AISC’s Approximate Second-Order Analysis B1 B2 Method

Lets look at a very simple building with a simple moment frame to resist lateral loads. We will complete the analysis using AISC’s approximate second-order analysis more commonly known as the B1 – B2 method. We will demonstrate this method with the use of an example on a very simple building. Before we begin lets discus the B1, B2 method. The B1-B2 Method is an approximate second-order analysis using the multipliers B1 and B2 (no way!). The procedure can be found in Appendix 8 of the 2010 AISC 350 Steel Construction Manual.

B1 Discussion:

     See AISC Eqn A-8-3

Where:

• coefficient assuming no lateral translation of the frame. See Eqn A-8-4 . However I would conservatively use 1.0 and be done with it!
• 1 (LRFD), 1.6 (ASD)
• = may be determined from a first-order estimate is permitted (for use in eqn A-8-3 only).
  • = axial load (using ASD/LRFD Load Combo’s) assuming the structure is “restrained against lateral translation” (first order).
  • = axial load due to lateral translation of the structure only (using ASD/LRFD Load Combo’s) (first order).
• = coefficient assuming no lateral translation of the frame.
  • = Flexural rigidity required by the analysis i.e because we are using DAM =
  • where is defined in AISC Section C2.3 (adjustments to stiffness). I prefer to avoid adjusting as it seems to become an iterative process. Therefore I add an additional notional load =
    • is defined in Section C2.2b (essentially the gravity load at level ‘i’) and is as defined above.
• is based on the assumption of no lateral translation so we conservatively use 1.0.

B1 accounts for for effects in compression non-sway compression members. These are the moments caused by local displacements due to axial load. The AISC commentary suggests that if B1 is > than 1.2 than a rigorous second order analysis should be undertaken. This is due to the fact that B1 captures the local effects of second order forces/deformation but does not capture what effect these ‘local’ deformations may have on the overall structure. This is semi obvious in the fact that the we are using results from a first order analysis and also none of the variables relate to the rest of the structure.

B2 Discussion:

      See AISC Eqn A-8-6

Where:

• is as defined above.
• total vertical load supported by the story (using ASD/LRFD load Combos) including loads in columns not part of the lateral force resisting system. This is essential the total gravity load on the story under evaluation.
• the “elastic critical buckling strength for the story in the direction of translation being considered, determined by buckling analysis” or
  • 
    • 
      • = total vertical load in columns (in the story under evaluation) that are part of the moment frames (=0 for braced frame systems).
    • L = height of story
    •  Inter-story drift. Use first order analysis and stiffness as required by analysis i.e For DAM use reduced stiffness as discussed above (see the B1 discussion). Where drift varies across the story, the maximum drift may be used conservatively or a weighted average based on vertical load. It is important to realize here that this drift or deflection should include the deflection of the columns in the frame and ALSO the diaphragm deflection. The commentary words this as any “horizontal framing system that increases over-turning effect”. This makes sense as columns that are not part of the frame (usually called leaning columns) will displace greater than the columns that are part of the moment frame. This displacement coupled with gravity load will increase the demand on the frame.
    • H = Story shear (lateral force), produced by the lateral force used to compute the inter-story drift. Once again, coordinate the use of total story shear or individual force on the frame. As mentioned for .
    • AISC provides a user note that says H and “may be based on any lateral loading that provides a representative value of the story lateral stiffness.” As you can see the equation is really using the lateral stiffness of the structure (kip/in).

B2 accounts for effects on forces and moments in all members. These effects are due to lateral displacement of the structure. We also notice that B2 uses several variables which related back to the overall structure, mainly the the story shear, gravity load and deflection. Furthermore we see that the deflection is based on not only on the deflection of the frame but also of the diaphragm which indirectly accounts for “leaning columns”.

Short Summary

We can see that B2 applies to all members part of the Lateral Force Resisting System (LFRS), meaning any member with (members not part of the LFRS will not have these forces) and B1 applies only to compression members of the LFRS. We see that by using the DA method we eliminate having to use sidesway alignment chart (fig C-A-7-2) to try and determine K (effective length). However the B1-B2 method can be trick when when B1 gets large and multiple members frame into a column. Why? Well because the moments should be balanced and thus the column (compression member) will be multiplied by B1 and the beams would not be. So this moment then needs to be distributed to these connecting elements. I would suggest reading the Summary at the end of the commentary to the B1-B2 method  (Comm 8 pg 16.1-526) they discuss the how to apply the method in more ‘global’ terms and will give you a better feel for applying the method. It is too long to repeat here.

