Chapter 15Member Subjected to Combined Forces

NBC 2016 (SP 7:2016) Volume 1

2016 · PDF page 1021

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15.1 General

This Section governs the design of members subjected to combined forces, such as shear force and bending, axial force and bending, or shear force, axial force and bending.

15.2 Combined Shear and Bending

15.2.1 No reduction in moment capacity of the section is necessary as long as the cross-section is not subjected to high shear force (factored value of applied shear force is less than or equal to 60 percent of the shear strength of the section as given in 14.4). The moment capacity may be taken as, $M_{\mathrm{d}}$ (see 14.2) without any reduction.

15.2.2 When the factored value of the applied shear force is high (exceeds the limit specified in 15.2.1), the factored moment of the section should be less than the moment capacity of the section under higher shear force, $M_{\mathrm{dv}}$ , calculated as given below:

a) Plastic or compact section

M _ {\mathrm{dv}} = M _ {\mathrm{d}} - \beta \left(M _ {\mathrm{d}} - M _ {\mathrm{fd}}\right) \leq 1. 2 Z _ {\mathrm{e}} f _ {\mathrm{y}} / \gamma_ {\mathrm{m0}}

where

\begin{array}{r c l} \beta & = & (2 V / V _ {\mathrm{d}} - 1) ^ {2}, \end{array}

$M_{d} =$ plastic design moment of the whole section disregarding high shear force effect (see 14.2.1.2) considering web buckling effects (see 14.2.1.1),

$V =$ factored applied shear force as governed by web yielding or web buckling,

$V_{d}$ = design shear strength as governed by web yielding or web buckling (see 14.4.1 or 14.4.2),

$M_{fd} =$ plastic design strength of the area of the cross-section excluding the shear area, considering partial safety factor $\gamma_{m0}$ , and

$Z_{c}$ = elastic section modulus of the whole section.

b) Semi-compact section

M _ {\mathrm{dv}} = Z _ {\mathrm{e}} f _ {\mathrm{y}} / \gamma_ {\mathrm{m0}}

15.3 Combined Axial Force and Bending Moment

Under combined axial force and bending moment, section strength as governed by material failure and member strength as governed by buckling failure shall be checked in accordance with 15.3.1 and 15.3.2 respectively.

15.3.1 Section Strength

15.3.1.1 Plastic and compact sections

In the design of members subjected to combined axial force (tension or compression) and bending moment, the following should be satisfied:

\left| \left(\frac {M _ {\mathrm{y}}}{M _ {\mathrm{ndy}}}\right) ^ {\alpha_ {1}} + \left(\frac {M _ {\mathrm{z}}}{M _ {\mathrm{ndz}}}\right) ^ {\alpha_ {2}} \leq 1. 0 \right|

Section 6 (k) Connections

(Md−Mfd)

≤

1.2Zefy/γm0

−

1

)2

,

m0

Mndy

My

)

α1

+

(MndzMz)α2

≤

1.0

Conservatively, the following equation may also be used under combined axial force and bending moment:

\frac {N}{N _ {\mathrm{d}}} + \frac {M _ {\mathrm{y}}}{M _ {\mathrm{dy}}} + \frac {M _ {\mathrm{z}}}{M _ {\mathrm{dz}}} \leq 1. 0

where

$M_{\mathrm{y}}, M_{\mathrm{z}}$ = factored applied moments about the

minor and major axis of the cross-section, respectively;

$M_{ndy}, M_{ndz} = \text{design reduced flexural strength under combined axial force and the respective uniaxial moment acting alone (see 15.3.1.2);}$

N = factored applied axial force (Tension, T or Compression, P);

$N_{d}$ = design strength in tension, $T_{d}$ as obtained from 12 or in compression due to yielding given by $N_{d}=A_{g}f_{y}/\gamma_{m0}$ ;

$M_{dy}, M_{dz}$ = design strength under corresponding moment acting alone (see 14.2);

$A_{\mathrm{g}}$ = gross area of the cross-section;

$\alpha_{1},\alpha_{2}$ = constants as given in Table 17; and

$\gamma_{\mathrm{m0}}$ = partial factor of safety in yielding.

