Prestressed Concrete Structures
- 2. A hello! I am Abhishek I am here to give you detail knowledge about pre-stressed concrete structures as per latest codal provisions IS 1343 2012. You can find me at abhishek.sharma98729@gmail.com
- 3. 3 Books & references: 1. Design of Pre-stressed Concrete by Raymond Ian Gilbert, Neil Colin Mickleborough, Gianluca Ranzi 2. Pre-stressed Concrete Building, Design, and Construction by Charles W. Dolan, H. R. (Trey) Hamilton 3. Design of Pre-stressed Concrete Structures by T. Y. Lin, Ned H. Burns 4. Pre-stressed Concrete by N. KRISHNA RAJU, 6TH EDITION A
- 4. 4 5. NPTEL COURSE MATERIAL, PRE-STRESSED CONCRETE STRUCTURES, DR. AMLAN K. SENGUPTA AND PROF. DEVDAS MENON 6. IS 1343 2012, IS 456 : 2000, IS 1343 : 1980 7. MECHANICS OF MATERIALS, R.C HIBBELER A
- 5. A 1 MODULE 1 a) Introduction b) Advantages and types of pre-stressing c) Pre-stressing systems d) Materials for pre-stressing E) PREREQUISITE OF SOM
- 6. 6 A) MAIN TOPIC a1) SUBTOPIC A
- 9. 9 Most concrete construction in the world is cast-in-place reinforced concrete. In R.C.C, steel reinforcement is placed into the concrete to provide the tensile resistance to flexural loads or to assist the concrete in carrying compressive loads. A
- 10. 10 A cracked reinforced concrete beam. (a) Elevation and section. (b) Free-body diagram, stress distribution and resultant forces C and T. A
- 11. 11 Practical reinforced concrete beams are usually cracked under day-to-day service loads. On a cracked section, the applied bending moment Mext Is resisted by = compression in the concrete above the crack = tension in the bonded reinforcing steel crossing the crack A
- 12. 12 steel reinforcement provides the cracked beam with flexural strength, but it does not prevent cracking, and it does not prevent a loss of stiffness when cracking occurs. Crack widths are approximately proportional to the strain, and hence stress, in the reinforcement. ‘‘Cracking cause lack in stiffness, and hence deflection increases’ ’ A
- 13. 13 SECTION IS NOT SERVICEABLE ANYMORE, WHEN DEFLECTION INCREASES THE `PERMISSIBLE VALUES For prismatic beams of rectangular sections and slabs of uniform thicknesses and spans up to 10 m, the limiting l/d ratios are specified by the Code (Cl. 23.2.1) of IS 456:2000 as: and the modification factors kt (which varies with pt and fst) and kc (which varies with pc) are as given in Fig. 4 and Fig. 5 of the Code IS 456: 2000 A
- 15. 15 Pre-stressed concrete is a particular form of reinforced concrete. Pre-stressing involves the application of an initial compressive load to the structure to reduce or eliminate the internal tensile forces and, thereby, control or eliminate cracking. A
- 17. 17 ‘‘The initial compressive load is imposed and sustained by highly tensioned steel reinforcement (tendons) reacting on the concrete’’ With cracking reduced or eliminated, prestress sections have following major advantages 1) a pre-stressed concrete section is considerably stiffer than the equivalent (usually cracked) reinforced concrete sections 2) Pre-stressing may also impose internal forces that are of opposite sign to the external loads and may, therefore, significantly reduce or even eliminate deflection A
- 18. 18 Now Let ’s See the behavior of a R.C.C and a PRESTRESSED Beam Under External Loads. R.C.C BEAM A
- 19. 19 A
- 21. 21 Behavior comparison of non pre-stressed and pre-stressed concrete beams A
- 22. 22 A3) NEED FOR HIGH STRENGTH MATERIALS A
- 23. 23 In the early applications, the strength of the mild steel and the strain during pre-stressing were less. The residual strain and hence, the residual pre-stress was only about 10% of the initial value. The following sketches explain the phenomena A
- 24. 24 The residual strain in steel = original tensile strain in steel – compressive strains corresponding to short-term and long-term losses Original tensile strain in steel = (L2 – L1)/L1 Compressive strain due to elastic shortening of beam (short term loss) = (L2 – L3)/L1 Compressive strain due to creep and shrinkage (long term loss) = (L3 – L4)/L1 Therefore, residual strain in steel = (L4 – L1)/L1 A
- 25. 25 The maximum original tensile strain in mild steel = Allowable stress / elastic modulus = 140 MPa / 2×105 MPa = 0.0007 The total loss in strain due to elastic shortening, creep and shrinkage was also close to 0.0007. Thus, the residual strain was negligible. The solution to increase the residual strain and the effective pre-stress are as follows. • Adopt high strength steel with much higher original strain. This leads to the scope of high pre-stressing force. • Adopt high strength concrete to withstand the high pre-stressing force A
- 26. 26 The normal loss of stress in steel is generally about 100 N/MM2 to 240 N/MM2 and it is apparent that if this loss of stress is to be a small portion of the initial stress, the stress in steel in the initial stages Must be very high. about 1200 to 2000 N/MM2. These high stress ranges are possible only with the use of high- strength steel The early attempts to use mild steel in pre- stressed concrete were not successful, as a working stress of 120 N/MM2 in mild steel is more or less completely lost due to elastic deformation, creep and shrinkage of concrete. ***NEED OF HIGH STRENGTH STEEL*** A
- 27. 27 ***NEED OF HIGH STRENGTH CONCRETE*** IT offers high resistance in tension. shear, bond and bearing. in the zone of anchorages, The bearing stresses being higher, high- strength concrete is invariably preferred to minimize COSTS arising due to anchorage reinforcements. High strength concrete is less liable to shrinkage cracks. And has a higher modulus of elasticity, smaller ultimate creep strains, resulting in smaller loss of pre-stress in steel The use of high-strength concrete results in a reduction in the cross-sectional dimensions of pre-stressed concrete structural elements. With a reduced dead-weight of the material, longer spans become technically and economically practicable.
