Fire Safety Engineering & Structures in Fire

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Workshop at Indian Institute of Science

9
-
13 August, 2010

Bangalore

India

Fire Safety Engineering & Structures in
Fire

Organisers:


CS Manohar and Ananth Ramaswamy

Indian Institute of Science

Speakers:


Jose Torero,
Asif

Usmani

and Martin Gillie

The University of Edinburgh

Funding and
Sponsorship:

Basic Structural Mechanics and
Modelling in Fire

Structural Mechanics at High
Temperature


The mechanics of restrained heated structures


Another look at strain


Behaviour of uniformly heated beams


Curvature


Behaviour of beams with thermal gradients


Behaviour of beams heated with thermal gradients


Another Look at Strain

T+
Δ
T

L

Δ
L

L
L



T
EA
P
L
T
L
EA
PL
L
L











For a rod…

thermal
mech
total





…or

more generally

Thermal strain

Mechanical strain

Ambient temperature=T

P

Stresses and Deflections

T+
Δ
T

L

Δ
L

Uniformly heated bar

0






TL
L
L

Δ
L

Bar with end load

A
P
EA
PL
L




P

In general:







total
mech
remembering

thermal
mech
total





Heated Restrained Beam (1)


Uniformly heated restrained beam


No deflections (unless buckling occurs)…


… but compressive stresses

TE
T
T
total













0

0
Thermal effects

Mechanical effects

T+
Δ
T

T

T

Heated Restrained Beam (2)

Problem: Determine
Δ
T



at failure

Assume elastic perfectly plastic material behaviour

then either plastic failure will occur at

TE
A
TEA
TEA
P













steel
for

C
115




E
T
y


T+
Δ
T

T

…or

T

Heated Restrained Beam (3)

T+
Δ
T

T

Problem: Determine
Δ
T



at failure

T

TE
A
TEA
TEA
P













…an Euler buckle will occur at

2
2
2
2
2
2
2















AL
I
EAL
EI
T
L
EI
TEA
P
cr
r
l


where

Thermal Buckling


Buckling temperature independent of
E


Buckling expression valid for other end conditions if
L

interpreted as an effective length


Buckling stable as end displacements defined


Combined yielding
-
buckling failure possible in reality
(as at ambient temperature)

Heating of Restrained Beam
-

Deflections

- 9 0
- 8 0
- 7 0
- 6 0
- 5 0
- 4 0
- 3 0
- 2 0
- 10
0
10
0
0.2
0.4
0.6
0.8
1
1.2
Temperature
Vertical Deflection
Stocky beam

Slender beam

Really stocky

beam!

Heating of Restrained Beam


Axial
Force

0
1000000
2000000
3000000
4000000
5000000
6000000
0
0.2
0.4
0.6
0.8
1
1.2
Temperature
Compressive Axial Force
Stocky beam

Slender beam

Yield

Heated Restrained Beam (3)

Mechanical strain or

temperature

Stress

Uniform heating then cooling

Compression during

heating

Tension during

cooling

Finish here!

Elastic/plastic

Expansion Against Finite Stiffness

T+
Δ
T

T

Problem: Determine
Δ
T



at failure

T

)
/
1
(
kL
EA
TE






K

If the stiffness of the support is comparable to the stiffness of the member,

the stress produced by thermal expansion will be reduced by a factor of

about 2


Curvature of Beams
-

Mechanical

R

θ

d

M

M

Uniform moment, M,

produces mechanical

curvature

EI
M
dz
y
d
mech


2
2

Curvature defined as

dx
d
dx
y
d
R





2
2
1
Curvature
-

a generalised strain

Curvature of Beams
-

Thermal

d
T
T
T
y
1
2


Hot (T
2)

Cold (T
1
)

R

θ

d

y
thermal
T



Thermal gradient

produces thermal curvature

Uniform thermal

gradient in beam
with uniform
moment

Length

of hottest fibre

L
T
T
)
(
0
2


L
T
T
)
(
0
1


Length

of coldest fibre

Curvature of beams

Hot (T
2)

Cold (T
2
)

R

θ

d

Uniform thermal
gradient in beam

with uniform
moment

Analogous relationship to that
for strains

Moments
s
Deflection
where
mech









total
thermal
mech
total
Shortening due to Thermal Curvature

Interpret shortening due to

curvature as a “strain”. From

geometry

2
/
)
2
/
sin(
1




L
L


Cold

Hot

Beam with thermal gradient

Note: shortening due to mechanical

curvature normally ignored because

of high stresses. Large curvature

possible with low stresses due to

thermal bowing. Problem nonlinear

Φ
total
= 0+
Φ
thermal


Beams with Pure Thermal Gradient

Cold

Hot

Simply supported: Curvature, no moment, contraction, no tension

Cold

Hot

Pin
-
ended: Deflections, tension, moment

P

P

Φ
total
=
Φ
mech
+
Φ
thermal

Φ
mech
-
ve

Φ
thermal

+ve

Beams with Pure Thermal Gradient

Cold

Hot

M

M

Built
-
in beam: End moments, moment in beam, no deflections

Φ
total
=
Φ
mech
+
Φ
thermal
=0

Φ
mech
=
-
Φ
thermal

Summary of Results so Far

stresses
deflection
where




mech
total
thermal
mech
total





moment
deflection
where
mech









total
thermal
mech
total
Simple
support

Pin
-
support

Built
-
in

Uniform

Heating

No (vertical)
deflection

No force

No moment

No deflection (or a
buckle)

