Learning Goal:

Consider a girder attached to two columns in a building frame $($Figure $1).$ In general, the columns are not perfectly rigid, so the girder ends are not fixed. The column connections also are not exactly equivalent to pin joints. Instead, they offer some resistance to rotation. An exact analysis of a load applied to the frame requires more advanced techniques; however, approximate methods exist that can be used under certain conditions for analysis.

Using an exact analysis of a fixed, supported girder of length $L$ subject to a uniform load shows that the points of zero moment are $0\mathrm{.}21L$ from the ends. A simply supported beam subject to a uniform load has zero moment at the end points. As an approximation, we can assume the zero moment point is $0\mathrm{.}1L$ from the ends of a girder supported by columns. Under these assumptions, there is no axial force in the girder.

The moment distribution in the girder can be approximated by modeling the system as a simply supported beam of length $0\mathrm{.}8L$ resting on two fixed supported cantilever beams of length $0\mathrm{.}1L($Figure $2).$ This forms a statically determinate system that can be analyzed using the standard methods for beams. This approximation provides the moment distribution in the girder, but does not provide information about the moment and force reactions on the frame.

Figure

$3$ of $4$

The girder in the frame $($Figure $3)$ is subject to a uniform load of $w=3800\frac{N}{m}$ over its length $L=8\mathrm{.}40m.$ Use this frame for Parts A and B below.

Part A $-$ Moment at the ends of the girder

What is the approximate moment in the girder at the face of the column at $C?$ Use the standard sign convention for beams.

Express your answer to three significant figures and include the appropriate units.

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Part B $-$ Maximum positive moment

What is the approximate maximum positive moment in the girder?

Express your answer to three significant figures and include the appropriate units.

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Part C $-$ Draw the approximate moment distribution