Problems for Chapter 10

Approximate Theories for Solids with Special Shapes:

Rods, Beams, Membranes, Plates and Shells

10.5.    Motion and Deformation of Thin Shells

10.5.1.    A spherical-polar coordinate system is to be used to describe the deformation of a spherical shell with radius R.Â  The two angles Â illustrated in the figure are to be used as the coordinate system Â for this geometry.

10.5.1.1.           Write down the position vector Â in terms of , expressing your answer as components in the basis Â shown in the figure.

10.5.1.2.           Calculate the covariant basis vectors Â in terms of Â Â and

10.5.1.3.           Calculate the contravariant basis vectors Â in terms of Â Â and

10.5.1.4.           Calculate the covariant, contravariant and mixed components of the metric tensor

10.5.1.5.           Calculate the covariant, contravariant and mixed components of the curvature tensor Â for the shell

10.5.1.6.           Find the components of the Christoffel symbol Â for the coordinate system

10.5.1.7.           Suppose that under loading the shell simply expands radially to a new radius r.Â  Find the components of the mid-plane Lagrange strain tensor Â and the components of the curvature change tensor

10.5.1.8.           Suppose that the shell is elastic, with Youngâ€™s modulus E and Poissonâ€™s ratio .Â  Calculate the contravariant components of the internal force Â and internal moment

10.5.1.9.           Find the physical components of the internal force and moment, expressing your answer as components in the spherical-polar basis of unit vectors

10.5.2.    A cylindrical-polar coordinate system is to be used to describe the deformation of a cylindrical shell with radius r.Â  The angles and axial distance Â illustrated in the figure are to be used as the coordinate system Â for this geometry.

10.5.2.1.           Write down the position vector Â in terms of , expressing your answer as components in the basis Â shown in the figure.

10.5.2.2.           Calculate the covariant basis vectors Â in terms of Â Â and

10.5.2.3.           Calculate the contravariant basis vectors Â in terms of Â Â and

10.5.2.4.           Calculate the covariant, contravariant and mixed components of the metric tensor

10.5.2.5.           Calculate the covariant, contravariant and mixed components of the curvature tensor Â for the shell

10.5.2.6.           Find the components of the Christoffel symbol Â for the undeformed shell

10.5.2.7.           Suppose that under loading the shell simply expands radially to a new radius , without axial stretch.Â  Find the covariant components of the mid-plane Lagrange strain tensor Â and the covariant components of the curvature change tensor

10.5.2.8.           Suppose that the shell is elastic, with Youngâ€™s modulus E and Poissonâ€™s ratio .Â  Calculate the contravariant components of the internal force Â and internal moment .

10.5.2.9.           Find the physical components of the internal force and moment, expressing your answer as components in the spherical-polar basis of unit vectors

10.5.3.    The figure illustrates a triangular plate, whose geometry can be described by two vectors a and b parallel to two sides of the triangle.Â  The position vector of a point is to be characterized using a coordinate system Â by setting Â where , .

10.5.3.1.           Calculate the covariant basis vectors Â in terms of a and b

10.5.3.2.           Calculate the contravariant basis vectors Â in terms of a and b

10.5.3.3.           Calculate the covariant, contravariant and mixed components of the metric tensor

10.5.3.4.           Suppose that the plate is subjected to a homogeneous deformation, so that after deformation its sides lie parallel to vectors Â and .Â  Find the mid-plane Lagrange strain tensor , in terms of a, b, Â and

10.5.3.5.           Suppose that the plate is elastic, with Youngâ€™s modulus E and Poissonâ€™s ratio .Â  Calculate the contravariant components of the internal force

10.5.4.    The figure illustrates a triangular plate. The position points in the plate is to be characterized using the height Â and angle Â as the coordinate system .

10.5.4.1.           Calculate the covariant basis vectors Â expressing your answer as components in the basis Â shown in the figure.

10.5.4.2.           Calculate the contravariant basis vectors Â Â as components in

10.5.4.3.           Calculate the covariant, contravariant and mixed components of the metric tensor

10.5.4.4.           Find the components of the Christoffel symbol Â for the undeformed plate

10.5.4.5.           Suppose that the plate is subjected to a deformation such that the position vector of a point that lies at Â in the undeformed shell has position vector Â after deformation Find the mid-plane Lagrange strain tensor , in terms of

10.5.4.6.           Suppose that the plate is elastic, with Youngâ€™s modulus E and Poissonâ€™s ratio .Â  Calculate the contravariant components of the internal force

10.5.4.7.           Find the physical components of the internal force T as components in the Â basis.

(c) A.F. Bower, 2008
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