{"id":2460,"date":"2024-08-05T13:18:29","date_gmt":"2024-08-05T07:48:29","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=2460"},"modified":"2025-07-16T15:23:39","modified_gmt":"2025-07-16T09:53:39","slug":"stress","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress","title":{"rendered":"What is Stress in Strength of Materials?"},"content":{"rendered":"<p style=\"text-align: justify;\">The fundamental concept of stress can be understood by considering a prismatic bar that is loaded by axial force P at the ends as shown. <img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2461 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/axial-stress.jpg\" alt=\"Axial stress\" width=\"275\" height=\"321\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/07\/axial-stress.jpg 275w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/07\/axial-stress-257x300.jpg 257w\" sizes=\"auto, (max-width: 275px) 100vw, 275px\" \/><\/p>\n<p style=\"text-align: justify;\">A prismatic bar is a straight structural member having constant cross-sectional area throughout its length. In the figure (a), axial force is acting away from the cross-section producing a uniform stretching of the bar, hence the bar is said to be in tension. Similarly in figure (c), axial force is acting towards the cross-section producing uniform compression of the bar, hence the bar is said to be in compression. To investigate the internal stresses produced<br \/>\nin the bar by axial forces, we make an imaginary cut at section mn as shown in figure (b) and (d). This section is taken perpendicular to the longitudinal axis of bar. Hence it is known as cross-section.<\/p>\n<p style=\"text-align: justify;\">Now isolating the part of the bar to the right of the cut and considering the right of the cut as a free body. The force P has a tendency to move free body in the direction of load, so to restrict the motion of bar an internal force is induced which is uniformly distributed over cross-sectional area. The intensity of force developed, that is, internal force per unit area is called the stress. <img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2462 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/07\/axial-load.jpg\" alt=\"Axial load\" width=\"116\" height=\"147\" \/><\/p>\n<p style=\"text-align: justify;\">Stress differs from pressure because pressure is defined as the externally applied force on unit area while stress is internal resistive force on unit area. To have better understanding of difference between externally applied force and internal resistance. Consider a bar suspended from a fixed end and a weight W is gradually applied at its free end as shown in figure.<\/p>\n<h2 style=\"text-align: justify;\"><strong>Case-I: Weight, W is applied gradually<\/strong><\/h2>\n<p style=\"text-align: justify;\">Gradual loading means that value of load is zero at the starting time and gradually increases to value of W. Here, the bar gradually elongates with the increasing value of load. With increase in elongation, resistance forces say R will also increase gradually.<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2465 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/external-load-.jpg\" alt=\"External Load \" width=\"399\" height=\"187\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/external-load-.jpg 399w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/external-load--300x141.jpg 300w\" sizes=\"auto, (max-width: 399px) 100vw, 399px\" \/><\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_1 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#Case-II-Weight-W-is-applied-suddenly\" >Case-II: Weight, W is applied suddenly<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#STRAIN\" >STRAIN<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#Stress-Strain-Curve-for-Tension\" >Stress Strain Curve for Tension<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#ELASTIC-CONSTANTS\" >ELASTIC CONSTANTS<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#Relationship-between-Elastic-Constants\" >Relationship between Elastic Constants<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#STRAIN-ENERGY\" >STRAIN ENERGY<\/a><ul class='ez-toc-list-level-4' ><li class='ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#Strain-Energy-Stored-in-Body-when-it-is-subjected-to-an-Axial-Load-Gradually\" >Strain Energy Stored in Body when it is subjected to an Axial Load Gradually<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#Strain-Energy-Stored-in-Body-when-the-Load-is-Applied-Suddenly\" >Strain Energy Stored in Body when the Load is Applied Suddenly<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-4'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/www.madeeasy.in\/study\/ce\/strength-of-material\/stress\/#Strain-Energy-Stored-in-a-Body-when-Load-is-Applied-with-Impact\" >Strain Energy Stored in a Body when Load is Applied with Impact<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Case-II-Weight-W-is-applied-suddenly\"><\/span>Case-II: Weight, W is applied suddenly<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Here, external load variation with elongation of bar is such as that its value instantly increases to W. This sudden load will result into elongation of bar say \u2206. When external load is applied suddenly, resistance force will be set up in bar, but unlike external load which is sudden, resistance force has always linear variation with elongation of bar.