Annex C(Clause 13.2.2)

NBC 2016 (SP 7:2016) Volume 1

2016 · PDF page 1069

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Determination of Effective Length of Columns

C-1 METHOD FOR DETERMINING EFFECTIVE LENGTH OF COLUMNS IN FRAMES

In the absence of a more exact analysis, the effective length of columns in framed structures may be obtained by multiplying the actual length of the column between the centres of laterally supporting members (beams) given in Fig. 27 and Fig. 28 with the effective length factor $K$ , calculated by using the equations given below, provided the connection between beam and column is rigid type:

a) Non-sway frames (Braced frame) [(see 10.1.2 (a)]

A frame is designated as non-sway frame if the relative displacement between the two adjacent floors is restrained by bracings or shear walls (see 10.1.2). The effective length factor, K, of column in non-sway frames is given by (see Fig. 27):

K = \frac {\left[ 1 + 0 . 1 4 5 \left(\beta_ {1} + \beta_ {2}\right) - 0 . 2 6 5 \beta_ {1} \beta_ {2} \right]}{\left[ 2 - 0 . 3 6 4 \left(\beta_ {1} + \beta_ {2}\right) - 0 . 2 4 7 \beta_ {1} \beta_ {2} \right]}

b) Sway frames (Moment Resisting Frames) [see 10.1.2 (b)]

The effective length factor $K$ , of column in sway frames is given by (see Fig. 28):

Chart

FIG. 27 COLUMN EFFECTIVE LENGTH FACTOR - NON-SWAY FRAME

Chart

FIG. 28 COLUMN EFFECTIVE LENGTH FACTOR — SWAY FRAME

K = \left[ \frac {1 - 0 . 2 \left(\beta_ {1} + \beta_ {2}\right) - 0 . 1 2 \beta_ {1} \beta_ {2}}{1 - 0 . 8 \left(\beta_ {1} + \beta_ {2}\right) + 0 . 6 \beta_ {1} \beta_ {2}} \right] ^ {0. 5}

where

\beta_ {1}, \beta_ {2} \text {are given,} \beta = \frac {\sum K}{\sum K _ {\mathrm{c}} + \sum K _ {\mathrm{b}}}

$K_{c}, K_{b} =$ Effective flexural stiffness of the columns and beams meeting at the joint at the ends of the columns and rigidly connected at the joints, and these are calculated by:

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

K = C (I / L)

$I =$ moment of inertia of the member about an axis perpendicular to the plan of the frame,

$L =$ length of the member equal to centre-to-centre distance of the intersecting member, and

C = correction factor as shown in Table 35.

Table 35Correction Factors for Effective Flexural Stiffness (Clause C-1)

Sl No.	Far End Condition	Correction Factor, C
Sl No.	Far End Condition	Braced Frame	Unbraced Frame
(1)	(2)	(3)	(4)
i)	Pinned	1.5(1- $\bar{n}$ )	1.5 (1- $\bar{n}$ )
ii)	Rigidly connected to column	1.0(1- $\bar{n}$ )

C-2 Method for Determining Effective Length for Stepped Columns (see 13.2.2)

C-2.1 Single Stepped Columns

Effective length in the plane of stepping (bending about axis z-z) for bottom and top parts for single stepped column shall be taken as given in Table 36.

NOTE — The provisions of C-2.1 are applicable to intermediate columns as well with stepping on either side, provided appropriate values of $I_{1}$ and $I_{2}$ are taken.

C-3 Effective 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}}}

+

β2

)

−

0.247

β1

β2

]

[1+0.145(β1+β2)−0.265β1β2]

+

β2

)

+

0.6

β1

β2

1−0.2(β1+β2)−0.12β1β2

]

0.5

c

+

∑

Kb

∑K

K1

=

1+t1+t2t1K12+(t2K22+K32)×(1+n2)2×Iav′I1

FIG. 29 EFFECTIVE LENGTH OF DOUBLE STEPPED COLUMNS

Table 36Effective Length of Singled Stepped Columns (Clause C-2.1)

Sl No.(1)	Degree of End Restraint	Effective Length Coefficients(3)	Column Parameters for all Cases(4)
i)	Effectively held in position and restrained against rotation at both ends	$K_{1} = \sqrt{\frac{K_{12}^{2} + K_{11}^{2} (\alpha - 1)}{\alpha}}�M1�K_{2} = \frac{K_{1}}{C_{1}} \leq 3$ where $K_{12}$ and $K_{11}$ are to be taken as per Table 38	$\alpha = \frac{P_{1} + P_{2}}{P_{2}}�M2�\frac{i_{2}}{i_{1}} = \frac{I_{2}}{L_{2}} \times \frac{L_{1}}{I_{1}}$
iii)	Effectively held in position and restrained against rotation at bottom end, and top end held against rotation but not held in position	$K_{1}$ to be taken as per Table 39 $K_{2} = \frac{K_{1}}{C_{1}} \leq 3$
iv)	Effectively held in position and restrained against rotation at bottom end, and top end neither held against rotation nor held in position	$K_{1}$ to be taken as per Table 40 $K_{2} = \frac{K_{1}}{C_{1}} \leq 3$

