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UNIT 1.1 – SIMPLE STRESS AND STRAIN 61. Define elasticity.

Elasticity is the property by virtue of which the entire deformation (strain) produced in the material by the load (stress), disappears when the loading is removed.

62. What is the main advantage we get, when we use ductile materials? Ductile materials show significant deformation before failure (unlike brittle materials which experiences sudden failure without warning). Hence we can identify and repair the structures before failure by regular inspection.

63. Define a rigid body. A rigid body does not undergo any deformation. But it is only a hypothesis. All bodies are elastic and deformable up to a certain limit.

64. Define an elastic body. A material body which comes back to its original position after the loads are removed is called an elastic body.

65. Define a plastic body. A material body is said to be plastic if permanent residual deformations exist even after the removal of external forces.

66. Define elastic limit. There is always a limiting value of load up to which the strain totally disappears on the removal of load – the stress corresponding to this load is called elastic limit.

67. Define deformation. Deformation of the material is the change in geometry (change in size and change in shape) created when stress is applied (in the form of force loading, gravitational field, acceleration, thermal expansion, etc.). Deformation is expressed by the displacement field of the material.

68. Define analysis and design. In analysis, the structural system is already defined. That is, we know about the material with which the structure is made of and the geometry and boundary conditions of the structure. Then we apply different types of loads and we observe the response behaviour of the structure under known loads. Thus, the method of determining the stresses, strains, deformations and load carrying capacity in an already existing structure is called (stress) analysis. Design is a much more difficult task. Design is the determination of the geometric configuration of a structure in order that it will fulfill a prescribed function. We determine the material the structure is made of and we determine the geometry and boundary conditions of the structure. That is, we must determine the properties of the structure in order that the structure will support the loads and perform its intended functions. Design is the inverse process of analysis.

69. Differentiate between loads and reactions. When analyzing or designing a structure, we refer to the forces that act on it as either loads or reactions. Loads are active forces that are applied to the structure by some external cause, such as gravity, water pressure, wind, and earthquake ground motion. Reactions are passive forces that are induced at the supports of the structure.

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70. Define a prismatic bar. Give examples. A prismatic bar is a straight structural member having the same cross-sectional area throughout its length. Examples: a prismatic solid circular cylindrical bar, a prismatic hollow rectangular cylindrical bar.

71. Give examples of a non-prismatic bar. Tapered bar, bar of uniformly varying cross-section.

72. Define strength, stiffness and stability. Strength refers to the capacity of a structure to resist loads. Stiffness refers to the ability of the structure to resist deformation (= change of size and/or change of shape) (for instance, to resist stretching, bending or twisting). Stability refers to the ability of the structure to resist buckling under compressive stress.

73. Define isotropy. (or) What are isotropic materials? Isotropy is the property of the materials by virtue of which the physical properties (like density, elastic modulus, electrical conductivity, index of refraction, etc.,) are independent of direction. In isotropic materials, physical properties are same in all directions. An isotropic material is one whose physical properties are independent of direction in space.

74. Define orthotropy. (or) What are orthotropic materials? 75. Define anisotropy. (or) what are anisotropic materials? 76. Define homogeneity. (or) What is a homogeneous material.

A homogeneous material is a material whose material composition does not vary from point to point in space.

77. What are axially loaded members? Structural components subjected to only tension or compression are known as axially loaded members.

78. Define axial force or normal force. An axial force is a load directed along the axis of a member, resulting in either tension or compression in the member. The load passes through the axis connecting the centroids of the two end sections.

79. Define shear force. Shearing forces are un-aligned forces pushing one part of a body in one direction, and another part the body in the opposite direction. (When the forces are aligned into each other, they are called compression forces.)

