Vlasov torsion theory - Revision history

Ok: Text replacement - "" to " $"$

2024-10-22T00:35:25Z Text replacement - "<tex>" to "<math>"

← Older revision		Revision as of 02:35, 22 October 2024
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	The rotation of the beam cross-section follows the following differential equation (Hoogenboom 2006):		The rotation of the beam cross-section follows the following differential equation (Hoogenboom 2006):

	<~~tex~~>\;\;\;EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;		<math>\;\;\;EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;
	GI_t\;\frac{d^2 \phi}{dx^2}\;=\;		GI_t\;\frac{d^2 \phi}{dx^2}\;=\;
	m_x</math>		m_x</math>
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	These stresses are combined with stresses due to axial, bending and shear loading. For example the total axial stress can be obtained as follows (Hoogenboom 2006):		These stresses are combined with stresses due to axial, bending and shear loading. For example the total axial stress can be obtained as follows (Hoogenboom 2006):

	<~~tex~~>\;\;\;\sigma_{xx}\;=\;		<math>\;\;\;\sigma_{xx}\;=\;
	\frac{N}{A}\;+\;		\frac{N}{A}\;+\;
	\frac{M_y}{I_y}\;z\;-\;		\frac{M_y}{I_y}\;z\;-\;

2024-10-22T00:10:42Z

Text replacement - "</tex>" to "</math>"

← Older revision		Revision as of 02:10, 22 October 2024
Line 10:		Line 10:
	<tex>\;\;\;EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;		<tex>\;\;\;EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;
	GI_t\;\frac{d^2 \phi}{dx^2}\;=\;		GI_t\;\frac{d^2 \phi}{dx^2}\;=\;
	m_x</~~tex~~>		m_x</math>

	Refer to [[Torsion#Notation\|torsion page]] for the notation used in the above equation.		Refer to [[Torsion#Notation\|torsion page]] for the notation used in the above equation.
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	\frac{M_y}{I_y}\;z\;-\;		\frac{M_y}{I_y}\;z\;-\;
	\frac{M_z}{I_z}\;y\;+\;		\frac{M_z}{I_z}\;y\;+\;
	\frac{B}{I_{\omega}}\;\omega</~~tex~~>		\frac{B}{I_{\omega}}\;\omega</math>

	== References ==		== References ==

2009-09-22T04:01:37Z

References

← Older revision		Revision as of 06:01, 22 September 2009
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	== References ==		== References ==

	* V.Z. Vlasov, "Thin-Walled Elastic Bars" (in Russian), 2nd ed., Fizmatgiz, Moscow, 1959.		* V.Z. Vlasov: "Thin-Walled Elastic Bars" (in Russian), 2nd ed., Fizmatgiz, Moscow, 1959.
			* Zdenek P. Bazant: Nonuniform torsion of thin-walled bars of variable cross section, Publications, IABSE, 25, p. 245-167, 1965.

	== External Links ==		== External Links ==

2009-09-22T03:46:13Z

External Links

← Older revision		Revision as of 05:46, 22 September 2009
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	\frac{M_z}{I_z}\;y\;+\;		\frac{M_z}{I_z}\;y\;+\;
	\frac{B}{I_{\omega}}\;\omega</tex>		\frac{B}{I_{\omega}}\;\omega</tex>

			== References ==

			* V.Z. Vlasov, "Thin-Walled Elastic Bars" (in Russian), 2nd ed., Fizmatgiz, Moscow, 1959.

	== External Links ==		== External Links ==

2009-09-22T03:36:37Z

← Older revision		Revision as of 05:36, 22 September 2009
Line 8:		Line 8:
	The rotation of the beam cross-section follows the following differential equation (Hoogenboom 2006):		The rotation of the beam cross-section follows the following differential equation (Hoogenboom 2006):

	<tex>EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;		<tex>\;\;\;EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;
	GI_t\;\frac{d^2 \phi}{dx^2}\;=\;		GI_t\;\frac{d^2 \phi}{dx^2}\;=\;
	m_x</tex>		m_x</tex>

2009-09-22T03:36:21Z

Stresses

← Older revision		Revision as of 05:36, 22 September 2009
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	# shear stresses due to restrained warping		# shear stresses due to restrained warping
	# normal stresses due to restrained warping		# normal stresses due to restrained warping

			These stresses are combined with stresses due to axial, bending and shear loading. For example the total axial stress can be obtained as follows (Hoogenboom 2006):

			<tex>\;\;\;\sigma_{xx}\;=\;
			\frac{N}{A}\;+\;
			\frac{M_y}{I_y}\;z\;-\;
			\frac{M_z}{I_z}\;y\;+\;
			\frac{B}{I_{\omega}}\;\omega</tex>

	== External Links ==		== External Links ==

2009-09-22T03:31:41Z

← Older revision		Revision as of 05:31, 22 September 2009
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	Refer to [[Torsion#Notation\|torsion page]] for the notation used in the above equation.		Refer to [[Torsion#Notation\|torsion page]] for the notation used in the above equation.

			== Stresses ==

			According to Vlasov to the theory, the applied torque causes the following three types of stresses:

			# shear stresses due to unrestrained torsion
			# shear stresses due to restrained warping
			# normal stresses due to restrained warping

	== External Links ==		== External Links ==

2009-09-22T03:23:10Z

← Older revision		Revision as of 05:23, 22 September 2009
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	== Differential Equation ==		== Differential Equation ==

			The rotation of the beam cross-section follows the following differential equation (Hoogenboom 2006):

	<tex>EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;		<tex>EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;

2009-09-22T03:21:52Z

← Older revision		Revision as of 05:21, 22 September 2009
Line 9:		Line 9:
	GI_t\;\frac{d^2 \phi}{dx^2}\;=\;		GI_t\;\frac{d^2 \phi}{dx^2}\;=\;
	m_x</tex>		m_x</tex>

			Refer to [[Torsion#Notation\|torsion page]] for the notation used in the above equation.

2009-09-22T03:19:23Z

← Older revision		Revision as of 05:19, 22 September 2009
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			== Assumptions ==

			* Warping can be constrained
			* Derivative of section rotation about the axis of the twist is not constant

			== Differential Equation ==

			<tex>EI_{\omega}\;\frac{d^4 \phi}{dx^4}\;-\;
			GI_t\;\frac{d^2 \phi}{dx^2}\;=\;
			m_x</tex>


	== External Links ==		== External Links ==