Strength of Materials/Mechanics of Solids (Short Answer Questions from B Tech Exams)

Simple Stresses and Strains

What is meant by free body diagram? Draw free body diagram for box, place on a table. (GTU 2020)

The free body diagram of an element of a member in equilibrium is the diagram of only that member of the element, as if made free from the rest, with all the external forces acting on it.

In many problems, it is essential to isolate the body under consideration from the other bodies in contact and draw all the forces acting on the body. For this, first the body is drawn and then applied forces, self-weight, and the reactions at the points of contact with other bodies are drawn. Such a diagram of the body in which the body under consideration is freed from all the contact surfaces and shows all the forces acting on it (including reactions at contact surfaces), is called a Free Body Diagram (FBD).

State’s Law of Parallelogram of forces. (GTU 2020, 2021, 2022)

If two forces acting at a point are represented both in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal of the parallelogram drawn from the same point.

Write the statement of Varignon's theorem or principle of moments. (GTU 2021, 2022)
Answer: The theorem states that the moment of a resultant of two concurrent forces about any point is equal to the algebraic sum of the moments of its components about the same point. In other words, "If many coplanar forces are acting on a body, then the algebraic sum of moments of all the forces about a point in the plane of the forces is equal to the moment of their resultant about the same point."

State and explain Lami’s theorem. (GTU 2021, 2022)

If three forces acting at a point are in equilibrium, the ratio of the forces to the sine of the angle between the remaining two forces is the same.

F₁/sinα = F₂/sinβ = F₃/sinγ

What is Hook's law? Explain. (AKTU 2017, 2019, GTU 2021)

For the stress strain curve above, it is evident that the relation between stress and strain is linear for comparatively small values of the strain. This linear relation between elongation and the axial force causing it was first noticed by Sir Robert Hooke and is called Hooke’s law.

s = Ee, where E denotes the slope of the straight-line portion OP in the curve.

Define yield stress. (PTU 2019)

Yield point (stress) is the point at which there is an appreciable elongation or yielding of material without any corresponding increase of load. The yield stress is the point at which transition from elastic to plastic deformation occurs. It is the stress level at which the material begins to yield or flow, rather than simply stretching or compressing.

State Hooke’s law and define Poisson’s Ratio. (UTU 2014, JNTUK 2020)

Hook's law is defined above.

Poisson’s ratio. When a bar is subject to a simple tensile loading, there is an increase in length of the bar in the direction of the load, but a decrease in the lateral dimensions perpendicular to the load. The ratio of the strain in the lateral direction to that in the axial direction is defined as Poisson’s ratio. It is generally denoted by μ or 1/m. Its values lie between 0.25 and 0.35.

State Hook’s law. Draw a stress strain curve for MS specimen and explain each point in detail. (UTU 2014, GTU 2021, 2022, JNTUK 2023, PTU 2019)
Hook's law, already defined.

If tensile force is applied to a steel bar, it will have some elongation. If the force is small enough, the ratio of the stress and strain will remain proportional. This can be seen in the graph as a straight line between zero and point A – also called the limit of proportionality. If the force is greater, the material will experience elastic deformation, but the ratio of stress and strain will not be proportional. This is between points A and B, known as the elastic limit.

Beyond the elastic limit, the mild steel will experience plastic deformation. This starts the yield point – or the rolling point – which is point B, or the upper yield point. As seen in the graph, from this point on, the correlation between the stress and strain is no longer on a straight trajectory. It curves from point C (lower yield point), to D (maximum ultimate stress), ending at E (fracture stress).

What do you understand by engineering stress-strain curve? (PTU 2020)

A stress-strain curve for a material is plotted by elongating the sample and recording the stress variation with strain until the sample fractures. The strain is set to horizontal axis and stress is set to vertical axis. It is often assumed that the cross-section area of the material does not change during the whole deformation process. This is not true, since the actual area will decrease while deforming due to elastic and plastic deformation. The curve based on the original cross-section and gauge length is called the engineering stress-strain curve, while the curve based on the instantaneous cross-section area and length is called the true stress-strain curve.

