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CONSULTINGMECHANICAL ENGINEERS

GEODESIC DOME
Structural Analysis

April 2005

P: +613 5337 5700

2 Valentine Street, Ballarat, Vic 3356, Australia
F: +613 5336 4524 E: sales@asema.com.au W: www.asema.com.au

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

Revision History
NumberA
B

Date
12 Jan 2005
5 April 2005

Comment
Original
Added modified node calculations

Author
Ben Lewis
Ben Lewis

Table of Contents
INTRODUCTION............................................................................... 1
DOME CHARACTERISTICS................................................................. 2
ASSUMPTIONS ................................................................................ 3
REFERENCED STANDARDS ................................................................ 3
FAILURE MODES.............................................................................. 4
STRUT BUCKLING ............................................................................. 4
BOLT FAILURE ................................................................................. 5
NODE RING DEFORMATION .................................................................. 6
LOADS ..........................................................................................10
WIND LOAD ..................................................................................10
DEAD LOAD...................................................................................13
COMBINED LOADING ........................................................................14
SAFE WORKING LOAD .......................................................................14
OVERTURNING................................................................................16
MODIFIED NODE ............................................................................16
FINITE ELEMENT ANALYSIS .................................................................17
COMBINED LOADING ........................................................................18
SAFE WORKING LOAD .......................................................................18
SUSPENDED MASS FROM CENTRE OF BEAM .......................................19
CONCLUSION.................................................................................20

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

INTRODUCTION
The purpose of this report is to analyse the 9 m geodesic dome manufactured by
Dome Dimensions to determine the safe working load of the structure. It has beenassessed in accordance with AS4100 for steel structures and AS1170 for loading
conditions.

Figure 1: Dome Dimensions geodesic dome

Figure 2: Dome Dimensions geodesic hub joint

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Geodesic Dome Structural Analysis

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DOME CHARACTERISTICS
The geodesic characteristics of the Dome Dimensions 9 m dome are listed in Table 1.
Table 1: Dome Dimensions 9 m dome characteristics

Item
Dome type
Base diameter
Spherical diameter
HeightFrequency
Class
Eccentricity
Horizontal projected area
Vertical projected area
Spherical angle
Spherical dome surface area
Spherical dome volume
Number of hubs
Number of base boundary points
Number of struts
Total length of struts
Longest strut
Shortest strut
Number of panels
Total area of panels
Largest panel
Smallest panel

Value
Icosahedron dome (modified base)
8.86 m
9.00 m3.704 m3
1
1.00
61.62 m2
24.68 m2
159.61°104.72 m2
140.72 m3
46
15
120
212.47 m
1.90 m1.55 m75
100.98 m2
1.51 m2
1.14 m2

The materials used in the construction of the Dome Dimensions 9 m dome are listed inTable 2.
Table 2: Construction materials

Item
Strut
Node ring
Profile plates
Fasteners

Value
Tubeline R.H.S. C350L0 75 x 50 x 2.0 mm
Tubeline C.H.S. C250L0 88.9 x 4.0 mm
Mild Steel 5.0 mm
M10, Grade 8.8, button head cap screw

The mass of the geodesic dome has been calculated using the items listed in Table 3.
Table 3: Dome mass

Item
RHS struts
CHS Rings
Profile Clamps
Locking washers
Fasteners
Tarp
Total

Unit Mass
3.72 kg/m
8.38 kg/m
0.180 kg
0.110 kg
0.010 kg
1.0 kg/m2

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Qty
212 – (0.07 x 2 x 120) = 195 m
0.045 x 46 = 2.07 m
4 x 2 x 120 = 960
46 x 2 = 92
120 x 3 = 360
100 m2

Total [kg]
726
17
173
10
4
100
1,030

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

ASSUMPTIONS
The following assumptions have been made in this analysis.
• The canvas canopy has been represented as a series of vertical loads applied ateach node. In reality there will be an additional horizontal component as thetarp tension acts in a direction normal to the dome surface.
• Wind loading is assumed to act in a direction parallel to the ground plane on thevertical projected area of the dome.
• It is assumed that working loads are suspended from the geometric centre of anode. In reality any mass suspended from a node will be attached via a bolt
thus offsetting the load from the node centre. This will create a small torquecomponent not accounted for in this analysis.
• The dome is assumed to be mounted on flat level ground• It is assumed that each member on the base of the dome is fixed to the ground
via a stake or similar means.
• The safe working load of this structure has been assessed against a
serviceability state criterion and an ultimate state criterion. The serviceability
state criterion is designed to ensure the structure does not experience any
permanent deformation of parts under normal operating conditions. The
ultimate state criterion is designed to ensure the structure does not collapse
under the worst-case conditions, although some members may experience local
yielding.

