Based on the experimental and numerical studies performed by NIST an Alternative Load Path Analysis (ALPA) guideline is being developed by the Disproportionate Collapse Technical Committee to provide design, analysis and modeling methods for engineers in practice to reduce the vulnerability of steel and concrete structures to disproportionate collapse. This article presents a summary of chapter 3 of this guideline, titled Simplified Analysis to Predict Collapse Resistance. Chapter 3 focuses on simplified analysis methods that can be applied in form of spreadsheets or closed form solutions to predict the collapse capacity of steel and concrete buildings.

1. SIMPLIFIED ANALYSIS TO
PREDICT COLLAPSE
RESISTANCE
April 2017
Ramon Gilsanz
Akbar Mahvashmohammadi

2. Participants
•
2
Ramon Gilsanz
Karl Rubenacker
Akbar Mahvashmohammadi
Joshua Peng
John Abruzzo
Ross Cussen
Joe Gannon
Nabil A. Rahman
Sarah Orton
Min Liu

3. Contents
• 1- Introduction
• 2- Simplified Analysis to Components
• 3- Dynamic Effects on Collapse Capacity of Components
• 4- Simplified Analysis to Systems

4. 1- Introduction
• The objective is to develop simplified formulas to predict
collapse capacity of structures:
 Steel and concrete moment frames
 Steel gravity frames
 Flat plates
• Wood, cold formed steel, and masonry are not considered in
this study

5. 2.1- Steel Frame Structures
• Equilibrium:
• Material:
 Elastic perfectly plastic with effective yield stress 𝐹𝑦,𝑒𝑓𝑓 = (𝐹𝑦 + 𝐹𝑢)/2
 Assume yield surface below
F(𝛥)=
4𝑀(𝛥)
𝐿
𝑐𝑜𝑠2
𝜃 + 2𝑃(𝛥) · sin(𝜃)
𝑃
𝑃𝑦
+
𝑀
1.18𝑀 𝑝
= 1.0
𝑀 = 𝑀 𝑝
𝑃𝑦 = A𝐹𝑦,𝑒𝑓𝑓
𝑀 𝑝 = 𝑍𝐹𝑦,𝑒𝑓𝑓

6. 2.1- Steel Frame Structures
• Assumption: The resistance at each point is the maximum
resistance achieved by using any pair of moment and axial
force on yield surface.
• 𝑙 𝑝 = 0.25𝑑 for WUF-B connections
• 𝑙 𝑝 = 0.6𝑑 for RBS connections
𝐹(𝛥) = ma x 4𝑀 𝛥
𝐿
𝐿2 + 𝛥2
+ 2𝑃𝑦 1 −
)𝑀(𝛥
1.18𝑀 𝑝
𝛥
𝐿2 + 𝛥2
𝐹(𝛥) =
4𝑀 𝑝
𝐿
𝐿2 + 𝛥2 + 0.3 · 𝑃𝑦
𝛥
𝐿2 + 𝛥2
, 𝛥 𝐿2 + 𝛥2 ≤ 2.353
𝑍𝐿
𝐴
2 · 𝑃𝑦
𝛥
𝐿2 + 𝛥2
, 2.353
𝑍𝐿
𝐴
< 𝛥 𝐿2 + 𝛥2 < 𝛥 𝑢
0 𝛥 ≥ 𝛥 𝑢
𝛥 𝑢 = 𝜃 𝑢
𝐿
2
= 𝜙 𝑢 𝑙 𝑝
L
2
=
ɛ 𝑢
0.5𝑑
𝑙 𝑝 𝐿

7. 2.1- Steel Frame Component
• Experimental vs estimated results for WUF-B and RBS tests
WUF-B RBS