Briefly explain about types of stresses? (AKTU 2019, GTU 2022, JUTUK 2021)

Shear Stress (tangential stress) It is the stress that results when a material is subjected to opposing forces that act parallel to each other, but in opposite directions.
Tensile Stress It is a measure of the internal forces that are acting on a material when it is subjected to an external load or force that tries to pull it apart or elongate it.
Bending Stress It is the stress that results from the application of a bending moment to a material, causing it to deform.

Explain Hooke’s law and differentiate between Young’s modulus and modulus of rigidity. (AKTU 2019)

Hook's law already explained.

The quantity E, i.e. ratio of the unit stress to the unit strain, is the modulus of elasticity of the material in tension, or, as it is often called, Young’s modulus.
Modulus of rigidity (G) is defined as the ratio of shearing stress to shearing strain.

What is the relationship between modulus of elasticity, modulus of rigidity and bulk modulus? (PTU 2020)

E = 9KN/(N + 3K)

where N is modulus of rigidity, K is bulk modulus.

Explain the significance of modulus of rigidity. (PTU 2019)

The modulus of rigidity is the elastic coefficient when a shear force is applied, resulting in lateral deformation. It gives us a measure of how rigid a body is.

A load of 5 kN is to be raised with help of a steel wire. Find the minimum diameter of the wire, if the stress is not to be exceeded 100 N/mm². (GTU 2020)
Answer:

100 = P/A = (5 x 10³)/[(π/4)d²]

= 6.366 x 10³/d²

Giving d = 7.28 mm

A material has a Young's Modulus of 1.25 x 10⁵ N/mm² and Poisson Ratio of 0.25. Calculate Bulk Modulus. (PTU 2020)
Answer: 0.83 × 105 N/mm²

A steel rod 10 mm in diameter and 1 m long is heated from 20°C to 120°C, E = 200 GPa and α =12 × 10^-6 per °C. If the rod is not free to expand, find the thermal stress developed in the steel rod? (AKTU 2018

E = 200 x 10³ MPa

α = 12 x 10^-6 Per ⁰C

Δt = (120 - 20) = 100⁰C

σ_thermal = EαΔt

= 200 x 10³ x 12 x 10^-6 x 100

= 240 MPa (compressive)

Compound stresses and Strains

What are principal stresses and strains? (AKTU 2017, PTU 2020)

Principal stresses are those stresses which are acting on the principal planes.

If a plane element is removed from a body, it will be subject to the normal stresses σ_x and σ_y together with the shearing stress σ_xy as shown in the Figure.

There are certain values of the angle θ that lead to maximum and minimum values of σ for a given set of stresses σ_x, σ_y, and σ_xy. These maximum and minimum values that σ may assume are termed principal stresses and are given by

Let us take a case of small element subjected to three dimensional principal stresses σ₁, σ₂ and σ₃ (see figure), acting on three mutually perpendicular planes at a point in an isotropic material.

Principal strains ε₁, ε₂ and ε₃

What are principal planes? (PTU 2019)

Principal planes are these planes within the material such that the resultant stresses across them are wholly normal stresses or planes across which no shearing stresses occur.

Write a note on Mohr’s circle of stresses, stating its applications. (AKTU 2018, GTU 2021, JNTUK 2022, PTU 2019, 2020)

In Mohr circle, normal stresses are plotted along the horizontal axis and shearing stresses along the vertical axis. The stresses σ_x, σ_y, and σ_xy are plotted to scale and a circle is drawn through these points having its centre on the horizontal axis. Figure shows Mohr’s circle for an element subject to general case of plane stresses.

When Mohr’s circle been drawn, the principal stresses are represented by the line segments og and oh.

