Strength of Materials Contents Sl. No. Topic Page No. Chapter - 1 Stress and Strain 1 • Theory at a Glance (for IES, GATE, PSU) 1 • Previous 20-Years GATE Questions 23 • Previous 20-Years IES Questions 27 • Previous 20-Years IAS Questions 37 • Answers with Explanation (GATE, IES, IAS) 45 • Conventional Questions with Answers 53 Chapter - 2 Principal Stress and Strain 64 • Theory at a Glance (for IES, GATE, PSU) 64 • Previous 20-Years GATE Questions 102 • Previous 20-Years IES Questions 105 • Previous 20-Years IAS Questions 112 • Answers with Explanation (GATE, IES, IAS) 118 • Conventional Questions with Answers 126 Chapter - 3 Moment of Inertia and Centroid 135 • Theory at a Glance (for IES, GATE, PSU) 135 • Previous 20-Years GATE Questions 146 • Previous 20-Years IES Questions 147 • Previous 20-Years IAS Questions 148 • Answers with Explanation (GATE, IES, IAS) 149 • Conventional Questions with Answers 150 Chapter - 4 Bending Moment and Shear Force 151 Diagram • Theory at a Glance (for IES, GATE, PSU) 151 • Previous 20-Years GATE Questions 194 • Previous 20-Years IES Questions 198 • Previous 20-Years IAS Questions 209 • Answers with Explanation (GATE, IES, IAS) 218 • Conventional Questions with Answers 228 India’s No 1 Strength of Materials IES Academy Contents Sl. No. Topic Page No. Chapter - 5 Deflection of Beam 240 • Theory at a Glance (for IES, GATE, PSU) 240 • Previous 20-Years GATE Questions 277 • Previous 20-Years IES Questions 277 • Previous 20-Years IAS Questions 281 • Answers with Explanation (GATE, IES, IAS) 283 • Conventional Questions with Answers 286 Chapter - 6 Bending Stress in Beam 289 • Theory at a Glance (for IES, GATE, PSU) 289 • Previous 20-Years GATE Questions 294 • Previous 20-Years IES Questions 295 • Previous 20-Years IAS Questions 300 • Answers with Explanation (GATE, IES, IAS) 304 • Conventional Questions with Answers 307 Chapter - 7 Shear Stress in Beam 313 • Theory at a Glance (for IES, GATE, PSU) 313 • Previous 20-Years GATE Questions 317 • Previous 20-Years IES Questions 317 • Previous 20-Years IAS Questions 320 • Answers with Explanation (GATE, IES, IAS) 323 • Conventional Questions with Answers 326 Chapter - 8 Fixed and Continuous Beam 327 • Theory at a Glance (for IES, GATE, PSU) 327 • Previous 20-Years IES Questions 332 • Previous 20-Years IAS Questions 333 • Answers with Explanation (GATE, IES, IAS) 334 • Conventional Questions with Answers 336 Chapter - 9 Torsion 337 • Theory at a Glance (for IES, GATE, PSU) 337 • Previous 20-Years GATE Questions 346 • Previous 20-Years IES Questions 347 • Previous 20-Years IAS Questions 356 • Answers with Explanation (GATE, IES, IAS) 360 • Conventional Questions with Answers 367 Chapter-10 Thin Cylinder 376 • Theory at a Glance (for IES, GATE, PSU) 376 • Previous 20-Years GATE Questions 379 • Previous 20-Years IES Questions 380 • Previous 20-Years IAS Questions 383 • Answers with Explanation (GATE, IES, IAS) 386 • Conventional Questions with Answers 390 www.iesacademy.com Email: [email protected] Page - i i ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290 India’s No 1 Strength of Materials IES Academy Contents Sl. No. Topic Page No. Chapter-11 Thick Cylinder 393 • Theory at a Glance (for IES, GATE, PSU) 393 • Previous 20-Years GATE Questions 402 • Previous 20-Years IES Questions 402 • Previous 20-Years IAS Questions 405 • Answers with Explanation (GATE, IES, IAS) 406 • Conventional Questions with Answers 410 Chapter-12 Spring 418 • Theory at a Glance (for IES, GATE, PSU) 418 • Previous 20-Years GATE Questions 422 • Previous 20-Years IES Questions 423 • Previous 20-Years IAS Questions 428 • Answers with Explanation (GATE, IES, IAS) 432 • Conventional Questions with Answers 438 Chapter-13 Theories of Column 442 • Theory at a Glance (for IES, GATE, PSU) 442 • Previous 20-Years GATE Questions 449 • Previous 20-Years IES Questions 450 • Previous 20-Years IAS Questions 454 • Answers with Explanation (GATE, IES, IAS) 