Stress linearization formulas for axisymmetric models

For an axisymmetric model, the components of stress expressed in terms of a local (x,y,z) coordinate system are as follows:

\sigma = \left[ {\begin{array}{*{20}{c}} {{\sigma _{xx}}}&{{\sigma _{xy}}}&0\ {{\sigma _{xy}}}&{{\sigma _{yy}}}&0\ 0&0&{{\sigma _{zz}}} \end{array}} \right]

where the directions are given as follows:

{\hat n_x}

Unit vector that is aligned with and has the same positive sense as the stress classification line (SCL)

{\hat n_z}

Unit vector in the hoop direction whose positive sense is:

• The same as the Y-axis of the absolute (X,Y,Z) coordinate system if the axisymmetric plane is the absolute XZ-plane

• The same as the Z-axis of the absolute (X,Y,Z) coordinate system if the axisymmetric plane is the absolute XY-plane

{\hat n_y}

Unit vector in the meridional direction obtained from the cross product of the unit vectors in the local x- and z-axes such that {\hat n_y} = {\hat n_z} \times {\hat n_x}

The following figure shows the absolute and local coordinate systems, the SCL, and the other physical distances that are used to calculate the linearized stresses.

Note:

To use stress linearization on axisymmetric models, you must enable the StressLinearization_AxisymmetricStructure early access feature. For more information, see Enable an early access feature.

Meridional membrane and bending stress formulas

The force and moment resultants in the meridional direction over angle \Delta \theta are:

{F_y} = \Delta \theta \int\limits_{ - t/2}^{t/2} {{\sigma _{yy}}Rdx}

{M_y} = \Delta \theta \int\limits_{ - t/2}^{t/2} {x{\sigma _{yy}}Rdx}

where R = R\left( x \right) = {R_c} + x,\cos \phi is the radial position of a point on the SCL and t is the length of the SCL.

Assume the normal stress in the meridional direction is distributed linearly along the SCL such that:

{\sigma _{y,f}}\left( x \right) = \sigma _y^m + \left( {\frac{{2\sigma _y^b}}{t}} \right)x

where \sigma _y^m and \sigma _y^b are the membrane and bending stresses in the meridional direction.

The force and moment resultants in the meridional direction over angle \Delta \theta for this stress distribution are:

{F_{y,f}} = \Delta \theta \left[ {\sigma _y^m{R_c},t + \sigma _y^b\left( {\frac{{{t^2}}}{6}} \right)\cos \phi } \right]

{M_{y,f}} = \Delta \theta \left[ {\sigma _y^m\left( {\frac{{{t^3}}}{{12}}} \right)\cos \phi + \sigma _y^b{R_c}\left( {\frac{{{t^2}}}{6}} \right)} \right]

To find the position of the neutral plane relative to the midsurface, {x_f}, equate the moment that {F_{y,f}} produces about the neutral plane to the moment {M_{y,f}} without the contribution from the bending stress. Rearranging and simplifying yields:

{x_f} = \frac{{{t^2}\cos \phi }}{{12{R_c}}}

However, if {R_c} < 0.001,t, the software calculates {x_f} as follows:

{x_f} = \frac{{{t^2}}}{{12\rho }}

The force resultant in the meridional direction acts over an area equal to {R_c},t,\Delta \theta . Thus, membrane stress in the meridional direction is:

\sigma y^m = \frac{1}{{{R_c}t}}\int\limits{ - t/2}^{t/2} {{\sigma _{yy}}Rdx}

The second moment of the area about which meridional bending occurs is:

{I_y} = \Delta \theta ,{R_c},t\left( {\frac{{{t^2}}}{{12}} - x_f^2} \right)

The meridional bending stress at the innermost fiber, point A, and outermost fiber, point B, are as follows:

\sigma _{y,B}^b = \frac{{{M_y}\left( {{x_B} - {x_f}} \right)}}{{{I_y}}}

\sigma _{y,A}^b = \frac{{{M_y}\left( {{x_A} - {x_f}} \right)}}{{{I_y}}}

Combining the formulas for the moment resultant in the meridional direction and the second moment of the area, yields the formulas for the meridional bending stress at the innermost and outermost fibers.

