Ae2_20_plastic analysis of slabs_yield lines

STRUCTURAL ANALYSIS II
PLASTIC ANALYSIS. KINEMATIC METHOD
PLASTIC ANALYSIS OF SLABS
SLAB: Structural element contained in a plane (the third
dimension is much smaller than the other two).
The load direction is perpendicular to the structure plane

PLASTIC ANALYSIS OF SLABS

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
MARCH
APRIL

YIELD LINES: líneas de rotura (plástica)
YIELD LINES = ROTATION LINE
YIELD LINES METHOD

ASSUMPTION: YIELD LINES = BISECTRICES
STUCTURAL MODEL OF A SLAB (PLAN)
MODELO ESTRUCTURAL DE LOSA (PLANTA)

YIELD LINES: BISECTRICES
YIELD LINES CAN BE POSSITIVE OR NEGATIVE
STUCTURAL MODEL OF A SLAB (PLAN)
MODELO ESTRUCTURAL DE LOSA (PLANTA)

VERY IMPORTANT: DRAW THE CROSS SECTION!!!!
Based in the VIRTUAL WORK PRINCIPLE: Wext = Wint
CROSS
SECTION
CROSS
SECTION
PLANPLAN

Wext = P1 x v/2 + P2 x v/2 = P1 x v = q L2 /2 x g1 x L/2
CROSS
SECTION
CROSS
SECTION
PLANPLAN
Wint= Mp x L x (g1 + g2) = 2 L Mp x g1
P1 = P2 = L/2 x L x q = q L2 /2
q L2 x g1 x L/4 = 2 L Mp x g1 Mp = q L2 /8
Wint= Wext

CROSS
SECTION
CROSS
SECTION
PLANPLAN
Wint= L x Mp x (g1 + g2) = 2 L Mp x g1
P1 = P2 = L/2 x L x q = q L2 /2
Wint= Wext
Wint= L x Mp + L x Mp x (g1 + g2) + L x Mp = 4L Mp x g1
P1 = P2 = L/2 x L x q = q L2 /2

YIELD LINES: BISECTRICES
YIELD LINES CAN BE POSSITIVE OR NEGATIVE
1.- Yield lines divide the slab into rigid regions which remain plane through the
collapse;
2.- Yield lines are straight;
3.- Axes of rotation generally lie along line of supports and pass over columns;
4.- Yield lines between adjacent rigid regions must pass through the intersection point
of the axes of rotation of those regions;
5.- Yield lines must end at a slab boundary;
6.- Continuous supports repel and a simple support attract yield lines.

EXTERNAL WORK
SQUARED SLAB
PINNED SUPPORTED
ALONG FOUR SIDES
REMEMBER:
TO MEASURE THE
EXTERNAL WORK,
REGIONS CROSS
SECTIONS MUST BE
PERPENDICULAR TO
ROTATION LINES
(rotation lines are
usually the sides of the
slab)

SQUARED SLAB
PINNED SUPPORTED
ALONG FOUR SIDES
INTERNAL WORK
REMEMBER: TO
MEASUSER INTERNAL
WORK, CROSS
SECTIONS MUST BE
PERPENDICULAR TO
YIELD LINES SO WE CAN
CONSIDER ITS TRUE
DIMENSION

SQUARED SLAB
PINNED SUPPORTED
ALONG FOUR SIDES
¿INTERNAL WORK?
TO MEASUSER
INTERNAL WORK, AND
USE THE ROTATION OF
THE REGION SIDE
INSTEAD OF THE
ROTATION OF THE
YIELD LINE, WE CAN
USE THE YIELD LINES
PROJECTIONS LENGTH

SQUARED SLAB
PINNED SUPPORTED
ALONG FOUR SIDES
INTERNAL WORK
TO MEASUSER
INTERNAL WORK, AND
USE THE ROTATION OF
THE REGION SIDE
INSTEAD OF THE
ROTATION OF THE
YIELD LINE, WE CAN
USE THE YIELD LINES
PROJECTIONS LENGTH

Wext = Wint Mp = q L3 / 24

SQUARED SLAB
FIXED SUPPORTED ALONG FOUR SIDES ….. DIFFERENCES?