15.3.1.2 For plastic and compact sections without bolts holes, the following approximations may be used for evaluating $M_{\mathrm{ndy}}$ and $M_{\mathrm{ndz}}$ :

a) Plates

M _ {\mathrm{nd}} = M _ {\mathrm{d}} (1 - n ^ {2})

b) Welded I or H-sections

\left| M _ {\mathrm{ndy}} = M _ {\mathrm{dy}} \left[ 1 - \left(\frac {n - a}{1 - a}\right) ^ {2} \right] \leq M _ {\mathrm{dy}}, \text {where} n \geq a \right|

M _ {\mathrm{ndz}} = M _ {\mathrm{dz}} (1 - n) / (1 - 0. 5 a) \leq M _ {\mathrm{dz}}

where

n = N / N _ {\mathrm{d}} \text {and} a = (A - 2 b t _ {\mathrm{f}}) / A \leq 0. 5

c) For standard I or H sections

M _ {\mathrm{ndz}} = 1. 1 1 M _ {\mathrm{dz}} (1 - n) \leq M _ {\mathrm{dz}}

for $n\leq 0.2$ ， $M_{\mathrm{ndy}} = M_{\mathrm{dy}}$

M _ {\mathrm{ndz}} = 1. 1 1 M _ {\mathrm{dz}} (1 - n) \leq M _ {\mathrm{dz}}

d) For rectangular hollow sections and welded box sections

When the section is symmetric about both axes and without bolt holes

\begin{array}{l} M _ {\mathrm{ndy}} = M _ {\mathrm{dy}} (1 - n) / (1 - 0. 5 a _ {\mathrm{f}}) \leq M _ {\mathrm{dy}} \\ M _ {\mathrm{ndz}} = M _ {\mathrm{dz}} (1 - n) / (1 - 0. 5 a _ {\mathrm{w}}) \leq M _ {\mathrm{dz}} \end{array}

where

\begin{array}{l} a _ {\mathrm{w}} = (A - 2 b t _ {\mathrm{f}}) / A \leq 0. 5 \\ a _ {\mathrm{f}} = (A - 2 h t _ {\mathrm{w}}) / A \leq 0. 5 \end{array}

e) Circular hollow tubes without bolt holes

M _ {\mathrm{nd}} = 1. 0 4 M _ {\mathrm{d}} \left(1 - n ^ {1. 7}\right) \leq M _ {\mathrm{d}}

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

Table 17Constants a_1 and 2 (Clause 15.3.1.1)

Sl No.(1)	Section(2)	$\alpha_{1}$ (3)	$\alpha_{2}$ (4)
i)	I and channel	$5n \geq 1$	2
ii)	Circular tubes	2	2
iii)	Rectangular tubes	$1.66/(1 - 1.13n^{2}) \leq 6$	$1.66/(1 - 1.13n^{2}) \leq 6$
iv)	Solid rectangles	$1.73 + 1.8n^{3}$	$1.73 + 1.8 n^{3}$
NOTE — $n = N/N_{\text{d}}$ .

15.3.1.3 Semi-compact section

In the absence of high shear force (see 15.2.1), semi-compact section design is satisfactory under combined axial force and bending, if the maximum longitudinal stress under combined axial force and bending, $f_{x}$ satisfies the following criteria:

f _ {\mathrm{x}} \leq f _ {\mathrm{y}} / \gamma_ {\mathrm{m0}}

For cross-section without holes, the above criteria reduces to,

\frac {N}{N _ {\mathrm{d}}} + \frac {M _ {\mathrm{y}}}{M _ {\mathrm{dy}}} + \frac {M _ {\mathrm{z}}}{M _ {\mathrm{dz}}} \leq 1. 0

where

$N_{\mathrm{d}}, M_{\mathrm{dy}}, M_{\mathrm{dz}}$ are as defined in 15.3.1.1.

