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Chapter 13 Buckling of Columns

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Buckling When designing a member one has to evaluate strength, deflection, and stability requirements When designing a member one has to evaluate strength, deflection, and stability requirements For a long and slender member subjected to compressive loadings, the loadings may be large enough to cause the member to deflect laterally or sidesway For a long and slender member subjected to compressive loadings, the loadings may be large enough to cause the member to deflect laterally or sidesway Long slender members subjected to an axial compressive force are called columns, and the lateral deflection that occurs is called buckling Long slender members subjected to an axial compressive force are called columns, and the lateral deflection that occurs is called buckling Buckling of a column can lead to a sudden and dramatic failure of a structure or mechanism Buckling of a column can lead to a sudden and dramatic failure of a structure or mechanism

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Relating Internal Moment in Column to Deflected Shape Assuming the ends are free to rotate (a free end or pinned support) and beginning with the same equation used in determining deflection Assuming the ends are free to rotate (a free end or pinned support) and beginning with the same equation used in determining deflection

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Determining the Critical Buckling Load (Euler Load)

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Mode Shapes Increasing the value of n changes the mode shape and increases the value of P needed to induce such a mode shape (higher order mode shapes would only result if additional constraints were present) Increasing the value of n changes the mode shape and increases the value of P needed to induce such a mode shape (higher order mode shapes would only result if additional constraints were present)

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Slenderness Ratio Recall the geometric quantity referred to as radius of gyration, Recall the geometric quantity referred to as radius of gyration,

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Other Types of Supports The Euler formula was developed for a column having ends that are pinned or free to rotate - L in the equation represents the unsupported distance between points of zero moment The Euler formula was developed for a column having ends that are pinned or free to rotate - L in the equation represents the unsupported distance between points of zero moment Euler's formula can be used to determine the critical load for other types of supports at the ends provided "L" represents the distance between the zero- moment points - this "L" is referred to as the effective length, L e = KL (where K is the effective-length factor) Euler's formula can be used to determine the critical load for other types of supports at the ends provided "L" represents the distance between the zero- moment points - this "L" is referred to as the effective length, L e = KL (where K is the effective-length factor) A more generalized form of the Euler formula A more generalized form of the Euler formula

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Effective-length Factors Problems, pg 671 Problems, pg 671

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