Circular glulam arched structures

118 Pages·2016·9.52 MB·English

by  Rafiullah Sherzad

Master Thesis in Structural Engineering

Buckling and Geometric
Nonlinear Stress Analysis
- Circular glulam arched structures

Authors: Rafiullah Sherzad; Awrangzib Imamzada
Supervisor: Sigurdur Ormarsson; Sara Florisson
Examiner: Björn Johannesson
Course Code: 4BY363/4BY35E
Semester: Spring 2016, 15 credits
Linnaeus University, Faculty of Building Technology

Abstract
An arched structure provides an effective load carrying system for large span
structures. When it comes to long span roof structures, timber arches are one of the
best solutions from both structural and aesthetical point of view. Glulam arched
structures are often designed using slender elements due to economic consideration.
Such slender cross-section shape increases the risk of instability.
Instability analysis of straight members such as beam and column are explicitly
defined in Eurocode. However, for instability of curved members no analytical
approach is provided in the code, thus some numerical method is required.
Nonetheless, an approximation is frequently used to obtain the effective buckling
length for the arched structures in the plane of arches.
In this master thesis a linear buckling analysis is carried out in Abaqus to obtain an
optimal effective buckling length both in-plane and out-of-plane for circular glulam
arched structures. The elastic springs are used to simulate the overall stiffness of the
bracing system.
The results obtained by the FE simulations are compared with a simple
approximation method. Besides, the forces acting on the bracings system is obtained
based on 3D geometric nonlinear stress analysis of the timber trusses.
Our findings conclude that the approximation method overestimates the effective
buckling length for the circular glulam arched structures. In addition, the study
indicates that the position of the lateral supports along the length of the arch is an
important design aspect for buckling behaviour of the arched structures. Moreover, in
order to acquire an effective structure lateral supports are needed both in extrados
and intrados.
Furthermore, instead of using elastic spring elements to simulate the overall stiffness
of the bracing system, a full 3D simulation of two parallel arches was performed. It
was shown that the springs are stronger than the real bracing system for the studied
arch.
Key words: ARCH, BUCKLING, EUROCODE, EXTRADOS, GLULAM,
INSTIBILTY, INTRADOS, LATERAL SUPPORTS, LINEAR ANALYSIS,
III
Acknowledgement
This master thesis completes our one year master program in structural engineering
at the Linnaeus University, Faculty of Technology, Sweden. The proposal was made
by Professor Sigurdur Ormarsson and supervised by Professor Sigurdur Ormarsson
and PhD Sara Florisson.
We would like to avail the opportunity to express our special thanks to Professor
Sigurdur Ormarsson, for his inspiring guidance and for giving us valuable ideas and
advice regarding the project.
We also wish to thank PhD Sara Florisson for all her supports, guidance throughout
the project. She has proofread our manuscript and given precious feedback,
especially on the Academic Writing.
Finally we would like to express our gratitude to our families and friends for their
inspiring and supports during the work performed.

Rafiullah Sherzad & Awrangzib Imamzada
Växjö 25thof May 2016

IV
Table of notations and abbreviations

𝐴 Cross-sectional area [𝑚2]
𝑟 Radius of the arch [m]
𝐶 Thermal coefficient
𝑡
𝐶 Exposure coefficient
𝑖
Modulus of elasticity [𝑃𝑎]
𝐸
Fifth percentile value of modulus of elasticity
𝐸
0.05 Moment of inertia [𝑚4]
𝐼
Compression strength [𝑃𝑎]
𝑓
𝑐 Design strength [𝑃𝑎]
𝑓
𝑑 Bending strength [𝑃𝑎]
𝑓
𝑚 Shear modulus [𝑃𝑎]
𝐺
Cross-sectional height [𝑚𝑚]
ℎ
Cross-sectional width[𝑚𝑚]
𝑏
𝑁
Linear spring stiffness [ ]
𝑘 𝑚
Reduction coefficient with respect to lateral
𝑘
𝑐𝑟𝑖𝑡
torsional buckling