Wext = Wint Mp = q L3 / 48

HEXAGONAL SLAB PINNED SUPPORTED AND FIXED SUPPORTED

VIRTUAL WORK PRINCIPLE: EXTERNAL WORK = INTERNAL WORK
Wext = 6 x q x L x a /2 x v/3 = q L a x g1
Wint = 6 Lx Mp x g1
q L a x g1 = 6 Lx Mp x g1 Mp = q a /6
Wext = 6 x q x L x a /2 x v/3 = q L a x g1
Wint = (6 Lx Mp x g1) x 2
q L a x g1 = 12 Lx Mp x g1 Mp = q a /12

CIRCULAR SLAB PINNED SUPPORTED AND FIXED SUPPORTED

VIRTUAL WORK PRINCIPLE: EXTERNAL WORK = INTERNAL WORK
Wext = pi R2 x q x v/3 = pi R2 x q x R g1 /3
Wint = 2 pi R x Mp x g1
pi R2 x q x R g1 /3 = 2 pi R x Mp x g1
Mp = q a /6
Wext = pi R2 x q x v/3 = pi R2 x q x R g1 /3
Wint = 2 x (2 pi R x Mp x g1)
pi R2 x q x R g1 /3 = 4 pi R x Mp x g1
Mp = q a /12

BISECTRICES = YIELD LINE IS AN ASSUMPTION ACCURATE ENOUGH

BISECTRICES = YIELD LINE IS AN ASSUMPTION ACCURATE ENOUGH
Wext = 2 x(P1 x v/3 + P2 x v/2) + P5 x v/3 =
= 5 q x L2 v /12 = 5 q x L3 x g1/24
Wint = 3 L Mp x g1
Mp = 5 q x L2 /72
P1 = q x L2 /8
P2 = q x L2 /4
P5 = q x L2 /4
5 q x L3 x g1/24 = 3 L Mp x g1

THE SLAB BREAKS FALLING OUTSIDE

A2 = 2 x 2 / 2 = 2 m2 its gravity center goes down 2v1/3
A3 = 2 x 5 = 10 m2 its gravity center goes down v1/2
v1 = 2 θ1 v2 = 2,5 θ1
Wext = 4 ( - A1 x v2 / 3 x q + A2 x 2 v1 x q + A3 x v2 x q) =
4 (-6,25 x 2,5 θ1 /3 x q + 2 x 2 x 2 θ1 /3 + 10 x 2 θ1 /2) = 4 (-5,208 + 2,67 + 10) q θ1 = 29,848 q θ1
Wint = 9 x 4 Mp θ1 = 36 Mp θ1
So when we equal external and internal work we get that: 36 Mp = 29,848 q Mp = 0,829 q

THE SLAB BREAKS FALLING INSIDE

Wext = 4 (A1 x v2 / 3 x q - A2 x 2 v1 x q - A3 x v2 x q) = 4 (6,25 x 2,5 θ1 /3 x q - 2 x 2 x 2 θ1 /3 - 10 x 2 θ1 /2)
= 4 (5,208 - 2,67 - 10) q θ1 = - 29,848 q θ1
A NEGATIVE EXTERNAL WORK DOES NOT MAKE SENSE
THIS IS NOT THE TRUE COLLAPSE PATTERN

THE SLAB INSIDE THE WALLS BREAK

A1 = 5 x 2,5 / 2 = 6,25 m2 its gravity center goes up v1/3 v1 = 2,5 θ1
Wext = 4 (A1 x v1 / 3 x q) = 8,33 q 2,5 θ1 = 20,83 q
Wint = 4 x 5 x Mp x θ1+ 4 x 5 x Mp x θ1 = 40 Mp θ1

THE WHOLE CANTILEVER BREAKS

A2 = 2 x 2 / 2 = 2 m2 its gravity center goes down 2v1/3
A3 = 2 x 5 = 10 m2 its gravity center goes down v1/2
v1 = 2 θ1
Wext = 4 (A2 x 2 v1 / 3 x q + A3 x v1 / 2 x q) = 4 ( 6,333 ) q 2 θ1 = 50,66 q
Wint = (2 x 8 + 4 x 5 ) Mp x θ1 = 36 Mp θ1

PART OF THE SLAB IN CANTILEVER BREAKS

A4 = 9 x 2 = 18 m2 its gravity center goes up v1/2 v1 = 2 θ1
Wext = A1 x v1 / 2 x q = 9 q 2 θ1 = 18 q
Wint = 9 x Mp x θ1 = 9 Mp θ1
So when we equal external and internal work we get that: 9 Mp = 18 q Mp = 2 q

WHICH PATTERN IS THE TRUE ONE?

A4 = 9 x 2 = 18 m2 its gravity center goes up v1/2 v1 = 2 θ1
Wext = A1 x v1 / 2 x q = 9 q 2 θ1 = 18 q
Wint = 9 x Mp x θ1 = 9 Mp θ1
So when we equal external and internal work we get that: 9 Mp = 18 q Mp = 2 q
THE HIGHEST Mp (or lowest qu)

SLAB OF ASSIGNMENT 17/18

SLAB OF ASSIGNMENT 18/19

Ae2_20_plastic analysis of slabs_yield lines

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