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- Strength Of Materials - SOM
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**Visvesvaraya Technological University Regional Center - VTU**- Civil Engineering
- 11 Topics
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Page-2

- Simple Stresses And Strains - ( 1 - 4 )
- Hooke's law - ( 5 - 5 )
- Young's modulus - ( 6 - 18 )
- Saint venant's priciple - ( 19 - 34 )
- Thin And Thick Cylinders - ( 35 - 44 )
- Thick cylinder theory - ( 45 - 51 )
- Bending Moment And Shear Forces - ( 52 - 107 )
- Columns And Struts - ( 108 - 120 )
- Typical failure of coloums - ( 121 - 139 )
- Euler's theory - ( 140 - 147 )
- Theories Of Failure - ( 148 - 160 )

Topic:

1.1 Introduction In civil engineering structures, we frequently encounter structural elements such as tie members, cables, beams, columns and struts subjected to external actions called forces or loads. These elements have to be designed such that they have adequate strength, stiffness and stability. The strength of a structural component is its ability to withstand applied forces without failure and this depends upon the sectional dimensions and material characteristics. For instance a steel rod can resist an applied tensile force more than an aluminium rod with similar diameter. Larger the sectional dimensions or stronger is the material greater will be the force carrying capacity. Stiffness influences the deformation as a consequence of stretching, shortening, bending, sliding, buckling, twisting and warping due to applied forces as shown in the following figure. In a deformable body, the distance between two points changes due to the action of some kind of forces acting on it. A weight suspended by two cables causes stretching of the cables. Cables are in axial tension. Inclined members undergo shortening, and stretching will be induced in the horizontal member. Inclined members are in axial compression and horizontal member is in axial tension. Bolt connecting the plates is subjected to sliding along the failure plane. Shearing Cantilever beam subjected to bending due to transverse loads results in shortening in the bottom half and stretching in the top half of the beam. Cantilever beam subjected to twisting and warping due to Buckling of long compression members due to axial load. torsional moments. forces are induced.

Such deformations also depend upon sectional dimensions, length and material characteristics. For instance a steel rod undergoes less of stretching than an aluminium rod with similar diameter and subjected to same tensile force. Stability refers to the ability to maintain its original configuration. This again depends upon sectional dimensions, length and material characteristics. A steel rod with a larger length will buckle under a compressive action, while the one with smaller length can remain stable even though the sectional dimensions and material characteristics of both the rods are same. The subject generally called Strength of Materials includes the study of the distribution of internal forces, the stability and deformation of various elements. It is founded both on the results of experiments and the application of the principles of mechanics and mathematics. The results obtained in the subject of strength of materials form an important part of the basis of scientific and engineering designs of different structural elements. Hence this is treated as subject of fundamental importance in design engineering. The study of this subject in the context of civil engineering refers to various methods of analyzing deformation behaviour of structural elements such as plates, rods, beams, columns, shafts etc.,. 1.2 Concepts and definitions A load applied to a structural member will induce internal forces within the member called stress resultants and if computed based on unit cross sectional area then they are termed as intensity of stress or simply stress in the member. The stresses induced in the structural member will cause different types of deformation in the member. If such deformations are computed based on unit dimensions then they are termed as strain in the member. The stresses and strains that develop within a structural member must be calculated in order to assess its strength, deformations and stability. This requires a complete description of the geometry, constraints, applied loads and the material properties of the member. The calculated stresses may then be compared to some measure of the strength of the material established through experiments. The calculated deformations in the member may be compared with respect limiting criteria established based on experience. The calculated buckling load of

the member may be compared with the applied load and the safety of the member can be assessed. It is generally accepted that analytical methods coupled with experimental observations can provide solutions to problems in engineering with a fair degree of accuracy. Design solutions are worked out by a proper analysis of deformation of bodies subjected to surface and body forces along with material properties established through experimental investigations. 1.3 Simple Stress Consider the suspended bar of original length L0 and uniform cross sectional area A0 with a force or load P applied to its end as shown in the following figure (a). Let us imagine that the bar is cut in to two parts by a section x-x and study the equilibrium of the lower portion of the bar as shown in figure (b). At the lower end, we have the applied force P It can be noted that, the external force applied to a body in equilibrium is reacted by internal forces set up within the material. If a bar is subjected to an axial tension or compression, P, then the internal forces set up are distributed uniformly and the bar is said to be subjected to a uniform direct or normal or simple stress. The stress being defined as ( ) ( ) ( ) Note i. This is expressed as N/mm2 or MPa. ii. Stress may thus be compressive or tensile depending on the nature of the load. iii. In some cases the stress may vary across any given section, and in such cases the stress at any point is given by the limiting value of P/A as A tends to zero.

1.4 Simple Strain If a bar is subjected to a direct load, and hence a stress, the bar will change in length. If the bar has an original length L and changes in length by an amount L as shown below, then the strain produced is defined as follows: ( ) ( ) This strain is also termed as longitudinal strain as it is measured in the direction of application of load. Note: i. Strain is thus a measure of the deformation of the member. It is simply a ratio of two quantities with the same units. It is non-dimensional, i.e. it has no units. ii. The deformations under load are very small. Hence the strains are also expressed as strain x 10 -6. In such cases they are termed as microstrain (). iii. Strain is also expressed as a percentage strain : (%) = (L/L)100. 1.5 Elastic limit – Hooke’s law A structural member is said to be within elastic limit, if it returns to its original dimensions when load is removed. Within this load range, the deformations are proportional to the loads producing them. Hooke's law states that, “the force needed to extend or compress a spring by some distance is proportional to that distance”. This is indicated in the following figure. Since loads are proportional to the stresses they produce and deformations are proportional to the strains, the Hooke‟s law also implies that, “stress is proportional to strain within elastic limit”. ( ) ( ) or / = constant

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