aisc (2005) specifications

7/29/2019 AISC (2005) Specifications

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Second Order Analysis

In the previous classes we looked at a method that

determines the load corresponding to a state of bifurcationequilibrium of a perfect frame by eigenvalye analysis

The system was assumed to be perfect, and there wereno lateral deflection until the load reached Pcr.

At Pcr, the original configuration of the frame becomesunstable, and with a slight perturbation the deflectionsstart increasing without bound.

However, if the system is not perfect, or it is subjected tolateral loads along with gravity loads, then the deflectionswill start increasing as soon as the loads are applied.

However, for an elastic frame, the maximum load capacity

will still be limited to that corresponding to Pcr


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To trace this curve, a complete load-deflection analysis of

the frame is necessary. A second order analysis willgenerate this load-deflection curve

In a second order analysis procedure, secondary effects asthe P- and P- effects, can be incorporated directly. As a

result, the need for the B1 and B2 factors is eliminated.

In a second-order analysis, the equilibrium equations areformulated with respect to the deformed geometry, whichis not known in advance and constantly changing, we

need to use an iterative technique to obtain solutions.

In numerical implementation, the incremental loadapproach is popular. The load is divided into smallincrements and applied to the structure sequentially.


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Discretize frame intobeam-column elements

For each beam-columnelement, formulate stiffnessmatrix k = ko + kg

Assemble structure stiffnessmatrix Kusing kand idarray

Begin Iteration for i load step

Ri = Ki1Di1Di1 = [Ki1]-1Ri

Where in the i load step,

Ri = incremental load vectorKi

1 = structure secant stiffness matrix

Di1= incremental displacement vector

Di1 = Di + Di1

For each element,

Extract di1 from Di

1

For each element,

Compute e, a, b

from di1 using Td

For each element,

compute P, a, b from e,

a, b using element local ke

For each element,

compute element end forces ri1

using TF and P, Ma, Mb

Form the structure internal force

vector Ri

1 using all element ri

1


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Second Order AnalysisForm the structure internal force

vector Ri1 using all element ri

1

Form the structure external force

vector Ri+1 = Ri + Ri

Evaluate unbalanced force vector

Qi1 = Ri+1 - Ri1

For each element, using the current

value of axial force P

Update element k

Assemble element kto form

updated structure stiffness matrix

Ki2

Di2=[Ki2]-1Qi1 Di2=Di + (Dik)

For each element,

Extract di2 from Di

2

For each element,

Compute e, a

, b

from di2 using Td

For each element,

compute P, a, b from e,

a, b using element local ke

For each element,compute element end forces ri

2

using TF and P, Ma, Mb

Form the structure internal force

vector Ri2 using all element ri

2

Re

peatuntilconvergence,whe

reQija

pprox.

0


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After convergence, Di+1 = Din = Di + Dik

Assumed another load increment, and go back to begining


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AISC (2005) Specifications

Chapter C Stability Analysis and Design


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C1. Stability Design Requirements

1. General Requirements

Stability shall be provided for the structure as a whole andfor each of its elements.

Any method that considers the effects of the following onthe stability of structure and elements is permitted:

Influence of second-order effects (P- and P-) produced byflexural, shear, and axial deformations

Geometric imperfections

Member stiffness reduction due to residual stress

All component and connection deformations that contributeto lateral displacements must be considered in the analysis

The methods prescribed in this Chapter C and Appendix 7satisfy these requirements


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C1. Stability Design Requirements

1. General Requirements

For structures designed by elastic analysis, individualmember stability and stability of the structure as a wholeare provided jointly by:

Calculation of the required load effects (Pr

, Vr

, Mr

) formembers and connection using one of the methods specifiedin C2.2

Designing members and connections using the specificationsof Chapters D, E, F, G, H, and I. e.g., for beam-column

members using the interaction equations of Chapter H.

For structures designed by inelastic analysis, theprovisions of Appendix 1 must be satisfied.


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C2 CALCULATION OF LOAD EFFECTS

C2.2b Design by first order analysis.

First order analysis does not account for the effects ofstructural deflections (due to applied loads) on the memberload effects, i.e., the second order effects. This is covered inCE474 or any matrix structural analysis class.

Required strengths (Pr, Vr, Mr) are permitted to bedetermined by a first-order analysis with all membersdesigned using K=1.0 provided that:

The required compressive strength (Pr) for all beam-columns

satisfy the limitation Pr < 0.5 PY

All load combinations must include an additional lateral loadNi applied in combination with other loads. Ni is applied ateach level (or story) of the structure and its value is:


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C2 Calculation of Load Effects

Ni = 2.1 (H/L) Yi 0.0042 Yi

Where, Yi = design gravity load applied at level i in kips

H/L = maximum ratio of story drift (H) to height (L) for allstories in the structure.

H = first order inter-story drift due to design loads

This notional load Ni must be considered independently intwo orthogonal directions of the structure.

The non-sway amplification of beam-column moment isconsidered by applying the B1 amplification to the moment

calculated from first order analysis.


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The amplified first-order elastic analysis method defined in

Section C2.1b is also an accepted method or second-orderelastic analysis of braced, moment, and combined frames.

