The Direct Analysis Method
AISC developed the Direct Analysis Method (DAM) as a solution to meeting stability analysis and design requirements in a modern way that is most suitable for implementation in analysis and design software. This method is not limited in its application, applies to all buildings when designing to US codes, and is the most general and accurate approach provided. The requirements for the DAM include:
- Second-order analysis: A second-order analysis which considers both P-Δ and P-δ effects is required.
- Initial imperfections: The effects of initial imperfections of the structure geometry are considered by applying notional loads, which are lateral loads proportional to the gravity loads applied at each framing level.
- Inelasticity: The axial and flexural stiffnesses of members that contribute to the stability of the structure are required to be reduced. This is to account for the effects of residual stresses that lead to inelastic softening before the members reach their design strength.
- Effective Length Factor: Setting K to be 1.0 can be allowed because the effects for which it was meant to compensate (initial imperfections and inelasticity) have already been accounted for in the method.
DAM implementation in Tekla Structural Designer
Tekla Structural Designer implements the Direct Analysis Method fully, using a rigorous second-order analysis. The following table describes how the AISC requirements have been implemented.
|AISC Required effect to be considered||Tekla Structural Designer implementation|
|Flexural, shear, and axial deformations||A general 3D analysis is performed which considers all required deformations.|
|All component and connection deformation that contribute to the lateral displacement of the structure||All moment connections are assumed to be fully restrained. As is typical of most analysis programs, it is rationalized that these deformations do not contribute significantly to the stability of the structure and are therefore not directly included in the analysis.|
(both P-Δ and P-δ)
|The Chen and Lui rigorous second-order analysis is implemented. The two-cycle iterative method automates a two-pass analysis procedure during which nodal displacements are used to determine ‘stress stiffening’ in structural elements. The resulting matrix accommodates the P-Δ and P-δ effects as well as accounting for ‘stress stiffness’. See also: What are the "2nd Order" Analysis options listed on the Analyze Ribbon in TSD? Are they the same as "P-Delta" analysis?|
|Geometric imperfections||Notional loads are calculated and applied wherever there's a column stack or wall panel division based on the gravity load for each appropriate load combination.|
|Member stiffness reductions due to residual stresses||All relevant members have reduced axial and flexural stiffnesses, the default for this being 0.8. The option to take tb = 1.0 and increase the notional load factor is exercised.|
Choice of analysis type
This option is not suitable for final design of steel members. A standard linear elastic static analysis is included within Tekla Structural Designer to assist the user with steel structures that prove to be unstable and fail to complete second-order analysis.
The results from the first-order analysis can be utilized to "pre-design" all members, after which you are more likely to have sections that will be sufficient for second-order analysis to solve.
The first-order analysis is also used to determine both building drifts and member deflections in order to determine structure stability, seismic drift and serviceability design. The stability coefficient (Δ2/Δ1) is based on an approximate calculation of the second-order deflections of the structures based on the first-order analysis results.
This option is suitable for use in the Direct Analysis Method. Second-order analysis is performed using a two step iterative method incorporating a geometric stress stiffness matrix. An initial linear static analysis is performed, the stiffness matrix is updated to account for forces derived from initial displacements and finally a second linear static analysis is performed using the modified stiffness matrix.
This second-order analysis accurately determines both:
- P-Δ (P-big delta) effects resulting from loads acting on the global deformation of the entire structure or a part of the structure
- P-δ (P-little delta) effects resulting from the line of action of the axial force in a member acting at an eccentricity due the deformed shape of the member.
The forces resulting from this analysis are utilized in the member design.
Also during the design process first-order analysis is used to determine both building drift and member deflections in order to determine structure stability, seismic drift and serviceability design.
The stability coefficient (Δ2/Δ1) is based on a combination of results from both the second and first-order analyses.Note: There is potential for buckling to occur during 2nd order analysis. See the following article for more details: What is 2nd Order Buckling Analysis? How and when would I use it?
Analysis model adjustments
Certain adjustments should be made to the structural model in order to prepare it for analysis – these are controllable by the user.
- For both first and second-order analysis
- concrete member section properties should be set as cracked, and suitable Modification Factors should be applied via Analysis Settings, (see ACI 318-05 10.11.1)
- For second-order analysis
- 2005 AISC Specification 7.3 (3) & (4) requires that for those
members whose stiffness is considered to contribute to the lateral
stability of the structure, a reduced flexural stiffness (EI) and a
reduced axial stiffness (EA) must be used in the analysis. In Tekla Structural Designer, a reduction factor of 0.8 (default) is automatically applied to the
stiffness (EI and EA) of all steel members. You are able to review this
default stiffness factor and adjust it if you require via Design Settings >
Analysis. Note: The above factor is applied in addition to any Modification Factors specified for steel members in Analysis Settings.
