Chapter 10Methods of Structural Analysis

NBC 2016 (SP 7:2016) Volume 1

2016 · PDF page 981

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10.1 Methods of Determining Action Effects

10.1.1 General

For the purpose of complying with the requirements of the limit states of stability, strength and serviceability specified in Section 6(e), effects of design actions on a structure and its members and connections, shall be determined by structural analysis using the assumptions

of 10.2 and 10.3 and one of the following methods of analysis:

a) Elastic analysis in accordance with 10.4,

b) Plastic analysis in accordance with 10.5,

c) Advanced analysis in accordance with Annex A, or

d) Dynamic analysis in accordance with 5 of Part 6 ‘Structural Design, Section 1 Loads, Forces and Effects’ of the Code.

The design action effects for design basis earthquake loads shall be obtained only by an elastic analysis. The maximum credible earthquake loads shall be assumed to correspond to the load at which significant plastic hinges are formed in the structure and the corresponding effects shall be obtained by plastic or advanced analysis. More information on analysis and design to resist earthquake is given in Section 6(n) and 5 of Part 6 'Structural Design, Section 1 Loads, Forces and Effects' of the Code.

10.1.2 Non-Sway and Sway Frames

For the purpose of analysis and design, the structural frames are classified as non-sway and sway frames as given below:

a) Non-sway frame — One in which the transverse displacement of one end of the member relative to the other end is effectively prevented. This applies to triangulated frames and trusses or to frames where in-plane stiffness is provided by bracings, or by shear walls, or by floor slabs and roof decks secured horizontally to walls or to bracing systems parallel to the plane of loading and bending of the frame.

b) Sway frame — One in which the transverse displacement of one end of the member relative to the other end is not effectively prevented. Such members and frames occur in structures which depend on flexural action

FIG. 4 MAXIMUM LENGTH OF COVERED BUILDING/SECTION WITH TWO BAYS OF BRACINGS

of members to resist lateral loads and sway, as in moment resisting frames.

c) A rigid jointed multi-storey frame may be considered as a non-sway frame if in every individual storey, the deflection $d$ , over a storey height $h_{s}$ due to the notional horizontal loading given in 10.3.6 satisfies the following criteria:

For clad frames, when the stiffening effect of the cladding is not taken into account in the deflection calculations:

\delta \leq \frac {h _ {\mathrm{s}}}{2 0 0 0}

For unclad frame or clad frames, when the stiffening effect of the cladding is taken into account in the deflection calculations:

\delta \leq \frac {h _ {\mathrm{s}}}{4 0 0 0}

A frame, which when analysed considering all the lateral supporting system does not comply with the above criteria, should be classified as a sway frame, even if it is braced or otherwise laterally stiffened.

10.2Forms of Construction Assumed for Structural Analysis

10.2.1 The effects of design action in the members and connections of a structure shall be determined by assuming singly or in combination of the following forms of construction (see 16.6.1).

10.2.1.1 Rigid construction

In rigid construction, the connections between members (beam and column) at their junction shall be assumed to have sufficient rigidity to hold the original angles between the members connected at a joint unchanged under loading.

10.2.1.2 Semi-rigid construction

In semi-rigid construction, the connections between members (beam and column) at their junction may not have sufficient rigidity to hold the original angles between the members at a joint unchanged, but shall be assumed to have the capacity to furnish a dependable and known degree of flexural restraint. The relationship between the degree of flexural restraint and the level of the load effects shall be established by any rational method or based on test results (see Annex E).

10.2.1.3 Simple construction

In simple construction, the connections between

members (beam and column) at their junction will not resist any appreciable moment and shall be assumed to be hinged.

10.2.2 Design of connections

The design of all connections shall be consistent with the form of construction, and the behaviour of the connections shall not adversely affect any other part of the structure beyond what is allowed for in design. Connections shall be designed in accordance with Section 6(k).

10.3Assumptions in Analysis

10.3.1 The structure shall be analysed in its entirety except as follows:

a) Regular building structures, with orthogonal frames in plan, may be analysed as a series of parallel two-dimensional sub-structures (part of a structure), the analysis being carried out in each of the two directions, at right angles to each other, except when there is significant load redistribution between the sub-structures (part of a structure). For earthquake loading three dimensional analysis may be necessary to account for effects of torsion and also for multi-component earthquake forces (see 5 of Part 6 ‘Structural Design, Section 1 Loads, Forces and Effects’ of the Code).

b) For vertical loading in a multi-storey building structure, provided with bracing or shear walls to resist all lateral forces, each level thereof, together with the columns immediately above and below, may be considered as a substructure, the columns being assumed fixed at the ends remote from the level under consideration.

c) Where beams at a floor level in a multi-bay building structure are considered as a substructure (part of a structure), the bending moment at the support of the beam due to gravity loads may be determined based on the assumption that the beam is fixed at the far end support, one span away from the span under consideration, provided that the floor beam is continuous beyond that support point.