- 29. SR. NO NAME OF SCIENTIST & PLACE CONTRIBUTION YEAR 1 Aspdin, J., (England) Patent for the manufacture of Portland cement 1824 2 Monier, J., (France) Introduced steel wires in concrete 1857 3 Jackson, P. H., (USA) Concept of tightening steel tie rods in arches 1886 4 Doehring, C. E. W., (Germany) Mfg. concrete slabs & small beams with tensile steel. 1888 5 Stainer, C. R., (USA) Recognised losses due to shrinkage and creep 1908 6 Emperger, F., (Austria) method of winding and pre- tensioning high tensile steel wires around concrete pipes 1923 7 Hewett, W. H., (USA) hoop-stressed horizontal reinforcement using turnback 1924 8 Dill, R. H., (USA) Used high strength unbonded steel rods in Post tension 1925 9 Eugene Freyssinet (France) Used high tensile steel wires, with ultimate strength as high as 1725 MPa , Father of Pre-stressed concrete 1926 10 Hoyer, E., (Germany) Developed ‘long line’ pre-tensioning method 1938 11 Magnel, G., (Belgium) anchoring system for post-tensioning, using flat wedges 1940
- 30. 30 B) ADVANTAGES AND TYPES OF PRE-STRESSING A
- 31. 31 B1) ADVANTAGES OF PRE-STRESSED CONCRETE A
- 32. 32 Section remains uncracked under service loads 👉Reduction of steel corrosion • Increase in durability. 👉Full section is utilised • Higher moment of inertia (higher stiffness) • Less deformations (improved serviceability) • Increase in shear capacity. • Suitable for use in pressure vessels, liquid retaining structures • Improved performance (resilience) under dynamic and fatigue loading A
- 33. 33 High span-to-depth ratios Non-prestressed slab 28:1 Prestressed slab 45:1 Typical values of span-to-depth ratios in slabs are given below 👉 For the same span, less depth compared to RC member. • Reduction in self weight • More aesthetic appeal due to slender sections • More economical sections. A
- 34. 34 Suitable for precast construction • Better quality control • Reduced maintenance • Suitable for repetitive construction • Multiple use of formwork ⇒ Reduction of formwork • Availability of standard shapes. 👉 Rapid construction A
- 36. 36 B2) LIMITATIONS OF PRE-STRESSED CONCRETE A
- 37. 37 • Pre-stressing needs skilled technology. Hence, it is not as common as reinforced concrete. • The use of high strength materials is costly • There is additional cost in auxiliary equipments. • There is need for quality control and inspection A
- 38. 38 B3) TYPES OF PRE-STRESSING A
- 39. 39 This classification is based on the method by which the prestressing force is generated. a_)Source of pre-stressing force There are four sources of prestressing force *Mechanical, *hydraulic, *electrical and *chemical A
- 40. 40 b)External or internal pre-stressing c)Pre-tensioning or post-tensioning d)Linear or circular pre-stressing e)Full, limited or partial pre- stressing(TYPE 1, TYPE 2, TYPE 3) f)Uni-axial, biaxial or multi-axial pre-stressing A
- 41. 41 This is the simplest type of pre-stressing, producing large pre-stressing forces. The hydraulic jack used for the tensioning of tendons, comprises of calibrated pressure gauges which directly indicate the magnitude of force developed during the tensioning. a_)Source of pre-stressing force *hydraulic, In this type of pre-stressing, the devices includes weights with or without lever transmission, geared transmission in conjunction with pulley blocks, screw jacks with or without gear drives and wire-winding machines. This type of pre-stressing is adopted for mass scale production. *Mechanical, A
- 42. 42 In this type of pre-stressing, the steel wires are electrically heated and anchored before placing concrete in the moulds. This type of pre-stressing is also known as thermoelectric pre-stressing. *electrical In the method of chemical pre-stressing or self-stressing, special types of cements, called 'Expanding cements', are used in concrete. A typical expanding cement may consist of around 75% Portland cement, 15% High-alumina cement and 10% gypsum. Using this cement, a linear expansion of around 3-4% may be achieved in concrete. When concrete expands, the tendons also expand in length and tensile pre-stresses are introduced in tendons. These tensile stresses are transferred to concrete in the form of compressive stresses. *chemical A
- 43. 43 *hydraulic pre-stressing *chemical pre- stressing A
- 44. 44 b)External or internal pre-stressing When the pre-stressing is achieved by elements located outside the concrete, it is called external pre-stressing. The tendons can lie outside the member (for example in I-girders or walls) or inside the hollow space of a box girder External pre-stressing A
- 45. 45 When the pre-stressing is achieved by elements located inside the concrete member (commonly, by embedded tendons), it is called internal pre-stressing. Most of the applications of pre-stressing are internal Pre-stressing internal pre-stressing A
- 46. 46 c)Pre-tensioning or post-tensioning pre-tensioning: the tension is applied to the tendons before casting of the concrete. the pre- compression is transmitted from steel to concrete through bond over the transmission length near the ends Post-tensioning: The tension is applied to the tendons (located in a duct) after hardening of the concrete. The pre-compression is transmitted from steel to concrete by the anchorage device (at the end blocks) A
- 49. 49 Linear Pre-stressing When the pre-stressed members are straight or flat, in the direction of pre-stressing, the pre- stressing is called linear pre-stressing. For example, pre- stressing of beams, piles, poles and slabs d)Linear or circular pre-stressing Circular Pre-stressing When the pre-stressed members are curved, in the direction of pre-stressing, the pre- stressing is called circular pre-stressing. For example, circumferential pre-stressing of tanks, silos, pipes and similar structures. A
- 50. 50 Linear Pre-stressed Sleepers Circular pre-stressed containment structure A
- 51. 51 e)Full, limited or partial pre-stressing (TYPE 1, TYPE 2, TYPE 3) CLAUSE 20.3.2 OF IS 1343:2012 ***Full Pre-stressing: When the level of pre-stressing is such that no tensile stress is allowed in concrete under service loads, it is called Full Pre-stressing (Type 1, as per IS:1343 - 2012) ***Limited Pre-stressing: When the level of pre-stressing is such that the tensile stress under service loads is within the cracking stress of concrete, it is called Limited Pre-stressing (Type 2). ***Partial Pre-stressing: When the level of pre-stressing is such that under tensile stresses due to service loads, the crack width is within the allowable limit, it is called Partial Pre- stressing (Type 3) A
