Strength of Materials

Strength of Materials
[For Engineering Degree, Diploma and A.M.I E. Students]
By
S. Ramamrutham
B.E., (Civilf M.l.C.E.
Principal, Modern College of Engineering, New Delhi
Author of Design of Reinforced Concrete Structures, Design of Steel Structutes,
Theory of Structures, Applied Mechanics etc.CONTENTS
Chapter * Pages

1. Simple Stresses and Strains *
   Introduction – Definitions, stress, strain, tensile and compres¬
   sive stresses – Sheaf stress – Plastic limit – Hooke’s law
   poisson’s ratio – Modulus of Elasticity – Modulus of Rigidity,
   Bulk Modulus – Bars of varying section – Extension of a
   tapering rod – Composite section -modular ratio- Bar of
   uniform strength – Equivalent area of composite sections –
   Temjierature stresses – Hoop stress-Stresses on oblique
   sections – State of simple shear – Relation between the Elastic
   constants – Volumetric Strain – Rectangular block subject to
   normal stresses – Diagonal tensile and diagonal compressive
   stresses – Solved problems I to 71 – Problems for exercise.
   1 – 100
2. Strain Energy – Impact Leading
   Strain Energy – Elastic, plastic and rigid members – Stresses
   due to different types of axial loading – Gradually applied
   loads – Suddenly applied load) – Impact loads – Solved pro¬
   blems 72 to 84 – Problems for exercise. 101 – 118
   3- Centre of Gravity and Moment of Inertia
   Centre of Gravity – Definition – Lamina – Moment of an
   area – Centroid of a uniform lamina – Centroids of laminae of
   various shapes – Triangle, circle, semicircle, trapeziumBuilt-in sections – Analytical and graphical methods – Moment
   of Inertia of a lamina – Definition – Parallel axes theoremPerpendicular axes theorem – Moment of Inertia of laminae of
   different shapes- Rectangalar, ciicular, triangular and com¬
   posite sections – Solved problems 85 to 104 – Problems for
   exercise. 119 – 157
3. Shear Forces and Bending Moments
   Definitions -Cantilevers, simply supported beam, fixed beam,
   continuous beams -Conception of Shear Force and Bending
   Moment – Sign conventions – shear force and Bending Mo¬
   ment diagrams for cantilevers, beams supported at ends.
   Beams with overhangs – Point of contraffexure-Member sub¬
   jected to couples – Members subjected to Oblique loading –
   Miscellaneous types of members and corresponding S.F.
   and B.M. diagrams – Inter-relation between S F. and B. M^
   diagrams – To obtain the B.M. diagrams from S.F. dia¬
   gram-Solved problems 105 to 130 -Problems for exercise.
   158-227
4. Stresses in Beams
   Definition – Pure or simple bending – Theory of simple bend¬
   ing-Netural layer – Neutral axis – Bending Stress distribu¬
   tion-moment of resistance – Assumptions in the theory ofan
   Chapter Pages
   simple binding – Practical application of bending equation
   Section modulus – Section moduli for different shapesRectangular, triangular, circular, I-section, T-section – Normal
   force on a partial area of a beam section – Moment of resis¬
   tance of a partial area of a btam section -Flitched beams –
   Equivalent section – Beams of uniform strength – Shear stress
   distribution on a beam section – Shear stress distribution on
   rectangular, circular, triangular, 1 and T sections – Shear
   stresses in bolts connecting components in laminated beams.
   Proportion of B M and S F. resisted by flange and web
   of anI section – Shrar centre – Solved problems 131 to 199 –
   Problems for exercise. 228 – 337
5. Direct and Bending Stresses
   Stress distribution of the section of an eccentrically loaded
   rectangular column. The middle third rule – Core or kernel of
   a section – Circular section – Hollow section – Structural
   section – Walls and pillars – Solved Problems 200 to 223 –
   Problems for exercise. 338 – 370
6. Masonary Dams
   Forces acting on a dam – Stress distribution on the base of a
   dam, Stability of a dam – Minimum bottom width of a dam
   section. Solved Problems 224 to 228. 371 – 388
7. Deflection of beams
   Member bending into a circular are – slope, deflection and
   ladius of curvature – Derivation of formulae for slope and
   deflection – Cantilever – Propped cantilevers- Beams – Macau¬
   lay’s Method – Beams subjected to couples – Moment area
   method – Mohr’s theorems – Relations between maximum
   bending stress and maximum deflection – Beams of varying
   . section – Strain energy stored due to bending – Law of reci¬
   procal deflections -Bette’s law – The first theorem of
   Castigliano – Impact loading on beams – Laminated SpringsConjugate beam method – Solved problems 229 to 312 Pro^
   blems for exercise. 389 – 527
8. Fixed and Continuous Beams
   Fixed beam -Relation between the free B.M. diagram and
   the fixed B.M. diagram-slope and deflection – Effect of sinking
   of supports – Fixed beam subjected to couple – Degree of
   fixing – Advantages and disadvantages of fixing beams –
   Continuous beam – Clapyron’s theorm of three moments –
   Solved problems 3)3 to 324 – Problems for exercise. 528 – 581
9. Torsion of Shafts
   Pure Torsion – Theory of Pure Torsion – Torsional mement of
   resistance. Assumptions in the theory of pure Torsion – Polar
   modulus – H.P. transmitted by a shaft – Torsional Rigidity –
   Stepped shafts – Composite shafts – Keys – Couplings – Shear
   and Torsional resilience – Shaftsof non-circular section – Close(fit)
   coiled helical springs – Torsion of a tapering rod. Solved
   Problems 325 to 361 – Problems for exercise. ’582 – 632
10. Principal stresses and strains
    Normal stresses – Tangential or shear streses – Principal stresses

