##### From Theory of Beam Columns: In-Plane Behavior and Design, Volume 1

## 3.3 SOLUTIONS FOR BEAM-COLUMNS (DEFLECTION PROBLEM)

Columns subjected to lateral loads or end moments in addition to axial compression are categorized as beam-columns. The lateral loads or end moments cause deflection which is further amplified by the axial compression. The main result of the above discussion is the elastic beam-column equation (3.19). The general solutions are given by Eq. (3.22), The problem here is to find the function *f*( *x*) which depends only on the lateral loading and the constants *A*, *B*, *C*, *D* that satisfy the boundary conditions at the clamped, free, or simply supported ends.

#### Beam-Column with Uniform Load

As an example. consider a beam-column which is subjected to uniform load *q* as well as an axial compression *P* as shown in Fig. 3.9(a). The beam-column has length *l* and bending rigidity *EI* which is assumed to be constant along the length.

Figure 3.9: Beam-column with uniform load

The governing equation (3.19) is

(3.23) | |

and the general solution is

(3.24) | |

The four constants *A*, *B*, *C* and *D* are determined from the four boundary conditions,

(3.25) | |

Actual calculation of Eq. (3.22) gives

(3.26) | |

Thus the deflection equation (3.24) yields

(3.27) | |

It is known from the above expression that the deflection is linear to the lateral load *q* but not to the axial compression *P* (or ). The deflection pattern also changes as the axial force changes.

The maximum deflection ? occurs...

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