Compressive force

No moment

No deflection (or a
buckle)

Compressive force

No moment

Pure thermal
gradient

Curvature

No force

No moment

Curvature

Tensile force

Moment

No curvature

No force

Moment

Combined Thermal Gradient and
Heating


Thermal expansion produces expansion strains


Thermal curvature produces contraction
“strains”


Behaviour depends on the interplay between
the two effects

An equivalent effective strain to
combine the two thermal effects

T
T




2
2
sin
1




l
l








T
eff
Combined Thermal Gradient and Heating

0
10
20
30
40
50
60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature
Deflection
Increasing

Thermal

gradient

Pin
-
ended beam

Constant centroidal temperature

Varying thermal gradient

Combined Thermal Gradient and
Heating

-3000000
-2000000
-1000000
0
1000000
2000000
3000000
4000000
5000000
6000000
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Temperature
Axial Force
Increasing

Thermal

gradient

Pin
-
ended beam

Constant centroidal temperature

Varying thermal gradient

Runaway in simple beams


Unrestrained as in furnace tests


Restrained as in large framed structures


Large displacement effects important


Runaway in beams

Runaway temperatures vs loads

Composite

beam
-
slab
moment
-
resisting

connections


Mean

temperature

Thermal
gradient


C

C

T

C

T

y

T

C

Gravity


load

EI


EA
T

M
load

E
vent 1:

local buckling

of beam bottom flange







Numerical Modelling of Heated
Structures


Needed for all but simple structures


Finite element models normal


Some “intermediate” analysis methods exist but
limited


Challenging!

Types of Analysis


Heat Transfer


Specify temperature of the surface


Numerically simple


conduction only


Does not require estimates of emissivity and heat transfer
coefficient


Useful for modelling experiments


Model radiation and convection


Numerically complex


Need to estimate parameters


tricky


Normally required for design


Model heat flux


Can be useful if using input from a CFD code

Descretization


Heat Transfer

Types of Analysis
-

Structural

Static

Quasi
-
static

Dynamic


implicit or explicit schemes

Coupled thermo
-
mechanical

Plasticity

Buckling

Geometric nonlinearity

Creep

Inertia effects


e.g collapse

Numerically more stable

Effects such as spalling

Currently a research area

I

N

C

R

E

A

S

I

N

G


C

O

M

P

L

E

X

I

T

Y

Geometric Non
-
Linearity

P

If deflections are large, axial forces produced

In the beam due to deflection


“Catenary” action

“Tensile membrane action” in 3
-
d


Geometric non
-
linearity must be modelled to

capture this effect

Tension due to deflections

Geometric Non
-
linearity


Many numerical codes allows for this


Must

be used for accurate results at high
temperature


Means analyses must be solved incrementally



therefore take longer and are more demanding

Material Behaviour
-

Ambient

Stress

Strain

Linear or

Elastic plastic

Often assumed at ambient temperature

Material Behaviour


High Temperature

Stress

Strain

Full non
-
linearity needed

Temperature dependence

T+

Von Mises Yield Surface
-

Steel

Drucker
-
Prager Yield Surface


Concrete
Compression

Element Choice


Detailed models computationally expensive


Simply models may miss phenomena


How to model a beam


Beam elements?


Shell elements


Solid elements?


It depends!

Benchmark 1

T

T

t

σ
T
/
σ
A

σ

ε

T+

Uniform load 4250N/m

Heating

800C

1000C

Elastic
-
plastic material

1m

35mm

35mm

75% axial stiffness of

beam

Purpose of Benchmark 1


Model not “real” but…


… shows if complex phenomena captured


Non
-
linear material behaviour


Temperature dependent


Plastic


Thermal expansion


Non
-
linear geometric behaviour


Boundary conditions important


Can be used for demonstrating


Software capability


Appropriate modelling techniques


Benchmark 1
-

Deflections

-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0
200
400
600
800
Temperature (C)
Deflection (m)
Abaqus Standard
Vulcan
Ansys
Abaqus Explicit
Simply
-
supported

Simply
-
supported

(Standard Fire

Test)

“Runaway”

Benchmark 1
-

Axial Force

-150000
-100000
-50000
0
50000
100000
150000
0
200
400
600
800
Temperature (C)
Force (N)
Abaqus Standard
Vulcan
Ansys
Abaqus Explicit
Simply
-
supported

Simply
-
supported

(Standard Fire

Test)

Buckling

Effect of BCs on Deflections

-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
0
0
200
400
600
800
Temperature (C)
Deflection (m)
Pinned
75% Stiffness (benchmark)
25% Stiffness
5% Stiffness
Simply
-
supported

Simply
-
supported

(Standard Fire

Test)

“Runaway”

Effect of BCs on Axial Force

-150000
-100000
-50000
0
50000
100000
150000
0
200
400
600
800
Temperature (C)
Force (N)
Pinned
75% Stiffness (benchmark)
25% Stiffness
5% Stiffness
Simply
-
supported

Simply
-
supported

(Standard Fire

Test)

Effect of Non
-
linear Geometry on
Deflections

-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0
200
400
600
800
Temperature (C)
Mid-span deflection (m)
Geometrically non-linear
Geometrically linear
Aside


Cardington Tests

Aside
-

Cardington Test 1

Example


Real structure


Based on Cardington test 1


Carefully conducted test on real structure (v. rare)


Has been extensively modelled


Experimental data available


Simplified so


Precisely defined


Practical to model


As challenging as many larger structures

Example

Model

Example Deflections

Example Axial Force