15.3.2 Overall Member Strength

Members subjected to combined axial force and bending moment shall be checked for overall buckling failure as given in this section.

15.3.2.1 Bending and axial tension

The reduced effective moment, $M_{\mathrm{eff}}$ , under tension and bending calculated as given below, should not exceed the bending strength due to lateral torsional buckling, $M_{\mathrm{d}}$ (see 14.2.2),

M _ {\text {eff}} = \left[ M - \Psi T Z _ {\text {ec}} / A \right] \leq M _ {\text {d}}

where

$M, T =$ factored applied moment and tension, respectively;

$A$ = area of cross-section;

$Z_{ec}$ = elastic section modulus of the section with respect to extreme compression fibre; and

$\psi = 0.8$ , if $T$ and $M$ can vary independently, or otherwise

= 1.0.

15.3.2.2 Bending and axial compression

Members subjected to combined axial compression and

biaxial bending shall satisfy the following interaction relationships:

\frac {P}{P _ {\mathrm{dy}}} + K _ {\mathrm{y}} \frac {C _ {\mathrm{my}} M _ {\mathrm{y}}}{M _ {\mathrm{dy}}} + K _ {\mathrm{LT}} \frac {M _ {\mathrm{z}}}{M _ {\mathrm{dz}}} \leq 1. 0

\frac {P}{P _ {\mathrm{dz}}} + 0. 6 K _ {\mathrm{y}} \frac {C _ {\mathrm{my}} M _ {\mathrm{y}}}{M _ {\mathrm{dy}}} + K _ {\mathrm{z}} \frac {C _ {\mathrm{mz}} M _ {\mathrm{z}}}{M _ {\mathrm{dz}}} \leq 1. 0

where

$C_{my}, C_{mz} = \text{equivalent uniform moment factor as per Table 18;}$

P = applied axial compression under factored load;

$M_{y}, M_{z}$ = maximum factored applied bending moments about y and z-axis of the member, respectively;

$P_{dy}, P_{dz} = \text{design strength under axial compression as governed by buckling about minor (y) and major (z) axis, respectively;}$

$M_{dy}, M_{dz} = \text{design bending strength about y (minor) or z (major) axis considering laterally unsupported length of the cross-section [see Section 6(h)];}$

\begin{array}{r l r} {K _ {\mathrm{y}}} & & {= 1 + (\lambda_ {\mathrm{y}} - 0. 2) n _ {\mathrm{y}} \leq 1 + 0. 8 n _ {\mathrm{y}};} \\ {K _ {\mathrm{z}}} & & {= 1 + (\lambda_ {\mathrm{z}} - 0. 2) n _ {\mathrm{z}} \leq 1 + 0. 8 n _ {\mathrm{z}}, \mathrm{and}} \end{array}

K _ {\mathrm{LT}} \quad = 1 - \frac {0 . 1 \lambda_ {\mathrm{LT}} n _ {\mathrm{y}}}{\left(C _ {\mathrm{mLT}} - 0 . 2 5\right)} \geq 1 - \frac {0 . 1 n _ {\mathrm{y}}}{\left(C _ {\mathrm{mLT}} - 0 . 2 5\right)}

where

$n_{y}, n_{z}$ = ratio of actual applied axial force to the design axial strength for buckling about the y and z-axis, respectively, and

$C_{mLT}$ = Equivalent uniform moment factor for lateral torsional buckling as per Table 18 corresponding to the actual moment gradient between lateral supports against torsional deformation in the critical region under consideration.

dz

≤

6

Ky

MdyCmyMy

+

KLTMdzMz≤

1.0

0.6KyMdyCmyMy+

KzMdzCmzMz≤

1.0

Kz

=1+(λy−0.2)ny≤1+0.8ny;=1+(λz−0.2)nz≤1+0.8nz,and

(CmLT−0.25)

0.1λLTny

≥

1−

(CmLT−0.25)0.1ny