• Principal planes – Graphical and analytical methods – Ellipse
  of stress – Determination of principal stresses and strains –
  Obliquity – Mohr’s circle of stress – Combined bending and
  Torsion – Strain energy in terms of principal stresses –
  Equivalent bending moment and equivalent torque – Principal
  strains – Criterion for failure – Ellipse of strain – Solved prob¬
  lems 362 to 388. Problems for exercise. 633 – 682
  12- Thin Cylinders and Spheres
  Thin cylinders – circumferential ond longitudinal Stresses –
  Riveted cylinderical boilers – Wire bound pipes. Thin spherical
  shells – Biaxial stresses in doubly curved walls of pressure
  vessels – Stresses in a conical tank. Solved problems 342 to

1. Problems for exercise. 683 – 705
2. Thick cylinders and Spheres
   Thick cylinders -Derivation of formulae – Lamme’s equations

• Hoop stresses and radial pressure distribution – compound
  cylinders – Thick spherical Shells- Solved problems 350 to
  413 – Problems for exercise. 706 – 726

1. Columns and Struts
   Introduction – Axially loaded compression members – Crush¬
   ing load – Buckling or critical load or crippling load – Euler’s
   theory of long columns – Different end conditions – Effective
   length of colums – Assumptions made in Euler’s theoryLimitations of Euler’s formula – Empirical formulae –
   Rankine’s formula – Straight line formula – Johnson’s para¬
   bolic formula – Formula given by the I S. code – Column
   •objected to eccentric loading – Euler’s method – Rankine’s
   method – Prof Perry’s formula – Columns with initial curva¬
   ture – Laterally loaded struts – Solved problems 413 to 429.
   Problems for exercise. 727 – 768
2. Riveted Joints
   Types of joints – Lap and butt joints – Failure of a riveted
   joint – Tearing strength, shearing strength, bearing strength –
   Efficiency of a joint – Riveted joints in structural steel work –
   Chain riveting and diamond riveting – Eccentric Riveted
   connections – Resistance of a rivet against translation and
   rotation. Solved problems 430 to 442. Problems for
   exercise. 769 – 798
3. Welded Connections
   The welding process – Advantages of welded connection –
   Disadvantages of welded connection – Types of weld – Mini¬
   mum sizes of weld – Effective length – Minimum length – Fillet(iv)
   weld applied tp the edge of a plate-Angle between fusion
   faces – Throat thickness-Intermittent fillet welds – Lap joints

• Fillet welds in slots or holes – End returns – Bending about
  a single fillet -Permissible stresses in welds – Combined
  stresses in welds -Eccentric welded connections, Solved
  problems 443 to- 462 799 – 830

