.26
• Steel: 0.27 – 0.30
• Stainless steel: 0.30 – 0.31
• Copper: 0.33
• Rubber: 0.5

For column, the buckling load is taken as,

\({\rm{P}} = \frac{{{\pi ^2}{\rm{E}}\:{{\rm{I}}_{{\rm{min}}}}}}{{{\rm{L}}_{\rm{e}}^2}}\)

Where

E = Modulus of Elasticity

I_min = Minimum moment of inertia

L_e = effective length of column.

Calculation:

In first case: Height = h, and cross-sectional area = a × 2a

\({\rm{P}} = \frac{{{\pi ^2}{\rm{E}}\:\left( {\frac{{2{\rm{a}} \times {{\rm{a}}^3}}}{{12}}} \right)\:}}{{{{\rm{h}}^2}}} = \frac{{{\pi ^2}{\rm{E}}{{\rm{a}}^4}}}{{6{{\rm{h}}^2}}}\)

In second case: Height = 1.5 h, and cross-sectional area = 3a × 0.5a

\({\rm{P'}} = \frac{{{\pi ^2}{\rm{E}}\:\left[ {\frac{{3{\rm{a}} \times {{\left( {0.5{\rm{a}}} \right)}^3}}}{{12}}} \right]}}{{{{\left( {1.5\:{\rm{h}}} \right)}^2}}} = \frac{1}{{12}} \times \frac{{{\pi ^2}{\rm{E}}{{\rm{a}}^4}}}{{6{{\rm{h}}^2}}} = \frac{{\rm{P}}}{{12}}\)

Levers of the first order (first-order lever): In this type of lever, the fulcrum lies between the load and effort.

A pair of scissors, a crowbar, a claw hammer, a wire cutter, a pair of pliers, Arm of man, Pair of clinical tongs, a pincer, a see-saw, a common balance, a hand pump, and a boat and oars are examples of a first-order lever.

A lever is a rigid rod that rotates about a fixed point called the fulcrum.

All levers are functioning on the following principle:

Load × Load arm = Effort × Effort arm

The distance of the load from the fulcrum is called the load arm and the distance of the effort from the fulcrum is called the effort arm.

Straight levers are classified as:

Two equal non-collinear parallel forces acting in opposite directions constitute a couple.

• The effect of a couple is to produce a tendency to rotation.
• The moment of a couple is the product of one of the forces and the perpendicular distance between the lines of action of the forces. It is independent of the position of the pole.
• The couple can be transferred to any desired position.

The M.I. of the hollow circular section about a central axis perpendicular to the section as compared to its M.I. about the horizontal axis is Double.

The moment of inertia is the sum of the product of the mass of each particle with the square of its distance from the axis of the rotation.

M.O.I for hollow circular cross-section

${I_{XX}} = \dfrac{\pi }{{64}}\left( {d_0^4 - d_i^4} \right)$

Where

d₀ = outer diameter of cross-section

d_i = inner diameter of cross-section

The moment of inertia about the axis perpendicular to the plane

$$\({I_{ZZ}} = \dfrac{\pi }{{32}}\left( {d_0^4 - d_i^4} \right)\)

According to question

\(\dfrac{{{I_{ZZ}}}}{{{I_{XX}}}} = \dfrac{{\dfrac{\pi }{{32}}\left( {d_0^4 - d_i^4} \right)}}{{\dfrac{\pi }{{64}}\left( {d_0^4 - d_i^4} \right)}}\)

I_ZZ = 2 × I_XX

The M.I. of the hollow circular section about a central axis perpendicular to the section as compared to its M.I. about the horizontal axis is Double.

• The virtual work concept is used in the application of the principle of least action to the study of forces and movement of a mechanical system.
• In a virtual work equation reaction is due to a rough surface on a body that rolls on it without slipping or a smooth surface with which body is in contact and reaction at a point or on an axis, fixed in space, around which a body is constrained to turn are neglected.