\sigma {y,A}^b = \frac{{\left( {{x_A} - {x_f}} \right)}}{{{R_c}t\left( {\frac{{{t^2}}}{{12}} - x_f^2} \right)}}\int\limits{ - t/2}^{t/2} {x{\sigma _{yy}}Rdx}

\sigma {y,B}^b = \frac{{\left( {{x_B} - {x_f}} \right)}}{{{R_c}t\left( {\frac{{{t^2}}}{{12}} - x_f^2} \right)}}\int\limits{ - t/2}^{t/2} {x{\sigma _{yy}}Rdx}

Hoop membrane and bending stress formulas

The force and moment resultants in the hoop direction over angle \Delta \phi are:

{F_z} = \Delta \phi \int\limits_{ - t/2}^{t/2} {{\sigma _{zz}}\left( {\rho + x} \right)dx}

{M_z} = \Delta \phi \int\limits_{ - t/2}^{t/2} {\left( {x - {x_h}} \right){\sigma _{zz}}\left( {\rho + x} \right)dx}

where {x_h} is similar to {x_f} and is given by:

{x_h} = \frac{{{t^2}}}{{12\rho }}

The force resultant in the hoop direction acts over an area equal to \rho t\Delta \phi . Thus, membrane stress in the hoop direction is:

\sigma z^m = \frac{1}{{\rho t}}\int\limits{ - t/2}^{t/2} {{\sigma _{zz}}\left( {\rho + x} \right)dx}

The second moment of the area about which hoop bending occurs is:

{I_z} = \rho t\Delta \phi \left( {\frac{{{t^2}}}{{12}} - x_h^2} \right)

The hoop bending stress at the innermost fiber, point A, and outermost fiber, point B, are as follows:

\sigma _{z,B}^b = \frac{{{M_z}\left( {{x_B} - {x_h}} \right)}}{{{I_z}}}

\sigma _{z,A}^b = \frac{{{M_z}\left( {{x_A} - {x_h}} \right)}}{{{I_z}}}

Combining the formulas for the moment resultant in the hoop direction and the second moment of the area, yields the formulas for the hoop bending stress at the innermost and outermost fibers.

\sigma {z,A}^b = \frac{{\left( {{x_A} - {x_h}} \right)}}{{\rho t\left( {\frac{{{t^2}}}{{12}} - x_h^2} \right)}}\int\limits{ - t/2}^{t/2} {\left( {x - {x_h}} \right){\sigma _{zz}}\left( {\rho + x} \right)dx}

\sigma {z,B}^b = \frac{{\left( {{x_B} - {x_h}} \right)}}{{\rho t\left( {\frac{{{t^2}}}{{12}} - x_h^2} \right)}}\int\limits{ - t/2}^{t/2} {\left( {x - {x_h}} \right){\sigma _{zz}}\left( {\rho + x} \right)dx}

Stress linearization formulas in the direction of the SCL

The average normal stress in the direction of the SCL is given by:

\sigma x^m = \frac{1}{t}\int\limits{ - t/2}^{t/2} {{\sigma _{xx}}dx}

Note:

When you display the stress linearization results in the Information window, the average normal stress results in the direction of the SCL are listed with the membrane stresses in the XX column of the table.

The linearized normal stress in the direction of the SCL at the innermost and outermost fibers are given by:

\sigma _{x,A}^b = {\sigma _{x,A}} - \sigma _x^m

\sigma _{x,B}^b = {\sigma _{x,B}} - \sigma _x^m

Note:

When you display the stress linearization results in the Information window, the linearized normal stress results in the direction of the SCL are listed with the bending stresses in the XX column of the table.

Shear stress effects

The average shear stress along the direction of the SCL is given by:

\sigma {xy}^m = \frac{1}{{{R_c}t}}\int\limits{ - t/2}^{t/2} {{\sigma _{xy}}Rdx}

Note:

When you display the stress linearization results in the Information window, the average shear stress results along the direction of the SCL are listed with the membrane stresses in the XY column of the table.

The software does not compute a linearized shear stress along the direction of the SCL because {\sigma _{xy}} is assumed to be zero at the innermost and outermost fibers.

Note:

When you display the stress linearization results in the Information window, the linearized shear stress result along the direction of the SCL of zero is listed with the bending stresses in the XY column of the table.

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