- 52. 52 f)Uni-axial, biaxial or multi-axial pre-stressing Uniaxial Pre-stressing When the pre-stressing tendons are parallel to one axis, it is called Uniaxial Pre- stressing. For example, longitudinal pre-stressing of beams Biaxial Pre-stressing When there are pre-stressing tendons parallel to two axes, it is called Biaxial Pre- stressing. Multiaxial Pre-stressing When the pre-stressing tendons are parallel to more than two axes, it is called Multiaxial Pre-stressing. For example, pre-stressing of domes A
- 54. 54 C1. Stages of Loading A
- 55. 55 The analysis of pre-stressed members can be different for the different stages of loading. The stages of loading are as follows 1) Initial : It can be subdivided into two stages. a) During tensioning of steel b) At transfer of pre-stress to concrete 2) Intermediate : This includes the loads during transportation of the pre-stressed members. 3) Final : It can be subdivided into two stages. a) At service, during operation. b) At ultimate, during extreme events. A
- 56. 56 C2. Pre-tensioning Systems and Devices A
- 57. 57 In pre-tensioning, the tension is applied to the tendons before casting of the concrete. The stages of pre- tensioning are described next 👉 Stages of Pre-tensioning 1. In pre-tensioning system, the high-strength steel tendons are pulled between two end abutments (also called bulkheads) prior to the casting of concrete. The abutments are fixed at the ends of a pre-stressing bed A
- 58. 58 2. Once the concrete attains the desired strength for pre-stressing, the tendons are cut loose from the abutments 3. The pre-stress is transferred to the concrete from the tendons, due to the bond between them. 4. During the transfer of pre-stress, the member undergoes elastic shortening. 5. If the tendons are located eccentrically, the member is likely to bend and deflect (camber). A
- 60. 60 👉 Advantages of Pre-tensioning *Pre-tensioning is suitable for precast members produced in bulk. *In pre-tensioning large anchorage device is not present. A
- 61. 61 👉 Disadvantages of Pre-tensioning *A pre-stressing bed is required for the pre- tensioning operation. *There is a waiting period in the pre-stressing bed, before the concrete attains sufficient strength *There should be good bond between concrete and steel over the transmission length. A
- 62. 62 👉Devices *Pre-stressing bed *End abutments *Shuttering / mould *Jack *Anchoring device *Harping device (optional) A
- 63. 63 👉HOYER SYSTEM OR LONG LINE METHOD This system is generally used for mass production. The end abutments are kept sufficient distance apart, and several members are cast in a single line. LONG LINE METHOD A
- 64. 64 LONG LINE METHOD REQUIRES end abutments have to be sufficiently stiff and have good foundations. This is usually an expensive proposition, particularly when large pre-stressing forces are required. It is possible to avoid transmitting the heavy loads to foundations, by adopting self-equilibrating systems. This is a common solution in load-testing. Typically, this is done by means of a ‘tension frame’ SOLUTION A
- 65. 65 The jack and the specimen tend to push the end members. But the end members are kept in place by members under tension such as high strength steel rods. TENSION FRAME A
- 66. 66 The frame that is generally adopted in a pre- tensioning system is called a stress bench. The concrete mould is placed within the frame and the tendons are stretched and anchored on the booms of the frame. STRESS BENCH A
- 67. 67 FREE BODY DIAGRAM OF STRESS BENCH A
- 68. 68 STRESS BENCH AFTER CASTING CONCRETE A
- 69. 69 The load applied by a jack is measured by the pressure reading from a gauge attached to the oil inflow or by a separate load cell. 👉JACKS The jacks are used to apply tension to the tendons. Hydraulic jacks are commonly used. These jacks work on oil pressure generated by a pump. The principle behind the design of jacks is Pascal’s law. A
- 71. 71 Anchoring devices are often made on the wedge and friction principle. In pre-tensioned members, the tendons are to be held in tension during the casting and hardening of concrete. Here simple and cheap quick-release grips are generally adopted 👉anchoring devices A
- 72. 72 Chuck assembly for anchoring tendons (Reference: Lin, T. Y. and Burns, N. H., Design of Prestressed Concrete Structures) A
- 73. 73 👉harping devices The tendons are frequently bent, except in cases of slabs-on-grade, poles, piles etc. The tendons are bent (harped) in between the supports with a shallow sag Harping of tendons A
- 74. 74 Hold-down anchor for harping of tendons (Reference: Nawy, E. G., Prestressed Concrete: A Fundamental Approach) A
- 75. 75 C3. Post-tensioning Systems and Devices A
- 76. 76 *Unlike pre-tensioning, the tendons are pulled with the reaction acting against the hardened concrete In post-tensioning systems, *the ducts for the tendons (or strands) are placed along with the reinforcement before the casting of concrete. *The tendons are placed in the ducts after the casting of concrete. *The duct prevents contact between concrete and the tendons during the tensioning operation. A
- 77. 77 *If the ducts are filled with grout, then it is known as Bonded post-tensioning *In un-bonded post-tensioning, as the name suggests, the ducts are never grouted and the tendon is held in tension solely by the end anchorages A
- 78. 78 Post-tensioning procedure. (a) Concrete cast and cured. (b) Tendons stressed and prestress transferred. (c) Tendons anchored and subsequently grouted A
- 79. 79 6) Cutting of the tendons. 👉 The various stages of the post-tensioning operation are summarised as follows. 1) Casting of concrete 2) Placement of the tendons. 3) Placement of the anchorage block and jack 4) Applying tension to the tendons. 5) Seating of the wedges. A
- 80. 80 👉 Advantages of post tensioning • The transfer of pre-stress is independent of transmission length • Post-tensioning is suitable for heavy cast-in-place members. • The waiting period in the casting bed is less. A
- 81. 81 👉 disadvantages of post tensioning The relative disadvantage of post-tensioning as compared to pre-tensioning is the requirement of anchorage device and grouting equipment. A
- 82. 82 👉 devices 1) Casting bed 2) Mould/Shuttering 3) Ducts 4) Anchoring devices 5) Jacks 6) Couplers (optional) 7) Grouting equipment (optional) A
- 83. 83 4) Anchoring devices A strand chuck is a mechanical device to grip the strand while stressing and to hold the strand through contact with the bulkhead until the force is transferred to the concrete Wedge action The anchoring device based on wedge action consists of an anchorage block and wedges. The strands are held by frictional grip of the wedges in the anchorage block. Some examples of systems based on the wedge-action are Freyssinet, Gifford-Udall, Anderson and Magnel-Blaton anchorage A
- 84. 84 Strand chuck and components A
- 85. 85 Freyssinet “T” system anchorage cones (Reference: Lin, T. Y. and Burns, N. H., Design of Prestressed Concrete Structures)A