456 • Conventional Questions with Answers 459 Chapter-14 Strain Energy Method 469 • Theory at a Glance (for IES, GATE, PSU) 469 • Previous 20-Years GATE Questions 473 • Previous 20-Years IES Questions 474 • Previous 20-Years IAS Questions 476 • Answers with Explanation (GATE, IES, IAS) 478 • Conventional Questions with Answers 480 Chapter-15 Theories of Failure 482 • Theory at a Glance (for IES, GATE, PSU) 482 • Previous 20-Years GATE Questions 489 • Previous 20-Years IES Questions 490 • Previous 20-Years IAS Questions 493 • Answers with Explanation (GATE, IES, IAS) 495 • Conventional Questions with Answers 498 Chapter-16 Riveted and Welded Joint 504 • Theory at a Glance (for IES, GATE, PSU) 504 • Previous 20-Years GATE Questions 506 • Previous 20-Years IES Questions 508 • Previous 20-Years IAS Questions 510 • Answers with Explanation (GATE, IES, IAS) 511 • Conventional Questions with Answers 513 www.iesacademy.com Email: [email protected] Page - i i i ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290 1. Stress and Strain Theory at a Glance (for IES, GATE, PSU) 1.1 Stress (σ) When a material is subjected to an external force, a resisting force is set up within the component. The internal resistance force per unit area acting on a material or intensity of the forces distributed over a given section is called the stress at a point. • It uses original cross section area of the specimen and also known as engineering stress or conventional stress. P Therefore, σ= A • P is expressed in Newton (N) and A, original area, in square meters (m2), the stress σ will be expresses in N/ m2. This unit is called Pascal (Pa). • As Pascal is a small quantity, in practice, multiples of this unit is used. 1 kPa = 103 Pa = 103 N/ m2 (kPa = Kilo Pascal) 1 MPa = 106 Pa = 106 N/ m2 = 1 N/mm2 (MPa = Mega Pascal) 1 GPa = 109 Pa = 109 N/ m2 (GPa = Giga Pascal) Let us take an example: A rod 10 mm ×10 mm cross-section is carrying an axial tensile load 10 kN. In this rod the tensile stress developed is given by P 10 kN 10×103N (σ )= = = =100N/mm2 =100 MPa t A (10mm×10mm) 100mm2 • The resultant of the internal forces for an axially loaded member is normal to a section cut perpendicular to the member axis. • The force intensity on the shown section is defined as the normal stress. ΔF P σ= lim and σ = ΔA→0ΔA avg A • Tensile stress (σ) t If σ > 0 the stress is tensile. i.e. The fibres of the component tend to elongate due to the external force. A member subjected to an external force tensile P and tensile stress distribution due to the force is shown in the given figure. Stress and Strain India’s No 1 IES Academy Chapter-1 • Compressive stress (σ ) c If σ < 0 the stress is compressive. i.e. The fibres of the component tend to shorten due to the external force. A member subjected to an external compressive force P and compressive stress distribution due to the force is shown in the given figure. • Shear stress (τ ) When forces are transmitted from one part of a body to other, the stresses developed in a plane parallel to the applied force are the shear stress. Shear stress acts parallel to plane of interest. Forces P is applied transversely to the member AB as shown. The corresponding internal forces act in the plane of section C and are called shearing P forces. The corresponding average shear stress (τ)= Area 1.2 Strain (ε) The displacement per unit length (dimensionless) is known as strain. • Tensile strain (ε) t The elongation per unit length as shown in the figure is known as tensile strain. ε = ΔL/ L t o It is engineering strain or conventional strain. Here we divide the elongation to original length not actual length (L + ΔL) o Let us take an example: A rod 100 mm in original length. When we apply an axial tensile load 10 kN the final length of the rod after application of the load is 100.1 mm. So in this rod tensile strain is developed and is given by ΔL L−L 100.1mm−100mm 0.1mm (ε)= = o = = =0.001(Dimensionless)Tensile t L L 100mm 100mm o o www.iesacademy.com Email: [email protected] Page-2 ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290 Stress and Strain India’s No 1 IES Academy Chapter-1 • Compressive strain (ε ) c If the applied force is compressive then the reduction of length per unit length is known as compressive strain. It is negative. Then ε = (–ΔL)/ L c o Let us take an example: A rod 100 mm in original length. When we apply an axial compressive load 10 kN the final length of the rod after application of the load is 99 mm. So in this rod a compressive strain is developed and is given by ΔL L−L 99mm−100mm −1mm (ε )= = o = = =−0.01(Dimensionless)compressive c L L 100mm 100mm o o • Shear Strain (γ): When a force P is applied tangentially to the element shown. Its edge displaced to dotted δ line. Where is the lateral displacement of the upper face of the element relative to the lower face and L is the distance between these faces. Then the shear δ strain is (γ)= L Let us take an example: A block 100 mm × 100 mm base and 10 mm height. When we apply a tangential force 10 kN to the upper edge it is displaced 1 mm relative to lower face. Then the direct shear stress in the element 10kN 10×103N (τ)= = =1N/mm2 =1MPa 100mm×100mm 100mm×100mm 1mm And shear strain in the element (γ) = = =0.1 Dimensionless 10mm 1.3 True stress and True Strain The true stress is defined as the ratio of the load to the cross section area at any instant. load (σ )= =σ(1+ε) T Instantaneous area σ ε Where and is the engineering stress and engineering strain respectively. www.iesacademy.com Email: [email protected] Page-3 ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290 Stress and Strain India’s No 1 IES Academy Chapter-1 • True strain (εT)= ∫L dll = ln⎛⎜⎜⎜⎜⎝LL ⎞⎠⎟⎟⎟⎟= ln(1+ε)=ln⎛⎜⎜⎜⎜⎝AAo⎞⎠⎟⎟⎟⎟=2ln⎛⎝⎜⎜⎜⎜ddo⎞⎠⎟⎟⎟⎟ L o o or engineering strain (ε) =eεT -1 The volume of the specimen is assumed to be constant during plastic deformation. [∵ A L = AL] It is valid till the neck formation. o o • Comparison of engineering and the true stress-strain curves shown below: • The true stress-strain curve is also known as the flow curve. • True stress-strain curve gives a true indication of deformation characteristics because it is based on the instantaneous dimension of the specimen. • In engineering stress-strain curve, stress drops down after necking since it is based on the original area. • In true stress-strain curve, the stress however increases after necking since the cross- sectional area of the specimen decreases rapidly after necking. • The flow curve of many metals in the region of uniform plastic deformation can be expressed by the simple power law. σT = K(εT)n Where K is the strength coefficient n is the strain hardening exponent n = 0 perfectly plastic solid n = 1 elastic solid For most metals, 0.1< n < 0.5 • Relation between the ultimate tensile strength and true stress at maximum load P The ultimate tensile strength (σ )= max u A o P The true stress at maximum load (σ ) = max u T A www.iesacademy.com Email: [email protected] Page-4 ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290 Stress and Strain India’s No 1 IES Academy Chapter-1 ⎛A ⎞ A And true strain at maximum load (ε)T =ln⎜⎜⎜⎜⎝ Ao⎠⎟⎟⎟⎟ or Ao =eεT P P A Eliminating Pmax we get , (σu)T = Amax = Amax× Ao =σueεT o Where P = maximum force and A = Original cross section area max o A = Instantaneous cross section area Let us take two examples: (I.) Only elongation no neck formation In the tension test of a rod shown initially it was A o = 50 mm2 and Lo = 100 mm. After the application of load it’s A = 40 mm2 and L = 125 mm. Determine the true strain using changes in both length and area. Answer: First of all we have to check that does the (If no neck formation member forms neck or not? For that check occurs both area and A L = AL or not? gauge length can be used o o Here 50 × 100 = 40 × 125 so no neck formation is for a strain