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The Uniform development Code is devoted to the advance of higher development development and bigger defense to the general public by means of uniformity in construction legislation. The code is based on broad-based ideas that make attainable using new fabrics and new development platforms.
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Extra info for 1997 Uniform Building Code, Vol. 2: Structural Engineering Design Provisions
L (83) where summation is implied for both / and;. Thus, the components of Vv are simply the various partial derivatives of the component functions with respect to the coordinates, that is, the component [Vv]y gives the rate of change of the ith component of v with respect to the ;th coordinate axis. We can interpret the gradient of a vector field geometrically by considering the construction shown in Fig. 23. , A^ is very small). The unit vector n points m the direction from a to b. The value of the vector field at a is v(x) and the value of the vector field v(x-f A5n)- v(x-^-Asn) Figure 23 Interpretation of the gradient of a vector field with the directional derivative Chapter 1 Vectors and Tensors 43 at b is v(x + Asn).
In this abbreviated notation, the divergence has the more compact expression div (V) = v„, with summation implied across the comma. It should be evident that the comma notation is convenient for index manipulation. Consider again the domain 95 with boundary Q shown in Fig. 17. The gradient of a vector field v(x) is a second-order tensor defined as the limit of the ratio of the flux v (g) n over the surface to the volume, as the volume shrinks to zero. To wit Vv = lim ^ ^ \®ndA (82) Again, T(95) is the volume of the region % Q is the surface of the region, and n is the unit normal vector field to the surface.
Show that this tensor can be expressed as Q = Gy [g, ® gy], that is, Q^ are the components of Q with respect to the basis [g,®gy]. Show that the tensor can also be expressed in the form Q = [e,®gJ. (c) We can define a rotation tensor Q^, such that g, = Q^e, (the reverse rotation from part (b). Show that this tensor can be expressed as Q^ = Qij[^j ® e j , that is, Q^ are the components of Q^with respect to the basis [ Cj ® e J.