Large span structures_new

1. Large span timber structures
Roberto Crocetti
Division of Structural Engineering
Lund University

2. Table of contents
- Material efficiency
- Shape efficiency
- Common shapes
- Trusses
- Arches
- Bracing

5. Buckling due to only self-weight

6. Trees can reach considerable heights

7. How tall can we build a column before it buckles? (due to its own
weight)

8. Critical buckling length due to self-weight
3
2
25,1

rE
Lcr


E: column’s E-modul
r: radius of column’s cross section
: column specific weight(r*g)

9. Specific strength and stiffness and material efficiency
(1) In case of compression, the values are usable only for members restrained against buckling
(2) Applies only for members in compression
(3) Applies only for member in tension

10. It is not a coincidence that among the largest spans, for roof
structures, are made by timber

11. Geodetisk kupol
Arena i Northern Michigan
University, Michigan, USA.
Diameter: 163 m
Pilhöjd: 49 m
Byggår: 1995
The superior dome

12. Two geodesic domes (coal power supply)
Brindisi, Italy
Diameter: 143 m (largest in Europe)
Rise: 44 m
Built in: 2014
The domes in Brindisi

13. Efficient shapes

14. Efficient shapes
Arch action Cable action

15. Efficient shapes
W
d
C
T
The efficiency of the beam is not too high because:
- The parts of the beam close to neutral axis are almost unstressed
- The lever arm is small (depth of the beam is approx. 1/20 of span)

16. Structural efficiency – Planar structures
Increasingstructuralefficiency
Beam
Truss
Arch
Suspension system

17. Structural efficiency – Spatial structures

18. 3D-system
2D-system

19. If structural efficiency is the goal, one should avoid
bending and choose forms where members work
only in tension or in compression

20. Show an example of bending stress vs axial stress

21. Suppose that a force “F” acts in between the two supports
L/2
F
Support 1 Support 2
L/2

22. Suppose that the force can be taken by means of the 2
structures below
L
f
F
b
h1
L
F b
h2
1 2
Hypothesis:
- Bending strength = compression strength = f
- Disregard bending in structure (1) (immovable
supports) and assume that buckling is not an issue
- For the beam case, assume L/(h2)=20
Determine :
1. The ratio of (Volume 1)/ (Volume 2) as a function of the
slope a
2. What is the value of that ratio when f/L=0,15?

23. Efficient shapes: observe this
Structural Engineering - Lund University 23
The shape of a hanging cable subjected to a set o load is similar to the shape of
the bending moment of a corresponding beam subjected to the same set of loads.

24. Efficient shapes: observe this
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25. Cable shape vs bending moment diagram
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26. Structural Engineering - Lund University 26
The shape of a hanging cable subjected to a set o load is
similar to the shape of the bending moment of a corresponding
beam subjected to the same set of loads.
OK, so what?

27. Structural Engineering - Lund University 27
If we give the structure the same shape as the hanging cable (or the same shape as the
bending moment of the corresponding beam), then we will have only tension in the
structure! (Or only compression if we turn the structure upside down)

28. Hanging rope subjected to “uniformly” distributed load
Thrust exerts
a “pull” in the
hands
This shape
gives no
bending
moments!

29. Upside down rope (arch) subjected to “uniformly” distributed
load
Thrust exerts a
“push” in the
hands
This shape
gives no
bending
moments
either!

30. Structural efficiency – Spatial structures

31. Pedestrian bridge, Essing, Germany

32. Grandview Heights Aquatic, BC, Canada

34. Attachment to the
concrete supports

35. Parabola-shaped trusses

36. Beams with ”constant stress”
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37. Parabola-shaped trusses

38. Parabola-shaped trusses

39. A few example from Nature…
(On Growth and Form, by Sir D’arcy Thompson, 1917)
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40. 40

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42. Trusses

43. Typical shapes –
truss with parallel chords

44. Typical shapes –
pitched trusses

45. Typical shapes –
curved trusses

46. Forces in the members
Ratio of:
lq
membertheinForce


47. Preliminary design

48. Preliminary design

49. Preliminary design
In preliminary design, consider a reduced area of the cross section of the members:
Ared  0,7 A

50. Preliminary design
Choose:
• Relatively wide cross sections (to allocate connection)
• Relatively shallow cross section of lower and upper chord (to reduce bending
stiffness and thus to reduce magnitude of bending moments)

51. Preliminary design
Avoid eccentricity at the nodes

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Clamps or hinges?