- 2005 AISC Specification 7.3 (3) & (4) requires that for those members whose stiffness is considered to contribute to the lateral stability of the structure, a reduced flexural stiffness (EI) and a reduced axial stiffness (EA) must be used in the analysis. In Tekla Structural Designer, a reduction factor of 0.8 (default) is automatically applied to the stiffness (EI and EA) of all steel members. You are able to review this default stiffness factor and adjust it if you require via Design Settings > Analysis.
Additionally, in order to adhere to the Direct Analysis Method, certain extra enhancements are made to the analysis model:
- For both first and second-order analysis
- All load combinations require a minimum lateral load at each level of the structure of
0.2% (2005 AISC Specification 7.3 (2)). In addition, Tekla Structural Designer assumes τb = 1.0, so an additional 0.1% is required
(2005 AISC Specification 7.3 (3)).
It is the user's responsibility to include notional loads in combinations using the combinations wizard. for these Tekla Structural Designer automatically calculates the additional notional load to be 0.3% of the gravity component in each combination. (Although the user is provided with an option in Model Settings to override this with their own percentage values if they require.)
Hence in combinations that have notional loads included by the user, Tekla Structural Designer automatically calculates the additional notional load to be 0.3% of the gravity component in each combination. (Although the user is provided with an option in Model Settings to override this with their own percentage values if they require.)
- All load combinations require a minimum lateral load at each level of the structure of 0.2% (2005 AISC Specification 7.3 (2)). In addition, Tekla Structural Designer assumes τb = 1.0, so an additional 0.1% is required (2005 AISC Specification 7.3 (3)).
- For second-order analysis
- For designs using ASD, prior to second-order analysis all loads are automatically factored up by 1.6 and before design, all design forces are reduced by a factor of 1.6 as required by 2005 AISC Specification 7.3 (1).
When applying the Direct Analysis Method the following approach is recommended. This helps to avoid spending unproductive time on analysis issues and a potentially overly conservative design:
- Never just construct a model (or import one e.g. from Revit) and attempt design using second-order analysis straight away (by setting Design > Settings > Analysis > Second-order).
- Always conduct a full First-order analysis and design for both gravity and
lateral combination to size members for these by setting Design > Settings
> Analysis > First-order. This should ensure that:
There are no mechanisms - troubleshoot and remove any mechanisms that come to light at this stage by following this article Guide to resolving Mechanisms.
Section sizes are sensible and adequate for First-order design forces. They will then usually be adequate for second-order analysis without buckling (though not always). See this article for more about this; I have analysis Warnings and/ or Errors about buckling. Why is this happening and what do I do about it?
- It is a good idea to assess displacements and Drift check results after this initial design also as these can highlight parts of the structure that are poorly braced or not well connected to the lateral force resisting system (LFRS). However bear in mind that these are not final results - second-order analysis may lead to increased section sizes which will moderate drift results.
- Only when all members are passing at this stage, and overall displacements and drift results appear reasonable, should you proceed to second-order analysis and design. Set Design > Settings > Analysis > Second-order and first perform a “check” design, by turning off auto-design for all members then running a Design (Static) command, and assess the results of this.
- Members that now fail at this stage are most likely to be members in the LFRS
since second-order magnification effects will tend to be largest in these. If
there are only a few failing members, consider skipping to step 6. Otherwise,
set ONLY the failing members to auto-design on, then run a full model design
When this is complete, assess the efficiency of the section sizes assigned to the auto-designed members by examining their utilization ratios (URs) (tip - Design review filters can help with this). If the UR’s are low, it may be that pre-size sections (which are assigned to all auto-design members for the first analysis run) are causing excessive second-order magnification of design forces leading to inefficient section selection.
Though it is rare, buckling issues may occur during the second-order analysis if the pre-size sections are inadequate. The "Prevent out of plane instability" setting can be used to mitigate this in some circumstances as described in this support article.
- If there are only a few members failing the second-order check-design - or if
excessive second-order magnification or buckling of pre-size sections is
occurring - we recommend conducting a 'manual' iterative second-order analysis
and design as follows:
- After setting second-order analysis for design and running a “check” design as described in step 4) above, add any failing members to a sub structure and set them to auto-design on.
- Run a design of this sub-structure ONLY from the Structure tree
context menu as shown in the picture below. Note that when this is complete
by default this will set auto-design off for the members in it (but check
that this is the case).
- Run a “check” design of the entire structure - this updates the global analysis results for the new sections which may produce a change in member forces so some members may now fail again.
Set any failing members (ONLY) back to auto-design, update the sub structure if necessary, then design it once more.
- Iterate steps c) and d) until all members are passing.
This "manual" iterative approach avoids the potential for pre-size sections to fail analysis due to buckling or cause excessive second-order amplification leading to a suboptimal design solution.
Note however that you may still need to manually size members to satisfy Drift checks or, for members of a SFRS, to pass specific seismic design checks (members are not auto-sized for these).