8. 2.2 Concrete Frame Structures
• Following the research done by
Park:
• Deformation geometry:
 ℎ − 𝑐′
− 𝑐 = −
𝛥
2`
−
𝛽𝐿2
2𝛥
(ɛ +
2𝑡/𝐿)
• Equilibrium:
 𝐶𝑐
′
+ 𝐶𝑠
′
− 𝑇′
= 𝐶𝑐 + 𝐶𝑠 + 𝑇
• Leads to:
 𝑐′
=
ℎ
2
−
𝛥
4
−
𝛽𝑙2
4𝛥
ɛ +
2𝑡
𝑙
+
𝑇′−𝑇−𝐶 𝑠
′+𝐶 𝑠
1.7𝑓𝑐
′ 𝛽1
 𝑐 =
ℎ
2
−
𝛥
4
−
𝛽𝑙2
4𝛥
ɛ +
2𝑡
𝑙
+
𝑇′−𝑇−𝐶 𝑠
′+𝐶 𝑠
1.7𝑓𝑐
′ 𝛽1
Depth of beam available for arching
Formation of plastic hinges
C.G
Compressive
stress
Deformation diagram of AB yielded section

9. 2.2 Concrete Frame Structures
𝛥 = 𝛥 𝑢𝛥 = ℎ − (𝑐′
+ 𝑐)/2
𝐹
𝛥
𝑀(𝛥) =
0.85𝑓𝑐
′
𝑏𝛽1 𝑐(0.5ℎ − 0.5𝛽1 𝑐) + 𝐶𝑠 (0.5ℎ − 𝑑′
) + 𝑇(𝑑 − 0.5ℎ) 𝛥 ≤ 𝑑 − (𝑐’ + 𝑐 ) 2
0 𝛥 > 𝑑 − (𝑐’ + 𝑐 ) 2
𝑃(𝛥) =
𝐶𝑐
′
+ 𝐶𝑠
′
− 𝑇′
= 𝐶 𝐶 + 𝐶𝑆 − 𝑇 𝛥 ≤ 𝑑 − (𝑐’ + 𝑐 ) 2
𝑇 + 𝑇′ 𝛥 > 𝑑 − (𝑐’ + 𝑐 ) 2
𝐹 𝛥 =
2 𝑀′ 𝛥 + 𝑀 𝛥 − 𝑃 𝛥 𝛥
𝛽𝐿
𝛥 ≤ 𝑑 − (𝑐′
+𝑐)/2
2 · 𝑃𝑦
𝛥 𝑢
𝐿2 + 𝛥 𝑢
2
𝛥 = 𝛥 𝑢

10. 2.3 Concrete Frame Component
• Experimental vs estimated results for IMF and SMF tests
IMF SMF

11. 2.3 Steel Gravity Frame Component
• Use Oosterhof and Driver (2016):
1- Localized deformation capacity: 𝛿 𝑚𝑎𝑥 = 0.7𝐿 𝑒
2- Calculate the rotation at initial failure, 𝜃 𝑢 using:
𝛿𝑠𝑝𝑟𝑖𝑛𝑔 = 𝛿 𝐶𝐿 + 𝑒𝑡𝑎𝑛(𝜃𝑐)
𝛿 𝐶𝐿 =
𝐿
2
(
1
cos 𝜃 𝑐
− 1)
3- Calculate the axial deformation at each bolt
location in the connection at initial failure
4- Determine the effective number of bolts in
catenary
𝑛 𝑒𝑓𝑓 =
𝐹𝑏𝑟
𝑅 𝑛
=
𝛿𝑠𝑝𝑟𝑖𝑛𝑔
𝛿 𝑦𝑖𝑒𝑙𝑑
5- Determine nominal resistance:
𝑅 𝑛 = min(0.6 2𝐿 𝑒 𝑡 𝜎 𝑦 + 𝜎 𝑢 2 , 3𝑡𝑑𝜎 𝑢,
1.5 𝐿 𝑒 −
𝑑
2
𝑡𝜎 𝑢, 0.6
𝜋𝑑2
4
𝜎 𝑢)

12. 2.3 Steel Gravity Frame Component
6- Plot the resultant force versus rotation diagram
7- Plot force vs deformation
𝐹 = 2𝐹𝑅sin (𝛾)
Δ = L. tan(𝜃𝑐)

13. 2.3 Steel Gravity Frame Component
• Component Example:

14. 2.4 Flat Plates
Force
KA = Factor of two-way behavior
KB = Factor of cracking
KC = Factor of edge conditions
𝛥 = 𝐾𝐴 𝐾 𝐵 𝐾𝐶
𝜔𝑙 𝑛
4
𝐷
𝐷 =
𝐸𝑐ℎ3
12 1 − 𝜈2
1. End of elastic flexural response:
2
)(*3
l
mbcmtc 

mtc: moment capacity of
top steel in column strip
mbc: moment capacity of
bottom steel in column strip
2. End of plastic flexural response:
3. End of tensile membrane response:
𝜔 =
2𝑇𝑥 𝑠𝑖𝑛 6𝜀 𝑥
𝐿 𝑥
+
2𝑇𝑦 𝑠𝑖𝑛 6𝜀 𝑦
𝐿 𝑦
𝐿 𝑥, 𝐿 𝑦= clear span in short
and long direction
𝑇𝑥, 𝑇𝑦 = force in reinforcement
in short and long direction
𝜀 𝑥, 𝜀 𝑦= clear span in short and long direction

15. 3 Dynamic Effects, Energy Balance
• 𝑊𝑒𝑥𝑡 = ∫ 𝐹𝑑𝑒𝑠𝑖𝑔𝑛 𝑑𝛥 =𝐹𝑑𝑒𝑠𝑖𝑔𝑛 𝛥
• 𝑊𝑖𝑛𝑡 = ∫ 𝑅 𝑛(𝛥)𝑑𝛥 =Area under pushover curve
• 𝐶𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝛥 = 𝑊𝑖𝑛𝑡 𝛥 /𝛥

16. 3 Dynamic Effects, Energy Balance
• Steel Examples
WUF-B Subsystem RBS Subsystem

17. 4 Simplified Analysis of Systems
• Lateral Load Resisting Column Removal in 10 Story Steel
Building

18. 4 Simplified Analysis of Systems
• Gravity Column Removal in 10 Story Steel Building

19. 4 Simplified Analysis of Systems
• Concrete Moment Frame-10 Story Frame or 4 Bay Slab

20. Conclusions
• Simple closed form formulas were presented for static force-
displacement behavior of steel and concrete moment resisting
systems and steel gravity systems.
• Dynamic effects were considered using energy balance
method.
• Building examples were analyzed using the proposed
formulas.

21. References
• Lew, H.S., Main, J. A., Robert, S.D., Sadek, F., Chiarito, V.P, (2013) “Performance of Steel
Moment Connections under a Column Removal Scenario. I: Experiments”, Journal of
Structural Engieering, 139:98-107.
• Oosterhof, S.A. and Driver, R.G. (2014). “Behavior of steel shear connections under
column-removal demands.” J. Struct. Eng., 141(4), 10.1061/(ASCE)ST.1943-541X.0001073,
04014126.
• Oosterhof, S.A. and Driver, R.G. (2016). “Shear connection modelling for column removal
analysis.” J. Constr.Steel Res., 117 (2), 227-242, doi:10.1016/j.jcsr.2015.10.015.
• Park, R., and Gamble, W.L., Reinforced Concrete Slabs, second edition, John Wiley & Sons,
2000, 716pp. `
• Sadek, F., Main, J.A., Lew, H.S., El-Tawil, Sh., “Performance of Steel Moment
Connections under a Column Removal Scenario. II: Analysis” , Journal of Structural
Engieering, 139:98-107.
• Sadek, F., Main, J.A., Lew, Robert, S.D., Chiarito, V.P, El-Tawil, Sh., “An Experimental
and Computational Study of Steel Moment Connections under a Column Removal
Scenario”, NIST Technical Note 1669.
• Lew, H.S., Bao, Y., Sadek, F., Main, J.A., Pujol, S., Mete, A.S., “An Experimental and
Computtaional Study of Reinforced Concrete Assemblies under a Column Removal
Scenario”, NIST Technical Note 1720.
• Uang, C.M, Ozkula, G., (2015) “Beam Plastic Hinge Length” University of California, San
Diego