1. Analysis of Framed Structures
   Perfect frame -Deficient frame – Redundant frame-Reactions
   at supports – Analysis of a truss – Method of joints-Method
   of section – Graphical method. Solved problems 463 to 489.
   831-914
2. Simple Mechanical Properties of Metals
   Yield or flow of material – Tensile stress – Stress – Strain
   diagrams for Mild Steel Specimen – Limit of proportionality –
   Ultimate stress-Working stress – Factor of safety – Measure¬
   ment of ductility -Unwin’s Method based on reduction of
   sectional area – Hardness – Scratch test – Indentation test –
   Brinnel’s method-Impact testing – Fatigue of metals –
   Endurance limit. Solved problems 449 to 491. 915 – 922
3. Elements of reinforced Concrete
   General principles of design – Assumptions – Singly reinforced
   beams-Netural axis -Lever arm – Moment of resistanceBalanced or economic or critical sections – Unbalanced
   sections – Under-reinforced and over-reinforced sections –
   Doubly reinforced beams-Shear in beams- Shear stresses –
   Diagonal tensile and diagonal compressive stresses in concrete-Stirrups-Diagonal reinforcement – Bond stresses –
   End anchora.e -Standard hook – Reinforcement-T and L
   beams-Axially Loaded Columns-Combined bending and
   direct stresses. Solved problems 492 to 520. Problems for
   exercises. 923 – 998
   Appendix Useful tables. 999-1035
   Index 1036 – 1038INDEX
   A
   Analysis of frames, 831
   Analysis of dams. 371
   Assumptionsin theory of bending, 228
   Axis, neutral, 229, 230 ‘
   B
   Balanced section, 928
   Bar of composite section, 27
   Bar of uniform stren th, 47
   Bar of varying section, 11
   Beams, 158
   Beams-deflection, 389
   Beams of uniform strength, 295
   Beams of varying section, 469
   Bearing value of rivets, 773
   Bending, 228
   Bending moment, >58, 160
   Beltrami theory, 678
   Bending stress, 228
   Bet’e’s law, 484
   Bond stress, 966, 967
   Boom, 727
   Bow’s notation, 891
   Brinnel’s method. 921
   Buckling load, 72k
   Built-m-beams, 158, 528
   Bulk m< dulus, 86 Butt-joint, 769 Butt weld, 801 C Cantilevers, 158 Cantilever-propped, 400 Carriage springs, 5<)0 Castigliano’s theorem, 485 Centre of gravity, 119 Centroid, t <9 Cham riveting, 787 Circum’erential stress, 72. 684 Clapeyron’s theorem, 561 Clear span, 158 Close coiled springs, 620 Columns, 727 Combined stresses, 338 Complementary shear stress, 81 Composite sections, 27 Composite shafts, 610 Compound cylinders, 713 Compoon stresses and strains, 633 Compressive siress, 31 Compressive strain, 31 Cbmpound section,27 Conjugate beam method, 505 Continuous beam, 158, 528, 560 Continuous columns, 987 Contraflexure-point of, 186 Core of a section, 342 Couples, 203 Couplings, 616 Cover, 973 CripplingI ad, 728 Critical load, 728 Critical section, 928 Crushing load, 727 Crushing s’ress, 727 D Dams, 371 Deficient frame, 832 Deflection of beams, 381 Deformation, 1 Diagonal compres ion, 84 Diagonal tension, 84 Diamond riveting, 787 Direct stress, 338 Direct and bending stresses, 338 Direct bond, 967 Doubly curved walls, 701 Doubly reinforced beams, 949 Ductility, 917 E Eccentricity, 339 Eccentric riveted connection, 791 Eccentric welded connections, 815 Economic section, 928 , Effective length of column, 735 Effective length of weld, 802 Effective span. 158 Efficiency of joints, 774 Elastic instability, 728 Elasic limit, 6 Elastic material, 1, 102 Elastic modulus, 102 Elapse of strain, 679 Ellipse of stress, 642 Encastred beams, 528 End Anchorage, 969 End returns, 804 Endurance limit, 922 Energy, strain, 47, 101, 102 Equivalent ana, 38 1036INDEX 1037 Euler’s theory, 728 F Factor of safety, 916 Failure of a riveted joint, 770 Fatigue of metals, 922 Fillet weld, 800 Fixed beam, 158, 52* Fixed ei d moment, 528 Flexural rigidity, 586 Flitched beams, 278 1rained structures, 831 Freely supported beams, 158 French formula, 774 G German formula, 774 Gradually applied loads, 103 Graphical methods, 838 Graphic statics, 891 Guest theory, <>77
   Gusset plate, 785
   Gyration-radius of, 137
   H
   Haigh’s theory, 678
   Hardness, 920
   Helical springs 620
   Hogging moment, 161
