Chapter 13Design of Compression Members

NBC 2016 (SP 7:2016) Volume 1

2016 · PDF page 993

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13.1 Design Strength

13.1.1 Common hot rolled and built-up steel members used for carrying axial compression, usually fail by

flexural buckling. The buckling strength of these members is affected by residual stresses, initial bow and accidental eccentricities of load. To account for all these factors, the strength of members subjected to axial compression is defined by buckling class a, b, c, or d as given Table 7.

13.1.2 The factored design compression, $P$ in members shall satisfy the following requirement:

| P < P _ {\mathrm{d}} |

where

P _ {\mathrm{d}} = A _ {\mathrm{e}} f _ {\mathrm{cd}}

where

$A_{e}$ = effective sectional area as defined in 13.3.2, and

$f_{cd}$ = design compressive stress, obtained as per 13.1.2.1.

13.1.2.1 The design compressive strength, $f_{\mathrm{cd}}$ of axially loaded compression members shall be calculated using the following equation:

f _ {\mathrm{cd}} = \frac {f _ {\mathrm{y}} / \gamma_ {\mathrm{m0}}}{\phi + \left[ \phi^ {2} - \lambda^ {2} \right] ^ {0 . 5}} = \chi f _ {\mathrm{y}} / \gamma_ {\mathrm{m0}} \leq f _ {\mathrm{y}} / \gamma_ {\mathrm{m0}}

where

\phi = 0. 5 [ 1 + \alpha (\lambda - 0. 2) + \lambda^ {2} ]

$\lambda =$ non-dimensional effective slenderness ratio

= \sqrt {f _ {\mathrm{y}} / f _ {\mathrm{cc}}} = \sqrt {f _ {\mathrm{y}} \left(\frac {K L}{r}\right) ^ {2} / \pi^ {2} E}

f _ {\mathrm{cc}} = \text {Euler buckling stress} = \frac {\pi^ {2} E}{(K L / r) ^ {2}}

where

KL/r = effective slenderness ratio or ratio of effective length, KL to appropriate radius of gyration, r;

$\alpha$ = imperfection factor given in Table 7;

$\chi$ = stress reduction factor (see Table 8) for different buckling class, slenderness ratio and yield stress

= \frac {1}{\left[ \phi + \left(\phi^ {2} - \lambda^ {2}\right) ^ {0 . 5} \right]}

$\gamma_{\mathrm{m0}}$ = partial safety factor for material strength.

NOTE — Calculated values of design compressive stress, $f_{cd}$ for different buckling classes are given in Table 9.

13.1.2.2 The classification of different sections under different buckling class a, b, c or d, is given in Table 10.

The stress reduction factor $\chi$ , and the design compressive stress $f_{cd}$ , for different buckling class, yield stress, and effective slenderness ratio is given in Table 8 for convenience. The curves corresponding to different buckling class are presented in non-dimensional form, in Fig. 8.

Table 7 Imperfection Factor, $\alpha$

(Clauses 13.1.1 and 13.1.2.1)

Buckling Class(1)	a(2)	b(3)	c(4)	d(5)
$\alpha$	0.21	0.34	0.49	0.76

13.2Effective Length of Compression Members

13.2.1 The effective length $KL$ , is calculated from the actual length $L$ , of the member, considering the rotational and relative translational boundary conditions at the ends. The actual length shall be taken as the length from centre-to-centre of its intersections with the supporting members in the plane of the buckling deformation. In the case of a member with a free end, the free standing length from the centre of the intersecting member at the supported end, shall be taken as the actual length.

13.2.2 Effective Length

Where the boundary conditions in the plane of buckling can be assessed, the effective length, KL can be calculated on the basis of Table 11. Where frame analysis does not consider the equilibrium of a framed structure in the deformed shape (second-order analysis

or advanced analysis), the effective length of compression members in such cases can be calculated using the procedure given in C-1. The effective length of stepped column in single storey buildings can be calculated using the procedure given in C-2.