REFERENCED STANDARDSAS 1170: 2002
AS 4100: 1998
AISC - ASD

Structural Design Actions
Steel Structures
Allowable Stress Design - Buckling of Compact Rolled Shapes

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Geodesic Dome Structural Analysis

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FAILURE MODES
This section examines the most likely cause of structural failure. The weakest
member in the structure is determined by comparing the magnitude of the force foreach failure mode.
Table 4 lists each failure mode examined here with thecorresponding strut force. It can be seen that the node is the weakest component in
the structure and is expected to yield at approximately 6.0 kN of applied load from thestrut.
Table 4: Failure mode with estimated force

Compression
Members [kN]
55
47
14
N/A
55
6.0

Buckling force about pinned axis
Buckling force about fixed axis
Bolt-strut normal force
Strut shear force
Bolt shear force
Ring force

TensionMembers [kN]
N/A
N/A
14
42
55
6.0

Strut Buckling
The buckling strength of the longest member has been assessed in accordance with
the AISC (American Institute of Steel Construction) standard using the allowablestress design (ASD) method for compact rolled shapes.
The characteristics of each of the struts are shown in Table 5.

Table 5: Strut characteristics

Item
Cross sectional area
Length A
Length B
Length C
Dead length
Max length pin to pin
Strength limit (sigma x A)

Value
384x10-6 m2
1.569 m1.816 m1.856 m0.015 m1.826 m134 kN

Figure 3: Strut buckling analysis

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

Table 6 shows the calculations used to determine the buckling strength of the longest
member in both the pined and fixed axes. The strut is weakest in buckling about its
fixed (Z) axis. Buckling is expected to occur at 47.2 kN.
Table 6: Strut buckling calculations

Buckling
about
pinned (Y)axis

Buckling
about fixed(Z) axis

252x10342.9x103

Radius of gyration [mm]

25.6

10.6

Slenderness ratio

71.2

86.4

Critical slenderness ratio

106

106

Inelastic

Inelastic

Inelastic bucking stress [MPa]

144

123

Nominal buckling force [kN]
Design buckling force [kN]

55.4
33.2

47.2
28.3

Item

FormulaMass moment of inertia [mm4]Buckling mode-> Inelastic-> Elastic

S.F = 1.67

Bolt Failure
Strut fasteners are M10, grade 8.8 button head cap screws. Table 7 shows thecalculations used to determine the shear capacity of these bolts. These calculations
have been completed in accordance with AS4100:1998. It can be seen that thenominal shear capacity of these bolts is 60 kN and the design capacity is 48 kN

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Geodesic Dome Structural Analysis

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Table 7: Bolt shear capacity

Item
Bolt minimum tensile strength (seeTable 9.3.1)
Bolted lap connection, length reduction
factor (see Table 9.3.2.1)
Number of shear planes with threads
intercepting the shear plane
Minor diameter area of the bolt (seeAS 1275 Table 3.3)
Number of shear planes without threads
intercepting the shear plane
Nominal plain shank area of the bolt
Capacity factor (see Table 3.4)

Formula

Value

fuf

830 MPa

kr

1.0

nn

2

Ac

58

nx

0

Ao
φ

78.5
0.8

Bolt nominal shear capacity

60 kN

Bolt design shear

48 kN

The strut may fail in the region around the bolt due to localised stresses. Two failuremodes have been investigated here. The strut may yield due to an excessive
compressive force on the bolt-bearing surface; it may also fail in shear as the bolt
tries to pull through the strut material holding it captive. The calculations in Table 8
have been used to determine the approximate forces at which these failures will
occur. It can be seen that the strut material will begin to yield in compression aroundthe bolt at approximately 14 kN. It should be noted that yielding in this region is not
likely to result in a catastrophic failure of the entire structure it will simply cause anelongation of the bolt hole. This would only be a problem if this loading wereexperienced repetitively over an ongoing period of time.
Table 8: Local strut failure around the bolted connection