<\/p>\n<p style=\"text-align: justify;\">Now, as clear from figure (a) and (b), intensity of pressure is not equal to stress induced in bar. Thus, stress can be defined as \u2013 \u201cStress is the internal resistance of a material offered against \u201cStress is the internal resistance of a material offered against deformation which is expressed in terms of force per unit area\u201d.<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2466 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/elongation.jpg\" alt=\"Elongation\" width=\"332\" height=\"182\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/elongation.jpg 332w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/elongation-300x164.jpg 300w\" sizes=\"auto, (max-width: 332px) 100vw, 332px\" \/><\/p>\n<p style=\"text-align: justify;\">Stress induced in material depends upon the nature of force, point of application and cross-sectional area of material. Stress can be tensile or tensile compressive in essive nature depending on the nature of load. Generally,<br \/>\nstress is represented by the Greek letter \u03c3. We can calculate stress mathematically as\u00a0\u03c3 = P\/A<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"STRAIN\"><\/span>STRAIN <img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2467 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/strain-in-bar.jpg\" alt=\"Strain in Bar\" width=\"212\" height=\"242\" \/><span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">An axially loaded bar undergoes a change in length, becoming longer when in tension and shorter when in\u00a0 compression. The elongation or shortening in axially loaded member per unit length is known as strain. Strain is represented by \u2208.<\/p>\n<p style=\"text-align: justify;\">Mathematically, strain can be calculated as \u2208 = \u2206L\/L<\/p>\n<p style=\"text-align: justify;\">Strain is dimensionless quantity and is always expressed in the form of number. If the member is in tension then the strain is called tensile strain. If the member is in compression, then the strain is called compressive strain.<\/p>\n<p style=\"text-align: justify;\">On the basis of length of member used in calculation of strain, strain can be of following two types:<\/p>\n<p style=\"text-align: justify;\"><strong>(a) Engineering or Nominal Strain<\/strong>: Engineering or nominal strain is strain calculated, when length of member is taken as original length<\/p>\n<p style=\"text-align: justify;\">Mathematically, \u2208<sub>0<\/sub> = \u2206l\/L<sub>0<\/sub> where, l<sub>0<\/sub> = original length of member<\/p>\n<p style=\"text-align: justify;\">(b) True or Actual Strain : True or actual or Actual Strain is strain calculated, when length of member is taken as<br \/>\nactual length of member at loading<\/p>\n<p style=\"text-align: justify;\">Mathematically \u2208<sub>a<\/sub> = \u2206l\/l<sub>a<\/sub> where, l<span style=\"font-size: 13.3333px;\">a<\/span> = actual length of member<\/p>\n<p style=\"text-align: justify;\">l<sub>a<\/sub> = l<sub>o<\/sub> \u00b1 \u2206l \u2018+\u2019 sign for tension; \u2018\u2013\u2019 sign for compression<\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Stress-Strain-Curve-for-Tension\"><\/span>Stress Strain Curve for Tension<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<ul style=\"text-align: justify;\">\n<li><strong>A is limit of proportionality:<\/strong> Beyond this linear variation ceases. Hooke\u2019s law is valid in OA.<\/li>\n<li><strong>B is elastic limit:<\/strong> The maximum stress upto which a specimen regains its original length on removal of applied load. For mild steel, B is very near to A. However, for other materials B may be greater than A.<\/li>\n<\/ul>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2468 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/starin.jpg\" alt=\"Strain\" width=\"408\" height=\"297\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/starin.jpg 408w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/starin-300x218.jpg 300w\" sizes=\"auto, (max-width: 408px) 100vw, 408px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>C\u2032 is upper yield point:<\/strong> The magnitude of the stress corresponding to C\u2032 depends on the cross sectional area, shape of the specimen and the type of the equipment used to perform the test. It has no practical significance.<\/p>\n<p style=\"text-align: justify;\"><strong>C is lower yield point:<\/strong> This is also called actual yield point. The stress at C is the yield stress (\u03c3<sub>y<\/sub> ) with a typical value of \u03c3<sub>y<\/sub> = 250 N\/mm<sup>2<\/sup> (for mild steel). The yielding begins at this stress.<\/p>\n<p style=\"text-align: justify;\"><strong>CD represents perfectly plastic region:<\/strong> It is the strain which occurs after the yielding point C, without any increase in stress. The strain corresponding to point D is about 1.4% and corresponding to C is about 0.12% for mild steel. Hence, plastic strain is 10 to 15 times of elastic strain.