- 86. 86 Anchoring devices (Reference: Collins, M. P. and Mitchell, D., Prestressed Concrete Structures) A
- 87. 87 Anchoring devices (Reference: VSL International Ltd) A
- 88. 88 Anchoring with button heads (Reference: Collins, M. P. and Mitchell, D., Prestressed Concrete Structures) A
- 89. 89 Most flat-slab building construction use monostrand, single- strand unbonded tendons. Monostrand tendons consist of an anchor assembly and strand, which is encapsulated in an extruded plastic sheath filled with a corrosion protectant filler. Encapsulation provides protection against intrusion of moisture, provides corrosion protection for the strand, and reduces friction during stressing. monostrand, A
- 90. 90 Monostrand anchor and jack. (a) Monostrand anchor and wedges (courtesy of VSL). (b) Monostrand jack (courtesy of DYWIDAG-Systems International USA Inc.) A
- 91. 91 Multistrand anchor systems. (a) DYWIDAG-Systems International USA Inc.; (b) Freyssinet Inc A
- 92. 92 Multistrand anchor systems. (c) Schwager Davis, Inc.; (d) VSL A
- 93. 93 Where large post-tensioning forces are necessary, such as post-tensioned beams and bridge girders, multiple strand tendons are used. Figure shows a multistrand anchor where all strands are stressed simultaneously using a single hydraulic jack. multistrand anchor Bar Anchors Prestressing bars systems are typically equipped with threads and nuts to facilitate anchorage,. The jack threads onto the bar and reacts against the anchor plate during stressing. Once the target stress level has been reached, the nut is tightened with a wrench system on the jack. This minimizes the seating loss and allows for the use of this system in short tendons A
- 94. 94 Hydraulic jacks for post-tensioning multistrand tendons (courtesy of Freyssinet Inc. (left) and Schwager Davis, Inc. (right) A
- 96. 96 Prestressing bar hydraulic jack. Prestressing bar anchorages after stressing A
- 97. 97 d) Materials for pre-stressing A
- 99. 99 Concrete is a composite material composed of gravels or crushed stones (coarse aggregate), sand (fine aggregate) and hydrated cement (binder). It is expected that the student of this course is familiar with the basics of concrete technology. *water 7%. A typical mix used for pre-stressed concrete by weight might be: *coarse aggregate 45%, *fine aggregate 30%, *cement 18% and A
- 100. 100 👉 ***aggregate CLAUSE 5.3 OF IS 1343 2012 The coarse aggregate are granular materials obtained from rocks and crushed stones. They may be also obtained from synthetic material like slag, shale, fly ash and clay for use in light-weight concrete. All aggregates shall comply with the requirements of IS 383 A
- 101. 101 The important properties of aggregate are as follows. 1) Shape and texture 2) Size gradation 3) Moisture content 4) Specific gravity 5) Unit weight 6) Durability and absence of deleterious materials A
- 102. 102 Classification Description Examples Rounded Fully water worn or completely shaped by attrition River or seashore gravels; desert, seashore and wind blown sands Irregular or Partly rounded Naturally irregular or partly shaped by attrition, having rounded edges Pit sands and gravels; land or dug flints; cuboid rock Angular Possessing well-defined edges formed at the intersection of roughly planar faces Crushed rocks of all types; talus; screes Flaky Material, usually angular, of which the thickness is small relative to the width and/or length Laminated rocks Shape of Particle A
- 103. 103 Surface Characteristics of Aggregate A
- 104. 104 Limits of Deleterious Materials (IS: 383-1970) A
- 105. 105 Average Test Values For Rocks of Different Groups A
- 106. 106 : the maximum size of aggregate that can be used in any given condition may be limited by the following conditions (CLAUSE 5.3.2) IS 1343 2012 For most work, 20 mm aggregate is suitable. The nominal maximum coarse aggregate size is limited by the lowest of the following quantities. 1) 1/4 times the minimum thickness of the member 2) Spacing between the tendons/strands minus 5 mm 3) 40 mm. A
- 107. 107 👉D 2) Cement (CLAUSE 5.1 OF IS 1343 2012) A
- 108. 108 In present day concrete, cement is a mixture of lime stone and clay heated in a kiln to 1400 - 1600ºC. a) 33 grade ordinary Portland cement conforming to IS 269, b) 43 grade ordinary Portland cement conforming to IS 8112, c) 53 grade ordinary Portland cement conforming to IS 12269, d) Rapid-hardening Portland cement conforming to IS 8041, The cement used shall be any of the following, and the type selected should be appropriate for the intended use: A
- 109. 109 e) Portland slag cement conforming to IS 455, f) Portland pozzolana cement (fly ash based) conforming to IS 1489 (Part 1), g) Portland pozzolana cement (calcined clay based) conforming to IS 1489 (Part 2), h) Hydrophobic cement conforming to IS 8043, j) Low heat Portland cement conforming to IS 12600, and k) Sulphate resisting Portland cement conforming to IS 12330. A
- 110. 110 👉 WATER REQUIREMENT TABLE 1 OF IS 456:2000 A
- 111. 111 👉 Chemical admixtures shall be used according to 5.5 of IS 456. (as per clause 5.5 of is 1343 2012) IS:9103 - 1999, Concrete Admixtures – Specification The admixtures can be broadly divided into two types: 1)chemical admixtures and 2) mineral admixtures. A
- 112. 112 The common chemical admixtures are as follows. 1) Air-entraining admixtures 2) Water reducing admixtures 3) Set retarding admixtures 4) Set accelerating admixtures 5) Water reducing and set retarding admixtures 6) Water reducing and set accelerating admixtures A
- 113. 113 The common mineral admixtures are as follows. 1) Fly ash 2) Ground granulated blast-furnace slag 3) Silica fumes 4) Rice husk ash 5) MetakAoline A
- 114. 114 👉D 3) PROPERTIES OF HARDENED CONCRETE A
- 115. 115 The concrete in pre-stressed applications has to be of good quality. It requires the following attributes 1) High strength with low water-to-cement ratio 2) Durability with low permeability, minimum cement content and proper mixing, compaction and curing 3) Minimum shrinkage and creep by limiting the cement content A
- 116. 116 1) Strength of Concrete For pre-stressed concrete applications, high strength concrete is required for the following reasons 1) To sustain the high stresses at anchorage regions. 2) To have higher resistance in compression, tension, shear and bond. 3) To have higher stiffness for reduced deflection. 4) To have reduced shrinkage cracks A
- 117. 117 **Compressive Strength The compressive strength of concrete is given in terms of the characteristic compressive strength of 150 mm size cubes tested at 28 days (fck). The characteristic strength is defined as the strength of the concrete below which not more than 5% of the test results are expected to fall This concept assumes a normal distribution of the strengths of the samples of concrete A
