Appendix: Geometry of surfaces A.O The need for geometry The design of a plate as a load-bearing element in a structure or machine requires that the deformation of the plate be understood. Initially a plate is plane, or nearly so. When loaded it deflects into some shape which is not plane; hence there is a need to describe this new, curved shape in order to compare with the unloaded plane shape and so evaluate the internal actions generated. For the purposes of this chapter the plate will be characterized by the "middle surface" in the case of a solid plate, or some other representative surface for composite plates. The middle surface, as the name implies, is the central layer equidistant from the faces of the plate. Thus the deformation of the plate will be characterized by the deformation of the middle surface. This is reasonable, since most plates are thin compared with the distances spanned by the plate and, as already discussed for beams (1.6), the plate will be assumed to obey the Euler-Bernoulli hypothesis of plane sections remaining plane. However, unlike beams whose geometry of deformation is com­ paratively simple, plates are by definition two-dimensional objects and their deformation presents other aspects which can usefully be studied in this chapter. A.1 Geometry of a plane curve-curvature A plane curve possesses shape by virtue of the curvature of the curve. Consider a small piece of the curve. Assuming this piece to be circular, of radius p (see Fig. A.O), then the curvature of this portion of curve is defined as K == 1/p. The quantity p is known as the radius of curvature. If the element of arc length is denoted by ds and dljl is the angle between adjacent radii of curvature then dljl K = l/p = Ts. (A.l.O) 146 APPENDIX: GEOMETRY OF SURFACES 147 dt/t R o o Figure A.O Geometry of a curve. Now rework this definition in a vector form. Consider a radius vector r to point P. Then for Q, the radius vector will be r+dr. Hence dr is a tangent vector. Further, dr/ds is a unit tangent vector (t) because I dr I = ds. Denote dr/ds by rs, and consider what meaning can be attached to rss( = d2r/ds2 ). Now (A.Ll ) From Fig. A.O we see that (rs)Q and (rs)p are inclined at dtfJ and the difference is directed along the inward normal n. Comparing (A.l.O) and (A.Ll) it is seen that K=n.rss' (A.l.2) where n is a unit inward normal, directed toward the centre of curvature, O. This is a convenient vectorial expression for curvature of a plane curve which will also be found useful in the later discussion of surfaces. Example: Derive an expression for the curvature (K) of a plane curve in terms of cartesian coordinates (Fig. A.1). Now r = xi+yj, where i, j are fixed unit vectors. The curve is a one-parameter object such that y = f(x), say, where f(x) is given. 148 BASIC PRINCIPLES OF PLATE THEORY y n j o x Figure A.I Plane curve-curvature. Hence r = xi+f(x)j. Now where or ds = J(1+~)dx. dr dr dx [. f 'J 1 . rs = -d = -d '-d = 1+ xl . ~' = umt tangent vector. s x s V 1 +f; . I -fxi+j n = umt norma vector = ~ v 1+f; and note that n. t = n. rs = 0, since n is the normal and rs the tangent to the curve. Then dt dt dx (fxi + j)fxx rss = ds = dx' ds = (l +f;f . Hence APPENDIX: GEOMETRY OF SURFACES 149 A.2 Length measurement on a surface­ first fundamental form Before any useful analysis of surface geometry can be undertaken, certain preliminaries must be gone through. The first of these is to discuss the measurement of lengths on a surface. Consider the surface to be covered with a coordinate system of two independent variables u, v. There is no need to be specific about u and v at the moment, except to note that each point on the surface can be identified by a (u, v) pair of coordinates. Consider a typical point P on the surface (Fig. A.2), and a neighbouring point Q. Choose a fixed origin, 0, and let the point P be located by the radius vector (r) from Osuch that r = r(u,v) (A.2.0) Distances measured along the surface, say from P to Q, will be denoted by arc length s, or in this particular case by ds since the points P, Q are regarded as infinitesimally separated. From the