13.2.3 Eccentric Beam Connection

In cases where the beam connections are eccentric in plan with respect to the axes of the column, the same conditions of restraint as in concentric connection shall be deemed to apply, provided the connections are carried across the flange or web of the columns as the case may be, and the web of the beam lies within, or in direct contact with the column section. Where practical difficulties prevent this, the effective length shall be taken as equal to the distance between points of restraint, in non-sway frames.

13.2.4 Compression Members in Trusses

In the case of bolted, riveted or welded trusses and braced frames, the effective length, KL, of the compression members shall be taken as 0.7 to 1.0 times the distance between centres of connections, depending on the degree of end restraint provided. In the case of members of trusses, buckling in the plane perpendicular to the plane of the truss, the effective length, KL shall be taken as the distance between the centers of intersection. The design of angle struts shall be as specified in 13.5.

13.3Design Details

13.3.1 Thickness of Plate Elements

Classification of members on the basis of thickness of

Chart

FIG. 8 COLUMN BUCKLING CURVES

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

Table 8(a) Stress Reduction Factor, \chi for Column Buckling Class a (Clauses 13.1.2.1 and 13.1.2.2)

KL/r	Yield Stress, $f_y$ (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
20	1.000	0.999	0.998	0.997	0.995	0.994	0.993	0.993	0.990	0.988	0.986	0.984	0.983	0.981	0.979	0.977	0.975	0.972	0.970
30	0.977	0.975	0.974	0.972	0.970	0.969	0.967	0.965	0.961	0.957	0.954	0.951	0.948	0.946	0.943	0.938	0.934	0.930	0.925
40	0.952	0.949	0.947	0.944	0.942	0.939	0.937	0.934	0.926	0.921	0.916	0.911	0.906	0.901	0.896	0.888	0.881	0.873	0.865
50	0.923	0.919	0.915	0.911	0.908	0.904	0.900	0.896	0.884	0.876	0.867	0.859	0.851	0.842	0.834	0.820	0.807	0.794	0.780
60	0.888	0.883	0.877	0.871	0.865	0.859	0.853	0.847	0.828	0.816	0.803	0.790	0.777	0.763	0.750	0.730	0.710	0.690	0.671
70	0.846	0.837	0.829	0.820	0.811	0.803	0.794	0.785	0.758	0.740	0.722	0.703	0.686	0.668	0.651	0.626	0.602	0.579	0.557
80	0.793	0.781	0.769	0.757	0.746	0.734	0.722	0.710	0.675	0.653	0.631	0.610	0.589	0.570	0.551	0.525	0.501	0.478	0.458
90	0.730	0.715	0.700	0.685	0.671	0.657	0.643	0.628	0.590	0.565	0.542	0.520	0.500	0.481	0.463	0.439	0.416	0.396	0.377
100	0.661	0.644	0.627	0.610	0.594	0.579	0.564	0.549	0.510	0.486	0.463	0.443	0.424	0.407	0.390	0.368	0.348	0.331	0.314
110	0.591	0.573	0.555	0.538	0.522	0.507	0.492	0.478	0.440	0.418	0.397	0.379	0.362	0.346	0.332	0.312	0.295	0.279	0.265
120	0.525	0.507	0.489	0.473	0.458	0.443	0.429	0.416	0.381	0.361	0.343	0.326	0.311	0.297	0.284	0.267	0.252	0.238	0.226
130	0.466	0.448	0.432	0.416	0.402	0.388	0.376	0.364	0.332	0.314	0.298	0.283	0.269	0.257	0.246	0.231	0.217	0.206	0.195
140	0.413	0.397	0.382	0.368	0.355	0.342	0.331	0.320	0.291	0.275	0.260	0.247	0.235	0.224	0.214	0.201	0.189	0.179	0.170
150	0.368	0.353	0.339	0.326	0.314	0.303	0.293	0.283	0.257	0.243	0.229	0.218	0.207	0.197	0.189	0.177	0.166	0.157	0.149
160	0.329	0.316	0.303	0.291	0.280	0.270	0.261	0.252	0.229	0.215	0.204	0.193	0.184	0.175	0.167	0.157	0.147	0.139	0.132
170	0.296	0.283	0.272	0.261	0.251	0.242	0.233	0.225	0.204	0.192	0.182	0.172	0.164	0.156	0.149	0.140	0.131	0.124	0.117
180	0.267	0.255	0.245	0.235	0.226	0.218	0.210	0.203	0.184	0.173	0.163	0.155	0.147	0.140	0.134	0.125	0.118	0.111	0.105
190	0.242	0.231	0.222	0.213	0.205	0.197	0.190	0.183	0.166	0.156	0.147	0.140	0.133	0.126	0.121	0.113	0.106	0.100	0.095
200	0.220	0.210	0.202	0.193	0.186	0.179	0.172	0.166	0.151	0.142	0.134	0.127	0.120	0.115	0.109	0.102	0.096	0.091	0.086
210	0.201	0.192	0.184	0.177	0.170	0.163	0.157	0.152	0.137	0.129	0.122	0.115	0.110	0.104	0.099	0.093	0.087	0.083	0.078
220	0.184	0.176	0.169	0.162	0.155	0.149	0.144	0.139	0.126	0.118	0.111	0.106	0.100	0.095	0.091	0.085	0.080	0.075	0.071
230	0.170	0.162	0.155	0.149	0.143	0.137	0.132	0.128	0.115	0.108	0.102	0.097	0.092	0.088	0.083	0.078	0.073	0.069	0.065
240	0.157	0.149	0.143	0.137	0.132	0.127	0.122	0.118	0.106	0.100	0.094	0.089	0.085	0.081	0.077	0.072	0.068	0.064	0.060
250	0.145	0.138	0.132	0.127	0.122	0.117	0.113	0.109	0.098	0.092	0.087	0.082	0.078	0.074	0.071	0.066	0.062	0.059	0.056

