# 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:

$B1=\frac{C_m}{1-\alpha P_r/P_{e1}}$      See AISC Eqn A-8-3

Where:

• $C_m =$ 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!
• $\alpha=$ 1 (LRFD), 1.6 (ASD)
• $P_r = P_{nt}+P_{lt}$ = may be determined from a first-order estimate is permitted (for use in eqn A-8-3 only).
• $P_{nt}$ = axial load (using ASD/LRFD Load Combo’s) assuming the structure is “restrained against lateral translation” (first order).
• $P_{lt}$ = axial load due to lateral translation of the structure only (using ASD/LRFD Load Combo’s) (first order).
• $P_{e1} = \frac {\pi EI* } { (K_1 L)^2 }$ = coefficient assuming no lateral translation of the frame.
• $EI*$ = Flexural rigidity required by the analysis i.e because we are using DAM =
• $EI* = 0.8 \tau_b EI$ where $\tau$ is defined in AISC Section C2.3 (adjustments to stiffness). I prefer to avoid adjusting $\tau$ as it seems to become an iterative process. Therefore I add an additional notional load =
• $N_{ \tau} = 0.001 \alpha Y_i \text{where} Y_i$ is defined in Section C2.2b (essentially the gravity load at level ‘i’) and $\alpha$ is as defined above.
• $K_1$ is based on the assumption of no lateral translation so we conservatively use 1.0.

B1 accounts for for $P-\delta$ 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:

$B2=\frac{1}{1-\frac{\alpha P_{story}}{P_{estory}}}$       See AISC Eqn A-8-6

Where:

• $\alpha$ is as defined above.
• $P_{story} =$ 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.
• $P_{estory} =$ the “elastic critical buckling strength for the story in the direction of translation being considered, determined by buckling analysis” or
• $P_{estory} = R_M \frac{H L} {\Delta_H}$
• $R_M = 1-0.15(P_{mf} / P_{story})$
• $P_{mf}$ = 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
• $\Delta_H =$ 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 $P_{story}$.
• AISC provides a user note that says H and $\Delta_H$ “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 $P-\Delta$ 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 $P_{lt} and M_{lt}$ (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.

• Dead load: Say 30 psf just to give it some weight.

• 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.

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; $w_{DL}=30\;psf \;x \;25 \;ft = 750\; plf$
Live load; $w_{LL}=100\; psf\; x\; 25\; ft = 2500\; plf$
Total load; $w = 3.25\; klf$
Shear; $V_r = [3.25\; klf\; x 25\; ft] / 2 = 40.6 kip$
Moment; $M_r = [3.25\; klf\; x (25\; ft)^2] / 8 = 254 kip-ft$
Unbraced length say 5ft. (practically fully braced for positive moment)

Required moment of inertia – dead load; $I_r=\frac {5wL^4}{(384E\Delta)} = \frac {5\;x\;3.25klf\;25ft^4\;1728}{(384\;x\;29000ksi\;x\;25\;x\;12/240)}= 788in^4$

Required moment of inertia – live load; $I_r=\frac {5\;x\;2.5klf\;25ft^4\;1728}{(384\;x\;29000ksi\;x\;25\;x\;12/360)}= 909in^4$ (Controls)

#### Typical Column – Exterior;

Dead load = $P_{DL}=\;30psf\;x\;25ft\;x\;12.5ft\;=9.375kip$
Live load = $P_{LL}=\;100psf\;x\;25ft\;x\;12.5ft\;=31.25kip$
Required axial load = $P_r=\;40.6kip$

#### Typical Column – Interior;

Dead load = $P_{DL}=\;30psf\;x\;25ft\;x\;25ft\;=18.75kip$
Live load = $P_{LL}=\;100psf\;x\;25ft\;x\;25ft\;=62.5kip$
Required axial load = $P_r=\;81.25kip$

To Be Cont….

## 2 comments for “Stability – AISC’s Approximate Second-Order Analysis B1 B2 Method”

1. Lane Wilder
November 24, 2014 at 2:02 pm

Newbie here. Is there a part 2 of this post? I’d be very interested. Thanks.

2. June 29, 2016 at 9:40 am

Dear Ryan,

I have two questions for you. I wonder your view on how to deal with it.
1) To run a second order analysis for ASD it is directed by AISC to multiply the loads by 1.6. But how to deal with it if for these loadcases a 1/3 increase is applicable. Should this factor by reduced to 1.2?
2) If the structure is braced with braces designed to take the lateral loads and the ratio of max second order drift to max first order drift is (say) <1.5, is a first order analysis not always adequate enough?

Thanks.