Page 1 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Axial Deformations Introduction Free body diagram - Revisited Normal, shear and bearing stress Stress on inclined planes under axial loading Strain Mechanical properties of materials True stress and true strain Poissons ratio Elasticity and Plasticity Creep and fatigue Deformation in axially loaded members Statically indeterminate problems Thermal effect Design considerations Strain energy Impact loading Page 2 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Axial Deformations Introduction Free body diagram - Revisited Normal, shear and bearing stress Stress on inclined planes under axial loading Strain Mechanical properties of materials True stress and true strain Poissons ratio Elasticity and Plasticity Creep and fatigue Deformation in axially loaded members Statically indeterminate problems Thermal effect Design considerations Strain energy Impact loading Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.1 Introduction An important aspect of the analysis and design of structures relates to the deformations caused by the loads applied to a structure. Clearly it is important to avoid deformations so large that they may prevent the structure from fulfilling the purpose for which it is intended. But the analysis of deformations may also help us in the determination of stresses. It is not always possible to determine the forces in the members of a structure by applying only the principle of statics. This is because statics is based on the assumption of undeformable, rigid structures. By considering engineering structures as deformable and analyzing the deformations in their various members, it will be possible to compute forces which are statically indeterminate. Also the distribution of stresses in a given member is indeterminate, even when the force in that member is known. To determine the actual distribution of stresses within a member, it is necessary to analyze the deformations which take place in that member. This chapter deals with the deformations of a structural member such as a rod, bar or a plate under axial loading. Top Page 3 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Axial Deformations Introduction Free body diagram - Revisited Normal, shear and bearing stress Stress on inclined planes under axial loading Strain Mechanical properties of materials True stress and true strain Poissons ratio Elasticity and Plasticity Creep and fatigue Deformation in axially loaded members Statically indeterminate problems Thermal effect Design considerations Strain energy Impact loading Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.1 Introduction An important aspect of the analysis and design of structures relates to the deformations caused by the loads applied to a structure. Clearly it is important to avoid deformations so large that they may prevent the structure from fulfilling the purpose for which it is intended. But the analysis of deformations may also help us in the determination of stresses. It is not always possible to determine the forces in the members of a structure by applying only the principle of statics. This is because statics is based on the assumption of undeformable, rigid structures. By considering engineering structures as deformable and analyzing the deformations in their various members, it will be possible to compute forces which are statically indeterminate. Also the distribution of stresses in a given member is indeterminate, even when the force in that member is known. To determine the actual distribution of stresses within a member, it is necessary to analyze the deformations which take place in that member. This chapter deals with the deformations of a structural member such as a rod, bar or a plate under axial loading. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.2 Free body diagram - Revisited The first step towards solving an engineering problem is drawing the free body diagram of the element/structure considered. Removing an existing force or including a wrong force on the free body will badly affect the equilibrium conditions, and hence, the analysis. In view of this, some important points in drawing the free body diagram are discussed below. Figure 1.1 At the beginning, a clear decision is to be made by the analyst on the choice of the body to be considered for free body diagram. Then that body is detached from all of its surrounding members including ground and only their forces on the free body are represented. The weight of the body and other external body forces like centrifugal, inertia, etc., should also be included in the diagram and they are assumed to act at the centre of gravity of the body. When a structure involving many elements is considered for free body diagram, the forces acting in between the elements should not be brought into the diagram. The known forces acting on the body should be represented with proper magnitude and direction. If the direction of unknown forces like reactions can be decided, they should be indicated clearly in the diagram. Page 4 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Axial Deformations Introduction Free body diagram - Revisited Normal, shear and bearing stress Stress on inclined planes under axial loading Strain Mechanical properties of materials True stress and true strain Poissons ratio Elasticity and Plasticity Creep and fatigue Deformation in axially loaded members Statically indeterminate problems Thermal effect Design considerations Strain energy Impact loading Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.1 Introduction An important aspect of the analysis and design of structures relates to the deformations caused by the loads applied to a structure. Clearly it is important to avoid deformations so large that they may prevent the structure from fulfilling the purpose for which it is intended. But the analysis of deformations may also help us in the determination of stresses. It is not always possible to determine the forces in the members of a structure by applying only the principle of statics. This is because statics is based on the assumption of undeformable, rigid structures. By considering engineering structures as deformable and analyzing the deformations in their various members, it will be possible to compute forces which are statically indeterminate. Also the distribution of stresses in a given member is indeterminate, even when the force in that member is known. To determine the actual distribution of stresses within a member, it is necessary to analyze the deformations which take place in that member. This chapter deals with the deformations of a structural member such as a rod, bar or a plate under axial loading. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.2 Free body diagram - Revisited The first step towards solving an engineering problem is drawing the free body diagram of the element/structure considered. Removing an existing force or including a wrong force on the free body will badly affect the equilibrium conditions, and hence, the analysis. In view of this, some important points in drawing the free body diagram are discussed below. Figure 1.1 At the beginning, a clear decision is to be made by the analyst on the choice of the body to be considered for free body diagram. Then that body is detached from all of its surrounding members including ground and only their forces on the free body are represented. The weight of the body and other external body forces like centrifugal, inertia, etc., should also be included in the diagram and they are assumed to act at the centre of gravity of the body. When a structure involving many elements is considered for free body diagram, the forces acting in between the elements should not be brought into the diagram. The known forces acting on the body should be represented with proper magnitude and direction. If the direction of unknown forces like reactions can be decided, they should be indicated clearly in the diagram. Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras After completing free body diagram, equilibrium equations from statics in terms of forces and moments are applied and solved for the unknowns. Top Page 5 Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras Axial Deformations Introduction Free body diagram - Revisited Normal, shear and bearing stress Stress on inclined planes under axial loading Strain Mechanical properties of materials True stress and true strain Poissons ratio Elasticity and Plasticity Creep and fatigue Deformation in axially loaded members Statically indeterminate problems Thermal effect Design considerations Strain energy Impact loading Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.1 Introduction An important aspect of the analysis and design of structures relates to the deformations caused by the loads applied to a structure. Clearly it is important to avoid deformations so large that they may prevent the structure from fulfilling the purpose for which it is intended. But the analysis of deformations may also help us in the determination of stresses. It is not always possible to determine the forces in the members of a structure by applying only the principle of statics. This is because statics is based on the assumption of undeformable, rigid structures. By considering engineering structures as deformable and analyzing the deformations in their various members, it will be possible to compute forces which are statically indeterminate. Also the distribution of stresses in a given member is indeterminate, even when the force in that member is known. To determine the actual distribution of stresses within a member, it is necessary to analyze the deformations which take place in that member. This chapter deals with the deformations of a structural member such as a rod, bar or a plate under axial loading. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.2 Free body diagram - Revisited The first step towards solving an engineering problem is drawing the free body diagram of the element/structure considered. Removing an existing force or including a wrong force on the free body will badly affect the equilibrium conditions, and hence, the analysis. In view of this, some important points in drawing the free body diagram are discussed below. Figure 1.1 At the beginning, a clear decision is to be made by the analyst on the choice of the body to be considered for free body diagram. Then that body is detached from all of its surrounding members including ground and only their forces on the free body are represented. The weight of the body and other external body forces like centrifugal, inertia, etc., should also be included in the diagram and they are assumed to act at the centre of gravity of the body. When a structure involving many elements is considered for free body diagram, the forces acting in between the elements should not be brought into the diagram. The known forces acting on the body should be represented with proper magnitude and direction. If the direction of unknown forces like reactions can be decided, they should be indicated clearly in the diagram. Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras After completing free body diagram, equilibrium equations from statics in terms of forces and moments are applied and solved for the unknowns. Top Strength of Materials Prof. M. S. Sivakumar Indian Institute of Technology Madras 1.3 Normal, shear and bearing stress 1.3.1 Normal Stress: Figure 1.2 When a structural member is under load, predicting its ability to withstand that load is not possible merely from the reaction force in the member. It depends upon the internal force, cross sectional area of the element and its material properties. Thus, a quantity that gives the ratio of the internal force to the cross sectional area will define the ability of the material in with standing the loads in a better way. That quantity, i.e., the intensity of force distributed over the given area or simply the force per unit area is called the stress. P A s = 1.1 In SI units, force is expressed in newtons (N) and area in square meters. Consequently, the stress has units of newtons per square meter (N/m 2 ) or Pascals (Pa). In figure 1.2, the stresses are acting normal to the section XX that is perpendicular to the axis of the bar. These stresses are called normal stresses. The stress defined in equation 1.1 is obtained by dividing the force by the cross sectional area and hence it represents the average value of the stress over the entire cross section.Read More

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