Table 8(b) Stress Reduction Factor, \chi for Column Buckling Class b (Clauses 13.1.2.1 and 13.1.2.2)

KL/r	Yield Stress, $f_y$ (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
20	1.000	0.998	0.996	0.994	0.993	0.991	0.990	0.986	0.983	0.981	0.978	0.975	0.972	0.970	0.967	0.963	0.960	0.956	0.953
30	0.963	0.961	0.958	0.955	0.953	0.950	0.948	0.943	0.938	0.933	0.929	0.924	0.920	0.915	0.911	0.904	0.898	0.892	0.886
40	0.925	0.921	0.917	0.913

Table 8(c) Stress Reduction Factor, \chi for Column Buckling Class c (Clauses 13.1.2.1 and 13.1.2.2)

KL/r	Yield Stress, $f_y$ (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
20	0.999	0.997	0.994	0.992	0.990	0.987	0.985	0.981	0.976	0.972	0.968	0.964	0.961	0.957	0.953	0.948	0.943	0.938	0.933
30	0.948	0.944	0.941	0.937	0.933	0.930	0.926	0.920	0.913	0.907	0.901	0.895	0.889	0.883	0.877	0.869	0.861	0.853	0.845
40	0.896	0.891	0.885	0.880

Table 8(d) Stress Reduction Factor, \chi for Column Buckling Class d (Clauses 13.1.2.1 and 13.1.2.2)

KL/r	Yield Stress, $f_y$ (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000	1.000
20	0.999	0.995	0.991	0.988	0.984	0.980	0.977	0.970	0.964	0.958	0.952	0.946	0.940	0.935	0.930	0.922	0.915	0.908	0.901
30	0.922	0.916	0.911	0.906	0.901	0.896	0.891	0.881	0.872	0.863	0.855	0.847	0.839	0.831	0.823	0.813	0.802	0.792	0.782
40	0.848	0.841	0.834	0.828

