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.  The two angles  illustrated in the figure are to be used as the coordinate system  for this geometry.

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

10.5.1.2.           Calculate the covariant basis vectors  in terms of   and  

10.5.1.3.           Calculate the contravariant basis vectors  in terms of   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  for the shell

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

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

10.5.1.8.           Suppose that the shell is elastic, with Young’s modulus E and Poisson’s ratio .  Calculate the contravariant components of the internal force  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.  The angles and axial distance  illustrated in the figure are to be used as the coordinate system  for this geometry.

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

10.5.2.2.           Calculate the covariant basis vectors  in terms of   and  

10.5.2.3.           Calculate the contravariant basis vectors  in terms of   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  for the shell

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

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

10.5.2.8.           Suppose that the shell is elastic, with Young’s modulus E and Poisson’s ratio .  Calculate the contravariant components of the internal force  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.  The position vector of a point is to be characterized using a coordinate system  by setting  where , .

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

10.5.3.2.           Calculate the contravariant basis vectors  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  and .  Find the mid-plane Lagrange strain tensor , in terms of a, b,  and  

10.5.3.5.           Suppose that the plate is elastic, with Young’s modulus E and Poisson’s ratio .  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  and angle  as the coordinate system .

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

10.5.4.2.           Calculate the contravariant basis vectors   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  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  in the undeformed shell has position vector  after deformation Find the mid-plane Lagrange strain tensor , in terms of  

10.5.4.6.           Suppose that the plate is elastic, with Young’s modulus E and Poisson’s ratio .  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  basis.

 

 

 

 

(c) A.F. Bower, 2008
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