The document discusses progressive collapse analysis of structures through elastic and plastic analysis methods. It describes the kinematic method and incremental method of plastic analysis to determine the formation of plastic hinges in frames as the load increases. The incremental method involves performing elastic analysis with an additional hinge at each load level until global collapse occurs. An example problem demonstrates applying the incremental method to calculate the load levels that result in formation of successive plastic hinges in a beam.

1. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
PROGRESSIVE COLLAPSE ANALYSIS

2. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
1.- ELASTIC ANALYSIS OF GRIDS
1.1.- COMPATIBILITY METHOD
1.2- SLOPE DEFLECTION METHOD
2.- PLASTIC ANALYSIS OF STRUCTURES
2.1- PLASTIC ANALYSIS OF BEAMS
2.2- PLASTIC ANALYSIS OF FRAMES
2.2.1- KINEMATIC METHOD
2.2.2- INCREMENTAL ANALYSIS. HINGE BY HINGE METHOD
3.- INTRODUCTION TO SECOND ORDER ANALYSIS OF STRUCTURES
2.3- PLASTIC ANALYSIS OF SLABS.
FEBRUARY
MARCHADNAPRIL
MAY

3. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
ELASTIC ANALYSIS PLASTIC ANALYSIS
≠ RESULTS*
Part of the structure
material is working beyond
the plastic limit
The structure should be
ductile enough to have a
plastic behaviour
All the structure material is
working below the plastic
limit
The structure can be
fragile or ductile to apply
elastic analysis
SAFER CHEAPER
Superposition Principle
cannot be applied
Superposition Principle can
be applied

4. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
KIMENATIC METHOD
1.- DRAW A POSSIBLE COLLAPSE PATTERN
2.- CALCULATE THE Mp THAT PRODUCES THAT COLLAPSE PATTERN
3.- CHECK IF THE COLLAPSE PATTERN + Mp VALUE ARE LINKED WITH A
BENDING MOMENT DIAGRAM IN EQUILIBRIUM
1.1. Number of hinges required?
1.2. Points where a hinge can be formed?
1.3. Combinations?
2.1. Application of Virtual Work Principle Wext = Wint
2.2. Calculation of Mp
3.1. Equilibrium (three equilibrium equations)

5. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
KINEMATIC METHOD INCREMENTAL METHOD
= RESULTS (both plastic analysis)
It let us know how the
frames behave step by
step
It leads us to the true
failure pattern directly. We
do not have to choose
among posible different
answers
It leads directly to the final
collapse pattern
We must guess and
choose among different
possible collapse patterns
Lot of numbers because we have
to consider elastic analysis
results in the proccess
Easy maths

6. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
INCREMENTAL METHOD (progressive collapse analysis)
1.- ELASTIC ANALYSIS P= 1
2.- ELASTIC ANALYSIS P=1 AND A HINGE AT THE POINT WHERE THE 1st
PLASTIC HINGE IS FORMED.
3.- …. REPEAT THE PROCCESS UNTIL THE GLOBAL COLLAPSE IS
PRODUCED
1.1. Where is the first plastic hinge formed?
1.2. Value of the load P that produces the first plastic hinge?
2.1. Where is the second plastic hinge formed?
2.2. Value of the load P that produces the second plastic hinge?
1.3. Bending moment diagram when the 1st plastic hinge is formed
2.3. Bending moment diagram when the 1st plastic hinge is formed

7. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS

8. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
ELASTIC ANALYSIS: RESULT OF SAP2000 OR SLOPE DEFLECTION METHOD

9. STRUCTURAL ANALYSIS II
CLASS 9

10. STRUCTURAL ANALYSIS II
CLASS 9

11. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
VALUE OF P WHEN THE FIRST HINGE IS FORMED
We have to choose
between these two values

12. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
VALUE OF P WHEN THE FIRST HINGE IS FORMED
THE LOWEST… WHY?

13. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
BENDING MOMENT DIAGRAM WHEN THE FIRST HINGE IS FORMED

14. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
BENDING MOMENT DIAGRAM WHEN THE FIRST HINGE APPEARS

15. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
TO CALCULATE WHERE THE SECOND HINGE
IS FORMED…
ELASTIC ANALYSIS: RESULT OF SAP2000 OR SLOPE DEFLECTION METHOD

16. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
THIS DIAGRAM IS NOT ENOUGH TO KNOW IT!

17. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
WE MUST CONSIDER THE FRAME BEAVIOUR FROM P = 0 TO P = 124,57 kN
AND THE FRAME BEHAVIOUR WHEN P IS BIGGER THAN 124,57 kN
WHERE DOES THE SECOND HINGE APPEAR?

18. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
WE MUST CONSIDER THE FRAME BEAVIOUR FROM P = 0 TO P = 124,57 kN
AND THE FRAME BEHAVIOUR WHE P IS BIGGER THAN 124,57 kN

19. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
P when the second hinge appears = 124,57 + 10,104 = 134,67 kN

20. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
BENDING MOMENT DIAGRAM WHEN THE SECOND HINGE IS FORMED

21. STRUCTURAL ANALYSIS II
CLASS 9
ELASTIC ANALYSIS: RESULT OF SAP2000 OR SLOPE DEFLECTION METHOD
We place another hinge
where the second plastic
hinge is formed
TO CALCULATE WHERE THE SECOND HINGE
IS FORMED…

22. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
WE NEED THESE TWO DIAGRAMS TO KNOW
WHERE THE THIRD PLASTIC HINGE WILL OCCUR

23. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
WE NEED THESE TWO DIAGRAMS TO KNOW
WHERE THE THIRD PLASTIC HINGE WILL OCCUR

24. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
P when the third plastic hinge appears = 134,67 + 36,24 = 170,91 kN

25. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
BENDING MOMENT DIAGRAM WHEN THE THIRD HINGE IS FORMED

26. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
AS IT IS STATICALLY DETERMINATE, WE DO NOT NEED SAP2000 TO SOLVE ELASTIC
ANALYSIS… BUT WE CAN USE IT.
We place a hinge where
the second plastic hinge is
formed
TO CALCULATE WHERE THE FOURTH HINGE
IS FORMED…

27. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
WE NEED THESE TWO DIAGRAMS TO KNOW
WHERE THE FOURTH PLASTIC HINGE WILL OCCUR

28. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
P when the fourth plastic hinge appears = 170,91 + 1,77 = 172,7 kN

29. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
BENDING MOMENT DIAGRAM WHEN THE FOURTH HINGE IS FORMED

30. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
BENDING MOMENT DIAGRAM EVOLUTION

31. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
33,23 = - 0,00008831 m x 124,57 - 0,0002 m x 10,104 - 0,000277 m x 36,24 - 0,00572 m x 1,775
Analysis of the vertical displacement at C as the hinges are formed

32. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
TEST 3 16/17
Variable data for the beam: a = 1 + 0,05·X m
HEB200 (Sx= 321,25 cm3; fy= 0,275 kN/mm2), Mp = 176,7 (kNm); EI = 11963,7 kNm2

33. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
Bending moment diagram for P=1
Mp = 176,7 (kNm)

34. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
Value of P that produces the FIRST PLASTIC HINGE
P1 = 176,7/5 = 35,34 kN

35. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
BENDING MOMENT DIAGRAM WHEN THE FIRST PLASTIC HINGE IS PRODUCED
P1 = 176,7/5 = 35,34 kN
3x35,34 kN
35,34/2 kN
35,34/2 kN

36. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
AUXILIARY SYSTEM TO CALCULATE THE SECOND PLASTIC HINGE

37. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
172,7 = 141,36 + D P x 14 D P = 2,24 kN

38. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
172,7 = 141,36 + D P x 14 D P = 2,24 kN Pu = 35,4 + 2,24 = 37,64 kN
3x37,64 kN
37,64/2 kN
37,64/2 kN
BENDING MOMENT DIAGRAM WHEN THE SECOND HINGE OCCURS

39. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
LET’S CHECK APPLYING KINEMATIC METHOD

40. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
LET’S CHECK APPLYING KINEMATIC METHOD

41. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
WE GET THE SAME RESULT!

42. STRUCTURAL ANALYSIS II
PROGRESSIVE COLLAPSE ANALYSIS
PROGRESSIVE COLLAPSE ANALYSIS