Table 37 Coefficients of Effective Lengths $K_{12}$ and $K_{11}$ for Columns with Both Ends Effectively Held in Position and Restrained Against Rotation

(Table 36)

$I_2/I_1$	Coefficients $K_{12}$ and $K_{11}$ for $L_2/L_1$ Equal to
$I_2/I_1$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	1.2	1.4	1.6	1.8	2.0
Coefficient $K_{12} (P_1 = 0)$
0.05	0.74	0.94	1.38	1.60	1.87	2.07	2.23	2.39	2.52	2.67	3.03	3.44	3.85	4.34	4.77
0.1	0.67	0.76	1.00	1.20	1.42	1.61	1.78	1.92	2.04	2.20	2.40	2.60	2.86	3.18	3.41
0.2	0.64	0.70	0.79	0.93	1.07	1.23	1.41	1.50	1.60	1.72	1.92	2.11	2.28	2.45	2.64
0.3	0.62	0.68	0.74	0.85	0.95	1.06	1.18	1.28	1.39	1.48	1.67	1.82	1.96	2.12	2.20
0.4	0.60	0.66	0.71	0.77	0.82	0.93	0.99	1.08	1.17	1.23	1.39	1.53	1.66	1.79	1.92
0.5	0.59	0.65	0.70	0.77	0.82	0.93	0.99	1.08	1.17	1.23	1.39	1.53	1.66	1.79	1.92
1.0	0.55	0.60	0.65	0.70	0.75	0.80	0.85	0.90	0.95	1.00	1.10	1.20	1.30	1.40	1.50
Coefficient $K_{11} (P_2 = 0)$
0.05	0.65	0.67	0.71	0.85	1.01	1.17	1.31	1.41	1.50	1.57	1.67	1.74	1.78	1.82	1.86
0.1	0.64	0.65	0.65	0.65	0.78	0.92	1.05	1.15	1.25	1.33	1.45	1.55	1.62	1.68	1.71
0.2	0.62	0.64	0.65	0.65	0.66	0.73	0.83	0.92	1.01	1.09	1.23	1.33	1.41	1.48	1.54
0.3	0.60	0.63	0.64	0.65	0.66	0.67	0.73	0.81	0.89	0.94	1.09	1.20	1.28	1.35	1.41
0.4	0.58	0.63	0.63	0.64	0.64	0.66	0.68	0.75	0.82	0.88	1.01	1.10	1.19	1.26	1.32
0.5	0.57	0.61	0.63	0.64	0.64	0.65	0.68	0.72	0.77	0.83	0.94	1.04	1.12	1.19	1.25
1.0	0.55	0.58	0.60	0.61	0.62	0.63	0.65	0.67	0.70	0.73	0.80	0.88	0.93	1.01	1.05
NOTE — Intermediate value may be obtained by interpolation

Table 38 Coefficients of Effective Lengths $K_{12}$ and $K_{11}$ for Columns with Both Ends Effectively Held in Position and Restrained Against Rotation at Bottom End Only

(Table 36)

$I_2/I_1$	Coefficients $K_{12}$ and $K_{11}$ for $L_2/L_1$ Equal to
$I_2/I_1$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	1.2	1.4	1.6	1.8	2.0
Coefficient $K_{12} (P_1 = 0)$
0.05	0.97	1.66	2.10	2.43	2.72	2.92	3.08	3.24	3.48	3.73	4.22	4.85	5.36	6.00	6.58
0.1	0.83	1.21	1.57	1.95	2.14	2.33	2.46	2.60	2.76	2.91	3.28	3.61	4.03	4.43	4.85
0.3	0.78	0.90	1.09	1.27	1.44	1.60	1.74	1.86	1.98	2.11	2.35	2.51	2.76	2.99	3.25
0.5	0.78	0.86	0.99	1.10	1.22	1.35	1.47	1.57	1.67	1.76	1.96	2.15	2.34	2.50	2.76
1.0	0.78	0.85	0.92	0.99	1.06	1.13	1.20	1.27	1.34	1.41	1.54	1.68	1.82	1.97	2.10
Coefficient $K_{11} (P_2 = 0)$
0.05	0.67	0.67	0.82	1.16	1.35	1.48	1.58	1.65	1.69	1.74	1.81	1.84	1.86	1.88	1.90
0.1	0.67	0.67	0.73	0.93	1.11	1.25	1.36	1.45	1.52	1.57	1.66	1.72	1.77	1.80	1.82
0.3	0.67	0.67	0.67	0.71	0.80	0.90	0.99	1.08	1.15	1.22	1.33	1.41	1.48	1.54	1.59
0.5	0.67	0.67	0.67	0.69	0.73	0.81	0.87	0.94	1.01	1.07	1.17	1.26	1.33	1.39	1.44
1.0	0.67	0.67	0.67	0.68	0.71	0.74	0.78	0.82	0.87	0.91	0.99	1.07	1.13	1.19	1.24

