Chapter 1 Mechanics of Materials Fundamentals In theoretical mechanics， bodies are assumed to be perfectly rigid.The deformations of bodies are important， however， as far as the resistance of the structures and machines to failure is concerned.Therefore， the bodies in mechanics of materials will no longer be assumed to be perfectly rigid as considered in theoretical mechanics. Mechanics of materials studies the ability of structures and machines to resist failure， and mainly involves the following tasks: ① strength， i.e.， the ability of members to support a specified load without experiencing excessive stresses; ② rigidity， i.e.， the ability of members to support a specified load without undergoing unacceptable deformations; ③ stability， i.e.， the ability of members to support a specified axial compressive load without causing a sudden lateral deflection. Any material dealt with in mechanics of materials is assumed to be: ① continuous， i.e.， the material consists of a continuous distribution of matter without voids; ② homogeneous， i.e.， the material possesses the same mechanical properties at all points in the matter; ③ isotropic， i.e.， the material has the same mechanical properties in all directions at any one point of the matter. The strength and rigidity of a material depend on its abilities to support a specified load without experiencing both excessive stresses and unacceptable deformations.These abilities are inherent in the material itself and must be determined by experimental methods.One of the most important tests to determine the mechanical properties of a material is the tensile or compressive test.This test is often used to determine the stress-strain relation of the material used. 1.1 External Forces Any external force applied to a body can be classified as either a surface force or a body force. 1.Surface Force An external force that is applied to the surface of a body is called a surface force. If the surface force is distributed over a finite area of the body， it is said to be a distributed load on a surface， Fig.1.1(a).If the surface force is applied along a narrow area， this force is defined as a distributed load along a line， Fig.1.1(b).If the area subjected to a surface force is very small， compared with the surface area of the body， then this surface force can be regarded as a concentrated load， Fig.1.1(c). Fig.1.1 2.Body Force An external force that is applied to every point within a body is called a body force.A gravitational force is an excellent example of the body force since it acts upon each of the particles forming the body. 1.2 Internal Forces When various external loads are applied to a member， the corresponding distributed internal forces will be developed at any point within the member.The distributed internal forces on any section within the member can be determined by using the method of sections. We imagine to use a plane ， Fig.1.2(a)， to section the member where the distributed internal forces need to be determined.For determination of the distributed internal forces on the cut plane， the portion of the member to the right of the cut plane is removed， and it is replaced by the distributed internal forces acting on the left portion， Fig.1.2(b). Fig.1.2 For equilibrium of the remaining portion of the member， the distributed internal forces can be determined by using the equations of static equilibrium.Although the exact distribution of internal forces may be unknown， we can use the equations of static equilibrium to relate the applied external loads to the resultant force R and resultant couple MO about point O on the cut plane， which are caused by the distributed internal forces， Fig.1.3(a). Fig.1.3 Generally speaking， the resultant force R and resultant couple MO have arbitrary directions，neither perpendicular nor parallel to the cut plane.However，we can resolve the resultant force and couple into six components， respectively along the x， y， and z axes， Fig.1.3(b). (1) Axial force.The normal component， along the x direction， of the resultant force is called the axial force (normal force)， N.It is developed when the external loads tend to pull or push the two segments of the member. (2) Shearing force.The tangential components， respectively along the y and z directions， of the resultant force are regarded as the shearing forces， denoted by Vy and Vz， which are developed when the external loads tend to cause the two segments of the member to slide over one another. (3) Torsional moment.The normal component， rotating about the x axis， of the resultant couple is called the torsional moment (twisting moment， or torque)， T， and developed when the external loads tend to twist one segment of the member with respect to the other. (4) Bending moment.The tangential components of the resultant couple tend to bend the member about the y and z axes， respectively.These t
