Fire Safety Engineering & Structures in
Fire
Workshop at Indian Institute of Science
9-13 August, 2010
Bangalore
India
Basic Structural Mechanics and
Modelling 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:
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
Ambient temperature=T
P
T+ΔT
L
ΔL
PL
 LT
L EA
P



 T
L
L
EA
Mechanical strain
…or more generally
L

L
For a rod…
Thermal strain
total  mech  thermal
Stresses and Deflections
P
L
Bar with end load
ΔL
T+ΔT
L
Uniformly heated bar
In general:
mech  
total  
ΔL
remembering
PL
L 
EA
P
 
A
L  TL
 0
total  mech  thermal
Heated Restrained Beam (1)
T
T+ΔT
T
total  T  T  0
  0  TE
Thermal effects
Mechanical effects
Uniformly heated restrained beam
 No deflections (unless buckling occurs)…
 … but compressive stresses

Heated Restrained Beam (2)
T
T+ΔT
T
Problem: Determine ΔT
at failure
P  TEA
TEA
 
 TE
A
Assume elastic perfectly plastic material behaviour
then either plastic failure will occur at
y

T 
 115 C for steel
E
…or
Heated Restrained Beam (3)
T
P  TEA
TEA
 
 TE
A
T+ΔT
T
Problem: Determine ΔT
at failure
…an Euler buckle will occur at
Pcr  TEA 
 2 EI
2
L
 2 EI  2 I
2
T 


2
EAL AL 2
where
l

r
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
10
0
Vertical Deflection
0
0.2
0.4
0.6
0.8
Really stocky
beam!
- 10
-20
Stocky beam
-30
-40
-50
-60
1
Slender beam
-70
-80
-90
Temperature
1. 2
Heating of Restrained Beam – Axial
Force
Compressive Axial Force
6000000
5000000
Yield
Stocky beam
4000000
3000000
2000000
1000000
Slender beam
0
0
0. 2
0. 4
0. 6
0. 8
Temperature
1
1. 2
Heated Restrained Beam (3)
Uniform heating then cooling
Elastic/plastic
Stress
Tension during
cooling
Mechanical strain or
temperature
Compression during
heating
Finish here!
Expansion Against Finite Stiffness
T
K
T+ΔT
T
Problem: Determine ΔT
at failure
 
TE
(1  EA / kL)
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
θ
Curvature - a generalised strain
Curvature defined as
1 d 2 y d
  2 
R dx
dx
R
M
M
d
Uniform moment, M,
produces mechanical
curvature
d2y
M
mech 

2
dz
EI
Curvature of Beams - Thermal
Uniform thermal
gradient in beam
with uniform
moment
θ
Thermal gradient
Length
of hottest fibre
R
Ty 
T 2 T1
d
Cold (T1)
Length
of coldest fibre
d
 (T2  T 0) L
 (T1  T 0) L
produces thermal curvature
thermal  Ty
Hot (T2)
Curvature of beams
Uniform thermal
gradient in beam
with uniform
moment
θ
Analogous relationship to that
for strains
R
Cold (T2)
Hot (T2)
d
total  mech  thermal
where
total  Deflection s
mech  Moments
Shortening due to Thermal Curvature
Beam with thermal gradient
Cold
Hot
Interpret shortening due to
curvature as a “strain”. From
geometry
sin( L / 2)
  1 
L / 2
Note: shortening due to mechanical
curvature normally ignored because
of high stresses. Large curvature
possible with low stresses due to
thermal bowing. Problem nonlinear
Beams with Pure Thermal Gradient
Cold
Hot
Simply supported: Curvature, no moment, contraction, no tension
Φtotal= 0+Φthermal
Cold
P
P
Hot
Pin-ended: Deflections, tension, moment
Φtotal= Φmech+Φthermal
Φmech -ve
Φthermal +ve
Beams with Pure Thermal Gradient
Cold
M
M
Hot
Built-in beam: End moments, moment in beam, no deflections
Φtotal= Φmech+Φthermal=0
Φmech=-Φthermal
Summary of Results so Far
total  mech  thermal
total  mech  thermal
where
where
total  deflection
mech  moment
total  deflection
mech  stresses
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
  1 
sin
l
2
l
2
 eff   T   
Combined Thermal Gradient and Heating
60
Pin-ended beam
Constant centroidal temperature
Varying thermal gradient
Deflection
50
40
Increasing
Thermal
gradient
30
20
10
0
0
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
Temperature
0. 7
0. 8
0. 9
1
Combined Thermal Gradient and
Heating
6000000
5000000
Increasing
Thermal
gradient
Axial Force
4000000
3000000
2000000
1000000
0
0
-1000000
-2000000
0. 1
0. 2
0. 3
0. 4
0. 5
0. 6
Pin-ended beam
Constant centroidal temperature
Varying thermal gradient
-3000000
Temperature
0. 7
0. 8
0. 9
1
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
Gravity
load
Mean
temperature
Thermal
gradient
T

C
Mload


C
C
EAT
T
y
T
C
EI
Event 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
I
N
C
R
E
A
S
I
N
G
C
O
M
P
L
E
X
I
T
Y
Static
Quasi-static
Plasticity
Buckling
Geometric nonlinearity
Creep
Dynamic – implicit or explicit schemes
Inertia effects – e.g collapse
Numerically more stable
Coupled thermo-mechanical
Effects such as spalling
Currently a research area
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
Tension due to deflections
Geometric non-linearity must be modelled to
capture this effect
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
Linear or
Elastic plastic
Often assumed at ambient temperature
Strain
Material Behaviour – High Temperature
Stress
Full non-linearity needed
Temperature dependence
T+
Strain
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
Uniform load 4250N/m
75% axial stiffness of
beam
35mm
1m
800C
T
35mm
σ
T+
t
Heating
σT/ σA
ε
Elastic-plastic material
T
1000C
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
Deflection (m)
-0.01
0
200
400
Simply-supported
(Standard Fire
Test)
-0.02
“Runaway”
-0.03
-0.04
-0.05
-0.06
-0.07
600
Simply-supported
Abaqus Standard
Vulcan
Ansys
Abaqus Explicit
Temperature (C)
800
Benchmark 1 - Axial Force
150000
Simply-supported
(Standard Fire
Test)
Force (N)
100000
50000
0
0
200
400
-50000
-100000
Buckling
600
Simply-supported
Abaqus Standard
Vulcan
Ansys
Abaqus Explicit
-150000
Temperature (C)
800
Effect of BCs on Deflections
0
Deflection (m)
-0.01
0
200
400
Simply-supported
(Standard Fire
Test)
-0.02
“Runaway”
-0.03
-0.04
-0.05
-0.06
-0.07
600
Simply-supported
Pinned
75% Stiffness (benchmark)
25% Stiffness
5% Stiffness
Temperature (C)
800
Effect of BCs on Axial Force
150000
Simply-supported
(Standard Fire
Test)
Force (N)
100000
50000
0
0
200
400
600
-50000
-100000
Simply-supported
Pinned
75% Stiffness (benchmark)
25% Stiffness
5% Stiffness
-150000
Temperature (C)
800
Effect of Non-linear Geometry on
Deflections
Mid-span deflection (m)
0
-0.05
0
200
400
600
-0.1
-0.15
-0.2
-0.25
-0.3
Geometrically non-linear
Geometrically linear
-0.35
Temperature (C)
800
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