To determine the normal and shearing stresses on a plane inclined at a counter-clockwise angle θ with the x-axis, we measure a counter-clockwise angle equal to 2θ from the diameter bd of Mohr’s circle.

Mohr's Circle has a large range of applications across several fields, including civil and mechanical engineering and material science. It helps in the analysis of soil under different loading conditions, material selection processes, and stress analysis of geological formations.

Define the term obliquity. (AKTU 2019)
Answer: It is the angle made by normal stress with the tangential stress.

Bending and Shear Stresses in Beams and Curved Beams

Define bending and shear stress. (PTU 2019)

Bending stress arises when external forces or moments cause an object to bend or deform. This stress can lead to both compressive and tensile forces within the material. This type of stress typically arises in structures or components like beams, bridges, and columns that are subjected to loads that cause them to bend.

Shear stress occurs when the material or fluid’s adjacent layers move relative to each other due to an applied force. This results in a component of the force per unit area acting parallel to the surface. This force attempts to deform the material by causing one portion of it to slide past another. It is typically represented by the symbol "τ or σ".

The Equations Expressing relationship between bending moment(M) acting at any section in a beam and the bending stress (σ) at any point in this same section.

σ/y = E/R

M/I = σ/y

Explain: (i) Section Modulus (ii) Modular ratio. (AKTU 2017, 2020)

Section Modulus is a geometric property used to calculate the bending stresses in a structural member. It is defined as the ratio of the moment of inertia of a section about its centroidal axis and to the distance of the extreme layer from the neutral axis.
The modular ratio is the ratio of the modulus of elasticity of two different materials. If E₁ and E₂ are Young's modulus of two materials, then the ratio E₁/E₂ or E₂/E₁ is known as modular ratio.

What do you mean by “simple/pure bending”? What are the assumptions made in the theory of simple bending? (AKTU 2018, PTU 2020)

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

Assumption in bending theory are

Normal sections of the beam, which were plane before bending, remain plane after bending.
The material is homogeneous and isotropic, so that it has the same elastic properties in all directions.
The beam is initially straight and of uniform cross section.
Modules of elasticity in tension and compression are equal.
It obeys Hook’s law i.e. the stress is proportional to strain within the elastic limit.
The radius of curvature of the bam before bending is very large in comparison to its transverse dimensions.
The resultant pull or push across transverse section is zero.

Draw Shear stress distribution for rectangular section. (AKTU 2019, PTU 2020)

Show that for a beam subjected to pure bending, neutral axis coincides with the centroid of the cross-section. (AKTU 2018)

The neutral axis is an axis in the cross-section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric centroid. All fibres on one side of the neutral axis are in a state of tension, while those on the opposite side are in compression.

Draw shear stress distribution diagram for : (1) I section (2) Circular section (3) Triangular section. (GTU 2020)

Shear stress distribution for I-Section

Shear stress distribution for circular section

Shear stress distribution for triangular section

Differentiate between bending of straight and curved beam. (AKTU 2020)

The neutral and centroidal axis of the straight beam is coincident. However, in curved beams, the neutral axis is shifted towards centre of curvature.
The bending stress in straight beams varies linearly with the distance from the neutral axis. However, in curved beams the stress distribution is hyperbolic.

Torsion

Define torsion. (PTU 2019)

When a bar rigidly clamped at one end and twisted at the other end by a torque (T₂ = P₂d₂) applied in a plain perpendicular to the axis of the bar as shown in figure. such a bar is said to be in TORSION.

Effects of the torsion are

to import on angular displacement of one end cross section with respect to the other
to set up shearing stresses on any cross section of the bar perpendicular to its axis.

What do understand by torsional rigidity and angle of twist? (AKTU 2020, JNTUK 2022)

Torsional rigidity or stiffness of the shaft is defined as the product of modulus of rigidity (C) and polar moment of inertia of the shaft (J). Hence mathematically, the torsional rigidity is given as, Torsional rigidity = C x J.
Torsional rigidity is also defined as the torque required to produce a twist of one radian per unit length of the shaft.
Torsional rigidity = TL/θ, where θ is in radian.

