D-3Effective length for double stepped columns

Effective lengths in the plane of steppings (bending about axis z-z) for bottom, middle and top parts for a double stepped column shall be taken as follows (see also Fig. 29):

Coefficient $K_{1}$ for effective length of bottom part of double stepped column shall be taken from the formula:

$K_{1}=\sqrt{\frac{t_{1} K_{1}^{2}+\left(t_{2} K_{2}^{2}+K_{3}^{2}\right) \times\left(1+n_{2}\right)^{2} \times \frac{I_{1}}{I_{\mathrm{av}}^{\prime}}}{1+t_{1}+t_{2}}}$

where

$K_{1}, K_{2}$, and $K_{3}$ are taken from Table 4

$=\frac{P_{1}}{P_{3}}, \quad t_{2}=\frac{P_{2}}{P_{3}} \quad n_{2}=\frac{L_{2}}{L_{1}}$

$\begin{aligned} I_{\mathrm{av}}^{\prime} & =\text { average value of moment of inertia for the } \\ & \text { lower and middle parts }\end{aligned}$

$=\frac{I_{1} L_{1}+I_{2} L_{2}}{L_{1}+L_{2}}$

$I_{\mathrm{av}}^{\prime \prime}=$ average value of moment of inertia for the middle and top parts

12L2+13L3

L₁₂+ L3

Value of coefficient Kz for middle part of column is given by formula:

$K_{2}=\frac{K_{1}}{C_{2}}$, and

coefficient $K_{3}$ for the top part of the column is given by:

$K_{3}=\frac{K_{1}}{c} \leq 3$

where

$C_{2}=\frac{L_{2}}{L_{1}} \sqrt{\frac{I_{1}\left(P_{2}+P_{3}\right)}{I_{2}\left(P_{1}+P_{2}+P_{3}\right)}}$, $C_{3}=\frac{L_{3}}{L_{1}} \sqrt{\frac{I_{1} P_{3}}{I_{3}\left(P_{1}+P_{2}+P_{3}\right)}}$

NOTE — The provisions of D-3 are applicable to intermediate columns as well with steppings on either side, provided appropriate values of $I_{1}, I_{2}$ and $I_{3}$ are taken.

Figure description:

The image contains engineering diagrams of a stepped column under axial loads.

Entities and Details:

• Stepped Column (Diagram a: A vertical structural member fixed at the base with three distinct segments.
  • Segments: Three sections labeled with lengths $L_1$ (bottom, $L_2$ (middle, and $L_3$ (top.
  • Moments of Inertia: Corresponding stiffness properties $I_1, I_2, I_3$ for each segment.
  • Applied Loads: Vertical axial loads $P_3$ at the top, $P_2$ at the first step, and $P_1$ at the second step.
• Cross-Section: Below the main diagram, showing a double I-beam configuration oriented along y-y and z-z axes.

Figure description:

• Subject: Structural diagram of a vertical column/member labeled (b.
• Base: Fixed support.
• Lower Section:
  • Height: $L_1$
  • Moment of Inertia: $I_1$
• Upper Section:
  • Height: $L_2 + L_3$
  • Average Moment of Inertia: $I''_{av}$
• Load: Downward point load $P_1$ applied at the junction between the lower and upper sections.

Figure description:

• Diagram Type: Structural mechanics diagram of a stepped column fixed at the base.
• Segments and Dimensions:
  • Lower segment: Length labeled $L_1 + L_2$ with property $I'_{av}$.
  • Upper segment: Length labeled $L_3$ with property $I_3$.
• Applied Load: Vertical downward force $P_2$ acting on the shoulder where the cross-section changes.
• Label: (c)

Figure description:

Key Information from FIG. 29 (d

This technical diagram illustrates a structural model for a double stepped column fixed at the base.

Key Entities and Parameters:

• Applied Load ($P_3$: Vertical force acting at the top of the column.
• Upper Section: Characterized by length $L_3$ and moment of inertia $I_3$.
• Lower Section: Represented by combined length $L_1 + L_2$ and average moment of inertia $I'_{av}$.
• Base: Indicated as a fixed-end support.

FIG. 29 EFFECTIVE LENGTH OF DOUBLE STEPPED COLUMNS