S2 : Stadium Roof Design Emirates Stadium Structural
Analysis
Cut out Model
In order to understand the loading on the Primary Girder
assemble the 3D model of the Emirates Stadium Roof.
1. Locate the pieces with slot number 1.
2. Cut around the black lines leaving
the slots till last.
3. Slot the pieces together and repeat
for slots up until number 33.
4. Once all the girders are connected,
assemble the outer ring.
5. Put the structure in place and,
working in one direction around the
ring, secure the structure in place.
P16
Tension Forces
Applying a tension force to a ruler has the
effect of pulling the two ends further apart.
The tension force causes the ruler to stretch
and straighten.
If the ruler was made out of a different
material, such as a metal, ceramic or
composite it would behave differently
depending on the strength of that material.
String is strong in tension and needs an
extremely high force to get it to break.
However if you separate a single strand and
pull either end it breaks easily.
By increasing the cross-sectional area of the string by using lots of strands
the string can take a larger tensile force.
So Tension force is proportional to cross-sectional area and material
strength
The material property for strength is called yield strength σy
P17
Tension = A x σy
Compression Forces
Applying a compression force attempts
to push the ends of the ruler closer
together.
When enough force is applied, the
ruler will deform out of the plane in
which the force is acting.
This deformation behaviour is known as
buckling
A shorter ruler will require a greater
compressive load to get it to buckle
So longer components are more likely to fail by
buckling than shorter ones.
P18
Moments
A force applied to the free end of a ruler resting on the
edge of a table will cause the ruler to rotate about the
edge of the table.
If the ruler is “fixed” to the table with a hand, the
force applied to the free end will cause the ruler to
bend as it cannot rotate anymore.
The force causes the ruler to deform because it has
created a “bending moment” in the ruler.
If you apply a greater force you get a greater deflection,
so the bending moment experienced by the ruler must
also be larger.
If the length of the overhanging ruler is increased and the
same force is applied the deformation will be greater, so
the bending moment created must also be greater.
Moment is proportional to Force and Distance
Moment = Force x Perpendicular distance
P19
Couples
If laid flat on a table the ruler will rotate if two
equal forces in opposite directions are applied
to either end.
“Fixing” the ruler in the centre with a hand
prevents the rotation, so when the forces are
applied bending moments are set up in the
ruler causing it to deform.
Looking at the equilibrium of the ruler shows
that the two forces must be equal,
F1 = F2 = F
F1
pivot
This sort of moment is called a Couple (C).
If the forces act a distance d apart we get
C = F1 x d/2 + F2 x d/2 = (F1 + F2) x d/2
= (F + F) x d/2 = 2F x d/2
P20
=Fxd
d/2
d/2
F2
Roof Loads
Roof Load on Tertiary Girders
= Roof covering (opaque and transparent)
+ Elemental loads (Wind, Rain and Snow)
+ Lighting and sound
= 65kN/m
Transparent roof
covering to let in light
to the grass pitch
Opaque roof covering
providing shelter from
wind, rain and snow
P21
Playing surface
Load Path
Tertiary girders support
the roof load.
Tertiary girders transfer
load to secondary girders.
Tertiary girders transfer
load to primary girders.
Secondary girders transfer
load to primary girders.
P22
Free body diagram of
Primary Girder
A “free body diagram” will show magnitude and
direction of all forces acting on the primary girder.
There are 10 Tertiary girders loading the Primary girder
The Secondary girders transfer an additional load, from T6,
T7 and T8.
R
T1
+ Secondary
T2
T3
T4
T5
T5
T4
T3
T1
T2
+ Secondary
R
All these forces act downwards, so there must be a reaction force
from the Tripods it sits on to keep the girder in equilibrium.
P23
Cutting the structure
The magnitude and type of the internal forces in the Primary Girder
will determine how much steel is required to withstand these forces.
It’s also important to locate where these forces will be the greatest
in the structure.
Having identified the position of interest, say the half way
point, the girder needs to be “cut” at this point in order to
calculate the internal forces acting here.
M
R
T1 +
T2
Secondary
T3
T4
T5
If the structure was cut for real, the structure would rotate
about a pivot at the tripod support. So the missing internal
force must be a moment.
P24
This can be calculated by considering moment equilibrium at
this point.
Moment into tension and
compression
We know the girder will be made out of steel circular tubes,
one on top and one on the bottom a distance d (m) apart
which can be measured from the model.
Cross-section
d (m)
M
The forces in the tubes must be either tension or compression,
so M must be carried inside the structure as a couple.
T
C
C
T
Couple M = T x d or M = C x d
P25
Compression force in
the top tube and
tension in the bottom
tube gives the correct
anti-clockwise
direction of rotation.
Steel tube in tension
Steel can withstand a stress of 325N/mm2 before it breaks in
tension. So to find the required cross-sectional area we can use:-
Tension Force = σy x Cross-sectional Area
Cross-Sectional Area required (in mm) = Tension Force
325
We know that the thickness of tube (t) to
be used is 40mm
r
Cross-sectional Area = Circumference x thickness
t
P26
So we can evaluate the required Radius r
Steel tube in Compression
The compression tube will also require the cross-sectional area of the
tension tube: however, the buckling that it will undergo needs to be
limited as this will also cause a catastrophic failure.
Two straws threaded at either end on a cocktail stick will behave in the same
way as the two tubes when put under a bending moment.
Out of plane
deformation
Buckling failure
The green straw is in tension and stretches in the same direction as the force
The yellow straw is in compression and “buckles” or deforms out of plane
eventually failing at a relatively low force.
To resist this out of plane movement of the yellow straw a triangular crosssectional is used to resist this out of plane movement.
Compression girder
Additional Struts Resisting
out of plane movement
P27
Two Tension Girders
Cross-section
Actual Design
Additional
struts for
bracing
www.cidect.com
P28