10.3.2 Span Length

The span length of a flexural member in a continuous frame system shall be taken as the centre-to-centre distance between the supports.

10.3.3 Arrangements of Imposed Loads in Buildings

For building structures, the various arrangements of imposed loads considered for the analysis, shall include at least the following:

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

a) Where the loading pattern is fixed, the arrangement concerned.

b) Where the imposed load is variable and not greater than three-quarters of the dead load, the imposed load may be taken to be acting on all spans.

c) Where the imposed load is variable and exceeds three-quarters of the dead load, arrangements of imposed load acting on the floor under consideration shall include the following cases:

Imposed load on alternate spans,
Imposed load on two adjacent spans, and
Imposed load on all the spans.

10.3.4 Base Stiffness

In the analysis of all structures the appropriate base stiffness about the axis under consideration shall be used. In the absence of the knowledge of the pedestal and foundation stiffness, the following may be assumed:

a) When the column is rigidly connected to a suitable foundation, the stiffness of the pedestal shall be taken as the stiffness of the column above base plate. However, in case of very stiff pedestals and foundations the column may be assumed as fixed at base.

b) When the column is nominally connected to the foundation, a pedestal stiffness of 10 percent of the column stiffness may be assumed.

c) When an actual pin or rocker is provided in the connection between the steel column and pedestal, the column is assumed as hinged at base and the pedestal and foundation may be appropriately designed for the reactions from the column.

d) In case of (a) and (b), the bottom of the pedestal shall be assumed to have the following boundary condition in the absence of any detailed procedure based on theory or tests:

When the foundation consist of a group of piles with a pile cap, raft foundation or an isolated footing resting on rock or very hard soil, the pedestal shall be assumed to be fixed at the level of the bottom of footing or at the top of pile cap.
When the foundation consists of an isolated footing resting on other soils, pedestal shall be assumed to be hinged at the level of the bottom of footing.
When the pedestal is supported by a single pile, which is laterally surrounded

by soil providing passive resistance, the pile shall be assumed to be fixed at a depth of 5 times the diameter of the pile below the ground level in case of compact ground or the top level of compact soil in case of poor soil overlying compact soil.

When the column is founded into rock, it may be assumed to be fixed at the interface of the column and rock.

10.3.5 Simple Construction

Bending members may be assumed to have their ends connected for shear only and to be free to rotate. In triangulated structures, axial forces may be determined by assuming that all members are pin connected. The eccentricity for stanchion and column shall be assumed in accordance with 13.3.3.

10.3.6 Notional Horizontal Loads

To analyse a frame subjected to gravity loads, considering the sway stability of the frame, notional horizontal forces should be applied. These notional horizontal forces account for practical imperfections and should be taken at each level as being equal to 0.5 percent of factored dead load plus vertical imposed loads applied at that level. The notional load should not be applied along with other lateral loads such as wind and earthquake loads in the analysis.

10.3.6.1 The notional forces should be applied on the whole structure, in both orthogonal directions, in one direction at a time, at roof and all floor levels or their equivalent. They should be taken as acting simultaneously with factored gravity loads.

10.3.6.2 The notional force should not be,

a) applied when considering overturning or overall instability;

b) combined with other horizontal (lateral) loads;

c) combined with temperature effects, and

d) taken to contribute to the net shear on the foundation.

10.3.6.3 The sway effect using notional load under gantry load case need not be considered if the ratio of height to lateral width of the building is less than unity.

10.4Elastic Analysis

10.4.1 Assumptions

Individual members shall be assumed to remain elastic under the action of the factored design loads for all limit states.

The effect of haunching or any variation of the cross-section along the axis of a member shall be considered, and where significant, shall be taken into account in the determination of the member stiffness.

10.4.2 First-Order Elastic Analysis

10.4.2.1 In a first-order elastic analysis, the equilibrium of the frame in the undeformed geometry is considered, the changes in the geometry of the frame due to the loading are not accounted for, and changes in the effective stiffness of the members due to axial force are neglected. The effects of these on the first-order bending moments shall be allowed for by using one of the methods of moment amplification of 10.4.3.2 or 10.4.3.3 as appropriate. Where the moment amplification factor $K_{y}$ , $K_{z}$ , calculated in accordance with 10.4.3.2 or 10.4.3.3 as appropriate, is greater than 1.4, a second-order elastic analysis in accordance with Annex A shall be carried out.