53. Hinges or clamps?

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Model 1: Diagonals hinged to chords

55. 55
Model 1: Diagonals hinged to chords
In this case the bending moment in the diagonals is obviously zero

56. 56
Model 2: Diagonals clamped to chords
In this case the bending moment in the diagonals is ≠ 0
N M

57. 57
Model 2: Diagonals clamped to chords
Note
1. The ratio above is not influenced by the width of the diagonal members
2. In timber structures the diagonals will never be completely clamped, thus bending
moment (and thus bending stresses) will be lower in practice
0,0
5,0
10,0
15,0
20,0
25,0
30,0
0 100 200 300 400 500 600 700
Depth of the diagonal h [mm]
 
   
100
 MN
M


h
F

58. Structural Engineering - Lund University 58
How do we design the truss in practice?
- Model the truss with continuous upper and lower chords and hinged web members
- Determine the Axial forces and the bending moments
- The nodes can be designed by considering pure axial force increase by approx. 15-20% in order to take into
consideration the presence of moment and shear

59. Buckling of trusses
In-plane
Out-of-plane
(a)
Out-of-plane
(b)

60. Buckling of trusses
In-plane
Out-of-plane
(a)
Out-of-plane
(b)

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Buckling

62. Structural Engineering - Lund University 62
How to increase stiffness and lateral stability

63. Structural Engineering - Lund University 63
How to increase stiffness and lateral stability

64. Structural Engineering - Lund University 64
Nodes

65. Structural Engineering - Lund University 65
Nodes

66. Structural Engineering - Lund University 66
Nodes

67. Olympic hall in Hamar, 1994 L=71m
Courtesy Moelven Limtre A/S
Fd7000 kN!

68. Lintel truss beam
Courtesy Moelven Töreboda AB

69. The Perkolo bridge
47.5 m

70. The Perkolo bridge –the failure

71. The Perkolo bridge – the critical joint
47.5 m

72. Truss node and force polygon

73. The design force assumed by the designer for the
dimensioning of the node in the lower chord

74. The actual design force in the lower chord

75. Ductile failure mode

76. Glulam GL30c
Dowel: fy=900 MPa
Report:
Kollapsen av Perkolo bru – hva gikk galt?
Bell, K. 2016
Cross section of the lower chord

77. Arches

78. Preliminary design

79. Support conditions – tied arch

80. 2 M30, 8.8
Support conditions – tied arch

81. Support conditions: directely on fundation

82. Support conditions: directely on fundation

83. Statical systems
Three-hinged
Two-hinged
Zero-hinged
In general:
- Concrete arches are usually “zero-hinged”
- Steel arches are usually “two or three-hinged”
- Timber arches are usually “three-hinged” (when they are massive)

84. Three-hinged arch

85. Bridge over railway, Haninge
Three-hinged arch, Span: 35 m
Erection: 25 march 2017

86. Two-hinged trussed arch
(span: 71m) for road traffic

87. Tynset bridge
pre- assembling of the
arches in the factory

88. Fixed (zero-hinged)
Skubbergsenga bridge, Norway
Total length 40 m
Arch span 32 m
Bridge width 4 m
Building year 1997

89. Geometry considerations
Common rise to span ratio:
f/L=0,15 (up to 0,20)
circular arches






 2
2
4
1 x
l
fzEquation of parabola,
origin in the middle















2
3
8
1
l
f
ls





 

l
f4
arctan
180

a

90. Forces in the cross section

91. Effect of symmetric and non-symmetric loading

92. Effect of non- uniformly distributed loads

93. How can we reduce the effect of bending moment in three-hinged
arches?

94. Bandy hall in Nässjö (75m)

95. Bandy hall in Nässjö: Erection

96. Buckling of arches
• In plane buckling
• Out-of-plane buckling

97. Buckling modes

98. Different types of buckling analysis
The buckling load can be determined by 2nd- order analysis, by giving the arch initial
imperfection and increasing the load stepwise until instability is reached
By using simplified formulas to determine the buckling load. In this case the arch is
considered as a compressed strut with an appropriate buckling length

