Seria: BUDO W NICTW O z. 109 N r kol. 1735
Gabriel KUBIENIEC
Cracow University o f T echnology
INVESTIGATION AND FEM SIMULATION OF ADHESIVE REINFORCED KNEE JOINTS
Summary. This study includes full and detailed analysis o f possibility o f using glues in reinforcement o f steel structures. The num erical calculations w ere done on the basis o f the experimental investigations perform ed at B TU C ottbus w ith use o f num erical program m e Abaqus based on the F inite E lem ent M ethod. The num erical results w ere com pared w ith the experimental ones.
BADANIA I ANALIZA MES NAROŻA RAMY WZMOCNIONEGO PRZY UŻYCIU KLEJÓW
Streszczenie. A rtykuł ten zaw iera p e łn ą i szczegółow ą analizę m ożliw ości zastosow ania nowoczesnych klejów do w zm acniania konstrukcji stalow ych. A naliza num eryczna przeprowadzona przy w ykorzystaniu program u A baąus została w ykonana na podstaw ie wyników badań eksperym entalnych przeprow adzonych na B TU C ottbus. W yniki num eryczne zostały następnie porów nane z w ynikam i badań eksperym entalnych.
1. Analysed structure
The analysed knee jo in t w as m ade from double - tee w elded section w ith very slender web. A dditional transverse and diagonal stiffeners w ere applied. The w eb was reinforced by glued additional plate or plates. The geom etry o f analysed knee jo in t is presented in Fig. la . The layouts o f reinforcem ent in every experim ent called V w ere presented in Fig. 2.
* Prof. Dr.-Ing. habil. H artm ut Pasternak, Fakultät für Architektur, Bauingenieurw esen und Stadtplanung, Brandenburgische Technische U niversität Cottbus, Cottbus, Germany
172 G. Kubieniec
Fig. 1. The geometries o f analysed models and experimental scheme Rys. 1. Geometrie analizowanych naroży oraz schem at eksperymentalny
V- 02 V - 04 V - 05 V - 03 V - 06 V- 07 V - 08
--- Plate ..— Glue
1 x 1.0 mm 2 x 0.5 mm 1 x 1.0 mm 2 x 0.5 mm 1 x 2.0 mm 2 x 1.0 mm
Polyurethane Glue Epoxy Glue
Fig. 2. Layouts o f reinforcem ent Rys. 2. Układy w zmocnienia
The m ain structure and additional plates w ere m ade from steel S235. The material properties o f steel such as Y oung’s m odulus and yield stress w ere determ ined on the basis of the tension tests perform ed for steel plates that the m ain structures and additional plates were made. The m aterial properties o f m ain structures and additional plates w ere set up in Tab. 1.
T he glue properties w ere determ ined on the basis o f experim ental investigations perform ed at B TU Cottbus. Two types o f glue w ere tested. The first o f them w as epoxy glue the second one w as polyurethane glue. The stress - strain diagram s for glues w ere presented in Fig. 3.
T able 1 M aterial properties o f knee join ts
Name of experiment
Properties of web
Properties of flanges and
stiffeners
Properties of additional plate{s) Properties of glue
E [GPa] [MPa]
E [GPa] fy
[MPa]
Number of plates
Thickness of plate
[mm]
E [GPa]
f*
[MPa]
Type of glue
Number of layers of glue
Thickness of layer of glue
[mm]
V-02 226.0 335.0 20S.0 301.0 0 - - - - - -
V-03 226.0 335.0 20S.0 301.0 1 1.0 208.0 189.0 Epoxy 1 0.47
i s 226.0 335.0 20S.0 301.0 1 1.0 208.0 1S9.0 Pohur ethane 1 0.66
V - 05 226.0 335.0 208.0 301.0 2 0.5 234.0 219.0 Polyurethane 2 0.S5
V-06 226.0 335.0 208.0 301.0 2 0.5 234.0 219.0 Epoxy 2 0.58
V- 07 226.0 335.0 208.0 301.0 1 2.0 210.0 161.0 Epoxy 1 0.49
V - OS 226.0 335.0 208.0 301.0 2 1.0 208.0 189.0 Epoxy 2 0.52
7 1 - 1 Fig. 3. Stress - strain diagrams for glues
Rys. 3. Zależność naprężenie - odkształcenie dla klejów
2. Experimental investigations
At B TU C ottbus several experim ents w ith knee jo in ts reinforced by additional plates glued to the w eb plate w ere conducted [2]. Every experim ent w as called V. Seven experiments w ere carried out. The first experim ent concerned not reinforced knee jo in t w hile the rest o f experim ents reinforced knee joints. Every experim ent distinguished from each other w ith a num ber and thickness o f the additional plate o r plates and the type o f glue and the thickness o f the layer o f the glue. In experim ental investigations these com ers w ere under bending due to concentrated force w hich w as applied in the right w ing (A rm 1) o f knee joint.
