# Static Analysis

- Static Analysis Types: First-Order, P-Delta, or AISC Direct Analysis Method, Nonlinear
- Run a Static Linear Analysis
- Analyze with One-way Elements
- P-Delta Analysis: Some details and theory
- AISC Direct Analysis Method (DAM): More details, theory, and application,
- Approximations in Static Analysis
- See Also: Nonlinear Analysis

## Static Analysis Types

VisualAnalysis supports three methods for static analysis. You define which type to use in the Project Settings settings in the Modify tab of Project Manager.

**First Order**

A first-order analysis is linear and does not account for 2nd-order effects such moment-magnification or buckling. It is the fastest type and is often a good place to start when debugging models to insure that they are basically stable. The fast, proven algorithms assume small-displacements, no buckling, no material yielding or plasticity, no cracking. The method will blindly calculate erroneous results given a crazy model or ridiculous loads. VisualAnalysis checks for common problems after the analysis and can warn about some things, but will not report all possible problems.

If you have one-way members or springs in the model, the analysis is iterated to determine which elements violate their constraints, this iteration causes the results to be nonlinear, but we still call it a first-order analysis.

**P-Delta (2nd Order)**

A second order analysis can account for a number of common structural behaviors and provide more accurate results than a first order analysis. This analysis is iterated to include moment magnification effects, P-Delta, that can result when a structure displaces under loading. This type of analysis can fail if members are inadequately sized or if loads are too large. VisualAnalysis will attempt to provide you with feedback regarding where the problems lie, but this is not always definitive guidance.

**AISC's Direct Analysis Method (DAM)**

AISC introduced a new type of analysis in their 13th edition of the Steel Design Manual. This 2nd order P-Delta analysis includes additional features to account for known problems in structures:

- Members are not initially straight
- Buildings are not exactly plumb
- Steel members have residual stresses from fabrication processes

All three of these can reduce the capacity of a structure and are accounted for within the Direct Analysis Method automatically--there are no options or settings to control the DAM in VisualAnalysis. (See below for more details and discussion about the Direct Analysis Method.)

Performing a Direct Analysis is not trivial and involves
a large number of generated load combinations in addition to iterating the
analysis, so it can be significantly slower. * Indeed, if you have 30 load combinations you will actually analyze 90 load combinations with the direct analysis method! And each is analyzed twice, with some additional processing for notional loads and stiffness reduction, so the time for analysis can be over 6 times slower!* You should probably use First-Order or P-Delta during preliminary design phases for performance.

## Run a Static Linear Analysis

After using the **Check Model for Errors** command an analysis can be
performed using **Analyze | Analyze Now **or clicking on the Result View tab.

The simplest and most common analysis is a static linear analysis. In this case, there are no one-way members, and instability effects are assumed to be negligible. The principle of superposition of results is valid for static linear analyses. Most properly designed structures acting under normal design loads will behave in a manner that is adequately predicted by a static linear analysis. The followings sections describe complicating factors to consider as you decide which type of static analysis to perform.

## Analyze with One-way Elements

Whenever a structure has one or more one-way (tension-only or compression-only) elements, the analysis becomes iterative and nonlinear. This means that results from separate load cases cannot be superimposed.

The iterative procedure performs automatically what you would do manually. For the first run on any load case all of the one-way elements are included in the model. Forces are then calculated for the one-way elements, and if any constraints are violated, those elements are removed. If elements have been removed, another analysis is performed on the modified structure. Again, forces are calculated to determine whether or not violations exist. and if so, these elements are removed. In addition, the removed elements are checked to see if they should be returned to the model by calculating what force they would have had. If changes are made the analysis is repeated for the modified model.

This process repeats until no changes are necessary or until the iteration limit is reached. This process can be very time consuming, especially for large models or many load cases. Choose your load cases carefully.

Use **Analyze | Performance vs. Accuracy** to adjust the limit on
iterations. The default value should be enough to determine whether the analysis
will converge to a solution or not.

There is no guarantee that this type of analysis will succeed. If too many elements are removed the model may become unstable. If too many elements are marked as one-way, the software may "bog down" trying all the possible combinations. If you have problems, we recommend that you take a semi-manual approach to modeling your project and keep the number of one-way elements to a minimum.

