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1Combined Forces Theory
Developed by Scott CivjanUniversity of Massachusetts, Amherst
DIRECT ANALYSIS
2Combined Forces Theory
DIRECT ANALYSIS METHOD
Analysis of Entire Structure Interaction
Include Lateral “Notional” Loads
All Members Must be Evaluated Under Combined Axial and Flexural
Load
No K values required
Reduce Stiffness of Structure
3Combined Forces Theory
Moment M
Axi
al F
orce
PDIRECT ANALYSIS METHOD
Initially consider a “traditional” analysis
PnKL
Py
Axial Strength is defined as PnKL which includes K factors
(Py indicates crushing)
Mp
Bending Strength is defined as Mn, assumed here to be Mp for
a laterally braced member
4Combined Forces Theory
Moment M
PnKL
Axi
al F
orce
P
Pu
MuMp
Elastic 2nd Order(Nominal Loads)
Actual ResponsePy
DIRECT ANALYSIS METHODTypical design accounts for interaction by calibrating the
member design to column curves
Actual response produces a higher internal moment in
the member. This is accounted for in calibrating the member check, but does
not get transferred into adjacent members and
connections 5Combined Forces Theory
Moment M
PnKL
Axi
al F
orce
P
Pu
MuMp
Elastic 2nd Order(Nominal Loads)
Actual ResponsePy
DIRECT ANALYSIS METHOD
6Combined Forces Theory
Moment M
PnL
Axi
al F
orce
P
Py
Mp
DIRECT ANALYSIS METHOD
Bending Strength is defined as Mn, assumed here to be Mp for
a laterally braced member
Axial Strength is defined as PnL which assumes K=1 for all cases
Now consider the “Direct” analysis
Design Curve is therefore shifted upwards from previous assumptions
PnKL
7Combined Forces Theory
Moment M
PnL
Axi
al F
orce
P
Py
Pu
Mu Mp
DIRECT ANALYSIS METHOD
Elastic 2nd Order (Direct Analysis includes Notional Loads and Reduced Stiffness)
Direct Analysis accounts for interaction by including lateral “notional” loads which increase moment, reducing stiffness and
calibrating the member design to K=1 analysis
Actual Response
Actual response should then match the internal
moment , transferring this moment into adjacent
members and connections during analysis
8Combined Forces Theory
Moment M
PnL
Elastic 2nd Order (Direct Analysis includes Notional Loads and Reduced Stiffness)
Actual Response
Axi
al F
orce
P
Py
Pu
Mu Mp
DIRECT ANALYSIS METHOD
9Combined Forces Theory
“Notional” Loads
Notional loads are a function of the gravity load being applied
Notional loads are applied as a lateral load at each floor level in the direction that adds to the destabilizing effects of the load
combination being considered
Notional loads can account for geometric imperfections, inelasticity of members, and other non-ideal conditions
DIRECT ANALYSIS METHOD
10Combined Forces Theory
H+P/L
L
Recall that a vertical load acting through a displacement is similar to the application of a horizontal load P/L
Therefore, a notional load can be considered the equivalent of an assumed geometric imperfection
“Notional” Loads
H
P
L
DIRECT ANALYSIS METHOD
11Combined Forces Theory
DIRECT ANALYSIS METHOD
Analysis and Calibration
With proper calibration design strength approaches the actual response
Calibration consists of a combination of notional load values and reduction in member stiffness
Analysis is referenced to K=1 member capacities
12Combined Forces Theory
Appendix 7: Direct Analysis Method
13Combined Forces Spec 13th Ed
K=1 for all analysis
Rigorous Second Order Analysis Required (P- and P-)
(Such as verified computer analysis or amplified first order analysis)
Direct Analysis Method
REQUIRED if 2nd Order/1st Order>1.5(B2>1.5) (Section C2.2)
Analysis
14Combined Forces Spec 13th Ed
Rigorous Second Order Analysis
Typically computer analysis performed
Direct Analysis Method
Many programs neglect P- analysisOften not a significant effect, but this
should be checked (low B1 factor from AISC Section C2
indicates it can be neglected)
15Combined Forces Spec 13th Ed
If Pr<0.15PeL analysis can neglect P-
Direct Analysis Method
Where:= 1.0 (LRFD), 1.6 (ASD)Pr= Required Axial Compressive StrengthPeL= Euler Buckling Strength in the Plane of Bending (K=1)
Equation A-7-1
16Combined Forces Spec 13th Ed
Apply Notional Loads
Reduce Flexural Stiffness EI*
Reduce Axial Stiffness EA*
Direct Analysis Method
Steps
17Combined Forces Spec 13th Ed
Apply Notional Loads
Ni=0.002Yi
Ni= Notional Lateral Load Applied at Level iYi= Gravity Load at Level i from Load Combinations
Direct Analysis Method
18Combined Forces Spec 13th Ed
Notional loads are applied to ALL load combinations unless second order to first order drift ratio is ≤ 1.5. Then apply as minimum lateral load per Appendix
7.3.
Reduce Flexural Stiffness EI*
EI*=0.8bEI
Required for all members who contribute to lateral stability of the structure
(safe to include for all members)
E= Modulus of ElasticityI= Moment of Inertia about Axis of Bendingb=Reduction Factor for Inelastic Action
Direct Analysis Method
19Combined Forces Spec 13th Ed
Reduce Flexural Stiffness EI*
b=Reduction Factor for Inelastic Action
for
for
Direct Analysis Method
50.α y
rP
P
50.α y
rP
P
0.1τb
y
r
y
rP
PP
P α1α4τb
Pr= Required Axial Compressive StrengthPy= AFy = Member Yield Strength= 1.0 (LRFD), 1.6 (ASD)
20Combined Forces Spec 13th Ed
Reduce Axial Stiffness EA*
EA*=0.8EA
E= Modulus of ElasticityA= Cross Sectional Member Area
Direct Analysis Method
21Combined Forces Spec 13th Ed
Required for all members who contribute to lateral stability of the structure
(safe to include for all members)