80. Sketch the action of normal force and shear force.

81. Classify structures based on the type of load they resist. i. Bars - resist axial loads (tensile and compressive)

ii. Beams – resist transverse loads iii. Truss – collection of bars – resist axial loads (tensile and compressive)

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iv. Shafts – resist torsional loads v. Columns – resist axial compressive loads only

82. Write static equilibrium equations. Force equilibrium equation i.e., 0F =∑ forces along x-direction = 0

Moment equilibrium equation i.e., 0M =∑ moments about x-axis = 0

83. Classify structures based on their solution methods. Determinate structures – structures which can be solved using only the static equations of equilibrium. Indeterminate structures – structures which cannot be solved using only the static equations of equilibrium.

84. How do you solve indeterminate structures? With the help of compatibility equations, in addition to static equilibrium equations.

85. State the assumptions made in the study of stresses and strains in bars. a. The material is linearly elastic, homogeneous and isotropic. b. The axial loads act through the centroids of the cross-sections.

86. Define stress.

Stress is defined as the internal resisting force per unit area. internal resisting forcestress = area

Unit of stress is 2 2 2 or or 1 1N N NMPa MPamm m mm

=

21 1 NPam

=

87. Define Normal stress. internal resisting force (normal force) Normal stress =

area perpendicular to the direction of force

PA

σ =

Tensile stresses and compressive stresses are normal stresses.

88. Define Shear stress or average shear stress.

internal resisting force (shear force)Shear stress = area parallel to the direction of force

VA

τ =

89. Define complementary shear stress. The applied shear forces produces unbalanced moment and this moment must be balanced by an equal and opposite moment resulting from shear stresses acting on the sides of the element. These shear stresses are called complementary shear stresses.

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90. Compare and contrast “stress” and “pressure”. Comparison:

1. Both act on an area. 2. Both have the units of N/mm2.

Contrast

Stress Pressure 1. Stress can act both perpendicular or parallel to

the area. 1. Pressure acts only perpendicular (or normal) to

the area. 2. Stress is the intensity of internal resisting forces

produced in a body. 2. Pressure is intensity of external forces acting at a

point. 3. Stress is produced in a solid body. 3. Pressure is exerted by a fluid on a solid. 4. Magnitude of stress at a point in all directions is

generally unequal. 4. Magnitude of pressure at a point in all direction

remains same. 5. Stress can't be measured directly using any

device. 5. Pressure can be measured using measuring

devices like pressure gauge. 6. Stress is a tensor quantity. 6. Pressure is a scalar quantity.

91. What are the types of axial stress? Or what are the types of normal stress? (i) Axial tensile stress and (ii) Axial compressive stress

92. Define normal strain.

Normal strain = change in lengthoriginal length

or Lδε =

93. 1 horse power = 747 Watts 1 kip = 1 kilo pound = 1000 pounds 1 psi = 1 pound per square inch 1 ksi = 1 kilo pound per square inch Unit of energy is Joules or N-m.

94. What are the types of strain? a. Tensile strain or compressive strain. They are generally termed normal strain. b. Shear strain c. Volumetric strain

95. Define tensile strain or normal strain.

Tensile strain and compressive strain, generally called normal strain ε change in lengthoriginal length L

δ= =

96. Define shear strain. Shear strain – γ (gamma) is defined as the angle of distortion. It is also defined as the change in the angle of 90 degrees between two initially mutually perpendicular sides.

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97. State Hooke’s law in tension (or compression). Within the linear elastic limit, stress is directly proportional to strain.

i.e., normal stress ∝ normal strain, or or E E σσ ε σ εε

∝ = = .

98. State Hooke’s law in shear.

Within the linear elastic limit, shear stress ∝ shear strain, or or G G ττ γ τ γγ

∝ = =

99. Define volumetric strain.

Volumetric strain volε change in volumeoriginal volume

VVδ

= = Volumetric strain is also called dilatation.