<\/p>\n<p style=\"text-align: justify;\"><strong>DE represents strain hardening region:<\/strong> In this range further addition of stress gives additional strain. However, strain increases with faster rate in this region. The material in this range undergoes change in its crystalline structure, resulting in increased resistance to further deformation. This portion is not used for structural design.<\/p>\n<p style=\"text-align: justify;\"><strong>E is ultimate point:<\/strong> is ultimate point: The stress corresponding to this point is ultimate stress (\u03c3u) and the corresponding strain is about 20% for mild steel.<\/p>\n<p style=\"text-align: justify;\"><strong>F is fracture point: <\/strong>Stress corresponding to this is called breaking stress and strain is called fracture strain. It is about 25% for mild steel.<\/p>\n<p style=\"text-align: justify;\"><strong>EF post ultimate stress region:<\/strong> In this range, necking occurs, i.e. area of cross-section is drastically decreased.<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"ELASTIC-CONSTANTS\"><\/span>ELASTIC CONSTANTS<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\"><strong>1. Young\u2019s Modulus (E):<\/strong> Young\u2019s modulus or modulus of elasticity is the slope of stress-strain curve under direct loading<\/p>\n<p style=\"text-align: justify;\">E = Direct \/ Axial stress \/ Direct \/Linear strain = \u03c3\/\u2208<\/p>\n<p style=\"text-align: justify;\"><strong>2. Shear Modulus (G):<\/strong> Shear modulus is defined as the ratio of shear stress to shear strain\u00a0 <img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2470 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/shear-strain-.jpg\" alt=\"Shear strain \" width=\"224\" height=\"132\" \/><\/p>\n<p style=\"text-align: justify;\">G = Shear stress\/ Shear strain = \u03c4\/\u03c6 It is also called modulus of rigidity.<\/p>\n<p style=\"text-align: justify;\">When a body is subjected to shearing stresses, the shape of the body gets distorted. The measurement of this distortion is done by angle of distortion. Under pure shear, the shape of the body get distorted but the volume remains same.<\/p>\n<p style=\"text-align: justify;\">If under the shear, the shear strain is \u03c6 then the linear strain in the diagonal of the specimen is given by \u2208 = \u03c6\/2 i.e. linear strain of diagonal is half of the shear strain in the body. It can be derived as given below:<\/p>\n<p style=\"text-align: justify;\">Consider a cube of side \u2018a\u2019 subjected to shear and complimentary shear as shown in figure. Assuming that the strain are small and the angle ACB is 45\u00b0 strain in diagonal OA is given as<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2471 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/shear-modulus.jpg\" alt=\"Shear Modulus\" width=\"596\" height=\"178\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-modulus.jpg 596w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/shear-modulus-300x90.jpg 300w\" sizes=\"auto, (max-width: 596px) 100vw, 596px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong> Bulk Modulus (K):<\/strong> Bulk modulus is defined as the ratio of direct stress to the volumetric strain.<\/p>\n<p style=\"text-align: justify;\">K = Direct stress\/Volumetric strain = \u03c3\/\u2208<sub>v<\/sub><\/p>\n<p style=\"text-align: justify;\">Significance of Bulk modulus is with respect to compressibility. In 3D hydrostatic loading, <img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2473 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/volumetric.jpg\" alt=\"Volumetric\" width=\"195\" height=\"198\" \/><\/p>\n<p style=\"text-align: justify;\">\u03c3x = \u03c3y = \u03c3z = p<br \/>\nand Volumetric strain, \u2208<sub>v<\/sub> = \u2206V\/V<br \/>\nthen Bulk modulus, <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-2472\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/bulk-modulus.jpg\" alt=\" Bulk modulus\" width=\"98\" height=\"57\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Poisson\u2019s Ratio:<\/strong> Poisson\u2019s ratio is defined as ratio of lateral strain to longitudinal strain within elastic limit under direct loading. It is represented by \u00b5 or 1\/m.