vector relations indicated on Fig. A.2 it is seen that ds is the modulus of dr. In order to evaluate ds the standard approach is to form ds 2 = dr .dr. (A.2.l ) But (A.2.2) v o Figure A.2 Surface-element of arc. 150 BASIC PRINCIPLES OF PLATE THEORY since r = r(u, v), and where Hence ds 2 = (ru du+rydv). (rudu+rydv), =Edu 2 +2Fdudv+Gdv2, (A.2.3) =1. Here the following definitions have been made: (A.2.4) The expression (A.2.3) is known as the First Fundamental Form (1), and E, F, G, as the coefficients of this form. These definitions and (A.2.3) are very important and can be interpreted as follows. Suppose dv = O. Then Q is at q and the change du produces an arc length Pq which is given by ds = JE du, from (A.2.3). These are scalar quantities. But ru is a vector, and a tangent vector to the surface at P along the direction u increasing, namely P to q. Similarly ry is a tangent vector to the surface at P but along the direction v increasing, as indicated in Fig. A.2. This is not too surprising, but it is very important: to repeat, ru' r yare tangent vectors along the u, v lines respectively. Hence dr, obtained when both u and v vary to a point such as Q, is itself a tangent vector and relates to ru, ry via (A.2.2). These tangent vectors however are often not unit vectors. Roughly speaking they are seldom unit vectors when the u, v lines are curved in any sense. The reason is that such lack of straightness usually means that u, v are not naturally chosen to be arc lengths and this destroys the unit quality. Consider the cylinder, Fig. A.3. Choose u, v to be arc lengths around and along the cylinder, then r = O.i+vj+Rk and di dk. dj de = - k, de = I, de = O. Hence -+ dr = PQ = Rde.i+dv.j+O.k. But =i and ry =j. APPENDIX: GEOMETRY OF SURFACES 151 Figure A.3 Geometry on the cylinder (u = RO, du = R dO). Hence E == fu' fu = i. i = 1, F == fu' fv = i.j = 0, G == fv' fv = j.j = 1. In this case, with this choice of coofdinates, E = G = 1, F = 0. However, suppose u = e, with v, as before, the (axial) arc length (Fig. A.3). Then fu = fe = Ri, fv = j (as before). Hence Hence E ~ G, though F = °still. Yet other coordinates might be chosen, such as a helix and an axial length. In such cases, E, F and G take on non-zero, and probably more complicated, values. Exercise: Consider the surface z = Kxy, where x, y, z are cartesian axes (that is all three axes are straight and mutually perpendicular). If x, yare chosen to be the independent coordinates, show that the coefficients of the first fundamental form are E = 1 + (Ky)2, F = K2xy, G = 1 + (KX)2. The surface z = Kxy is known as a hyperbolic paraboloid (H.P.). The discussion thus far allows quite general choice of u, v coordinates. In the examples above the coordinates were orthogonal, and then F = 0, since fu and f v are orthogonal. If ¢ is the angle between the u, v lines, then by definition of the various quantities, fu.fv == F = JEG.cos¢. Hence 2 H sin¢ = J1-cos2¢ = Ffd1- - = fTir'{' where H2 == EG-F2. EG yEG Exercise: For the H.P. show that H2 = 1 + K2. (x2 +y2). 152 BASIC PRINCIPLES OF PLATE THEORY A.3 The normal to a surface At every point P on a smooth surface there exist at least two tangent vectors, for example ru and r y' The plane of these two vectors is the tangent plane at P in which all such tangent vectors lie. The vector normal to this plane at Pis the normal to the surface at P. It is convenient to define this normal vector to be of unit length as follows: ru x ry 0==---=--'--- (A.3.0) I ru x ry I The denominator in this vector product expression is given by jEGsin¢ = H, since ru, ry are inclined at ¢ and I ru I = )E, I ry I = .)G. Since the surface is smooth, every point on the surface possesses a unique normal, the only ambiguity being the direction in which 0 is considered positive. Once u and v are defined, then (A.3.0) selects the direction for 0 positive. When adjacent points such as P and Q of Fig. A.2 are considered, each has an associated normal. Generally these normals will be oriented in non­ parallel directions, though the vector difference between the two normals is infinitesimal by virtue of smoothness.