- 118. 118 ratio of cylinder to cube strength varies from 0.8 for lower grades of m30 to 0.85 for higher grade of m90 A
- 119. 119 1 In the designation of concrete mix M refers to the mix and the number to the specified compressive strength of 150 mm size cube at 28 days, expressed in N/mm2. 2 For pre-tensioned prestressed concrete, the grade of concrete shall be not less than M 40. 3 Except where specifically mentioned otherwise, for concrete grade greater than M 60, design parameters given in the standard may not be applicable and the values may be obtained from specialized literatures and experimental results. NOTES: A
- 120. 120 The minimum grades of concrete for pre-stressed applications are as follows. • 30 MPa for post-tensioned members • 40 MPa for pre-tensioned members ****MINIMUM CEMENT CONTENT VARIES FROM 300 TO 360 KG/ CUM ***MAX. CEMENT CONTENT TO CONTROL EXCESSIVE SHRINKAGE IS 450 KG / CUM (Clause 8.2.4.2) 1343 2012 A
- 121. 121 1 Cement content prescribed in this table is irrespective of the grades and types of cement and it is inclusive of additionsmentioned in 5.2. The additions such as fly ash or ground granulated blast furnace slag may be taken into account in the concrete composition with respect to the cement content and water-cement ratio if the suitability is established and as long as the maximum amounts taken into account do not exceed the limit of pozzolana and slag specified in IS 1489 (Part 1) and IS 455 respectively. 2 The minimum cement content, maximum free water-cement ratio and minimum grade of concrete are individually related to exposure A
- 122. 122 PERMISSIBLE STRESSES IN PRESTRESSED CONCRETE PERMISSIBLE COMPRESSIVE STRESS UNDER FLEXURE (24.3.2.1) The following sketch shows the variation of allowable compressive stresses for different grades of concrete at transfer. The cube strength at transfer is denoted as fci. A
- 123. 123 Variation of allowable compressive stresses at service loads Zone I represents the locations where the compressive stresses are not likely to increase. Zone II represents the locations where the compressive stresses are likely to increase, such as due to transient loads from vehicles in bridge decks fci ≥ 0.5 fck A
- 124. 124 Allowable Compressive Stresses under Direct Compression (24.3.2.2) Except in the parts immediately behind the anchorages, the maximum stress in direct compression shall be limited to 0.8 times the permissible stress obtained from 24.3.2.1 . A
- 125. 125 **TENSILE Strength The tensile strength of concrete can be expressed as follows. 3) Direct tensile strength: It is measured by testing rectangular specimens under direct tension. 2) Splitting tensile strength: It is measured by testing cylinders under diametral compression 1) Flexural tensile strength: It is measured by testing beams under 2 point loading (also called 4 point loading including the reactions). A
- 126. 126 (CLAUSE 6.2.2) 1343 2012: The flexural and splitting tensile strength shall be obtained as per IS 516 and IS 5816 respectively. When the designer wishes to use an estimate of the flexural strength from the compressive strength, the following formula may be used: Here, fcr = flexural tensile strength in N/mm2 fck = characteristic compressive strength of cubes in N/mm2. A
- 127. 127 Splitting tensile strength is 2/3 of flexural tensile strength Splitting tensile strength is useful in design of Structural concrete members subjected to direct Tension. A
- 128. 128 The stiffness of concrete is required to estimate the deflection of members. The stiffness is given by the modulus of elasticity. For a non-linear stress (fc) versus strain (εc) behaviour of concrete the modulus can be initial, Tangential secant ***Stiffness of Concrete
- 129. IS:1343 - 2012 , recommends a secant modulus at a stress level of about 0.3fck. (clause 6.2.3.1) The modulus of elasticity for short term loading (neglecting the effect of creep) is given by the following equation Ec = short-term static modulus of elasticity, in N/mm2 fck = characteristic compressive strength of concrete, in N/mm2 ***The modulus is expressed in terms of the characteristic compressive strength and not the design compressive strength.
- 130. 130 Actual measured values may differ by ±20 percent from the values obtained from the above expression.
- 131. 131 The durability of concrete is defined as its ability to resist weathering action, chemical attack, abrasion, or any other process of deterioration. The common durability problems in concrete are as follows. ***durability (clause 8 of is 1343: 2012) The factors influencing durability include a) the environment; b) the cover to embedded steel c) the type and quality of constituent materials; d) the cement content and water-cement ratio of the concrete; e) workmanship, to obtain full compaction and efficient curing; and f) the shape and size of the member.
- 132. 132 CLAUSE 24.2 Limit State of Serviceability: Cracking Allowable Tensile Stresses under Flexure Type 1 No tensile stress Type 2 3 N/mm2. This value can be increased to 4.5 N/mm2 for temporary loads. Type 3 Table 10 of IS 1343 2012 provides hypothetical values of allowable tensile stresses.
- 133. 133A
- 134. 134 ****Stress-strain Curves for Concrete The stress versus strain behaviour of concrete under uniaxial compression is initially linear (stress is proportional to strain) and elastic (strain is recovered at unloading). With the generation of micro- cracks, the behaviour becomes nonlinear and inelastic. After the specimen reaches the peak stress, the resisting stress decreases with increase in strain. FIG. 3 OF IS 1343 2012 STRESS-STRAIN CURVE FOR CONCRETE A
- 135. 135 For concrete under compression due to axial load, the ultimate strain is restricted to 0.002. From the characteristic curve, the design curve is defined by multiplying the stress with a size factor of 0.67 and dividing the stress by a material safety factor of γm = 1.5. The design curve is used in the calculation of ultimate strength. A
- 136. 136 👉D 4) defrormation of concrete A
- 137. The deformation of a loaded concrete specimen is both instantaneous and time-dependent. If the load is sustained, the deformation of the specimen gradually increases with time and may eventually be several times larger than the instantaneous value. Concrete strain versus time for a specimen subjected to constant sustained stress. A
- 138. 138 The gradual development of strain with time is caused by creep and shrinkage. Creep strain is produced by sustained stress. Shrinkage is independent of stress and results primarily from the loss of water as the concrete dries and from chemical reactions in the hardened concrete A
