Verification of assumptions in dynamics of lattice structures B.Błachowski and W.Gutkowski Warsaw, Poland 37th SOLD MECHANCS CONFERENCE, Warsaw, Poland September 6 1, 21
Outline of presentation 1. Motivation 2. Examples of lattice structures 3. Joints in lattice structures 4. Pin joint assumption 5. Real structural behavior in dynamics 6. Examples 7. Conclusions
1. Motivation Methods in Structural Health Monitoring 1.1. Global methods Vibration-based methods (Operational Modal Analysis) Transformation to state space Measured quantities 1.2. Local methods Mq D q Kq w y C q C q C q x y q Ax v q x q w Cx Dw a v Ultrasonic waves propagation, Vibrothermography dentified mass, damping and stiffness matrices
2. Examples of lattice structures A large number of structures are build of system of rods (beams) commonly called trusses, frames and lattice structures. Truss bridges Double layer grids Transmission towers
3. Joints in lattice structures n most cases in such structures elements are rigidly connected. FF-System Free form Novum System Products Classical welded joint KK-System Kugel knoten
4. Pin joint assumption The origin of the pin joint assumption t t t t
Equilibrium equations - for pin joint - for rigid connection 2 1 2 2 1 1 P t t P t t C C C C 2 1 2 2 1 1 P t t P t t K K C t - K C - K t C t - K C - K t 2 2 1 1 4 2 2 1 i l l E EA K C l EA C 3 l E K
Where 1 and are of the same order then the product can be neglected comparing with t K t K t 1 t 1 C 1 Ct1 t 1 Neglecting bending term is equivalent to the assumption that two elements are pin joint.
For structures when displacements are caused by bending only, pin joint assumption can not be applied.
5. Real structural behavior in dynamics The static pin joint assumption has been incorporated in truss dynamics. Equivalent masses of beams are attached to pin joints. Problem is reduced to vibration of concentrated masses connected with massless springs.
n real, rigidly connected structure, beam mass is distributed along its length and transversal motion assumed.
(rad/s) First eigenfrequency for both cases 4 35 3 First eigenfrequency 25 2 15 1 5 R7x4 R82x4 R11x4 R133x4 R159x4 R177x5 R193x5 Cross section area Truss Frame
25 bar structure 1.9m 2.5m 1.9m Cross section areas: Circular hollow section Ø159x8 1.9m z 2.5m y 5.m x 5.m
First mode shape of 25 bar structure pin-joint model f=7.56 Hz rigid-joint model f=38.43 Hz
First mode shape animation pin-joint model f=7.56 Hz rigid-joint model f=38.43 Hz
Second mode shape of 25 bar structure pin-joint model f=73.63 Hz rigid-joint model f=39.9 Hz
Truss tower Cross section areas: CHS Ø159/8 Material: Steel Ex=25 1 6 kpa 1 m ρ = 785 kg/m 3 pin-joint model rigid-joint model
First mode shape of tower truss Pin-joint model 1st bending mode Rigid-joint model 1st bending mode
Third mode shape of truss tower Pin-joint model 1 st torsional mode Rigid-joint model 1 st torsional mode
7th mode shape of truss tower Pin-joint model 2 nd torsional mode Rigid-joint model Local bending mode of bracings
9th mode shape of truss tower Pin-joint model 2 nd bending mode Rigid-joint model 2 nd torsional mode
Pin-joint model Correlation of modes MAC ij T 2 φi φ j T φ φ T φ φ i i j j φ i mode shapes of pin-joint model φ i mode shapes of rigid-joint model Rigid-joint model
Damaged joint modeling Mq Kq Mr K ΔK r K λ M 2 φ i i K K 2 Δ Mψ l l i 1,2,, N m ΔK N k j1 k j B j l 1,2,, N Stiffness correction f
i q j q 1 1 1 1 B j n DOFs j i 2 1 n DOFs j i 2 1 Damaged joint modeling continued
-35W truss bridge Floor truss according to construction plans *) Astaneh-Asl, A., Progressive Collapse of Steel Truss Bridges, the Case of -35W Collapse, 7th nternational Conference on Steel Bridges, Guimarăes, Portugal, 4-6 June, 28.
Collapse On August 1, 27, the 4 years old - 35W steel deck truss bridge in Minneapolis, suddenly and without almost any noticeable warning collapsed. Joint detail
Computational model Elevation of truss Section near pier 7 3D FEM model A 1 =.1258 m 2 A 2 =.17632 m 2 A 3 =.4211 m 2
1 = 13.2 rad/s, f 1 = 2.1 Hz 1st mode out-of-plane bending 1 = 13.5 rad/s, f 1 = 2.15 Hz Rigid-joint model f 1 =2.15 Hz Pin-joint model f 1 =2.1 Hz A 1 =. A 2 =. A 3 =. A 1 =.1258 m 2 A 2 =.17632 m 2 A 3 =.4211 m 2
6th mode shape 6 = 27.9 rad/s, f 6 = 4.44 Hz Rigid-joint model f 6 =4.34 Hz 6 = 27.3 rad/s, f 6 = 4.34 Hz n-plane bending Pin-joint model f 6 =4.44 Hz Out-of-plane bending A 1 =.1258 m 2 A 2 =.17632 m 2 A 3 =.4211 m 2
A 1 =. A 2 =. A 3 =. 7th mode shape 7 = 28.2 rad/s, f 7 = 4.49 Hz 7 = 28.1 rad/s, f 7 = 4.48 Hz Rigid-joint model f 7 =4.48 Hz Out-of-plane bending Pin-joint model f 7 =4.49 Hz n-plane bending A 1 =.1258 m 2 A 2 =.17632 m 2 A 3 =.4211 m 2
7. Conclusions Theoretical background, together with presented examples show, that the pin-joint assumption in truss dynamics can lead to considerable errors in finding eigenfrequencies and eigenmodes. The pin-joint assumption doesn t allow to monitor, real structural joints, which are more often subjected to damages than prismatic rolled structural elements.
Acknowledgments The authors would like to thank: The Polish Ministry of Scientific Research and nformation Technology for the financial support of this work by the grant 126/B/T2/29/36
Thank you for your attention.