## P-Delta Analysis

VisualAnalysis has the ability to consider geometric stiffness effects for both members and plate elements. These P-Delta results are only available when geometric or material nonlinearity is not present from elements such as one-way springs or semi-rigid end connections without constant rigidity.

P-Delta analysis is handled by adding the geometric stiffness to the structure’s total stiffness. The process is iterative because geometric stiffness depends on member forces, which are not known until after the analysis.

When P-Delta results are presented, two effects are incorporated into the results, P-big-Delta (P-D), and P-little-delta (P-d). The first effect applies to members only and involves a modified bending moment acting along the length of the member. P-D is demonstrated in the sketch below. You can easily imagine how the columns will experience not just an axial loads (1st order analysis shows this) but also a moment equal to the vertical force, P, multiplied by the displacement, D.

The P-d effect is generally less important and
results from the bent shape of a member that is carrying an axial force.
In the above sketch, the columns are no longer straight after the initial
analysis, resulting in not just the P-D moment, but
also a P-d
moment that varies over the length of the member. **VisualAnalysis does
not calculate p-****d****
directly**, but this is not a serious problem because the largest moments
in a frame under lateral force sway generally occur at the column ends where p-d
is zero. If p-d is a concern, dividing each
column into several segments between floor levels will introduce the effect.

**The VisualAnalysis Procedure for P-Delta Analysis**

First, a standard first-order analysis is performed to calculate the deflection, D. Then, additional passes of "deflected" analyses are made with the deflected shape of the structure, resulting in potentially more moments. The iterative process is continued until either results converge, some maximum iteration limit is reached, or results start to diverge (some element's stiffness goes 'negative' indicating buckling). The results are based on the final "deflected" run.

In all cases when d or D are "large" you should carefully examine the results (as you should with any computer program). The potential problems resulting from large deflections and rotations have been discussed above. If these assumptions are violated, the results are no longer valid.

## Direct Analysis Method (DAM)

**Calculating Direct Analysis**

Chapter C of the American Institute of Steel
Construction (AISC) **Manual 13th edition** is a radical
deviation from previous AISC provisions for several reasons which affect how
structural engineers must design steel frame systems. Probably the most
difficult of the new provisions arises from the moment magnification effect due
to steel member out-of-plumbness
and out-of-straightness. In previous AISC
codes this was handled indirectly though LRFD phi
factors
and ASD safety factors. With Direct
Analysis these effects are handled through the application of
notional loads. VisualAnalysis automatically will calculate where
notional loads should be applied, in which directions, their magnitudes, and
will also generate new load combinations to include them with the correct sets
of loads. This is not a trivial task, even in software. It is something nobody
will ever do by hand for a non-trivial structural model!

The second consideration that AISC now emphasizes is member stiffness reduction arising from residual stress. This is accomplished in the DAM directly through stiffness reduction (EA and EI) applied to members automatically during the analysis. With previous AISC codes, or when not using DAM, it is handled indirectly by effective length factors that are extremely complex to calculate (and therefore often poorly approximated in practice). Again, VisualAnalysis will automatically incorporate the reduced stiffness values for members during the DAM.

**Using Direct Analysis**

What do these requirements mean to an engineer using a computer program like VisualAnalysis? The amount of moment magnification must be categorized as either moderate or severe by calculating the story drifts at each level in your structure. AISC requires you to calculate the ratio of 2nd-Order drift to 1st-Order drift. AISC Section C2.2 states that when the second/first order drifts exceed 1.5, the Direct Analysis Method of Appendix 7 is required otherwise, less intensive methods described in sections C2.2a and C2.2b are allowed.

To help you determine if DAM is necessary, VisualAnalysis now has a report to
show the 2nd-Order/1st-Order story drift ratios for comparison, called **
Diaphragm Drift Results**.