Item

FormulaValue

Bolt bearing normal force

14.0 kN

Strut shear force

42.0 kN

Node Ring Deformation
FEA analysis identified the ring to be the weakest component in the hub. The load
carrying capacity of this ring was investigated using three methods, hand calculations,
FEA analysis and experimentation. Table 4 below shows the results obtained fromeach method. It can be seen that no hand calculations or experimental data wasavailable for the node with locking washers but it can be seen from the results of theanalysis without locking rings that the FEA method was conservative. A nominal valueof 6.0 kN was selected as the maximum allowable strut force.
Figure 4: Node ring deformation estimates using various methods

MethodHand calculation
FEA analysis
Experimentation

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Strut Force [kN]
W/O Locking
With Locking
Washers
Washers
2.2
NA
2.0
6.0
4.0
NA

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Geodesic Dome Structural Analysis

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Hand Calculations
Basic formulas for curved beams indicate the load carrying capacity of the ring is
2.2 kN applied at two points on opposing sides of the ring, as shown in the left handexample of Figure 5. The majority of nodes in the dome are subject to combined
loads of compression and tension acting at right angles to each other as shown in theright hand example of Figure 5. The nominal force in each strut to cause deformationof the ring is therefore approximately 1.1 kN, or half that determined with curved
beam theory.

Figure 5: Combined effect of tension and compression on hub

Table 9: Node ring calculations

Item

FormulaValue

Ring force

2.2 kN

Diametral deflection, coaxial with load0.53 mm

Diametral deflection, transverse to load

0.48 mm

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

FEA Analysis
To analyses the geodesic structure a 3D geodesic dome was modelled using simulated
beams. Beam features simulated include material type, cross-section, orientation andend releases. Three nodes were modelled in the dome to assess what stresses were
induced in the ring under various loading scenarios. The location of the three nodes isshown in Figure 6. The top centre node was chosen because it is unique in that all
attached members are in equally spaced and in compression. The two side nodes
where chosen at the bottom of the structure because they carry the self weight of thestructure, one of the nodes has five struts while the other has six.

top node
5 strut
symmetric loadingside node
6 strut
asymmetric loading
corner node
5 strut
asymmetric loading
Figure 6: Location of nodes analysed on the structure

The dome model was loaded with masses at each node until the stress in the rings
was in the vicinity of 210 to 240 MPa. The left hand column of Figure 7 shows a forceof 2.0 kN in the struts was required to cause this level of stress. The addition oflocking washers to the model raised the allowable force to between 6.0 and 12.0 kN
depending on the node. This illustrates the importance of using the locking washers
and also suggests that the ring will benefit from a permanent gusset welded to its
centre. A nominal value of 6.0 kN was selected as the strut force maximum.

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Geodesic Dome Structural Analysis

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Without Locking Washers

Side Node

7.8 kN

0.5 kN

1.9 kN

With Locking Washers

11 kN

2.6 kN
2.2 kN

1.7 kN

1.0 kN

9.3 kN

5.2 kN

6.2 kN12 kN

2.4 kN

Corner Node

2.3 kN

9.3 kN

2.0 kN
1.8 kN

9.4 kN

4.4 kN

0.6 kN

1.1 kN

6.0 kN

6.0 kN

Top Node

6.0 kN

6.0 kN

6.0 kN

6.0 kN

4.5 kN

6.0 kN

6.0 kN

6.0 kN

6.0 kN

Figure 7: Node analysis

Experimentation
A node ring was compressed between two flat surfaces to obtain a force deflection
curve, the results can be seen in Figure 8. It can be seen that the ring yielded at
4.0 kN and its ultimate strength was in excess of 6.0 kN.

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Geodesic Dome Structural Analysis

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Geodesic Node Ring
7
6

Force [kN]5

Test cut shortat this point

Yield
4
3
2
1
0
0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Displacement [mm]Figure 8: Node ring force - deflection curve

LOADS
This section determines the maximum allowable loads on the dome so as not toexceed the allowable strut for of 6.0 kN determined in the previous section. Loads
investigated are wind loads and dead loads.

Wind Load
Figure 9 below shows the various wind regions throughout Australia. Region A wasselected as the most appropriate for the dome. This means it can be erected
anywhere in Australia, excluding cyclone affected coastal areas during cyclonicconditions (Regions B, C and D of AS1170.2: 2002).