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

Table 9(a) Design Compressive Stress, f_{\mathrm{cd}} (MPa) For Column Buckling Class a (Clause 13.1.2.1)

KL/r	Yield Stress, $f_y$ (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	182	191	200	213	218	227	236	255	273	291	309	327	345	364	382	409	436	464	491
20	182	191	200	208	217	226	235	252	270	287	305	322	339	357	374	400	425	451	476
30	178	186	195	203	212	220	229	245	262	279	295	311	328	344	360	384	408	431	454
40	173	181	189	197	205	213	221	237	253	268	283

Table 9(b) Design Compressive Stress, f_{\mathrm{cd}} (MPa) for Column Buckling Class b (Clause 13.1.2.1)

KL/r	Yield Stress, $f_y$ (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	182	191	200	209	218	227	236	255	273	291	309	327	345	364	382	409	436	464	491
20	182	190	199	208	217	225	234	251	268	285	302	319	336	353	369	394	419	443	468
30	175	183	192	200	208	216	224	240	256	271	287	302	318	333	348	370	392	414	435
40	168	176	183	191	198	206	213	228	242	256	270

Table 9(c) Design Compressive Stress, f_{\mathrm{cd}} (MPa) for Column Buckling Class c (Clause 13.1.2.1)

KL/r	Yield Stress, $f_y$ , (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	182	191	200	209	218	227	236	255	273	291	309	327	345	364	382	409	436	464	491
20	182	190	199	207	216	224	233	250	266	283	299	316	332	348	364	388	412	435	458
30	172	180	188	196	204	211	219	234	249	264	278	293	307	321	335	355	376	395	415
40	163	170	177	184	191	198	205	218	231	244	256

Table 9(d) Design Compressive Stress, f_{\mathrm{cd}} (MPa) for Column Buckling Class d (Clause 13.1.2.1)

KL/r	Yield Stress, $f_y$ (MPa)
↓	200	210	220	230	240	250	260	280	300	320	340	360	380	400	420	450	480	510	540
10	182	191	200	209	218	227	236	255	273	291	309	327	345	364	382	409	436	464	491
20	182	190	198	206	215	223	231	247	263	279	294	310	325	340	355	377	399	421	442
30	168	175	182	189	197	204	211	224	238	251	264	277	290	302	314	332	350	367	384
40	154	161	167	173	179	185	191	203	214

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

Table 10Buckling Class of Cross-Sections (Clause 13.1.2.2)

Cross-Section(1)		Limits(2)	Buckling About Axis(3)	Buckling Class(4)
Rolled I-SectionsI-y		$h/b_{\mathrm {f}} > 1.2 : t_{\mathrm {f}} \leq 40 \mathrm {mm}$

Table 11Effective Length of Prismatic Compression Members (Clause 13.2.2)

Boundary Conditions				Schematic Representation	Effective Length
At One End		At the Other End		Schematic Representation	Effective Length
Translation(1)	Rotation(2)	Translation(3)	Rotation(4)	(5)	(6)
Restrained	Restrained	Free	Free
Restrained	Restrained	Free	Free	1	2.0L
Restrained	Free	Free	Restrained
Restrained	Free	Restrained	Free		1.0L
Restrained	Restrained	Free	Restrained		1.2L
Restrained	Restrained	Restrained	Free		0.8L
Restrained	Restrained	Restrained	Restrained		0.65L
NOTE — L is the unsupported length of the compression member (see 13.2.1).

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

constituent plate elements shall satisfy the width-thickness ratio requirements specified in Table 2.

13.3.2 Effective Sectional Area, $A\{mathrm\{e\}\}$

Except as modified in 9.7.2 (Class 4), the gross sectional area shall be taken as the effective sectional area for all compression members fabricated by welding, bolting and riveting so long as the section is semi-compact or better. Holes not fitted with rivets, bolts or pins shall be deducted from gross area to calculate effective sectional area.

