unit2_notes

# unit2_notes - 16.21 Techniques of Structural Analysis and...

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Unformatted text preview: 16.21 Techniques of Structural Analysis and Design Spring 2005 Unit #2 - Stress and Momentum balance Ra´ul Radovitzky February 6, 2005 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 Δ S is defined as: f t = lim (1) Δ S 0 Δ 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 ( 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 e i (this can be done for any vector) as (see Fig. 2 ): t ( i ) = σ ij e j (2) 1 Δ s f surface forces body forces n body forces n Figure 1: Surface force f on area Δ S of the cross section by plane whose normal is n σ ij is the component of the stress vector t ( i ) along the e j direction. Stress tensor We could keep analyzing different planes passing through the point with different normals and, therefore, different stress vectors t ( n ) and one might wonder if there is any relation among them or if they are all independent. The answer to this question is given by invoking equilibrium on the (shrinking) tetrahedron of material of Fig. 3 . The area of the faces of the tetrahedron are ....
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unit2_notes - 16.21 Techniques of Structural Analysis and...

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