100. Define elastic constant. The material properties of a structure that describe the relation between stresses and strains in axial loading, bending loading and torsional loading are called elastic constants. Examples: E, G and K

101. What are the types of elastic constants? (OR) Give examples of elastic constants. a. Young’s modulus E b. Rigidity modulus G c. Bulk modulus K d. Poisson’s ratio ν e. Tangent modulus Et f. Secant modulus Es

102. Define Young’s modulus. The ratio of normal stress to normal strain within the linear elastic region is called Young’s modulus.

Young’s modulus E = normal stressnormal strain

σε

=

Unit of Young’s modulus is GPa (=109 N/m2)

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103. Define Rigidity modulus. What are its other names? The ratio of shear stress to shear strain within the linear elastic region is called Rigidity modulus. It is also

called Shear modulus. Rigidity modulus G = shear stressshear strain

τγ

= . Unit is GPa (=109 N/m2)

104. Define Bulk modulus. The ratio of volumetric stress to volumetric strain is called Bulk modulus of elasticity. Other name is

volume modulus of elasticity. Bulk modulus K = volumetric stressvolumetric strain

vol

vol

σε

= Unit is GPa (=109 N/m2)

105. A prismatic bar of area of cross-section A, Young’s modulus E and length L is fixed at the left end and is subjected to an axial tensile force P at the right end. Derive the equation for the elongation of the bar.

P L P L PLE E LA L A L AE

δ δσ ε σ ε δ= = = = =

106. Define stiffness of a spring? State the unit. Stiffness of a spring is defined as the force required to produce unit deflection. Unit is N/mm.

107. Write the simple spring equation. F Kδ= . F- force acting on the spring in N, δ - stretching, elongation or shortening of the spring in mm.

108. Compare the behavior of a bar and a spring. 1. Both structures elongate under the action of a tensile load and shorten under the action of a

compressive force. 2. When the applied loads are removed, both structures come back to their original position. 3. In both structures, the displacements are directly proportional to the applied loads. 4. That is, both structures are linearly elastic in nature.

109. What is the stiffness of a bar?

Equation PLLAE

δ = can be rewritten as AEP LL

δ =

. Comparing this equation with the spring stiffness

equation F Kδ= , stiffness of the bar is AEL

. This stiffness is specifically called axial stiffness.

110. Write the relation between E, G and K.

( )

( )

9 9 3 12 1 3

3 1 2

KGE G EK G E G K

E K

ν

ν

= + = = ++

= −

111. Define Poisson’s ratio. The ratio of lateral strain to longitudinal strain is called Poisson’s ratio. It is denoted by ν (nu).

Poisson’s ratio Lateral strainLongitudinal strain

ν =

If x-direction is the longitudinal direction and y- and z-directions are lateral directions, then

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Poisson’s ratio or Lateral strain

Longitudinal strainy z

x

ε εν

ε= =

112. Explain lateral strain with a simple sketch.

When a prismatic bar is subjected to a tensile load, the length of the bar increases and the lateral dimensions decrease.

change in width change in depthLateral strain original width original depthlateralε = =

In the figure shown, dotted line represents the original bar and the solid line represents the deformed bar. The reductions in the width and depth are clearly visible.

113. What are the maximum and minimum values of Poisson’s ratio? Why? Maximum value = 0.5. Minimum value = 0.

114. Mention the values of Poisson’s value for few metals. For most metals, Poisson’s ratio ν ranges from 0.25 to 0.30.

115. Which material is used as a bottle stopper? Why? Cork has a Poisson’s ratio of Zero. Hence cork is good for bottle stoppers because the cork will not expand when one is trying to push it into a bottle opening. Think of a cylindrical piece of rubber used for corking a bottle. When you try to push the rubber into the bottle, the part of the rubber right above the lip of the bottle will expand radially because of the non-zero Poisson’s ratio (rubber has a Poisson’s ratio close to 0.5). This expansion causes interference between the cork and the bottle, making it more difficult to push in the cork. Now envision the same cylindrical part made out of cork. Pushing on the cork will not cause any radial expansion, and the cork will slide into the bottle once you overcome surface friction between the cork and glass. No additional force is required to overcome any cork/bottle interference.