Deformation coefficient
𝑘
𝑑𝑒𝑓 Material modification coefficient
𝑘
𝑚𝑜𝑑 Reduction coefficient with respect to buckling
𝑘
𝑐 Length
𝐿
Length coefficient
𝑙_𝑛
Effective length
𝑙𝑒𝑓 Bending moment [𝑁𝑚]
𝑀 Critical bending moment regarding instability
𝑀 [𝑁𝑚]
𝑐𝑟
Critical axial concentrated load [𝑁]
𝑃 Distributed load [𝑁]
𝑐𝑟
𝑚
𝑞 𝑁
Snow load on the arch[ ]
𝑞 𝑚
i,𝑘 𝑁
𝑠 Snow load [ ]
𝑚2
𝑊 Elastic section modulus [𝑚3]
𝛽 Euler effective length factor
𝛾 Partial coefficient for wood material
𝑀
𝛾 Safety factor according to Swedish annex
𝑑
𝜀 Strain
Slenderness ratio
𝜆
Poisson’s ratio
𝜐
Stress [Pa]
𝜎
Compression stress [Pa]
𝜎
𝑐 Bending stress [Pa]
𝜎
𝑚 Critical stress regarding instability [Pa]
𝜎
𝑐𝑟𝑖𝑡 Degree of freedom
DOF
Eurocode 5
EC5
Finite element
FE
Finite element method
FEM Glued laminated timber
Glulam

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VI
Table of contents
1. INTRODUCTION.......................................................................................................... 1
1.1 BACKGROUND ....................................................................................................................................... 2
1.2 AIM AND PURPOSE ................................................................................................................................ 2
1.3 HYPOTHESIS AND LIMITATIONS ............................................................................................................. 3
1.4 RELIABILITY, VALIDITY AND OBJECTIVITY ............................................................................................ 4
2. LITERATURE REVIEW ............................................................................................. 5
2.1 HISTORY OF ARCHES ............................................................................................................................. 5
2.2 LITERATURE RESEARCH ON STABILITY OF STRUCTURES ........................................................................ 6
3. THEORY ........................................................................................................................ 8
3.1 A SHORT INTRODUCTION OF TREES ........................................................................................................ 8
3.2 MATERIAL STRUCTURE OF WOOD .......................................................................................................... 8
3.3 NATURAL CHARACTERISTICS OF WOOD ............................................................................................... 10
3.3.1 Knots ........................................................................................................................................... 10
3.3.2 Spiral grain angle ....................................................................................................................... 10
3.3.3 Reaction wood ............................................................................................................................ 11
3.3.4 Juvenile wood ............................................................................................................................. 12
3.4 PHYSICAL PROPERTIES OF WOOD ......................................................................................................... 12
3.4.1 Wood and moisture ..................................................................................................................... 12
3.4.1.1 Moisture content ................................................................................................................ 12
3.4.1.2 Influence of moisture content ............................................................................................. 12
3.4.2 Distortion of timber (Dimensional changes) .............................................................................. 13
3.4.3 Density ........................................................................................................................................ 13
3.5 MECHANICAL PROPERTIES OF WOOD AND TIMBER ............................................................................... 14
3.5.1 Strength and stiffness of wood .................................................................................................... 14
3.5.2 Strength and stiffness of structural timber .................................................................................. 14
3.5.3 Influence of moisture .................................................................................................................. 15
3.5.4 Influence of loading time ............................................................................................................ 16
3.5.5 Influence of temperature ............................................................................................................. 16
3.5.6 Influence of cross-section size .................................................................................................... 16
3.5.7 Long term deformations .............................................................................................................. 17
3.6 ORTHOTROPIC ELASTIC BEHAVIOUR OF WOOD .................................................................................... 18
3.7 ENGINEERED WOOD PRODUCT (EWP) ................................................................................................ 18
3.7.1 Engineered Wood Product based on sawn timber boards .......................................................... 19
3.8 GLULAM ARCHES ................................................................................................................................ 19
3.8.1 Lateral bracing ........................................................................................................................... 20
3.8.2 Lateral stability ........................................................................................................................... 20
3.9 INSTABILITY OF STRUCTURES .............................................................................................................. 21
3.10 BUCKLING ......................................................................................................................................... 22
3.10.1 Column buckling ....................................................................................................................... 22
3.11 BUCKLING OF A UNIFORMLY COMPRESSED CIRCULAR ARCH ............................................................. 24
3.12 LATERAL BUCKLING .......................................................................................................................... 26
3.12.1 Lateral-torsional buckling of beam .......................................................................................... 26
3.12.2 Lateral-torsional buckling of arches ........................................................................................ 27
4. METHOD ..................................................................................................................... 28
4.1 FINITE ELEMENT METHOD (FEM) ....................................................................................................... 28
4.1.1 Linear buckling analysis ............................................................................................................. 28
4.1.1.1 Buckling length based on FE method ................................................................................ 29
4.1.2 Buckling length based on approximation ................................................................................... 29
4.1.3 Geometric non-linear stress analysis.......................................................................................... 30
4.2 TYPE OF ELEMENTS ............................................................................................................................. 30
4.2.1 Elements characterizing ............................................................................................................. 30
VII
5. GEOMETRY, MATERIAL AND ACTION ON THE STUDIED ARCHES ........ 32
5.1 GEOMETRY .......................................................................................................................................... 32
5.2 MATERIAL DATA ................................................................................................................................. 32
5.3 LOADING ............................................................................................................................................. 32
5.3.1 Design Loads .............................................................................................................................. 32
5.3.1.1 Permanent Load ................................................................................................................. 33
5.3.1.2 Snow Load ......................................................................................................................... 33
5.3.1.3 Wind Load .......................................................................................................................... 37
5.3.2 Load Combination ...................................................................................................................... 39
6. ABAQUS ....................................................................................................................... 41
6.1 MODELLING IN ABAQUS® ................................................................................................................... 41
6.1.1 Geometry .................................................................................................................................... 41
6.1.2 Partitions .................................................................................................................................... 42
6.1.3 Material properties ..................................................................................................................... 42
6.1.4 Boundary conditions ................................................................................................................... 43
6.1.5 Hinges ......................................................................................................................................... 44
6.1.6 Lateral support ........................................................................................................................... 44
6.1.7 Loads .......................................................................................................................................... 45
6.1.8 Element mesh .............................................................................................................................. 45
6.1.9 Visualization of the simulation results ........................................................................................ 45
7. RESULTS AND ANALYSIS ...................................................................................... 46
7.1 EFFECTS OF LATERAL SUPPORTS AND HINGES ON BUCKLING MODES ................................................... 46
7.2 IN-PLANE BUCKLING ............................................................................................................................ 50
7.2.1 Buckling length based on a simple approximation ..................................................................... 50
7.2.2 Buckling length based on buckling simulations .......................................................................... 50
7.2.3 Strength Verification ................................................................................................................... 53
7.4 OUT-OF-PLANE BUCKLING ................................................................................................................... 57
7.4.1 Strength verification for out-of-plane buckling .......................................................................... 62
7.5 GEOMETRIC NONLINEAR STRESS ANALYSIS ......................................................................................... 68
8. 3D-SIMULATION OF THE CIRCULAR GLULAM ARCHES ............................ 71
8. DISCUSSION ............................................................................................................... 76
9. CONCLUSIONS AND RECOMMENDATIONS ..................................................... 78
10. FURTHER STUDIES ................................................................................................ 79
REFERENCES ................................................................................................................. 80
APPENDIX 1: FIRST BUCKLING MODES FOR FIXED, TWO-AND THERE-
HINGED CIRCULAR ARCHES (ISOMETRIC AND TOP VIEWS) ......................... 1