13.4Column Bases

13.4.1 General

Column bases should have sufficient stiffness and strength to transmit axial force, bending moments and shear forces at the base of the columns to their foundation without exceeding the load carrying capacity of the supports. Anchor bolts and shear keys should be provided wherever necessary. Shear resistance at the proper contact surface between steel base and concrete/grout may be calculated using a friction coefficient of 0.45.

The nominal bearing pressure between the base plate and the support below may be determined on the basis of linearly varying distribution of pressure. The maximum bearing pressure on concrete under the limit states design load should not exceed the bearing strength

equal to $0.45f_{\mathrm{ck}}\left[\sqrt{(A_1 / A_2)}\right]$ and $0.9f_{\mathrm{ck}}$ , whichever is smaller, where $f_{\mathrm{ck}}$ is the smaller of characteristic cube strength of concrete or bedding material, $A_{1}$ is the area of concrete pedestal on which the base plate is sitting, and $A_{2}$ is the surface area of the steel base plate transferring the load to the concrete pedestal.

13.4.1.1 If the size of the base plate is larger than that required to limit the bearing pressure on the base support, an equal projection $c$ of the base plate beyond the face of the column and gusset may be taken as effective in transferring the column load as given in Fig. 9, such that bearing pressure on the effective area does not exceed bearing capacity of concrete base.

13.4.2 Gusseted Bases

For stanchion with gusseted bases, the gusset plates, angle cleats, stiffeners, fastenings, etc, in combination with the bearing area of the shaft, shall be sufficient to take the loads, bending moments and reactions to the base plate without exceeding specified strength. All the bearing surfaces shall be machined to ensure perfect contact.

13.4.2.1 Where the ends of the column shaft and the gusset plates are not faced for complete bearing, the welds, fastenings connecting them to the base plate shall be sufficient to transmit all the forces to which the base is subjected.

13.4.2.2 Column and base plate connections

Where the end of the column is connected directly to the base plate by means of full penetration butt welds, the connection shall be deemed to transmit to the base all the forces and moments to which the column is subjected.

13.4.3 Slab Bases

Columns with slab bases need not be provided with gussets, but sufficient fastenings shall be provided to retain the parts securely in place and to resist all moments and forces, other than direct compression, including those arising during transit, unloading and erection.

13.4.3.1 The minimum thickness, $t_{\mathrm{s}}$ , of rectangular slab bases, supporting columns under axial compression shall be:

t _ {\mathrm{s}} = \sqrt {2 . 5 w (a ^ {2} - 0 . 3 b ^ {2}) \gamma_ {\mathrm{m0}} / f _ {\mathrm{y}}} > t _ {\mathrm{f}}

13.5Angle Struts

13.5.1 Single Angle Struts

The compression in single angles may be transferred

either concentrically to its centroid through the end connections or eccentrically by connecting one of its legs to a gusset or adjacent member.

13.5.1.1 Concentric loading

When a single angle is concentrically loaded in compression, the design strength may be evaluated using 13.1.2, by calculating the non-dimensional slenderness ratio, $\lambda_{\mathrm{vv}}$ , for buckling about the minor axis, $v - v$ , as given by:

\lambda_ {\mathrm{vv}} = \frac {\left(\frac {l _ {\mathrm{vv}}}{r _ {\mathrm{vv}}}\right)}{\varepsilon \sqrt {\frac {\pi^ {2} E}{2 5 0}}}

$l_{vv}$ = Laterally unsupported length preventing translation normal to the minor axis

$r_{vv}$ = radius of gyration of the angle section about the minor axis

13.5.1.2 Loaded through one leg

The design strength of single angle members loaded in compression through one of its legs is affected by flexural torsional buckling and bending. The design compressive strength of such eccentrically loaded angle members, in lieu of a more exact second order design under combined bending axial compression, may be evaluated approximately as given below.