116. Define volumetric strain or cubical dilatation.

The unit volume change ( )increase in volume 1 2 or unit volume x y z x y z

Ve eV Eδ νε ε ε σ σ σ−

= = = + + = + +

e is called cubical dilatation or volumetric strain. 117. What are the types of stress occurring in structures?

Normal stress and shearing stress. Any type of stress can be brought under any one of these two stresses. 118. State the units of stress and strain.

Unit of stress – MPa, 2

Nmm

1 Pascal = 21 Nm

1 MPa = 6210 N

m 1 GPa = 9

210 Nm

Unit of strain = actually no unit, but it is expressed in microstrains ( )με

119. Give examples of normal stress.

Stresses due to axial loads in bars PA

σ =

Bending normal stresses in beams bM yI

σ =

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120. Give examples of shear stress.

a. Shear stress in shafts subjected to torque TrJ

τ =

b. Bending shear stresses in beams bVQIb

τ =

c. Single shear in riveted structures d. Double shear in riveted structures

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121. Define bearing stress. In a bolted connection, the bar and clevis will press against the bolt in bearing. The contact stresses between (i) the bar and the bolt and (ii) the clevis and the bolt are called bearing stresses. The bearing area is taken as the projected area of the curved bearing surface (d×t in the diagram).

122. Define bearing stresses. Bearing stresses (compressive normal stresses) occur on the surface of contact between two interacting members.

123. Define pure shear. An element that is subjected to only shear strains but no normal strains is said to be in pure shear.

124. Define factor of safety. ultimate stress ultimate loadFactor of safety

allowable stress allowable load= =

125. Define load factor. Factor of safety is otherwise called the load factor. Ultimate load = service load × load factor.

126. A prismatic bar is subjected to a tensile stress along the axial (or longitudinal) direction. In what directions the strains will occur? Why? A tensile normal stress in longitudinal direction produces tensile normal strain along the longitudinal direction (x-direction) and compressive normal strains in the lateral directions (y-direction and z-direction). This is due to Poisson’s ratio effect.

127. A rod of length L is tapered with a smaller diameter d1 at the left end and larger diameter d2 at the right end. Write an expression for the diameter of the rod at any distance from the left end and from the right end.

2 11x

d dd d xL− = +

from the smaller diameter end. 2 1d d

L−

is the gradient or slope.

2 12x

d dd d xL− = −

from the larger diameter end. compare it with y mx c= +

dx d2 d1

L

x

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128. An aluminum rod is surrounded by a steel tube, the ends being firmly fastened together. When subjected to a compressive load, write the relation between the stresses produced in each rod. Both rods are of the same length and both rods are compressed to the same amount. That is, shortening or contraction of both the bars is the same.

Aluminium Aluminum Steel SteelAluminium steel

Aluminum Aluminum Steel Steel

P L P LA E A E

δ δ= =

aluminium steel

aluminium steelE Eσ σ

=

129. Define axial rigidity, bending rigidity or flexural rigidity and torsional rigidity. Axial rigidity = AE Bending rigidity or flexural rigidity = EI Torsional rigidity = GJ

130. Write the stress-strain equations for tri-axial state of stress.

yx zx

yx zy

yx zz

E E E

E E E

E E E

σσ σε ν ν

σσ σε ν ν

σσ σε ν ν

= − −

= − + −

= − − +

131. Differentiate between moment and shear force. a. Two equal and opposite parallel forces whose lines of actions are separated by a large distance

constitutes a moment or couple. (Example: turning of a steering wheel, thread cutting in pipes using pipe wrench)

b. Two equal and opposite parallel forces whose lines of actions are separated by a small distance (but not zero) constitutes a shear force. (example: shearing action of a scissor, shearing of bolts and rivets, cutting of a thick rod using impact force)

132. Write the expression for the elongation of a straight bar due to its self-weight. 2

2glE

ρδ =

133. Write the expression for the elongation of a conical bar due to its self-weight. 2

6glE

ρδ =

134. Write the expression for the elongation of a tapered bar subjected to axial load.

1 2

4PlEd d

δπ

=

xσ

yσ

zσ

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135. What is residual stress? If a bar in tension is loaded beyond the elastic limit and then the load is removed, some stresses will remain in the bar. Such stresses are called residual stresses.

