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)
Midspan deflection (m)
Geometrically nonlinear
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
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