Angle of twist. If a shaft of length L is subject to a constant twisting moment T along its length, then the angle θ through which one end of the bar will twist relative to other is

θ = TL/GJ

Where G is modulus of elasticity in shear (shear stress/shear strain).

Write Torsion equation. (JNTUK 2021, 2 marks)

τ/R = q/r = Gθ/L = T/J

where

τ = Shear stress at a distance R from the axis of shaft
q = Shear stress at a distance r from the axis of the shaft

L = Length of shaft
θ = Angle of twist in radians
C or G = Modulus of rigidity of the material of the shaft
T - Torque applied
J = Polar moment of inertia of the shaft.

What will be the torque transmitted by solid circular shaft? (PTU 2020)
τ/R = T/J

τ/(D/2) = T/[(π/32)D⁴]

Hence, T = (π/16)τD³

Why a hollow shaft is preferred over a solid shaft? (PTU 2019)

Hollow shafts are preferred over solid shafts on account of their more power transmitting capacity for the same weight of material.
The strength of circular members is mainly contributed by their outer surface. The core of a solid shaft is essentially wasted material in some cases.

Hollow shaft is not always better compare to a solid shaft. It depends on our uses. When there is no space constraint, then for the same mass, the hollow shaft is better than the solid shaft. But if we have a radial space constraint then the solid shaft is preferred since, in this case solid shaft has more power transfer capacity than the hollow shaft.

State assumptions made in theory of torsion. (GTU 2021)

Plane normal sections of the shaft remain plane even after twisting.
Torsion is uniform along the rod.
Stress is proportional to strain.
Material is homogeneous and isotropic.
Diameter in the section before deformation remains a diameter, or straight line after deformation.

Shear Force and Bending Moment Diagrams

What is the difference between UDL and point load? (PTU 2020)

Concentrated load (or point load): If a load is acting on a beam over a very small length. It is called point load.

Uniformly Distributed Load: For finding reaction, this load may be assumed as total load acting at the center of gravity of the loading (Middle point).

Define: (1) Shear Force (2) Bending Moment (3) Points of contraflexure (GTU 2020, JNTUK 2023)

Shear Force. The algebraic sum of all the vertical forces at any section of a beam to the right or left of the section is known as shear force.
Bending Moment. The algebraic sum of all the moment of all the forces acting to the right or left of the section is known as bending Moment.
Point of Contraflexure. The point of zero bending moment, i.e. where the type of bending changes from sagging to hogging is called a point of inflection or contraflexure.

Springs

What is spring? What are different types of springs? (AKTU 2017)

Spring may be regarded as a device for storing up energy in the form of resilience. they are used either as a storage of energy or else are used to absorb excess energy, depending upon the function of particular spring under consideration. a spring in the clock-work, for example, is used to store work in the form of strain energy which is regained when the spring takes its original shape. We may divide spring chiefly into two categories.

Torsion Spring
Bending Spring

A torsion spring is the one which is subjected to a twisting moment and the resilience is mainly due to torsion. Closely-coiled helical springs subjected to axial pull fall under this category.
A bending spring is the one which is subjected to bending only and the resilience is mainly due to bending. The leaf or plate springs fall under this category.

Thin and Thick Cylinders

Differentiate between thin cylinder and thick cylinder. (AKTU 2017)

Thin Cylinders

The wall thickness is less than 1/20 of the inner radius of the cylinder.
The radial shear stress is neglected.
The hoop stress is assumed to be uniformly distributed over the thickness.
Examples are tyres, gas storage tanks.
Analytical treatment for stresses is simple and approximate.
A thin cylinder is statically determinate.
State of stress is membrane i.e. biaxial.