<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2474 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/poisson-ratio.jpg\" alt=\"Poisson\u2019s Ratio\" width=\"279\" height=\"50\" \/><\/p>\n<p style=\"text-align: justify;\">In general, for engineering purposes, Poisson\u2019s ratio varies from 0 to 0.5 but actual range is \u20131 to 0.5 The value of Poisson\u2019s ratio of various materials are tabulated below:<\/p>\n<table class=\"table table-striped table-bordered table-condensed\" style=\"margin: 0 auto; width: 99%;\">\n<tbody>\n<tr>\n<th width=\"134\">Material<\/th>\n<th width=\"113\">Poisson\u2019s ratio<\/th>\n<\/tr>\n<tr>\n<td width=\"134\">\n<p style=\"text-align: center;\">Aluminium<\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"113\">0.33<\/td>\n<\/tr>\n<tr>\n<td width=\"134\">\n<p style=\"text-align: center;\">Brass<\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"113\">0.34<\/td>\n<\/tr>\n<tr>\n<td width=\"134\">\n<p style=\"text-align: center;\">Bronze<\/p>\n<\/td>\n<td width=\"113\">\n<p style=\"text-align: center;\">0.34<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"134\">Cast Iron<\/td>\n<td width=\"113\">\n<p style=\"text-align: center;\">0.2-0.3<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td width=\"134\">\n<p style=\"text-align: center;\">Concrete<\/p>\n<\/td>\n<td style=\"text-align: center;\" width=\"113\">0.1-0.2<\/td>\n<\/tr>\n<tr>\n<td width=\"134\">\n<p style=\"text-align: center;\">Copper<\/p>\n<\/td>\n<td width=\"113\">\n<p style=\"text-align: center;\">0.33-0.36<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"134\">Copper<\/td>\n<td width=\"113\">\n<p style=\"text-align: center;\">0.05-0.1<\/p>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: center;\" width=\"134\">Pure rubber and perfectly plastic material<\/td>\n<td width=\"113\">\n<p style=\"text-align: center;\">0.5<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Relationship-between-Elastic-Constants\"><\/span>Relationship between Elastic Constants<span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2476 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/elastic-constants.jpg\" alt=\"Elastic Constants\" width=\"376\" height=\"61\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/elastic-constants.jpg 376w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/elastic-constants-300x49.jpg 300w\" sizes=\"auto, (max-width: 376px) 100vw, 376px\" \/><\/p>\n<p style=\"text-align: justify;\">where, E = Young\u2019s modulus of elasticity, G = Modulus of rigidity, K = Bulk modulus, \u00b5 = Poisson\u2019s ratio<\/p>\n<h3 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"STRAIN-ENERGY\"><\/span>STRAIN ENERGY<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Whenever a load is applied at the body, a strain is produced in the body due to which energy is absorbed in the body. The energy thus absorbed is known as strain energy. It is taken equal to work done by applied load in producing the strain. The strain produced can be due to:<\/p>\n<p style=\"text-align: justify;\">(a) Gradually applied load<br \/>\n(b) Suddenly applied load<br \/>\n(c) Load with impact<\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Strain-Energy-Stored-in-Body-when-it-is-subjected-to-an-Axial-Load-Gradually\"><\/span>Strain Energy Stored in Body when it is subjected to an Axial Load Gradually <img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2477 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/load-deflection.jpg\" alt=\"Load Deflection\" width=\"185\" height=\"175\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\">Consider the figure showing the load extension diagram of bar upto elastic limit when a tensile load \u2018P\u2019 is applied to bar. The \u2018P\u2019 increase gradually from zero to value of \u2018P\u2019 and with the increase in load P, deflection of body also increases gradually.<\/p>\n<p style=\"text-align: justify;\">The load P will produce strain in body so strain energy will be stored in body which is equal to area of load deflection curve upto elastic limit.<\/p>\n<p style=\"text-align: justify;\">Strain energy = Work done by load P in straining the body<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2478 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/tensile.jpg\" alt=\"Tensile\" width=\"536\" height=\"167\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/tensile.jpg 536w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/tensile-300x93.jpg 300w\" sizes=\"auto, (max-width: 536px) 100vw, 536px\" \/><\/p>\n<p style=\"text-align: justify;\">Here, \u03c32\/2E is total strain energy per unit volume and known as modulus of resilience. Hence, strain energy per unit<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2480 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/prismatic-bars.jpg\" alt=\" Prismatic bars\" width=\"213\" height=\"270\" \/> volume is equal to area of stress-strain curve upto elastic limit.<\/p>\n<p style=\"text-align: justify;\"><strong>Case-1:<\/strong> Strain energy of prismatic bars with varying cross-sections<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2479 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/prismatic.jpg\" alt=\"Prismatic\" width=\"374\" height=\"51\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/prismatic.jpg 374w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/prismatic-300x41.jpg 300w\" sizes=\"auto, (max-width: 374px) 100vw, 374px\" \/><\/p>\n<p style=\"text-align: justify;\"><strong>Case-2:<\/strong> Strain energy of non-prismatic bar with varying axial load.