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

Figure 9: Wind speed regions

Table 10 shows the calculations used to determine wind loading in accordance with
AS1170.2: 2002. It can be seen that for the serviceability limit state 550 N ofcompressive loading and 410 N of tensile loading must be allowed for. While for theultimate limit state criteria 910 N of compressive loading and 680 N of tensile loading
must be allowed for.

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Geodesic Dome Structural Analysis

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Table 10: Serviceability state site wind speed calculation

Ultimate
Limit State

See Figure 9

ServiceabilityLimit StateA

R

100 years

100 years

32 m/s

41 m/s

Md

1.0

1.0

Terrain/height multiplier(AS1170.2: 2002 Clause 4.2)

Mz,cat1.05(Category 1,
5 m)

1.05
(Category 1,
5 m)

Shielding multiplier(AS1170.2: 2002 Clause 4.3)

Ms

1.0

1.0

Mt

1.0
(H/[2Lu] <
0.05)

1.0
(H/[2Lu] <
0.05)

34

43

See Figure 10

550 N

910 N

See Figure 11

410 N

680 N

Item

FormulaWind region
Average recurrence interval(AS1170.2: 2002 Clause 3.2)
Regional wind speed (3 sgust)
(AS1170.2: 2002 Clause 3.2)
Wind directional multiplier(AS1170.2: 2002 Clause 3.3)

Topographic multiplier(AS1170.2: 2002 Clause 4.4)
Site wind speed(AS1170.2: 2002 Clause 2.2)
Max compressive force(FEA analysis)
Max tensile force(FEA analysis)A

Figure 10 shows the forces induced in each member under the serviceability limit
state criteria in region A wind conditions.
max compression
550 N
max tension
410 N

Wind

Figure 10: Beam forces under serviceability limit state

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Figure 11 shows the forces induced in each member under the ultimate limit state
criteria in region A wind conditions.
max compression
910 N

max tension
680 N

Wind

Figure 11: Beam forces under ultimate limit state

Dead Load
Figure 12 shows the forces induced in each member due to the structures own self
weight.
max tension
530 N

max compression
420 N

Figure 12: Beam forces due to structure dead weight

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Geodesic Dome Structural Analysis

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Combined Loading
Table 11 shows the calculations used to determine the available strut capacity of thestructure. It can be seen that the dome members have 5.1 kN of capacity to carry
the safe working load.
Table 11: Structure loads

Serviceability [kN]
6.0
-0.5 x 0.8 = -0.4
-0.5
5.1

Maximum allowable
Dead
Wind
Available Strut capacity

Ultimate [kN]
20.0
-0.5
-0.9
18.6

Safe Working Load
Table 12 summarises the safe working load of the structure so as not to exceed thecapacity of the members determined in the previous section. It can be seen that
200 kg can be suspended from all nodes or 400 kg can be suspended from a single
node.
Table 12: Safe working loads

Nodes
All
One

Safe Working Load
[kg]
200
400

Figure 13 shows the maximum tension and compression force in the dome when
2000 N is suspended from each node. It can be seen that the allowable force of
5.1 kN is not exceeded.

max tension
4.4 kNmax
compression
3.2 kN

Figure 13: 2,000 N suspended from all nodes

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

Figure 14 to Figure 19 shows the affect on the structure of suspending a mass of
400 kg from a single node. It can be seen that the allowable force of 5.1 kN isequalled in the case of node 1 and not exceeded in all other cases.

3.7 kN
compression

2.6 kNtension

4.6 kN
compression
5.2 kNtensionFigure 14: 4,000 N suspended from node 1

3.1 kN
compression

Figure 15: 4000 N suspended from node2

2.6 kNtension
2.5 kNtensionFigure 16: 4,000 N suspended from node 4

3.4 kN
compression

2.8 kNtensionFigure 18: 4,000 N suspended from node 7

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3.3 kN
compression

Figure 17: 4,000 N suspended from node 5

2.9 kN
compression

2.5 kNtensionFigure 19: 4,000 N suspended from node 815International

Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

Overturning
To determine if overturning is likely to be a problem moments are taken about oneedge of the dome. The overturing force is found to be 35.6 kN

The wind velocity required to create a force of this magnitude is approximately
100 m/s which is well below the 30 to 40 m/s wind speeds used in region A windcalculations so overturing should not be a problem with this structure in region A
winds.