   Homogeneity, 23.’
   Hook’s law, 6
   Hoop stress, 72, 684
   I
   Impact loading, 101
   Impact testing. 101, 105, 498, 921
   I ‘ entation test, 921
   Inertia-moment of, 136
   1< lens ty of stress, 2 Isotropy, 233 Izod test, 921, 922 J Johnson’s parabolic formula, 749 Joints-riveted, 769 Joiots-weided. 799 K Kernel of a Section, 342 Keys, 616 L L-boams, 974 Lame’s equatioa, 70* Laminated spriaM 900 Lap joint, 769. <04 Lateral strain, 74 Laterally loaded strata, 743 Leaf springs, *00 Lever arm, 927 Limit-elastic, 6 Load, I Loading-gradual, sudden, impact, 101 Local bond, 969 Long columns, 986 Longitudinal stress, 684 M Macaulay’s method, 423 Masonry dams, 371 Maxwell’s law, 483 Mechanical properties of metals, 915 Membrane stresses 701 Meridional stress, 702 Method of joints, 838 Method of resolution, 838 Method of section, 838 864 Method of substitution, 903 Middle third rule, 341 Modular rati >, 27, 279,923
   Modulus bulk, 86
   Modulus of elasticity, 6
   Modulus of rigidity, 6
   Modulus ol section. 2M
   Mohr’s Circle, 645
   Mohr’s theorems, 459, 460
   Moment area method, 459
   Mou. nt of Inertia, 136, 928
   MorneM of resistance, 231
   N
   Neutral axis. 229, 230, 926
   Neutral layer, 229
   Neutral suifa^e, 229
   Normal stresses, 83
   O
   Oblique loading on beams, 210
   Obliquity, 6 5
   Over reinforced sectian, 951
   P
   Parabolic formula, 749
   Parallel axes the rem, 138
   Perfect frame, 831
   Perpendicular au« theorem, 137
   P-rry’s f. rmula, 758
   PUUra, 364
   Plastic member, 1, 102
   Poet of contrafcxure, 186
   Poiet of laflextion. 86
   Pvkacw’a ami, 741038 Polar modulus, ^84 Port, 727 . Principal planes, 633, 634 Principal strains, 613. 675 Principal stresses, 633, 634 Propp,d cantilevers, 400 Pure bending, 228 R Radius of curvature, 390 Radius of gyration, 137 Rankine’s formula, 740 Rankine’s theory, 677 Reinforced < oncrete, 923 Reciprocal deflection theorem, 482 Reinforcement, 970 Redundant frame, 833 Relation between elastic constants, Resistance, I, 101 Resistance-moment of, 231 Rigid material, ], 102 Rigidity modulus 6 Riveted boilers, 693 Riveted joints, 769 Roller support, 833 Safety-factor of, 916 Sagging moment, 160 Scratch test, 921 Second moment of area, 136 Section-composite, 27 Section-method of, 833, 864 Section modulus, 234 Shafts, 582 Shear centre, 330 Shear delormalion, 5, 6 Shear force, 158 Shear mo ulus, 6 Sheaf reinforcement, 944 Shear resistance, 618 Shear strain, 4 Shear stress, 4, 297, 954, 957 Shear stress distribution, Shear value of rivets, 773 Shells, 683 Simple bending. 228 Simple stresses and strains, 1 Simply supported beams, 158 Singly reinforced beams, 924 Sieucer ess ratio, 736
   Slope, 390
   Span, 158
   Spherical shells. 698
   Spnngs-coiled, 620
   Spr ngs-lamrnated, 500
   Stability of dams, 373
   Stannard book, 969
   Stare of simple shear, 81
   Strffnem.502 621
   STRENGTH OF MATERIALS
   Stirrups, 960
   Straight line formula, 749
   Strain, 1, 2, 3
   Strain energy, 47, 101, 102
   Strength, i
   Stress, 1, 2, 3
   Stress intensity, 2
   Stresses in beams, 228
   Struts, 727
   St. Venant’s theory, 677
   Subs itution method, 903
   Suddenly applied loads, 104
   T
   T-beams, 973
   Tapering shafts, 628
   Tearing strength, 771
   Temperature stresses, 55
   Tensile strain, 3
   Tensile stress, 3
   Theorem of three moments, 561
   The ry of bending. 228
   Thick cylindeis, 706
   Thick sph-res, 7?t
   Thin cylinders, 683
   Thin spheres, 698
   Thr at of weld, 800, 802
   Thrust diagram, 11,212
   Torsion of shafts, 582
   Torsion of non circular section, 619-
   Torsional resilience, 618
   Torsional rigidity, 586
   Twist of a shaft, 583
   U
   U-Butt weld, 801
   Ultimate stress, 9 6
   Unbalanced setion, 931
   Under reinforced section 931
   Unit stress, 2
   Unwin’s formula, 774
   V
   V-Butt weld, 801
   Volumetric strain, 7*
   W
   Walls, 364
   Welded joints, 799
   Wind pressure, 364
   Wire bound pipes, 695
   Wohler’s experiments, 922.
   Work, 101..481
   Working stresses, 916
   Y
   Yield point, 915
   Young’s modulus, 6

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