The design compressive strength, of single angles loaded through gusset connection to one leg parallel to the $a - a$ axis ( $a - a$ axis being either the $z - z$ or the $y - y$ axis, depending on which leg is connected to the gusset) of the member may be calculated as:

13.5.2 Double Angle Struts

13.5.2.1 For double angle discontinuous struts, connected back to back, on opposite sides of the gusset or a section, by not less than two bolts or rivets in line along the angles at each end, or by the equivalent in welding, the load may be regarded as applied axially. The effective length, $KL$ , in the plane of end gusset shall be taken as between 0.7 and 0.85 times the distance between intersections, depending on the degree of the restraint provided. The effective length, $KL$ , in the plane perpendicular to that of the end gusset, shall be taken as equal to the distance between centers of intersections. The calculated average compressive stress shall not exceed the values based on 13.1.2. The angles shall be connected together over their lengths so as to satisfy the requirements of 13.8 and 16.2.5.

13.5.2.2 Double angle discontinuous struts connected back-to-back, to one side of a gusset or section by one or more bolts or rivets in each angle, or by the equivalent in welding, shall be designed in accordance with 13.5.1 and the angles shall be connected together over their lengths so as to satisfy the requirements of 13.8 and 16.2.5.

13.5.3 Continuous Members

Double angle continuous struts such as those forming the flanges, chords or ties of trusses or trussed girders, or the legs of towers shall be designed as axially loaded compression members, and the effective length shall be taken in accordance with 13.2.4.

13.5.4 Combined Stresses

In addition to axial loads, if the struts carry loads which cause transverse bending, the combined bending and axial stresses shall be checked in accordance with 14.3. For determining the permissible axial and bending stresses, the effective length shall be taken in accordance with the 13.2 and 14.3.

Sl No.	End Connection	Gusset/ Connecting Member $Fixity^{1)}$	$k_{1}$	$k_{2}$	$k_{3}$
(1)	(2)	(3)	(4)	(5)	(6)
i)	Fully welded or connected with two or more bolts	$\left\{ \begin{array}{l} \text{Fixed} \\ \text{Hinged} \end{array} \right.$	0.798	0.563	-2.072
i)	Fully welded or connected with two or more bolts		0.401	0.420	-1.040
ii)	Single bolt	$\left\{ \begin{array}{l} \text{Fixed} \\ \text{Hinged} \end{array} \right.$	0.418	0.547	-1.400
ii)	Single bolt		0.374	0.415	-2.072

Chapter 13Design of Compression Members

13.1 Design Strength

13.2Effective Length of Compression Members

13.2.2 Effective Length

13.2.3 Eccentric Beam Connection

13.2.4 Compression Members in Trusses

13.3Design Details

13.3.1 Thickness of Plate Elements

13.3.2 Effective Sectional Area, $A\{mathrm\{e\}\}$

13.3.3Eccentricity for Stanchions and Columns

13.3.4Splices

13.4Column Bases

13.4.1 General

13.4.2 Gusseted Bases

13.4.2.2 Column and base plate connections

13.4.3 Slab Bases

13.5Angle Struts

13.5.1 Single Angle Struts

13.5.1.1 Concentric loading

13.5.1.2 Loaded through one leg

13.5.2 Double Angle Struts

13.5.3 Continuous Members

13.5.4 Combined Stresses

13.1 Design Strength

13.2.2 Effective Length

13.2.3 Eccentric Beam Connection

13.2.4 Compression Members in Trusses

13.3.1 Thickness of Plate Elements

13.3.2 Effective Sectional Area, A{mathrm{e}}A\{mathrm\{e\}\}A{mathrm{e}}

13.4.1 General

13.4.2 Gusseted Bases

13.4.2.2 Column and base plate connections

13.4.3 Slab Bases

13.5.1 Single Angle Struts

13.5.1.1 Concentric loading

13.5.1.2 Loaded through one leg

13.5.2 Double Angle Struts

13.5.3 Continuous Members

13.5.4 Combined Stresses

13.3.2 Effective Sectional Area, $A\{mathrm\{e\}\}$