16.21 Techniques of Structural Analysis and sig Spring 2003 Unit #2 - Stress and momentum balance Stress at a point We are going to consider the forces exerted on a material. These can be external or internal. External forces come in two flavors: body forces(given per unit mass or volume) and surface forces(given per unit area). If we cut a body of material in equilibrium under a set of external forces along a plane as shown in fig. 1. and consider one side of it we draw two conclusions: 1 the equilibrium provided by the loads from the side taken out is provided by a set of forces that are distributed among the material particles adjacent to the cut plane and that should provide an equivalent set of forces to the ones loading the part taken out, 2)these forces can now be considered as external surface forces acting on the part of material under consideration The stress vector at a point on As is defined as s-0△S If the cut had gone through the same point under consideration but along a plane with a different normal, the stress vector t would have been different Let's consider the three stress vectors t () acting on the planes normal to the coordinate axes. Let's also decompose each to) in its three components in the coordinate system e:(this can be done for any vector)as( see Fig. 2) )=可ej16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #2 - Stress and Momentum balance Stress at a point We are going to consider the forces exerted on a material. These can be external or internal. External forces come in two flavors: body forces (given per unit mass or volume) and surface forces (given per unit area). If we cut a body of material in equilibrium under a set of external forces along a plane, as shown in Fig.1, and consider one side of it, we draw two conclusions: 1) the equilibrium provided by the loads from the side taken out is provided by a set of forces that are distributed among the material particles adjacent to the cut plane and that should provide an equivalent set of forces to the ones loading the part taken out, 2) these forces can now be considered as external surface forces acting on the part of material under consideration. The stress vector at a point on ΔS is defined as: t = lim ΔS→0 f (1) ΔS If the cut had gone through the same point under consideration but along a plane with a different normal, the stress vector t would have been different. Let’s consider the three stress vectors t(i) acting on the planes normal to the coordinate axes. Let’s also decompose each t(i) in its three components in the coordinate system ei (this can be done for any vector) as (see Fig.2): t(i) = σijej (2) 1