- 140. 140 The magnitude of the instantaneous strain εce(t) caused by either compressive or tensile stress depends on the magnitude of the applied stress, the rate at which the stress is applied, the age of the concrete when the stress was applied and the stress-instantaneous strain relationship for the concrete. Consider the uniaxial instantaneous strain versus compressive stress curve shown in Figure Typical compressive stress-instantaneous strain curve. A
- 141. 141 In concrete structures, compressive concrete stresses caused by the day-to-day service loads rarely exceed half of the compressive strength. It is therefore, reasonable to assume that the instantaneous behaviour of concrete at service loads is linear- elastic and that instantaneous strain is given by Instantaneous strain A
- 142. 142 👉D 4) CREEP OF CONCRETE A
- 143. 143 Creep of concrete is defined as the increase in deformation with time under constant load. Due to the creep of concrete, the prestress in the tendon is reduced with time. Hence, the study of creep is important in prestressed concrete to calculate the loss in prestress. The creep occurs due to two causes 1. Rearrangement of hydrated cement paste (especially the layered products) 2. Expulsion of water from voids under load A
- 144. 144 If a concrete specimen is subjected to slow compressive loading, the stress versus strain curve is elongated along the strain axis as compared to the curve for fast loading. This can be explained in terms of creep. Stress-strain curves for concrete under compression A
- 145. 145 Creep is quantified in terms of the strain that occurs in addition to the elastic strain due to the applied loads. If the applied loads are close to the service loads, the creep strain increases at a decreasing rate with time. The ultimate creep strain is found to be proportional to the elastic strain. The ratio of the ultimate creep strain to the elastic strain is called the creep coefficient θ. A
- 146. 146 Clause 6.2.5 of IS 1343 2012 As long as the stress in concrete does not exceed one-third of characteristic compressive strength, creep may be assumed to be proportional to the stress. the ultimate creep strain is given as follows A
- 147. 147 There is reduction of strain due to creep recovery which is less than the creep strain. There is some residual strain which cannot be recovered Recoverable and irrecoverable creep components. (a) Creep strain history. (b) Stress history. If a concrete specimen is unloaded after a long period under load, the magnitude of the recoverable creep is of the order of 40% to 50% of the elastic strain (between 10% and 20% of the total creep strain) A
- 148. 148 The creep strain depends on several factors. It increases with the increase in the following variables 1) Cement content (cement paste to aggregate ratio) 2) Water-to-cement ratio 3) Air entrainment 4) Ambient temperature A
- 149. 149 The creep strain decreases with the increase in the following variables 1) Age of concrete at the time of loading 2) Relative humidity 3) Volume to surface area ratio. A
- 150. 150 CLAUSE 6.2.5.1 OF IS 1343 2012 IS USED TO CALCULATE CREEP OF PRESTRESSED MEMBERS Where end results are not sensitive to precise values calculated as given IN CLAUSE 6.2.5.1, the values given in FOLLOWING table can be considered as final creep co-efficient for design for normal weight concrete of grades between M 30 and M 60, subject to condition compressive stress does not exceed 0.36 fck at the age of loading, mean temperature of concrete is between 10°C and 20°C with seasonal variation between –20°C to 40°C. temperature greater than 40°C the co-efficient given may be increased by 10 percent, in the absence of accurate data A
- 151. 151 TABLE GIVEN IN CLAUSE 6.2.5.1 OF IS 1343 2012 FINAL VALUES OF CREEP COEFFICIENTS A
- 152. 152 👉D 5) SHRINKAGE OF CONCRETE A
- 153. 153 Shrinkage of concrete is the time-dependent strain in an unloaded and unrestrained specimen at constant temperature. Shrinkage is often divided into several components, including plastic shrinkage, Chemical shrinkage, thermal shrinkage and drying shrinkage A
- 154. 154 Plastic shrinkage occurs in the wet concrete before setting, whereas chemical, thermal and drying shrinkage all occur in the hardened concrete after setting. Plastic shrinkage cracking occurs due to capillary tension in the pore water and is best prevented by taking measures during construction to avoid the rapid evaporation of bleed water. A
- 155. 155 Drying shrinkage is the reduction in volume caused principally by the loss of water during the drying process. It increases with time at a gradually decreasing rate and takes place in the months and years after setting. The magnitude and rate of development of drying shrinkage depend on all the FOLLOWING factors that affect the drying of concrete, including A
- 156. 156 ***relative humidity, ***the size and shape of the member ***the mix characteristics, ***the type and quantity of the binder, ***the water content and ***water-to-cement ratio, ***ratio of fine-to-coarse aggregate ***Type of aggregate. A
- 157. 157 Chemical shrinkage results from various chemical reactions within the cement paste and includes hydration shrinkage, which is related to the degree of hydration of the binder in a sealed specimen with no moisture exchange. Chemical shrinkage (often called autogenous shrinkage) occurs rapidly in the days and weeks after casting and is less dependent on the environment and the size of the specimen than drying shrinkage. A
- 158. 158 Thermal shrinkage is the contraction that results in the first few hours (or days) after setting as the heat of hydration gradually dissipates. The term endogenous shrinkage is sometimes used to refer to that part of the shrinkage of the hardened concrete that is not associated with drying ENDOGENOUS SHRINKAGE = autogenous & THERMAL. A
- 159. 159 Clause 6.2.4.1 of is 1343 2012 STATES The total shrinkage strain is composed of two components, the autogenous shrinkage strain and the drying shrinkage strain. The value of the total shrinkage strain, εcs is given by: εcs = εcd + εca A
- 160. 160 Concrete strain versus time for a specimen subjected to constant sustained stress. Shrinkage increases with time at a decreasing rate, as illustrated in Figure A
- 161. 161 The shrinkage strain increases with the increase in the following variables. 1) Ambient temperature 2) Temperature gradient in the members 3) Water-to-cement ratio 4) Cement content A
- 162. 162 The shrinkage strain decreases with the increase in the following variables. 1) Age of concrete at commencement of drying 2) Relative humidity 3) Volume to surface area ratio A
- 163. 163 εcs = εcd + εca εcs = total shrinkage strain; εcd = drying shrinkage strain; and εca = autogenous shrinkage strain. autogenous shrinkage strain. Clause 6.2.4.2 of is 1343 2012 A
