L Ei

where kL sin kL kL 2 cos kL

and kL 2 kL sin kL

are referred to as the stability functions.

Equations 2.308 and 2.309 are the slope deflection equations for a beam-column that is not subjected to transverse loading and relative joint translation. When P approaches zero, kL = (/P/EI)L approaches zero, and by using L'Hopital's rule, it can be shown that sij = 4 and sij = 2. Values for sii and sij for various values of kL are plotted as shown in Figure 2.118. Equations 2.309 and 2.310 are valid if the following conditions are satisfied.

1. The beam is prismatic.

2. There is no relative joint displacement between the two ends of the member.

3. The member is continuous, that is, there is no internal hinge or discontinuity in the member.

4. There is no in-span transverse loading on the member.

5. The axial force in the member is compressive.

If these conditions are not satisfied, some modifications to the slope deflection equations are necessary.

FIGURE 2.118 Plot of stability functions.

FIGURE 2.119 Beam-column subjected to end moments and side sway. Member Subjected to Side Sway

If there is a relative joint translation, D, between the member ends, as shown in Figure 2.119, the slope-deflection equations are modified as

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