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IX
1. Introduction
Glulam arched structures are used to overcome large spans in the range of
48 to 120 meters. These spans are possible because of the advantageous
combination of material and geometric properties (Saje, et al., 2013, p.
1350040-2). Glulam arches are broadly used in the roof structure of halls,
bridges, and viaducts because of their favourable shapes.
Glulam arched structures are often designed using a slender cross-section
due to economic consideration (Crocetti, et al., 2011, p. 3.39). These
structures are characterised by their slender design, which is economic
because it can be produced in curved shapes with varying depth without a
great increase in production cost. The upper edge of the curved beam is the
edge that is most frequently braced. This bracing also fulfils the function of
roof carrier, which explains their position at the top surface. Consequently,
the bottom edge of the curved beam is the edge that has no bracing. This
side can experience compression stresses due to internal forces, making it
vulnerable to out-of-plane buckling. As a result the system can become
unstable and has a possibility to buckle.
To increase the out-of-plane load carrying capacity an additional bracing
system needs to be introduced (Crocetti, et al., 2011, p. 6.9). Normally, these
elements also serve to increase the buckling strength of the arches.
Design of straight timber members have been thoroughly investigated both
theoretically and experimentally, when it comes to their buckling behaviour
and in specific their buckling lengths. However design of curve timber
members such as arches are not will defined theoretically regarding
calculation of buckling length needed for calculation of instability factors.
In most cases, when making a practical design, the effective buckling length
in the arches plane is assumed to be as follows, where s is half the length of
the arch in [m].
𝑙 = 1.25∙𝑠 ( 1 )
𝑒
The effective buckling length can be estimated according to the previously
introduced method as long as the cross-section is constant and the load is
uniformly distributed (Blass, et al., 1995, p. B7/5). Additionally, in reality,
there are many other factors that should also be included while computing
the effective buckling length such as stiffness, curvature, boundary condition
etc.
Arches can be categorised based on depth as deep or shallow arches. The
depth is indicating how far a shape of the arch is changed from the flat
beam. The behaviour of a shallow arch is quite similar to flat beam whereas
the behaviour of deep arch is very different. In the shallow arch the coupling
between the membrane and bending behaviour is week while in the deep
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Rafiullah Sherzad and Awrangzib Imamzada