The geom etries o f analysed m odels and experim ental schem e w ere presented in Fig. lb and the layouts o f reinforcem ent in Fig. 2. M aterial properties o f every experim ent w ere set up in Tab. 1.
In every experim ent the m easured quantities w ere applied force and displacem ent betw een external points o f knee jo in t called II and III (cf. Fig. lc ). N ext w ith use o f specific equations
174 G. Kubieniec
the bending m om ent and the rotation o f internal point o f knee jo in t called I w ere calculated (cf. Fig. lc ). These equations are:
<p = arccos
'W + w 22 -(w 3 - S f '
2 ■ wt ■ w2 - arccos
f 2 2 2 \
W l + W 2 ~ W 3
2 - wr w2 J
h = . w,2 w3 —S
M = Ph
(1)
(2)
(3)
w here: S = displacem ent betw een points II and III, wi, W3 = distances betw een points I, II, III (cf. Fig. lc ), (p = rotation o f point I, h = arm o f force, M = bending m om ent.
O n the basis o f m easured in experim ents values o f force and displacem ent and equations (1) to (3) the bending m om ent - rotation curves w ere determ ined for every experim ent.
The bending m om ent - rotation curves for not reinforced structures and for reinforced ones w ere presented in Fig. 4a, b, c respectively. The results for all experim ents are set up in Tab. 2.
a) b) c)
Fig. 4. Bending m om ent - rotation curves Rys. 4. Zależność m om ent zginający - kąt obrotu
T able 2 T he results for all experim ents
Name Experim ental resnlts of experiment M ,
IkNm]
Ma fkNm]
V - 0 2 - 30.56
V - 03 28.30 29.76 I
V - 04 31.16 32.97
V - 05 31.16 34.38
V - 06 34.54 34.86
V - 0 ? 37.86 35-34
V - 08 41.25 39.82
3. Numerical calculations
The num erical calculation w ere carried out w ith use o f program m e A B A Q U S [1], The calibration o f the m odels w as done on th e basis o f the results o f the experim ental investigations perform ed for knee jo in ts at B TU C ottbus in 2005. The m ain structures w as modelled w ith use o f 4 node shell finite elem ents called S4R. The layer o f glue w as m odelled with use o f 8 node linear brick, hybrid, linear pressure, incom patible m odes elem ents called C3D8IH. The m odel w as analysed as a sim ple supported one. A dditional boundary conditions were applied to prevent the m odel from global buckling in out o f plane o f w eb direction. The layer o f glue w as connected w ith th e m ain structures by option *TIE based on the theory o f slave and m aster surfaces. A tie constraint ties tw o separate surfaces together so that there is no relative m otion betw een them . T his type o f constraint allow s to use together tw o regions even though the m eshes created on the surfaces o f the regions m ay be dissim ilar. It is possible to define a tie constraint betw een edges o f a w ire or betw een faces o f a solid o r shell. Two surfaces can be tied together. Each node on the first surface (the slave surface) w ill have the same m otion as the point on the second surface (the m aster surface) to w hich it is closest [1], An appropriate introduction o f geom etrical im perfections into the num erical m odel should have been carefully analysed. The first step o f introduction o f im perfections w as based on the solution o f the linear buckling problem . A s a result several local buckling m odes w ere obtained (cf. Fig. 6c). In the second step a linear com bination o f chosen buckling (local) modes m ultiplied by adequate factors w as created and in this w ay a new configuration o f geom etry o f analysed m odel w as achieved.
176 G. K ubieniec
T he geom etrical im perfection has th e follow ing form [1]:
n
(4)
1=1
w here <j>, - i h m ode shape, w, - associated scale factor.
