# Fire Safety Engineering & Structures in Fire

Urban and Civil

Nov 29, 2013 (4 years and 5 months ago)

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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

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

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

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

produces thermal curvature

Uniform thermal

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

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

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

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

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

Curvature

No force

No moment

Curvature

Tensile force

Moment

No curvature

No force

Moment

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

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

Pin
-
ended beam

Constant centroidal temperature

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

Pin
-
ended beam

Constant centroidal temperature

Runaway in simple beams

Unrestrained as in furnace tests

Restrained as in large framed structures

Large displacement effects important

Runaway in beams

Composite

beam
-
slab
moment
-
resisting

connections

Mean

temperature

Thermal

C

C

T

C

T

y

T

C

Gravity

EI

EA
T

M

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

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+

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