<br \/>\nTotal strain energy <img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2481 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/axial-load.jpg\" alt=\"Axial Load\" width=\"107\" height=\"40\" \/><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-2482 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/non-prismatic-bar.jpg\" alt=\"Non-prismatic bar\" width=\"272\" height=\"235\" \/><\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Strain-Energy-Stored-in-Body-when-the-Load-is-Applied-Suddenly\"><\/span>Strain Energy Stored in Body when the Load is Applied Suddenly <img loading=\"lazy\" decoding=\"async\" class=\"alignright wp-image-2485 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/resistance.jpg\" alt=\"Resistance\" width=\"223\" height=\"382\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/resistance.jpg 223w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/resistance-175x300.jpg 175w\" sizes=\"auto, (max-width: 223px) 100vw, 223px\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\">In this case, the value of load, P is constant throughout the deformation while deformation increases from zero to value \u2206 as shown in figure (a)<br \/>\nHowever, the resistance force in the body is zero when deformation is zero and is equal to R when deformation reaches to its value \u2206 as shown in figure (b).<\/p>\n<p style=\"text-align: justify;\">So, work done by load, P = P \u00d7 \u2206<br \/>\nWork done by resistance force W = 1\/2\u00d7R\u00d7\u2206<br \/>\nwhere resistance force R = \u03c3 \u00d7 A<br \/>\n1\/2\u00d7R\u00d7\u2206 = P \u00d7 \u2206<br \/>\n1\/2\u00d7\u03c3\u00d7A = P<br \/>\n\u03c3 = 2P\/A<br \/>\nNow, strain energy stored in bar = \u03c32\/2E\u00d7Volume of bar<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2486 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/strain-energy.jpg\" alt=\"Strain Energy\" width=\"176\" height=\"60\" \/><\/p>\n<h4 style=\"text-align: justify;\"><span class=\"ez-toc-section\" id=\"Strain-Energy-Stored-in-a-Body-when-Load-is-Applied-with-Impact\"><\/span>Strain Energy Stored in a Body when Load is Applied with Impact <img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-2487 alignright\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/prismatic-1.jpg\" alt=\"Prismatic\" width=\"246\" height=\"246\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/prismatic-1.jpg 246w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/prismatic-1-150x150.jpg 150w\" sizes=\"auto, (max-width: 246px) 100vw, 246px\" \/><span class=\"ez-toc-section-end\"><\/span><\/h4>\n<p style=\"text-align: justify;\">Consider a prismatic bar suspended freely with a collar at one end and a load P at height h from collar as shown in<br \/>\nfigure. Let, when load \u2018P\u2019 is dropped on collar it elongates the bar by \u2018\u03b4L<\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2488 size-full aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/load-falling.jpg\" alt=\"Load falling\" width=\"479\" height=\"303\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/load-falling.jpg 479w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/load-falling-300x190.jpg 300w\" sizes=\"auto, (max-width: 479px) 100vw, 479px\" \/><\/p>\n<p style=\"text-align: justify;\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2489 size-full aligncenter\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2024\/08\/energy-stored-.jpg\" alt=\"Energy Stored \" width=\"497\" height=\"266\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/energy-stored-.jpg 497w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2024\/08\/energy-stored--300x161.jpg 300w\" sizes=\"auto, (max-width: 497px) 100vw, 497px\" \/><\/p>\n<p style=\"text-align: justify;\">By using equation (ii) and value of \u03c3<sub>max<\/sub>, we can find out strain energy.<br \/>\n<strong>Special case:<\/strong> When h = 0 i.e., it is analogues to case of sudden loading,<br \/>\nthen, \u03c3 = P\u00d72\/ A as proved earlier.<\/p>\n<p style=\"text-align: center;\"><a class=\"btn btn-danger\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/strength-of-material\/euler-theory\/\" target=\"_blank\" rel=\"noopener\">&lt;&lt; Previous<\/a> | <a class=\"btn btn-success\" role=\"button\" href=\"https:\/\/study.madeeasy.in\/ce\/strength-of-material\/loading-diagram\/\" target=\"_blank\" rel=\"noopener\"> Next &gt;&gt;<\/a><br \/>\n<strong> Must Read: <\/strong> <a href=\"https:\/\/study.madeeasy.in\/subjects\/what-is-strength-of-material\/\" target=\"_blank\" rel=\"noopener\"><strong>What is Strength of Material?<\/strong><\/a><\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>The fundamental concept of stress can be understood by considering a prismatic bar that is loaded by axial force P<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[685,2],"tags":[686,688,687,689],"class_list":["post-2460","post","type-post","status-publish","format-standard","hentry","category-strength-of-material","category-ce","tag-axial-load","tag-elongation","tag-external-load","tag-strain"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2460","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=2460"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/2460\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=2460"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=2460"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=2460"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}