In every experim ent the m easured quantities w ere applied force P and displacem ent U betw een external points o f knee jo in t called II and III (cf. Fig. lb ). N ext w ith use o f equations (1), (2) and (3) (cf. Paragraph 2) the bending m om ent and rotation o f internal point o f knee jo in t called I w ere calculated (cf. Fig. lb ). A ll m aterial and geom etrical properties used in num erical calculations w ere taken from Table 1. The FEM m odel and graphical representation o f slave and m aster surfaces theory w ere presented in the Fig. 6a and 6b respectively.
Experim ental investigations show ed that tw o m axim a appeared on the bending m om ent - rotation curve (cf. Paragraph 2). The first m axim um o f bending m om ent called M a is com bined w ith destruction o f bonds betw een glue and additional plate and the w eb plate. The second m axim um o f bending m om ent called M u is com bined w ith the C a rd iff M odel and describes ultim ate capacity o f knee joint.
It is very difficult in num erical calculations to describe behaviour o f glue in elastic - plastic range thus only elastic behaviour o f glue characterized by shear m odule G and P oisson’s ratio v (cf. Fig. 3) was taken into account and that this w ay in this analysis only first m axim um called M a as a sufficient estim ation o f carrying capacity w as calculated. The com parison o f statical paths o f equilibrium from experim ent and from num erical analysis for chosen cases w ere presented in Fig. 8. The com parison o f results for all cases are set up in Tab. 3.
a) ■>) c)
Fig. 5. Comparison o f results o f numerical and experimental analyses Rys. 5. Porównanie wyników analiz numerycznej i eksperymentalnej
Table 3 C om parison o f results
Name Experimental results Numerical results Me x p' Ma b a q u s
o f experiment Ma [kNm]
M„
[kNm]
Ma b a q u s
[kNm]
[%]
V - 0 2 - 30.56 28.10 8
V - 0 3 28.30 29.76 28.83 1
V —04 31.16 32.97 32.61 5
V - 0 5 31.16 34.38 33.77 8
V - 0 6 34.54 34.86 35.03 1
V - 07 37.86 35.34 39.25 4
V - 08 41.25 39.82 41.34 1
Fig. 6. FEM m odel o fk n e e joint Rys. 6. Model num eryczny naroża ramy
4. Conclusions
The experim ental results show ed that in case o f epoxy glue the b etter w ay o f reinforcem ent is to use tw o additional plates o f thickness 0.5 m m instead o f one plate o f thickness 1.0 m m (cf. Fig. 4b). Identical conclusion can be draw n in case o f experim ents V - 07 (one additional plate o f thickness 2.0 m m ) and V - 08 ( tw o plates o f thickness 1.0 m m ) (cf. Fig. 4c). A nother situation happens in case o f use polyurethane glue. H ere tw o curves from different experim ents superim posed thus the effect o f use one additional plate is identical w hat in case o f use tw o additional plates (cf. Fig. 4a).
The increase o f value o f bending m om ent M a o f reinforced knee jo in t in com parison w ith not reinforced one is significant. In case o f experim ents V - 03 to V - 06 the increase varies from 43% - 62% and for experim ents V - 07 and V - 08 from 151% to 355% (cf. Fig. 4).
Com parison o f bending m om ent - rotation curves in case o f value M a and corresponding rotation (p for all cases are presented in Fig. 4.
178 G. Kubieniec
T he num erical calculations show ed th at the analysed knee jo in t m ay be effectively m odelled w ith use o f shell finite elem ents called S4R in case the m esh in w ebs is dense enough, also proved that the layer o f glue can be m odelled w ith use o f solid finite elements called C3D 8IH w hat can be an alternative for another very expensive m ethod o f modelling glues com bined w ith creating finite spring elem ents. The difference betw een num erical and experim ental results is low er than 8% in every case (cf. Tab. 3).
REFER EN C ES
1. H ibbit, K arlsson & Sorensen, Inc: A baqus (standard) R eference M anuals ver. 6 ,2 0 0 1 . 2. P asternak H., M einz J.: V ersuche zu geklebten V erstärkungen im Stahlhochbau,
B auingenieur 8 1 ,2 0 0 6 , H eft 5.
Recenzent: D r hab. inż. M arian G wóźdź, prof. Politechniki Krakowskiej
