tailieunhanh - The Equilibrium Equations

Các mối quan hệ động học được mô tả trong phần 8 là hoàn toàn hình học, và không liên quan đến việc xem xét hành vi của vật chất. Các mối quan hệ cân bằng được thảo luận trong mô-đun này có độc lập tương tự từ vật liệu này. Họ chỉ đơn giản là Newton của pháp luật về chuyển động, nói rằng trong trường hợp không có khả năng tăng tốc tất cả các lực lượng hành động trên một cơ thể (hoặc một phần của nó) phải cân đối | The Equilibrium Equations David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA 02139 September 26 2000 Introduction The kinematic relations described in Module 8 are purely geometric and do not involve considerations of material behavior. The equilibrium relations to be discussed in this module have this same independence from the material. They are simply Newton s law of motion stating that in the absence of acceleration all of the forces acting on a body or a piece of it must balance. This allows us to state how the stress within a body but evaluated just below the surface is related to the external force applied to the surface. It also governs how the stress varies from position to position within the body. Cauchy stress A l Figure 1 Traction vector. In earlier modules we expressed the normal stress as force per unit area acting perpendicularly to a selected area and a shear stress was a force per unit area acting transversely to the area. To generalize this concept consider the situation depicted in Fig. 1 in which a traction vector T acts on an arbitrary plane within or on the external boundary of the body and at an arbitrary direction with respect to the orientation of the plane. The traction is a simple force vector having magnitude and direction but its magnitude is expressed in terms of force per unit of area lim aa Aq T 1 1 where AA is the magnitude of the area on which AF acts. The Cauchy1 stresses which are a generalization of our earlier definitions of stress are the forces per unit area acting on the Cartesian x y and z planes to balance the traction. In two dimensions this balance can be written by drawing a simple free body diagram with the traction vector acting on an area of arbitrary size A Fig. 2 remembering to obtain the forces by multiplying by the appropriate area. ơx A cos ớ Txy A sin ớ TxA Txy A cos ớ ơy A sin ớ TyA Canceling the factor A this can be written in matrix form as x .