- 164. 164 Drying shrinkage strain. The final value of the drying shrinkage strain, εcd, ∞ may be taken equal to kh. εcd Values of εcd may be taken from the table given below for guidance. These values are expected mean values, with a coefficient of variation of about 30 percent. Clause 6.2.4.3 of is 1343 2012 A
- 165. 165 kh is a coefficient depending on the notional size h0, as given below: Clause 6.2.4.2 of is 1343 2012 A
- 166. 166 👉D 6) grout properties A
- 167. 167 Grout is a mixture of water, cement and optional materials like sand, water-reducing admixtures, expansion agent and pozzolans. The water-to-cement ratio is around 0.5. Fine sand is used to avoid segregation. A
- 168. 168 The desirable properties of grout are as follows 1) Fluidity 2) Minimum bleeding and segregation 3) Low shrinkage 4) Adequate strength after hardening 5) No detrimental compounds 6) Durable A
- 169. 169 Clause 13.3 of IS 1343 2012 grouting 1) The sand should pass 150 µm Indian Standard sieve. 2) The compressive strength of 100 mm cubes of the grout shall not be less than 17 N/mm2 at 7 days. 3) The mass of sand in the grout shall not be more than 10 percent of the mass of cement, unless proper workability can be ensured by addition of suitable plasticizers 4) When an expanding agent is used, the total unrestrained expansion shall not exceed 10 percent. 5)The capacity of the grout pump should be such as to achieve a forward speed of grout of around 5 to 10 m/min A
- 170. 170 6) The grouting equipment should contain a screen having a mesh size of IS Sieve No. 106 (IS Sieve No. 150, if sand is used) 7) Water-cement ratio should be as low as possible, consistent with workability. This ratio should not normally exceed 0.45. 8) The compressive strength of 100 mm cubes of the grout shall be not less than 27 MPa at 28 days 9) Chlorides from all sources, that is, cement, water, sand, fillers and admixture should not exceed 0.1 percent by mass of the cement A
- 171. 171 D 7) pre-stressing steel A
- 172. 172 👉Forms of Pre-stressing Steel It is an alloy of iron, carbon, manganese and optional materials. The following material describes the types and properties of prestressing steel In addition to prestressing steel, conventional non pre- stressed reinforcement is used for flexural capacity (optional), shear capacity, temperature and shrinkage requirements A
- 173. 173 Wires A pre-stressing wire is a single unit made of steel. The nominal diameters of the wires are 2.5, 3.0, 4.0, 5.0, 7.0 and 8.0 mm. The different types of wires are as follows. 1) Plain wire: No indentations on the surface. 2) Indented wire: There are circular or elliptical indentations on the surface. A
- 174. 174 Strands A few wires are spun together in a helical form to form a pre-stressing strand. The different types of strands are as follows Two-wire strand THREE-wire strand Seven-wire strand A
- 175. 175 TENDONS A steel element, such as a wire, cable, bar, rod or strand, or a bundle of such elements used to impart pre-stress to concrete when the element is tensioned The strands are placed in a duct which may be filled with grout after the post-tensioning operation is completed Cross-section of a typical tendon A
- 176. 176 CABLES A group of tendons form a pre-stressing cable. The cables are used in bridges. A tendon can be made up of a single steel bar. The diameter of a bar is much larger than that of a wire. Bars are available in the following sizes: 10, 12, 16, 20, 22, 25, 28 and 32 mm. BAR A
- 177. 177 D 8) TYPES OF PRE-STRESSING SEEL A
- 178. 178 Cold working (cold drawing) The cold working is done by rolling the bars through a series of dyes. It re-aligns the crystals and increases the strength. Stress relieving The stress relieving is done by heating the strand to about 350º C and cooling slowly. This reduces the plastic deformation of the steel after the onset of yielding. Strain tempering for low relaxation This process is done by heating the strand to about 350º C while it is under tension. This also improves the stress-strain behaviour of the steel by reducing the plastic deformation after the onset of yielding. In addition, the relaxation is reduced
- 179. 179 CLAUSE 5.6.1.1 OF IS 1343 2012 a) Plain hard-drawn steel wire (cold-drawn stress relieved wire) conforming to IS 1785 (Part 1), b) Indented wire conforming to IS 6003, c) High tensile steel bar conforming to IS 2090, d) Uncoated stress relieved strand conforming to IS 6006, and e) Uncoated stress relieved low relaxation seven ply strand conforming to IS 14268. A
- 180. 180 Properties of Prestressing Steel 1) High strength 2) Adequate ductility 3) Bendability, which is required at the harping points and near the anchorage 4) High bond, required for pre- tensioned members 5) Low relaxation to reduce losses 6) Minimum corrosion.A
- 181. 181 Nominal Diameter (mm) 2.50 3.00 4.00 5.00 7.00 8.00 Minimum Tensile Strength fpu (N/mm2) 2010 1865 1715 1570 1470 1375 Elongation (percent) 2.5 2.5 3 4 4 4 Cold Drawn Stress-Relieved Wires (IS: 1785 Part 1) The proof stress (defined later) should not be less than 85% of the specified tensile strength A
- 182. 182 Nominal Diameter (mm) 3.00 4.00 5.00 Minimum Tensile Strength fpu (N/mm2) 1765 1715 1570 As-Drawn wire (IS: 1785 Part 2) The proof stress should not be less than 75% of the specified tensile strength A
- 183. 183 Nominal Diameter (mm) 3.00 4.00 5.00 Minimum Tensile Strength fpu (N/mm2) 1865 1715 1570 Indented wire (IS: 6003) The proof stress should not be less than 85% of the specified tensile strength. A
- 184. 184 MECHANICAL PROPERTIES OF HIGH TENSILE STEEL BARS CHARACTERISTIC TENSILE STRENGTH (MINIMUM) 980 N/ MM2 PROOF STRESS NOT LESS THAN 80 % OF THE MINIMUM TENSILE STRENGTH ELONGATION AT RUPTURE ON A GAUGE LENGTH OF 5.65 A A= X SECTIONAL AREA 10 % IS 2090 1983 A
- 185. 185 STIFFNESS OF STEEL A steel element, such as a wire, cable, bar, rod or strand, or a bundle of such elements used to impart pre-stress to concrete when the element is tensioned Modulus of elasticity (IS: 1343 - 1980) Type of steel Modulus of elasticity Cold-drawn wires 210 kN/mm2 High tensile steel bars 200 kN/mm2 Strands 195 kN/mm2 A
- 186. 186 At the time of initial tensioning, the maximum tensile stress, fpi immediately behind the anchorages shall not exceed 76 percent of the ultimate tensile strength, fpu of the wire or bar or strand. CLAUSE 19.5.1 Maximum Initial Pre-stress fpi ≤ 0.76 fpu A
- 187. 187 D 8) STRESS STRAIN CURVES FOR PRESTRESSING STEEL A
- 188. 188 The stress versus strain behaviour of prestressing steel under uniaxial tension is initially linear (stress is proportional to strain) and elastic (strain is recovered at unloading). Beyond about 70% of the ultimate strength the behaviour becomes nonlinear and inelastic. There is no defined yield point. A