136. What is permanent strain? Permanent deformation in structures after the loads are removed is called permanent strain or residual strain. It is also called permanent set.

137. What is hydrostatic stress? What is differential stress?

If x y z pσ σ σ= = = − , then the state of stress is called hydrostatic stress. If x y zσ σ σ≠ ≠ , then the state of stress is called a differential stress.

138. Define optimization.

Optimization is the task of designing the best structure to meet a particular goal, such as designing a structure with the least weight.

139. Define factor of safety for ductile materials. yield stress yield loadFactor of safety

allowable stress allowable load= =

140. Define factor of safety for brittle materials. ultimate stress ultimate loadFactor of safety

allowable stress allowable load= =

141. Define stress concentration. 142. What are the causes of stress concentration? 143. Give examples of stress concentration. 144. How do you minimize stress concentration? 145. What type of stress is produced in a rivet in a riveted joint? 146. If a composite bar of steel and copper is heated, what type of stresses will be induced? 147. What are Luder’s bands? (or) what are Piobert’s bands? 148. State Saint Venant’s principle with an example.

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UNIT 1.2 – THERMAL STRESSES

149. Define thermal stress and thermal strain. Change in temperature produce expansion or contraction of structural materials. The stresses developed due to change in temperature are called thermal stresses or temperature stresses. The strains developed due to change in temperature are called thermal strains or temperature strains.

150. Write the unit of coefficient of thermal expansion?

o

mmmm C

= change in length in a length of 1 mm due to 1 degree change in temperature.

151. Derive the equation for the extension of a bar due to thermal stress. For 1 degree change in temperature, the change in length occurring in a length of 1 mm is α mm. For 1 degree change in temperature, the change in length occurring in a length of L mm is αL mm. For dT degree change in temperature, the change in length occurring in a length of L mm is αLdT mm.

Then thermal strain change in lengthoriginal lengththermal

LdT dTL

αε α= = = microstrains ( )µε .

Then thermal stress thermal E E dTσ ε α= = N/mm2. 152. List the terms involved in the thermal stresses. (OR) List the factors affecting thermal stresses.

1. Young’s modulus, E, of the material of the specimen 2. Coefficient of thermal expansion, α 3. Change in temperature, dT

153. When a prismatic bar is subjected to an increase in temperature and its deformation is prevented at both ends, what type of stress is developed in the bar? Compressive stresses are developed in the bar.

154. Can there be stress without strain? Can there be strain without stress? Yes. Stress can exist without strain and vice versa in the case of Thermal stresses. Free expansion – thermal strain without thermal stress. Constrained thermal heating – thermal stress without thermal strain.

155. A bar is subjected to an increase in temperature. One end of the rod is fixed and the other end is free. What types of stresses are developed in the bar? This is the case of free expansion. No thermal stresses will be developed in the bar but thermal strain is produced in the bar.

156. What is a bimetallic strip? A bar made up of two materials with different coefficients of thermal expansion is called a bimetallic bar or bimetallic strip.

157. Explain the effect of change of temperature in a composite bar. Assume a composite bar made of copper and steel subjected to a change of temperature. Since the coefficient of thermal expansion of copper is greater than that of steel, the copper rod will try to pull the steel rod and the steel rod will try to push the copper rod. Finally they will become stable at a certain position after compromise.

158. Give examples of thermal stresses.