Example

Alright so lets define some parameters and loads.

Lets use a 1-story, 3-bay x 4-bay rectangular building. Bay size is 25’x25′. The columns are pinned at the base and have a moment connection from beam to column. For this analysis we will use a “Wind Only” moment frame or flexible moment connection. There is definitely some debate on using this type of system. Essentially beams are designed as simply supported for gravity loads and fixed for lateral loads. For a more complete discussion on “wind only” or flexible moment connection- moment frames see – Wind Only Moment Frames Discussion.

Size: 1-Story- 3×4 – 25’x25′ Bays (75′ x 125′ Building). Height is 15′ columns with 5′ parapet.

Gravity Loads:

• Dead load: Say 30 psf just to give it some weight.
• Live load: Say 100 psf again, too add some weight.

Lateral load:

• Check seismic, but lets use wind for now.
• Wind: say 20 psf. Most people will forget that there is a 1.5 multiplier on the parapet when designing for the LFRS so lets use 30 psf on  the parapet.

Analysis

We have 2 separate analysis to perform.

• Gravity load on analysis
• Lateral load only analysis

Gravity Load Analysis

Lets place the moment frame on grid lines 2 and 4 in the north-south direction.
For the gravity load analysis we would use all load combinations and assume the frame is restrained against lateral movement. Therefore we would not have moments due to lateral forces at the moment connections. For simplicity we will use our dead and live load. This would typically be snow load as this is a one story roof and remember to account for drift load as well.

Typical Beam;

Dead load;
Live load;
Total load;
Shear;
Moment;
Unbraced length say 5ft. (practically fully braced for positive moment)

Required moment of inertia – dead load;

Required moment of inertia – live load; (Controls)

Typical Column – Exterior;

Dead load =
Live load =
Required axial load =

Typical Column – Interior;

Dead load =
Live load =
Required axial load =

To Be Cont….

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Stability – AISC’s Direct Analysis Method

Intro

In the 14th Edition of the American Steel Construction Manual the Direct Analysis Method (DAM) is moved into the main specification from the appendix. I know many are not used to this new approach and some will say that if it’s not broke don’t fix it but I like the DA Method. Like anything else you need to put some time into learning it but it’s really not so bad in the end. It is a very interesting approach in that most design codes / manuals do not get into the analysis meaning that they don’t tell you how to get the required/design forces but rather they give an allowable/capacity of the member to which you are designing. However in DAM, AISC is assisting you in your analysis to make the design simpler. Essentially we need to address two different types of deflection/deformation associated with axial load that amplify moments in the structure. The first is P-δ (I may refer to this as P-d) which are moments associated with the axial load and deflection due to column curvature (Think of these as local displacements). The second is P-Δ (I may refer to this as P-D) moments which are caused by axial load and the translation of the end of the column (i.e. interstory drift) think of these as global displacements. Neither of these moments will show up in a first order elastic analysis. Well they may…sorta. I don’t want to get off track so I will explain what I mean later (this refers to common FEM models and placing multiple nodes along a member). We will now cover the DAM for a computer based approach and a simplified hand method.

Pd and PD Sketch

Overview

The direct analysis method is basically accounting for (3) issues:

1. Effects of initial geometric imperfections
2. Second-Order effects – Axial-Displacement Moments P-D and P-d (as shown above).
3. Effects of material non-linearity – In-elasticity due to residual stresses.

AISC actually states that there are (5) requirements. Below these requirements are listed and how they are addressed (AISC C-C1.1).

Considerations:

1. Consider all deformations
   • Note that this says ‘consider’ not necessarily include, i.e. column shears deformation, in-plane ‘rigid’ diaphragm displacement.
   • The model or analysis shall ‘consider’ all deformations.
2. Consider P-d and P-D
   • Perform a rigerous second order analysis
   • Use B1, B2 Method
3. Consider geometric imperfections
   • This typically this stems from column out of plumbness
   • This may be directly modeled in the analysis
   • A notional load may be applied to the analysis
   • Use KL = L
4. Consider stiffness due to inelasticity. This is typically due to residual stresses in framing members. Therefore some elements may soften ‘inelastically’  prior to reaching their design strength.
   • Apply a stiffness reduction factor
   • Use KL = L
5. Consider uncertainty in strength and stiffness
   • Apply a stiffness reduction factor.
   • Use KL = L

 Applying the Direct Analysis Method

First we will look at applying this method in a strict sense and assuming the use of a computer model. Then we will get to a more conservative hand calc method.