MODIFIED NODEThe capacity of the dome increases substantially with the following changes, (seeFigure 20).
• Gusset welded to centre of ring
• Locking washes spaced further apart

Figure 20: Differences between original and modified nodes

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Finite Element Analysis
The modified node was compared to the original node using FE analysis. The results
are shown in Figure 21. The modified node shows considerably less stress under asimilar loading scenario raising the allowable strut force to around 9.0 kN.
Original NodeSide Node

7.8 kN

Modified Node
8.0 kN

2.3 kN

2.9 kN

6.8 kN11 kN

5.5 kN

9.3 kN

5.2 kN

8.7 kN

6.2 kN12 kN

9.8 kN11 kN

9.3 kN

Corner Node

5.5 kN
9.4 kN

4.4 kN

5.6 kN

4.5 kN
9.0 kN

6.0 kN

9.1 kN

9.5 kN

9.5 kN

Top Node

6.0 kN
6.0 kN

6.0 kN

6.0 kN

9.5 kN

9.5 kN

9.5 kN

Figure 21: Comparison between strength of original and modifed nodes

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

Combined Loading
Table 13 shows the calculations used to determine the available strut capacity of thestructure. It can be seen that the dome members have 8.1 kN of capacity to carry
the safe working load.
Table 13: Structure loads

Maximum allowable
Dead
Wind
Available Strut capacity

Serviceability [kN]
9.0
-0.5 x 0.8 = -0.4
-0.5
8.1

Ultimate [kN]
20.0
-0.5
-0.9
18.6

Safe Working Load
Table 14 summarises the safe working load of the structure so as not to exceed thecapacity of the members determined in the previous section. It can be seen that
300 kg can be suspended from all nodes or 600 kg can be suspended from a single
node.
Table 14: Safe working loads for modified node

Nodes
All
One

Safe Working Load
[kg]
300
600

Figure 22 shows the maximum tension and compression force in the dome when
3,000 N is suspended from each node. It can be seen that the allowable force of
8.1 kN is not exceeded.
max tension
7.1 kNmax
compression
5.2 kN

Figure 22: 3,000 N suspended from all nodes

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Leith Atchison – Dome Dimensions
Geodesic Dome Structural Analysis

Taking the worst case single node loading found in the previous load cases Figure 23
shows the affect on the structure of suspending a mass of 600 kg from a single node.
It can be seen that the allowable force of 8.1 kN is not exceeded in any of themembers.

max
compression
7.2 kNmax tension
4.2 kN

Figure 23: 6,000 N suspended from centre node (worst case single node loading)

SUSPENDED MASS FROM CENTRE OF BEAM
Horizontal members in the dome are typically in tension and therefore can carry acentre hanging load without concerns of weakening these members due to buckling.
The allowable load to be suspended from the centre of a horizontal beam is given by:

F=

0.6σ y I− FselfWeightyd
0.6σ y I=− Lρ gLy2
0.6 × 350 ×106 × 42.9 × 10−9− (1.9 × 3.72 × 9.81)
1.9
0.025 ×2
= 310 N≈ 30 kg=

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Geodesic Dome Structural Analysis

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The safe working load to be suspended from the centre of any horizontal beam is
30 kg. This assumes the beam is not weakened in any way by the method used toattach the mass. For example this does not allow for holes to be drilled into beam toattach fixing points.

CONCLUSION
The 9 m Dome Dimensions dome has been analysed in accordance with the relevant
sections of AS 1170: 2002 (Structural Design Action), AS 4100: 1998 (Steel
Structures) and AISC for buckling of compact rolled shapes. Two versions of thedome have been considered depending on the node type. One node is the original
design with no centre gusset and closely spaced locking washers; the other is themodified node with a welded centre gusset and widely spaced locking washers. The
safe working loads for the dome with the two types of nodes are listed below in Table
15.
Table 15: Safe working loads

Nodes

Safe Working Load [kg]Original Node Design Modified Node Design

All

200

300

One

400

600

Centre beam mass
(Horizontal beams only)

30

30

It is important to note that these safe working loads only apply when the locking
washers used in the centre of the nodes are all in place as these are essential to thestrength of the node.
The structure is suitable for region A winds which means it can be erected anywherein Australia, excluding cyclone affected coastal areas during cyclonic conditions(Regions B, C and D of AS1170.2: 2002).

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