NOTE — Intermediate value may be obtained by interpolation

Table 39Coefficients of Effective Lengths K_{1} for Columns Effectively Held in Position and Restrained Against Rotation at Bottom End and Top End Held Against Rotation but not Held in Position (Table 36)

$C_1$	Coefficients $K_1$ for $I_2/I_1$ Equal to
$C_1$	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	1.2	1.4	1.6	1.8	2.0
0	2.0	1.8	1.7	1.67	1.6	1.55	1.50	1.4	1.43	1.4	1.37	1.3	1.1	1.10	1.0
0.5	2.0	1.9	1.8	1.74	1.6	1.65	1.61	1.5	1.55	1.5	—	—	—	—	—
1.0	2.0	2.0	2.0	2.00	2.0	2.00	—	—	—	—	—	—	—	—	—
1.5	2.0	2.2	2.3	2.48	—	—	—	—	—	—	—	—	—	—	—
2.0	2.0	2.6	2.9	—	—	—	—	—	—	—	—	—	—	—	—
2.5	2.5	3.1	3.5	—	—	—	—	—	—	—	—	—	—	—	—
3.0	3.0	3.7	4.1	—	—	—	—	—	—	—	—	—	—	—	—
NOTE — Intermediate value may be obtained by interpolation

119

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

Table 40Coefficients of Effective Lengths K_{1} for Columns with Top Ends Free and Bottom End Effectively Held in Position and Restrained Against Rotation (Table 36)

$C_1$	Coefficients $K_1$ for $I_2/I_1$ Equal to
$C_1$	0.1	0.2	0.4	0.6	0.8	1.0	1.2	1.4	1.6	1.8	2.0	2.5	5.0	10	20
0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0	2.0
0.5	2.0	2.14	2.24	2.36	2.47	2.57	2.67	2.76	2.85	2.94	3.02	—	—	—	—
1.0	2.0	2.73	3.13	3.44	3.74	4.00	—	—	—	—	—	—	—	—	—
1.5	3.0	3.77	4.35	4.86	—	—	—	—	—	—	—	—	—	—	—
2.0	4.0	4.90	5.67	—	—	—	—	—	—	—	—	—	—	—	—
2.5	5.0	6.08	7.00	—	—	—	—	—	—	—	—	—	—	—	—
3.0	6.0	7.25	—	—	—	—	—	—	—	—	—	—	—	—	—

Table 41 Values of $K_1$ , $K_2$ and $K_3$ (Clause C-3)
Sl No.(1)	Degree of End Restraint(2)	Sketch(3)	$K_1$ (4)	$K_2$ (5)	$K_3$ (6)	Column Parameters for all Cases(7)
i)	Effectively held in position and restrained against rotation at both ends		$K_1 = K_{11}$ where $K_{11}$ is taken from Table 37	where is taken from Table 37
ii)	Effectively held in position at both ends and restrained against rotation at bottom end only		$K_1 = K_{11}$ where $K_{11}$ is taken from Table 38	where is taken from Table 38
iii)	Effectively held in position and restrained against rotation at bottom end, and top end held against rotation but not held in position		$K_1 = K_1$ where $K_1$ is taken from Table 39 with
iv)	Effectively held in position and restrained against rotation at bottom end, and top end neither held against rotation nor against translation		$K_1 = 2$	$K_2 = 2$	$K_3 = K_1$ where is taken from Table 40 with

where

$K_{1}, K_{2}$ , and $K_{3}$ are taken from Table 41.

t _ {1} = \frac {P _ {1}}{P _ {3}}, t _ {2} = \frac {P _ {2}}{P _ {3}}, n _ {2} = \frac {L _ {2}}{L _ {1}}

$I'_{av} =$ average value of moment of inertia for the lower and middle parts

= \frac {I _ {1} L _ {1} + I _ {2} L _ {2}}{L _ {1} + L _ {2}}

$I''_{av} =$ average value of moment of inertia for the middle and top parts

= \frac {I _ {2} L _ {2} + I _ {3} L _ {3}}{L _ {2} + L _ {3}}

Value of coefficient $K_{2}$ for middle part of column is

given by formula:

K _ {2} = \frac {K _ {1}}{C _ {2}}, \text {and}

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

K _ {3} = \frac {K _ {1}}{C _ {3}} \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 C-3.1 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.

+

P2

�

M

2�

i1i2

=

L2I2×

I1L1

1

K1

≤

3

1

K1

≤

3

K_{2} = K_{11}

K_{11}

K_{2} = K_{11}

K_{11}

C_{1} = 0

K3

=

K1

K_1

C_1 = \frac{L_3}{L_1 + L_2} \sqrt{\frac{I_{av}}{I_3}}

3

)

I1(P2+P3)

(P1+P2+P3)

I1P3

I3Iav