Thick Cylinders

The wall thickness is more than or equal to 1/20 of the inner radius of the cylinder.
The radial shear stress is considered.
The hoop stress varies parabolically over the wall thickness.
Examples are gun barrels, high-pressure vessels in oil-refining industry.
Analytical treatment is complex and accurate.
A thick cylinder is statically indeterminate.
State of stress is triaxial.

Write an expression for hoop stress in a thin cylinder closed at both ends and subjected to inner fluid pressure. Indicate what the various notations stand for? (PTU 2020)

See figure.

Force on projected area (inner) = force developed in resisting section

p x (d x l) = s_c x (2 x t x l)

where l = length of the cylinder

σ_c = pd/2t

Derive the longitudinal stress of thin cylinder. (JNTUK 2023)

See the following figure.

Force on projected area (inner) = Force developed in resisting section

p(π/4)d² = σ₁πdt

Hence, σ_l = pd/4t

Explain why ‘wire wound cylinders are more efficient than ordinary thin cylinders'. (JNTUK 2023)

In order to resist large internal pressures in a thin cylinder, it will be wound closely with a wire. These are called wire bound pipes. The tension in the wire, binding the cylinder will create initial compressive stresses in the pipe before applying the internal pressure. Thus, on application of the internal pressure, the cylinder and the wire jointly resisted the bursting force. The final stresses in the wire or the cylinder will be the sum of the initial stresses due to winding and the stresses induced due to the application of internal pressure. The relation between the stresses in the wire and cylinder arc obtained by considering the strain at the common surface of the shell and the wire which should be the same for both.

Columns and Struts

Differentiate between short and long column. (PTU 2020)

Short columns are those whose slenderness ratio is less than 32 or length to diameter ratio is less than 8. Such columns are always subjected to direct compressive stress only.
Long columns have a slenderness ratio more than 120 or length to diameter ratio more than 30.
A short column fails by crushing while a long column fails by buckling.
Complete strength of the material is utilized in a short column while in long column gives way out of instability.
The maximum stress a short column can reach during failure is ultimate compressive stress.
In case of long columns, the maximum stress level is limited to yield stress.
Euler’s formula is applicable to long columns only.
Sometimes, the columns whose slenderness ratio is more than 80, are known as long columns, and those whose slenderness ratio is less than 80 are known as short columns.

Differentiate between column and beam. (PTU 2020)

Columns and beams differ through their uses. Columns transfer loads vertically into a foundation whilst beams transfer loads horizontally into columns.
Beams are horizontal, columns are vertical.
Beams are designed to resist bending and shear forces, ensuring the even distribution of loads across their span. Beams are essential for providing structural support to prevent excessive sagging or deflection of floors and roofs.
The main function of columns is to support the vertical loads and ensure the safe transfer of these loads to the ground. Columns are designed to resist axial loads, which are the forces that act along the longitudinal axis of the column.

What is slenderness ratio and equivalent length of a column? (AKTU 2017, 2019, PTU 2019)

In Euler's formula, the ratio l/k is known as slenderness ratio. It may be defined as the ratio of the effective length of the column to the least radius of gyration of the section.

where

   E = Modulus of elasticity or Young's modulus for the material of die column,
   A = Area of cross-section,
   k = Least radius of gyration of the cross-section,
   l = Length of the column, and
   C = Constant, representing the end conditions of the column or end fixity coefficient.

If the slenderness ratio is less than 80, Euler's formula for a mild steel column is not valid.

Equivalent length. Sometimes, the crippling load according to Euler's formula may be written as

where L is the equivalent length or effective length of the column.
The relation between the equivalent length and actual length for the given end conditions is shown in the following table.