calculation.) there. Therefore true strain L dl ⎛125⎞ (εT)=∫ l =ln⎜⎜⎜⎝100⎠⎟⎟⎟⎟=0.223 Lo (εT)=ln⎜⎜⎝⎜⎜⎛AAo⎠⎞⎟⎟⎟⎟=ln⎜⎝⎜⎜⎛5400⎠⎞⎟⎟⎟⎟=0.223 (II.) Elongation with neck formation A ductile material is tested such and necking occurs then the final gauge length is L = 140 mm and the final minimum cross sectional area is A = 35 mm2. Though the rod shown initially it was Ao = 50 mm2 and L = 100 mm. Determine the true strain using o changes in both length and area. Answer: First of all we have to check that does the (After necking, gauge member forms neck or not? For that check AL =AL length gives error but o o www.iesacademy.com Email: [email protected] Page-5 ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290 Stress and Strain India’s No 1 IES Academy Chapter-1 or not? area and diameter can Here AoLo = 50 × 100 = 5000 mm3 and AL=35 × 140 be used for the = 4200 mm3. So neck formation is there. Note here calculation of true strain A L > AL. at fracture and before o o Therefore true strain fracture also.) (εT)=ln⎜⎝⎜⎜⎛⎜AAo⎠⎞⎟⎟⎟⎟=ln⎝⎜⎜⎛⎜5305⎠⎞⎟⎟⎟⎟=0.357 L dl ⎛140⎞ But not (εT)=∫ l =ln⎜⎜⎜⎝100⎠⎟⎟⎟⎟=0.336 (it is wrong) Lo 1.4 Hook’s law According to Hook’s law the stress is directly proportional to strain i.e. normal stress (σ) α normal strain (ε) and shearing stress (τ)αshearing strain (γ ). σ = Eε and τ =Gγ The co-efficient E is called the modulus of elasticity i.e. its resistance to elastic strain. The co- efficient G is called the shear modulus of elasticity or modulus of rigidity. 1.5 Volumetric strain (ε ) v A relationship similar to that for length changes holds for three-dimensional (volume) change. For P volumetric strain (ε ), the relationship is (ε ) = (V-V )/V or (ε )= ΔV /V = v v 0 0 v 0 K • Where V is the final volume, V is the original volume, and ΔV is the volume change. 0 • Volumetric strain is a ratio of values with the same units, so it also is a dimensionless quantity. • ΔV/V= volumetric strain = ε +ε + ε = ε +ε + ε x y z 1 2 3 • Dilation: The hydrostatic component of the total stress contributes to deformation by changing the area (or volume, in three dimensions) of an object. Area or volume change is called dilation and is positive or negative, as the volume increases or decreases, p respectively. e= where p is pressure. K www.iesacademy.com Email: [email protected] Page-6 ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290 Stress and Strain India’s No 1 IES Academy Chapter-1 PL σ 1.6 Young’s modulus or Modulus of elasticity (E) = = Aδ ∈ τ PL 1.7 Modulus of rigidity or Shear modulus of elasticity (G) = = γ Aδ Δp Δp 1.8 Bulk Modulus or Volume modulus of elasticity (K) =− = Δv ΔR v R 1.10 Relationship between the elastic constants E, G, K, µ 9KG E = 2G(1+μ)= 3K(1−2μ)= [VIMP] 3K +G Where K = Bulk Modulus, μ= Poisson’s Ratio, E= Young’s modulus, G= Modulus of rigidity • For a linearly elastic, isotropic and homogeneous material, the number of elastic constants required to relate stress and strain is two. i.e. any two of the four must be known. • If the material is non-isotropic (i.e. anisotropic), then the elastic modulii will vary with additional stresses appearing since there is a coupling between shear stresses and normal stresses for an anisotropic material. Let us take an example: The modulus of elasticity and rigidity of a material are 200 GPa and 80 GPa, respectively. Find all other elastic modulus. 9KG Answer: Using the relation E =2G(1+μ)=3K(1−2μ)= we may find all other elastic 3K +G modulus easily E E 200 Poisson’s Ratio (μ): 1+μ= ⇒μ= −1= −1=0.25 2G 2G 2×80 E E 200 Bulk Modulus (K) : 3K = ⇒K = = =133.33GPa 1−2μ 3(1−2μ) 3(1−2×0.25) 1.11 Poisson’s Ratio (µ) Transverse strainor lateral strain −∈ = = y Longitudinal strain ∈ x (Under unidirectional stress in x-direction) www.iesacademy.com Email: [email protected] Page-7 ________________________________________________________________________ 25, 1st Floor, Jia Sarai, Near IIT. New Delhi-16 Ph: 011-26537570, 9810958290