1. Model your structure and apply all loads. Set up your load combinations according to LRFD or ASD (Most likely see IBC load combo’s).
2. Run a first-order analysis and determine deflections.
   • Amplify the ASD loads x 1.6
   • Modify the stiffness of all members. For a first trial run use a 0.8 factor. This would be applied to axial (0.8*EA) and flexural (0.8EI) stiffness.
     • AISC states that the stiffness reduction need only be applied to members that contribute to the stability of the structure however they can be applied to all members to prevent artificial distortion.
3. Run a second-order analysis.
   • Amplify the ASD loads x 1.6
   • Modify the stiffness of all members. For a first trial run use a 0.8 factor. This would be applied to axial (0.8*EA) and flexural (0.8EI) stiffness.
     • AISC states that the stiffness reduction need only be applied to members that contribute to the stability of the structure however they can be applied to all members to prevent artificial distortion.
   • So this is just a mouse click away right? Well not quite. You should really know what your analysis software is doing. It is difficult if not impossible in some situations for software programs to perform a rigorous second-order analysis. For the program to perform this analysis it usually needs to run an iterative process on many nodes which may not be realistic. Therefore the program may use a geometric stiffness method which only accounts for P-D moments. Therefore P-d moments are still unaccounted for. However these moments may be “semi” captured if the column element is broken into several nodes. This way the deflection between nodes is captured in the analysis. AISC recognizes this practical problem and states that the P-d effects on the structure may be neglected if the second order drift to first order drift ratio (also known as B2) is equal or less than 1.7, also no more than 1/3 of the total gravity load on the structure is supported by columns that are part of the moment-resisting frames in the direction of translation being considered.
   •  In the commentary they equate the 1.7 to a 1.5 limit with no stiffness reduction.
   • Furthermore P-d effects must be considered to individual members subject to compression and flexure. In this case B-1 could be used.
4. Find the drift ratio (B2) of second order to first order drift. This will be used to determine what sort of notional loads will need to be applied.
5. Notional loads – Initial Imperfections
   • Initial imperfections may be directly applied in the model. Typically an out of plumbness of 1/500 is used the maximum specified in the Code of Standard Practice.
   • If not modeled directly notional loads may be applied. These are lateral loads Ni =0.002*α*Yi. These loads are distributed over the level in the same maner as the gravity load.
     • Ni = notional load at level i
     • α = 1.0 (LRFD); α = 1.6 (ASD)
     • Yi = gravity load applied at level i under each respective load combination
   • If B2 (drift ratio) is <= 1.7 then the notional loads may be applied as a minimum. Meaning that they are applied to gravity only load combinations but are not applied if the ‘actual’ lateral loads i.e. wind/EQ forces are greater than the notional load.
6. Adjust stiffness
   • For all stiffness that contribute to the stability of  the structure a 0.8 factor shall be applied i.e. EI, AE, etc.
   • Additionally for flexure the stiffness should be multiplied by 0.8:
     • For
     • For
       • α = 1.0 (LRFD); α = 1.6 (ASD)
       • Pr = required axial compressive strength of the member
       • Py = axial yield strength = Fy*Ag (yield stress x gross area)
   • In lieu of using taub a notional load of 0.001*α*Yi may be applied to the structure in similar fashion as the notional loads for initial imperfections. However these notional loads are additive for all load combinations.
7. Rerun second order analysis and check drift ratio, B2. Update any parameters based on the new drift ratio.
8. Design members using K=1. No Alignment Chart Required, Yeah!!!

 Applying the Direct Analysis Method

Now for a simplified hand calc to demonstrate the use of B1 and B2.
This post got to be a tad long so I’m going to break this into a separate post here.

References:

AISC 14th Edition CSC’s “Simple Guide to Direct Analysis” and webinar. Note that CSC’s Fastrak software does perform a rigorous second order analysis. RISA’s Practical Analysis with the AISC 13th Edition by Josh Plummer AISC Engineering Journal 3th Q 2008 “A comparison of Frame Stability Analysis Methods”

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