**Validating and Reporting on DAM**

There are two new report tables that can be used with either 2nd-Order analysis (P-Delta or DAM). The first report table is the Diaphragm Drift Results. This report includes the ratio of second order to first order drift in each of the global in-plane directions for each diaphragm level in the structure. The report includes the node and second order drift for the location where the maximum second/first order ratio value listed. Note that AISC requires the Direct Analysis Method whenever the second/first order ratio exceeds 1.5.

A second report item available is the Member Moment Magnification table. This table lists the 2nd-Order and 1st-Order moments about for each member and the ratio of 2nd-Order/1st-Order moments. The result case where the largest ratio occurred is also indicated in parenthesis, keyed to a Result Cases table you may also include in the report. This table is useful for determining the second order P-Delta effects on your structural members.

Finally, while notional loads are automatic, and generally invisible, you can see their effects in an indirect way. VisualAnalysis reports a Statics Check for each result case (shown in the tab of , or in reports). This statics check will show total loads ad reactions in the lateral directions for your gravity load combinations. You can use this to help verify the magnitudes of the loads and to see that they were actually applied.

**DAM Notional Loads**

What is a “notional” load? Simply put, it is a percentage of the gravity load applied laterally to the structure, typically 0.2% of gravity. AISC C2.2 requires the application of this load in both building plan directions for each gravity-load-only load case therefore the analysis must have 2x the number of gravity load cases it used to have. If you have selected the DAM, VisualAnalysis now creates these new load cases automatically and applies the loads (behind the scenes) during analysis.

The way we set the notional load direction is to look at applied forces in each load case and if the resultant in a coordinate direction is a positive direction we put the notional in the positive direction and vice versa for negative. If there is no lateral load in the notional direction we just put it in the positive direction. If you apply loads in both horizontal directions in different load cases, you will get notional loads in both directions.

**Is DAM "Required" by AISC?**

An interesting note is that the load cases used to calculate ratios needs to be at strength level, i.e. using LRFD equations. ASD equations are allowed but must be multiplied by 1.6 which effectively shuts down ASD as an option. VisualAnalysis only uses LRFD equations in its reporting, and does not support ASD design checks with DAM.

Whether or not DAM is necessary, AISC requires that notional loads must be applied to model the crookedness affects occurred during fabrication and erection. In the case of the C2.2 provisions, AISC also throws the residual stress stiffness loss into the notional loads. The DAM requires the notional loads be applied to all load cases used for design. If C2.2 provisions are allowed, notional loads need only be applied to the gravity load cases.

So far, little has been said about the provisions in AISC (sections C2.2a and C2.2b) that might let you avoid DAM, which are allowed when the drift ratio is less than or equal to 1.5. It is the opinion of the developers of VisualAnalysis that these provisions have become obsolete for several reasons.

First, notional loads and their additional load cases **are still
necessary,**
albeit they may not be necessary for load cases with directional loads. These
notional load cases can control the design of structures with heavy gravity
loads and light lateral loads which might have a lower drift ratio.

Sections C2.2a and C2.2b represent methods which use either a) second order analysis results, or b) magnified first order results. In the case of magnified first order results, the B1 and B2 factors of previous codes must be calculated. These are not calculated by VisualAnalysis and can require the more difficult evaluation of the K2 effective length factor per the commentary equations C-C2-5 and C-C2-8, and all the ugly caveats described in that section of the commentary.

It is our opinion that the effort required in these provisions can easily outweigh using the DAM and the allowed K=1 in design, and that in practice, AISC is pushing the DAM method very hard!

**DAM Summary**

In summary, the changes to VisualAnalysis to accommodate AISC 13th edition have involved implementing the Direct Analysis Method described in Appendix 7. The generation of notional load cases (LRFD) is automatic for this method. Stiffness reductions required in the DAM are also handled in the program during analysis. Finally, the P-D requirements are taken care of using geometric stiffness in our second order analysis. VisualAnalysis will automatically utilize effective length factor K = 1 in your steel member designs with the DAM and will meet the Chapter C provisions.

## Approximations in Static Analysis

When modeling with Plate Elements it is important to recognize that these elements are approximations (unlike members which offer "true" solutions to the elasticity equations). You will need to use mesh refinement or other techniques to determine if your results are good. Please see the plate element topic for more details.