The required second-order flexural strength Mr and axialstrength Pr must be determined as:

Mr = B1 Mnt + B2 Mlt

Pr = Pnt + B2 Plt

where, B1 = Cm/ (1-Pr/Pe1) 1.0 Here, Pr=Pnt+Plt

B2 = 1/(1-Pnt/Pe2) 1.0

where, Pe1= elastic buckling resistance of member in plane ofbending calculated on the basis of no sidesway = p2 EI/(K1L)

2

K1 = effective length factor in plane of bending assuming nosway and set equal to 1.0 unless analysis indicates otherwise


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Where, Pe2= elastic buckling resistance for the story

determined by sidesway buckling analysis For moment frames, Pe2 can be determined as the column

elastic buckling resistances for in-plane sway buckling

K2 = column effective length in plane of bending calculated

based on sidesway buckling analysis For all other types of lateral load resisting systems, it is

permitted to use:

Pe2 = RmHL/H

where, Rm =1.0 for braced, and 0.85 for moment frames

H = first-order inter-story drift due to lateral forces

H = story shear produced by same lateral forces


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C2 Calculation of load effects

Design by second-order analysis

Any second-order elastic analysis method that considersboth P- and P- effects may be used.

All gravity-only load combinations must include a minimumlateral load applied at each level of the structure of 0.002 Yi,

where Yi is the design gravity load applied at level i (kips).

This notional lateral load will be applied independently inboth orthogonal directions of the structure.

The notional lateral load limits the error caused by

neglecting initial out-of-plumbness and member stiffnessreduction due to residual stresses in the analysis.


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C2 Calculation of load effects

Design by second-order analysis (cont.)

Calculate the ratio of second-order drift to first-order drift.This ratio can be calculation from analysis or using theequation for B2 (presented later).

If this ratio is less than 1.1 then members are permitted to

be designed using effective length K=1.0.

If this ratio is greater than 1.1 but less than 1.5, then themembers must be designed using K factor determined froma sidesway buckling analysis of the structure. Stiffness

reduction factor due to column inelasticity is permitted in thedetermination of the K factor. For braced frames K=1.0

If this ratio is greater than 1.5, then must use the directanalysis approach given in Appendix 7.


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App. 7 Direct Analysis Method

The available strength (capacity) of members can beobtained from the provisions of applicable chapters E, F,

G, H, or I.

For beam-columns designed using Chapter H, the nominalcolumn strengths Pn can be determined using K=1.0.

The required strength for members, connections, and

other structural elements shall be determined from asecond-order elastic analysis.

This second-order analysis must include P- and P- effects,and must include all deformation contributions from

members and connections. It is permitted to use any general second-order analysis

method or even the B1, B2 factor method provided thesefactors are based on reduced member stiffness defined later.

Methods of analysis that neglect the P- (B1) effect are

permitted if Pr < 0.15 Pe in the plane of bending.


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While conducting the second order analysis, notional loads

must be applied to the framing system to account for theeffects of geometric imperfections, inelasticity, or both.

Notional lateral loads must be applied at each framing leveland specified in terms of the gravity loads at that level.

Ni=0.002 Yi

The gravity loads used to determine the notional loads will beassociated with the load combination being evaluated.

Notional loads will be applied in the direction that adds to thedestabilizing effects under the specified load combination

The notional load coefficient of 0.002 is based on assumed

initial out-of-story plumbness=H/500. As mentioned earlier, if the ratio of second order-to-first

order interstory drift is < 1.5, then notional lateral loads canbe applied to only the gravity loading combinations.

Additionally, it is permitted to use assumed out-of-plumbnessgeometry in the analysis instead of the notional load


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While conducting the second-order analysis, a reduced

flexural stiffness EI*

must be used for all the members.EI* = 0.8 t E I

where, t = 1.0 , if Pr/PY 0.5

and, t = 4[Pr/PY(1-Pr/PY)] , if Pr/PY>0.5

You can choose to not use the t factor but then anadditional 0.001 Yi load must added to the notional load Ni

A reduced axial stiffness EA

*

must be used for memberswhose axial stiffness will contribute to lateral stability.


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Checking member strength (Chapter H)

Design of members under flexure and axial force (H1)

For doubly and singly symmetric members in flexure andcompression with moments primarily in one plane it ispermitted to consider two independent limit states:

In-plane instability

Out-of-plane buckling

For the limit-state of in-plane instability:

For Pr/Pc0.2 Pr/Pc+8/9[Mrx/Mcx]1.0

For Pr/Pc


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Checking member strength (Chapter H)

For the limit-state of out-of-plane instability:

Pr/Pco + (Mr/Mcx)21.0where, Pco = column strength for out-of-plane buckling

Mcx = beam flexural-torsional buckling strength

If bending occurs only about the weak axis then themoment ratio in the above equation must be neglectedbecause flexural-torsional buckling is not possible.

If biaxial bending occurs with Mr/Mc >0.05 in both axes.

For Pr/Pc 0.2 Pr/Pc + 8/9 (Mrx/Mcx+Mry/Mcy) 1.0

For Pr/Pc 0.2 Pr/2Pc + Mrx/Mcx+Mry/Mcy1.0

where Pc is column strength according to Ch. E

Mc is beam strength according to Ch. F.

aisc (2005) specifications

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