AISC (2005) Specifications
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Transcript of 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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Second Order Analysis
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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Second Order Analysis
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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Second Order Analysis
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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C2 Calculation of Load Effects
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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C2 Calculation of Load Effects
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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App. 7 Direct Analysis Method
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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App. 7 Direct Analysis Method
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.