99. Buckling factor (according to Timoshenko)

100. &
for  = 32, the buckling length factor becomes: b  1,17
Buckling Load at ¼ of the span

101. Inclined suspenders increase both in plane buckling
(and they also reduce bending moments in the arch)

102. After: Kolbein Bell
Bending in the arch with different hanger configuration

103. Buckling of the arch with different hanger configuration

104. Non-linear analysis

105. Out-of-plane buckling
Students at LTH getting
acquainted with 1. lateral
buckling and 2. bracing of
arches

106. Out-of-plane buckling

107. Out-of-plane buckling
Buckling analysis is conducted as for a compressed bar with relevant buckling lengths

108. Bridge in Hägernäs, total length: 42m, arch span: 34m
Out-of-plane buckling

109. Bracing of the bottom part of the arch
Extra compression strut

110. Bracing

111. Improper bracing during construction

112. Conversion of an unstable structure into a stable structure
Unstable configuration Stable configurations
or
(a)
(b)
(c)
(d)
(e)
(d’)

113. Bracing system’s main functions
• Transmission of horizontal loads
• Reduction of lateral deformations
• Enhancing buckling strength

114. Complete bracing

115. Bracing system for heavy timber structures

116. Bracing elements
• Longitudinal wall bracing (A)
• Transversal roof bracing (B)
• End wall bracing (C)
• Longitudinal roof bracing (D)

117. Stable or unstable?

118. Stable or unstable?

119. Stable or unstable?

120. Stable or unstable?

121. Stable or unstable?
Equilibrium?

122. Stable or unstable?

123. Stable or unstable?

124. Prerequisites for stability
• Wall bracings must be able to resist horizontal
forces along three different directions in the plane
• The three directions shall not converge in the same
point
• At least two of the three directions shall not be
parallel one to another

125. Brace stiffness

126. Brace stiffness
Hp: Neglect the axial
deformation of beam and
column

127. Bracing of high walls
System (b) is in general more efficient than system (a), due to ab < aa
40o<a<55o is a good compromise between economy and efficiency
System (b) is however more expensive than system (a)

128. Strength and stiffness requirements for bracing systems

129. Perfectly straight column

130. Perfectly straight column
Equilibrium about “C”:

131. Equilibrium about “C”:
This tends to
unstabilise- Munst
This tends to
stabilise- Mst
Perfectly straight column

132. Critical stiffness “CE”
The maximum axial load that the column can resist cannot be larger
than the buckling load of the column

133. Column with several braces
a
P
C E
E  4,3
(for 3 brace points)

134. Column with several braces
(for more than 4 braces)
a
P
C E
E  4

135. Beam bracing

136. Beam bracing








H
M
N d
d
3
2

137. Lateral forces in beams -model
R
q
Nd
R
q
Nd
Rq
L
H
R
N
qdRqdN d
rrd  aa
Nd
R
Nd
qr
da
dl=R·da

138. What is “R”

139. Assumed initial deformation: parabolic shape
DT
x
y
L
R
2
4 





D
L
x
y T
Assumed shape
2
2
1
dx
yd
R

2
2
2
2
2
2
8
4
1
LL
x
dx
d
dx
yd
R
T
T
D















D

140. Lateral forces
Nd
R
q
Nd
Rq
R
N
q d
r  2
81
LR
TD

T
d
r
LN
q
D


8
2
&

141. Estimation of lateral loads for glulam structures
• Initial out-of-straightness: D0=L/500
• Additional deformation D (e.g. due to wind load) shall not exceed
L/500.
• This means that the final deformation shall be (maximum value)
250
)( 0
L
T DDD

142. Lateral forces
Nd
R
q
Nd
Rq
T
d
r
LN
q
D


8
2








H
M
N d
d
3
2
HL
M
q d
r


20
250
L
T D

143. Brace forces for a series of bent members

144. EC5 approach
 crit
f
d
h k
LHk
M
nq 

 1
3,
Md= design moment in the beam
H = depth of beam
L= span of the beam
n= number of laterally braced beams
kf,3= modification factor (kf,3=30-80)
kcrit= reduction factor for lateral buckling when the beam is unbraced

145. Erection of timber structures

146. Timber struts as bracing system

147. Erection: pair of fully braced trusses

148. Bracing by diaphragm action