What is the limitation of Euler's formula for calculating the crippling load on the columns? (PTU 2020)

Crippling load

W_cr = Cπ²EA/(l/k)²

Crippling stress

W_cr = W_cr/A = Cπ²E/(l/k)²

Taking mild steel with E = 0.21 x 10⁶ N/mm² and crushing stress as 330 N/mm².
Now taking C as 1,

l/k = 80

Hence if the slenderness ratio is less than 80, Euler's formula for a mild steel column is not valid.

What is the difference between column and strut? (AKTU 2017, 2019, PTU 2019)

Vertical members of a building frame or any structural system which carry mainly axial compressive loads are called columns. The compression member of a truss is called strut. The common feature of columns and struts is that they are subjected to compressive forces.

What do you mean by Buckling Load in case of column? (AKTU 2020, 2019)

Lateral bending of column is known as buckling. The least axial load under which the buckling starts is known as critical or buckling load. This load is obviously less than crushing load. A column may be considered as failed, if it starts buckling.
The lateral bending of the column gives the feeling of insecurity and aesthetically ugly look. Since lateral buckling causes shortening of the column, it may break some of floor finishing material, failure of slabs and beams. Hence it is necessary to know the critical / buckling load, so that this value is maintained more than expected load on column strut.

Illustrate Euler's bucking load for column. (PTU 2019)

where

Describe assumptions in Euler’s column theory. (AKTU 2018)

The column is initially perfectly straight and the load is applied axially.
The cross-section of the column is uniform throughout its length.
The column material is perfectly elastic, homogeneous and isotropic and obeys Hooke’s law.
The length of the column is very large as compared to its lateral dimensions.
The direct stress is very small as compared to the bending stress.
The column will fail by buckling alone.
The self-weight of column is negligible.

Discuss the effect of eccentricity on columns. (PTU 2019)

Eccentricity refers to the offset between the centroid of the column's cross-section and the line of action of the applied axial load. This offset can occur due to various reasons, such as construction imperfections, lateral loads, or the presence of eccentric connections.

When a column experiences eccentric loading, it undergoes both axial compression and bending. The combination of these two effects can lead to the following consequences:

Increased Stresses: The eccentricity of the load introduces additional bending stresses in the column, which are added to the axial compressive stresses.
Reduced Axial Load Capacity: The presence of eccentricity reduces the column's ability to resist axial loads. The reduction in axial load capacity is proportional to the magnitude of the eccentricity.
Increased Lateral Deflections: The bending moment caused by the eccentric loading can lead to lateral deflections of the column.
Increased Buckling

Unsymmetrical Bending

Define shear centre and its importance. (AKTU 2019, 2020)

Shear centre is a point (in or outside a section) through which the applied shear force produces no torsion or twist of the member. If the load is not applied through the shear centre, there will be twisting of the beam due to unbalanced moment caused by the shear force acting on the section.

The shear centre lies on the axis of symmetry, if the section is symmetrical about one axis. The shear centre does not coincide with the centroid in this case.
For sections having two axes of symmetry, the shear centre lies on the intersection of these axes and thus coincides with the centroid.
Shear centre is also known as centre of twist.

When designing a beam, engineers must take into account the shear center to ensure the structural member can efficiently carry the load without experiencing twisting or other undesired deformations. This is especially important in beams with thin-walled open cross-section like angles and T-beams.

What is difference between pure bending and unsymmetrical bending? (AKTU 2019)
If a beam is loaded in such a fashion that the shear forces are zero on any cross-section perpendicular to the longitudinal axis of the beam and hence it is subjected to constant bending moment then the beam is said to be in the state of pure bending.

If the plane of loading or the plane of bending does not lie in (or parallel to) a plane that contains the principal centroidal axis of the cross-section, that bending is known as unsymmetrical bending.

In case of unsymmetrical bending, the neutral axis is not perpendicular to the plane of bending.
The unsymmetrical bending will be when

section is symmetrical (such as rectangular, circular, I-sections) but load line is inclined to both the principal axes.
the section is unsymmetrical (such as angle section or channel section etc.,) and load line is along any centroidal axis.

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