10.4.3 Second-Order Elastic Analysis

10.4.3.1 The analysis shall allow for the effects of the design loads acting on the structure and its members in their displaced and deformed configuration. These second-order effects shall be taken into account by using either:

a) A first-order elastic analysis with moment amplification in accordance with 10.4.2, provided the moment amplification factors, $K_{y}$ and $K_{z}$ are not greater than 1.4; or

b) A second-order elastic analysis in accordance with Annex A.

10.4.3.2 Moment amplification for members in non-sway frames

For a member with zero axial compression or a member subject to axial tension, the design bending moment is that obtained from the first-order analysis for factored loads, without any amplification.

For a braced member with a design axial compressive force $P_{\mathrm{d}}$ as determined by the first-order analysis, the design bending moment shall be calculated considering moment amplification as in 15.3.2.2.

10.4.3.3 Moment amplification for members in sway frames

The design bending moment shall be calculated as the product of moment amplification factor [see 15.3.2.2 $(K_{\mathrm{y}}, K_{\mathrm{z}})]$ and the moment obtained from the first-order analysis of the sway frame, unless analysis considering second order effects is carried out (see 10.4.3).

10.4.3.4 The calculated bending moments from the first order elastic analysis may be modified by redistribution up to 15 percent of the peak calculated moment of the member under factored load, provided that,

a) the internal forces and moments in the members of the frame are in equilibrium with applied loads.

b) all the members in which the moments are reduced shall belong to plastic or compact section classification (see 9.7).

10.5Plastic Analysis

10.5.1 Application

The effects of design action throughout or on part of a structure may be determined by a plastic analysis, provided that the requirements of 10.5.2 are met. The distribution of design action effects shall satisfy equilibrium and the boundary conditions.

10.5.2 Requirements

When a plastic method of analysis is used, all of the following conditions shall be satisfied, unless adequate ductility of the structure and plastic rotation capacity of its members and connections are established for the design loading conditions by other means of evaluation:

a) The yield stress of the grade of the steel used shall not exceed $450\mathrm{MPa}$ .

b) The stress-strain characteristics of the steel shall not be significantly different from those obtained for steels complying with the accepted standard [6-6(6)] or equivalent and shall be such as to ensure complete plastic moment redistribution. The stress-strain diagram shall have a plateau at the yield stress, extending for at least six times the yield strain. The ratio of the tensile strength to the yield stress specified for the grade of the steel shall not be less than 1.2. The elongation on a gauge length complying with the accepted standard [6-6(6)] shall not be less than 15 percent; and the steel shall exhibit strain-hardening capability. Steels conforming to the accepted standard [6-6(6)] shall be deemed to satisfy the above requirements.

c) The members used shall be hot-rolled or fabricated using hot-rolled plates and sections.

d) The cross-section of members not containing plastic hinges should be at least that of compact section (see 9.7.2), unless the members meet the strength requirements from elastic analysis.

e) Where plastic hinges occur in a member, the proportions of its Cross-section should not exceed the limiting values for plastic section given in 9.7.2.

f) The cross-section should be symmetrical about its axis perpendicular to the axis of the plastic hinge rotation.

g) The members shall not be subject to impact loading, requiring fracture assessment or

PART 6 STRUCTURAL DESIGN — SECTION 6 STEEL

fluctuating loading, requiring a fatigue assessment [see Section 6(p)].

10.5.2.1 Restraints

If practicable, torsional restraint (against lateral buckling) should be provided at all plastic hinge locations. Where not feasible, the restraint should be provided within a distance of $D / 2$ of the plastic hinge location, where $D$ is the total depth of section.

The torsional restraint requirement at a section as above, need not be met at the last plastic hinge to form, provided it can be clearly identified.

Within a member containing a plastic hinge, the maximum distance $L_{m}$ from the restraint at the plastic hinge to an adjacent restraint should be calculated by any rational method or the conservative method given below, so as to prevent lateral buckling.

Conservatively $L_{\mathrm{m}}$ (in mm) may be taken as:

L _ {\mathrm{m}} \leq \frac {3 8 r _ {\mathrm{y}}}{\left[ \frac {f _ {\mathrm{c}}}{1 3 0} + \left(\frac {f _ {\mathrm{y}}}{2 5 0}\right) ^ {2} \left(\frac {x _ {\mathrm{t}}}{4 0}\right) ^ {2} \right] ^ {1 / 2}}

where

$f_{c}$ = actual compressive stress on the cross-section due to axial load, in N/mm $^{2}$ ,

$f_{\mathrm{y}}$ = yield stress, in $\mathrm{N / mm^2}$ ;

$r_{y}$ = radius of gyration about the minor axis, in mm;

$x_{\mathrm{t}}$ = torsional index, $x_{\mathrm{t}} = 1.132 (AI_{\mathrm{w}} / I_{\mathrm{y}}I_{\mathrm{t}})0.5;$

$A$ = area of cross-section; and

$I_{w}, I_{y} = \text{ warping constant, second moment of the } I_{t} \text{ cross-section about the minor axes and St. Venant's torsion constant, respectively.}$

Where the member has unequal flanges, $r_{y}$ should be taken as the lesser of the values of the compression flange only or the whole section.