- 189. 189 The yield point is defined in terms of the proof stress or a specified yield strain. IS:1343 - 2012 recommends the yield point at 0.2% proof stress. A
- 190. 190 The stress-strain curves are influenced by the treatment processes. The following figure shows the variation in the 0.2% proof stress for wires under different treatment processes. Variation in the 0.2% proof stress for wires under different treatment processes A
- 191. 191 The design stress-strain curves are calculated by dividing the stress beyond 0.76fpu by a material safety factor γm =1.15. Characteristic and design stress-strain curves for prestressing steelA
- 192. 192 D 9) RELAXATION OF STEEL A
- 193. 193 Relaxation of steel is defined as the decrease in stress with time under constant strain. Due to the relaxation of steel, the pre-stress in the tendon is reduced with time. Hence, the study of relaxation is important in pre-stressed concrete to calculate the loss in pre-stress The relaxation depends on the type of steel, initial pre-stress and the temperature. The following figure shows the effect of relaxation due to different types of loading conditions A
- 194. 194 Effect of relaxation due to different types of loading conditions A
- 195. 195 VARIOUS CODE PROVISIONS FOR THE RELAXATION OF STRESS IN STEEL ARE BASED ON RESULT OF 1000 HOUR RELAXATION TEST ON WIRE. IT HAS BEEN OBSEREVED THAT THE LOSS RECORDED OVER PERIOD OF ABOUT 1000 HOURS FROM AN INITIAL PRESTRESS OF 70 PERCENT OF TENSILE STRENGTH IS ABOUT SAME AS THAT OVER A PERIOD OF 4 YEARS FROM INITIAL STRESS OF 60 PERCENT OF TENSILE STRENGTH RELAXATION OF STRESS IN 1000 HOURS IF fpi = 0.70 fpu RELAXATION OF STRESS IN 4 years IF fpi = 0.60 fpu= A
- 196. 196 Variation of stress with time for different levels of pre-stressing A
- 197. 197 Cold drawn stress- relieved wires 5% of initial prestress Indented wires 5% of initial prestress Stress-relieved strand 5% of initial prestress Bars 49 N/mm Relaxation losses at 1000 hours (IS:1785, IS:6003, IS:6006, IS:2090) A
- 198. 198 When experimental values are not available, the relaxation losses may be assumed as given in Table 6 OF IS 1343 2012 LONG TERM RELAXTION LOSS = VALUES IN TABLE 6 X 3 A
- 199. 199 D10) STRESS CORROSION & HYDROGEN EMBRITTLEMENT A
- 200. 200 Stress corrosion is another form of corrosion that is important to many fields including civil structures. Stress-corrosion occurs when a material exists in a relatively inert environment but corrodes due to an applied stress. The stress may be externally applied or residual. This form of corrosion is particularly dangerous because it may not occur under a particular set of conditions until there is an applied stress. The corrosion is not clearly visible prior to fracture and can result in catastrophic failure. A
- 201. 201 Stress corrosion cracking will usually cause the material to fail in a brittle manner, It can lead to unexpected sudden failure of normally ductile metal alloys subjected to a tensile stress, especially at elevated temperature A
- 202. 202A
- 203. 203 CAUSES OF Stress corrosion INTERNAL METALLURGICAL STRUCTURE INFLUENCED BY HEAT TREATMENT AND MECHANICAL PROCESSING IN POST TENSIONED, IF DUCTS ARE NOT GROUTED A
- 204. 204 IT is a metal’s loss of ductility and reduction of load bearing capability due to the absorption of hydrogen atoms or molecules by the metal. The result of hydrogen embrittlement is that components crack and fracture at stresses less than the yield strength of the metal. Hydrogen embrittlement A
- 205. 205 For hydrogen embrittlement to occur, a combination of three conditions are required: 1) the presence and diffusion of hydrogen 2) a susceptible material 3) stress A
- 206. 206A
- 207. 207 CAUSES OF HYDROGEN EMBRITTLEMENT USE OF BLAST FURNACE SLAG CEMENT AND HIGH ALUMINA CEMENT BECAUSE THESE ARE RICH IN SULPHIDES USE OF ALUMINIUM AND ZINC (DISSIMILIAR METALS) FOR SHEATH MINUTE TRACES OF SULPHUR COMES IN CONTACT WITH HIGH TENSILE STEEL WIRES DUE TO PRESENCE OF MOISTURE A
- 208. Durability Prestressing steel is susceptible to stress corrosion and hydrogen embrittlement in aggressive environments. Hence, prestressing steel needs to be adequately protected. 1) Epoxy coating 2) Mastic wrap (grease impregnated tape) 3) Galvanized bars 4) Encasing in tubes. For unbonded tendons, corrosion protection is provided by one or more of the following methods. For bonded tendons, the alkaline environment of the grout provides adequate protection. A
- 209. 209 E) Pre requisite of s.o.m A
- 210. 210 Stresses calculated from the flexure formula are called bending stresses or flexural stresses. This equation, called the flexure formula, shows that the stresses are directly proportional to the bending moment M and inversely proportional to the moment of inertia I of the cross section. Also, the stresses vary linearly with the distance y from the neutral axis, as previously observed. FLEXURE FORMULA AA
- 211. 211 Combined Direct and Bending Stresses The member is now subjected to a compressive load P, which is centric, and a BM Pe. it is possible to shift the load P to the centre of the section,. We introduce two equal and opposite forces P at 0 as shown in Fig.(b). The given load P and the equal and opposite force at 0 result in a couple of magnitude Pe. IF WE RECALL MECHANICS OF MATERIALS The net effect of shifting the load P to the centre is to cause a couple of Pe, which acts as a bending moment about the Y-axis, as shown in Fig. (c). A
- 212. 212 Resultant Stresses in Rectangular Section A
- 213. 213 PROBLEM 1: A steel plate of dimensions 200 x 25 mm carries an eccentric tensile force of 500 KN, as shown in Fig. (a). Find the maximum and minimum stresses in the section.
- 214. 214 SOLUTION: Maximum tensile stress = 145 N/mm2 Minimum tensile stress = 55 N/mm2 RESULTANT STRESS DIAGRAMA
- 215. 215 PROBLEM 2: A force of 300 KN is applied to the edge of the member shown in Fig. Neglect the weight of the member and determine the state of stress at points B and C. A
- 216. 216 A The member is sectioned through B and C, Fig. b. For equilibrium at the section there must be an axial force of 300 KN acting through the centroid and a bending moment of 45.0 KNm about the centroidal principal axis, Fig.b. SOLUTION: Internal Loadings.
- 217. 217 Normal Force. The uniform normal-stress distribution due to the normal force is shown in Fig.c. Stress Components. A
- 218. 218 The normal-stress distribution due to the bending moment is shown in Fig. d. The maximum stress is Bending Moment. A
- 219. 219 Superposition. Algebraically adding the stresses at B and C, we get
- 220. 220 thanks!
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