Ae2_19_plastic analysis_kinematic method

STRUCTURAL ANALYSIS II
PLASTIC ANALYSIS. KINEMATIC METHOD
PLASTIC 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. EQUILIBRIUM METHOD
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

When the punctual load is not applied in the middle of the span…
Or when we are analysing frames…
We cannot apply the expressions of Plastic Moments in simple beams
We apply the KIMENATIC METHOD

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)

1.- DRAW A POSSIBLE COLLAPSE PATTERN
1.1. Number of hinges required?
(degree of indeterminacy + 1)
1.2. Points where a hinge can be formed?
1.3. Combinations?
1 + 1 = 2
A, B, C
AB, AC, BC

2.- COLLAPSE PATTERN AB. VIRTUAL WORK PRINCIPLE
External Work: Load · displacement
Internal Work: Mp · rotation
Where there is a hinge in the collapse pattern,
there is a Mp in the bending moment diagram
BENDING MOMENT DIAGRAM

2.- COLLAPSE PATTERN AC. VIRTUAL WORK PRINCIPLE
External Work: Load x displacement
Internal Work: Mp x rotationExternal Work: Load · displacement

2.- COLLAPSE PATTERN AC. VIRTUAL WORK PRINCIPLE
Internal Work: Mp x rotation
External Work: Load · displacement

2.- COLLAPSE PATTERN BC. VIRTUAL WORK PRINCIPLE
Internal Work: Mp x rotationExternal Work: Load · displacement

2.- CHOOSE THE TRUE COLLAPSE PATTERN ¿?
Mp….. The highest
Pu….. The lowest
So the correct collapse pattern should be AC
Mp = 1,5 P
Pu = 0,666 Mp

2.- WHAT HAPPENS IF I MAKE A MISTAKE OR I FORGET A COLLAPSE
PATTERN THAT IS THE CORRECT ONE?
THE SOLUTION IS UNIQUE AND IF THE COLLAPSE PATTERN IS THE
CORRECT ONE AND IF THE Mp IS THE CORRECT ONE THE BENDING
MOMENT DIAGRAM WILL BE IN EQUILIBRIUM
and no moment value in the diagram will be higher than Mp

DETERMINE THE PLASTIC ANALYSIS OF THE
FRAME APPLYING THE KIMENATIC METHOD

1.- DEGREE OF INDETERMINACY?
2.- NUMBER OF HINGES REQUIRED TO PRODUCE THE GLOBAL COLLAPSE?
3.- POINTS WHERE THE MOMENT CAN BE MAXIMUM?

3
3+1
A, B, D, G, E

ADGE, ADEB, AGEB, DGEB
4.- COMBINATIONS

ADGE, ADEB, AGEB, DGEB4.- COMBINATIONS
IS THERE ANY POSSIBLE PARTIAL COLLAPSE PATTERN?
BEAM PARTIAL COLLAPSE?
CANTILEVER PARTIAL COLLAPSE?

4.- COMBINATIONS
ADGE and DGEB INCLUDE THE BEAM PARTIAL FAILURE

4.- COMBINATIONS
ADEB produces a 0 external work, so it cannot be the true one

4.- COMBINATIONS
AGEB could be possible

AGEB could be possible

LET’S TRY THE BEAM PARTIAL COLLAPSE

Mp = 0,5 P < Mp = 3P/4
As the true collapse pattern is a partial collapse, we cannot
calculate all the diagram values

3

3
3+1

3
3+1
A, B, D, E

ADEB
4.- COMBINATIONS

ADEB4.- COMBINATIONS
IS THERE ANY POSSIBLE PARTIAL COLLAPSE PATTERN?
BEAM PARTIAL COLLAPSE?
CANTILEVER PARTIAL COLLAPSE?

4.- COMBINATIONS
ADEB is the only likely one
What happens if
the beam can
bear up to 2Mp
and the support
only up to Mp?

IF WE CAN DRAW A BENDING MOMENT DIAGRAM IN EQUILIBRIUM AND
NO MOMENT IS BIGGER THAN Mp
WE CAN BE SURE THAT THE COLLAPSE PATTERN IS THE CORRECT ONE

WHICH OPTION IS MORE LIKELY?
ARE THERE OTHER OPTIONS (partial collapse patterns) possible?
DO NOT FORGET TO CHECK THEM!
DETERMINE THE PLASTIC ANALYSIS OF THE
FRAME APPLYING THE KIMENATIC METHOD

The cantilever
breaks: partial
collapse

The beam breaks:
beam collapse

The frame moves
horizontally: panel
collapse

The collapse is produce by
the combination of
horizontal and vertical load:
combined mechanism

RULES TO DRAW A COLLAPSE PATTERN
1.- BARS CANNOT GET LONGER OR SHORTER
2.- JOINT ANGLES CAN ONLY CHANGE IF THERE IS A HINGE
(plastic or real)
= bars can only rotate about hinges
3.- WE ASSUME THAT PERPENDICULAR DISPLACEMENTS ARE AS
CIRCUMFERENCE ARCS (small-angle theory)

Wext = P · u + P/2 · v= P · 4 · γ1 + P/2 · 2 γ1 = 5 · P · γ1
Wint = Mp ·γ1 + Mp · (γ3+ γ3) = 3 Mp · γ1
u = 4 ·γ1
v = 2 ·γ3
γ1 = γ3
u = 4 ·γ3
5 · P · γ1 = 3 Mp · γ1 Mp = 5P / 3

WHICH OF THESE COLLAPSE PATTERN IS MORE LIKELY?

WHY ARE THESE COLLAPS PATTERNS NOT LIKELY?

Mp =13P/7
Mp =5P/3Mp =P
Mp =4P/3

2.- CHOOSE THE TRUE COLLAPSE PATTERN ¿?
Mp….. The highest
Pu….. The lowest
So the correct collapse pattern should be ….
THE SOLUTION IS ALWAYS UNIQUE AND WE CAN BE SURE ABOUT IT

ONLY IF WE CAN DRAW A BENDING MOMENT DIAGRAM IN
EQUILIBRIUM THE COLLAPSE PATTER IS THE TRUE ONE

Ae2_19_plastic analysis_kinematic method

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