Where the cross-section of the member varies within the length $L_{\mathrm{m}}$ , the maximum value of $r_{\mathrm{y}}$ and the minimum value of $x_{\mathrm{t}}$ should be used.

The spacing of restraints to member lengths not containing a plastic hinge should satisfy the recommendations of section on lateral buckling strength of beams (see 14.2.2). Where the restraints are placed at the limiting distance $L_{m}$ , no further checks are required.

10.5.2.2 Stiffeners at plastic hinge locations

Web stiffeners should be provided where a concentrated load, which exceeds 10 percent of the shear capacity of the member, is applied within $D / 2$ of a plastic hinge

location, (see 14.2.1.2). The stiffener should be provided within a distance of half the depth of the member on either side of the hinge location and be designed to carry the applied load in accordance with 14.7.4. If the stiffeners are flat plates, the outstand width to the thickness ratio, b/t, should not exceed the values given in the plastic section (see 9.7 and Table 2). Where other sections are used, the ratio $\left(\frac{I_{\mathrm{so}}}{I_{\mathrm{t}}}\right)^{1/2}$ , should not exceed the values given for plastic section (for simple outstand, as in 9.7);

where

$I_{so} = \text{second moment of area of the stiffener about the face of the element perpendicular to the web; and}$

$I_{\mathrm{t}} =$ St. Venant's torsion constant of the stiffener.

10.5.2.3 The frame shall be adequately supported against sway and out-of-plane buckling, by bracings, moment resisting frame or an independent system such as shear wall.

10.5.2.4 Fabrication restriction

Within a length equal to the member depth, on either side of a plastic hinge location, the following restrictions should be applied to the tension flange and noted in the design drawings. Holes, if required, should be drilled or else punched 2 mm undersize and reamed. All sheared or hand flame cut edges should be finished smooth by grinding, chipping or planning.

10.5.3 Assumptions in Analysis

The design action effects shall be determined using a rigid-plastic analysis.

It shall be permissible to assume full strength or partial strength connections, provided the capacities of these are used in the analysis, and provided that,

a) In a full strength connection, the moment capacity of the connection shall be not less than that of the member being connected;

b) In a partial strength connection, the moment capacity of the connection may be less than that of the member being connected; and

c) In both cases the behavior of the connection shall be such as to allow all plastic hinges necessary for the collapse mechanism to develop, and shall be such that the required plastic hinge rotation does not exceed the rotation capacity at any of the plastic hinges in the collapse mechanism.

In the case of building structures, it is not normally necessary to consider the effect of alternating plasticity.

10.5.4 Second-Order Elastic Analysis

Any second-order effects of the loads acting on the structure in its deformed configuration may be neglected, provided the following are satisfied:

a) For clad frames, provided the stiffening effects of masonry infill wall panels or diaphragms of profiled wall panel is not taken into account, and where elastic buckling load factor, $\lambda_{cr}$ (see 10.6) satisfies $\lambda_{cr}/\lambda_{p} \geq 10$ .

If $10 > \lambda_{\mathrm{cr}} / \lambda_{\mathrm{p}} \geq 4.6$ the second-order effects may be considered by amplifying the design load effects obtained from plastic analysis by a factor $\delta_{\mathrm{p}} = \{0.9 \lambda_{\mathrm{cr}} / (\lambda_{\mathrm{cr}} - 1)\}$ .

If $\lambda_{\mathrm{cr}} / \lambda_{\mathrm{p}} < 4.6$ , second-order elastoplastic analysis or second order elastic analysis (see 10.4.3) is to be carried out.

b) for unclad frames or for clad frames where the stiffening effects of masonry infill or diaphragms of profiled wall panel is taken into account, where elastic buckling load factor, $\lambda_{\mathrm{cr}}$ (see 10.6) satisfies $\lambda_{cr}/\lambda_{p} \geq 20$ .

If $20 > \lambda_{cr}/\lambda_{p} \geq 5.75$ the second-order effects may be considered by amplifying the design load effects obtained from plastic analysis by a factor $\delta_{\mathrm{p}} = \{0.9 \lambda_{\mathrm{cr}} / (\lambda_{\mathrm{cr}} - 1)\}$ .

If $\lambda_{\mathrm{cr}} / \lambda_{\mathrm{p}} < 5.75$ , second-order elastoplastic analysis or second-order elastic analysis (see 10.4.3) shall be carried out.