a) The critical loads for columns with various kinds of support conditions can be determined from the differential equation of the deflection curve by following the same procedure as above

__Step 1__** **With the column assumed to be in the buckled state, we obtain an ** expression for the bending moment** in the column.

__Step 2__** **Set up the ** differential equation** of the deflection curve, using the bending-moment equation

__Step 3__** **** Solve the equation** and obtain its general solution, which contains two constants of integration plus any other unknown quantities

*.*

__Step 4__** **** Apply boundary conditions** pertaining to the deflection v and the slope v

^{’}and obtain a set of simultaneous equations.

__Step 5__** **Obtain the equation of deflection curve for buckled column

** Step 6** Solve those equations to obtain the

*critical load*Fixed-Free column | Fixed-fixed Column | Fixed-pinned column | |

Step 1 | |||

Step 2 | |||

Step 3 | |||

Step 4 | |||

Step 5 | Applying third boundary condition | Applying third boundary condition | Applying third boundary condition |

Step 6 | Lowest critical load | Lowest critical load | Lowest critical load |

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