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The stiffness of laser stake welded T-joints in web-core sandwich structures
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Thin-Walled Structures 45 (2007) 453–462 www.elsevier.com/locate/tws
The stiffness of laser stake welded T-joints in web-core sandwich structures Jani Romanoffa,, Heikki Remesa, Grzegorz Sochab, Mikko Jutilaa, Petri Varstaa a
Ship Laboratory, Department of Mechanical Engineering, Helsinki University of Technology, P.O. Box 5300, 02015 TKK, Finland b Institute of Fundamental Technological Research PAS, Swietokrzyska 21, 00-049 Warsaw, Poland Received 20 July 2006; accepted 16 March 2007 Available online 23 May 2007
Abstract The purpose of this paper is to describe experiments carried out on laser stake welded T-joints of web-core steel sandwich structures. A special test setup was developed to measure the shear-induced rotation at the T-joint. The ratio of the shear force to rotation angle gave the joint stiffness. This stiffness was measured for specimens with two different face-plate thicknesses. The influence of weld thickness, root gap and occurrence of contact were further investigated with finite element simulations. Finally, the shear stiffness of the sandwich structure transverse to the web plate direction was determined using the experimentally obtained average joint stiffness value. The validation of the shear stiffness was carried out by considering a beam in four-point bending. The agreement between calculated deflection and stress and experimental results was found to be good. r 2007 Elsevier Ltd. All rights reserved. Keywords: T-joint stiffness; Laser stake weld; Web-core sandwich panel
1. Introduction The demand for lighter and safer structures has stimulated the need to study new materials and new structural configurations in ship structures. All-metal sandwich panels offer an option that can fulfill these requirements. These panels are composed of face plates separated by a core. The simplest core geometry of allmetal sandwich structures is built from flat web plates perpendicular to the face plates. In this web-core structure, the web plates run only in one direction and are usually spaced 10–100 times the thickness of the face plate. The core produces continuous support to the face plates in the direction of the web plate, together with discrete support in the transverse direction. Thus, the web-core panel is highly orthotropic. The structural analysis of these panels can be carried out using either the finite element method (FEM) or analytical Corresponding author. Tel.: +358 9 451 4171; fax: +358 9 451 4173.
E-mail address: jani.romanoff@tkk.fi (J. Romanoff). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.03.008
methods. The FEM is very flexible and well known, but it suffers from long analysis time and fails to offer deeper understanding of the problem. Therefore, analytical methods are more promising. In order to model the periodic core of the sandwich panel analytically, the stiffness properties of the panel are homogenized. In this homogenisation, the initial periodic structure is transformed into a homogenous continuum, where, for example, the beam- or plate-bending problem is solved, see Refs. [1,2]. The stiffness models for web-core sandwich structures have been developed mainly by assuming that the T-joint between the web and face plates remains undeformed when loaded. Thus, the joint stiffness is assumed to be infinite, see, as an example, Refs. [2,3]. Due to the laser stake welding, this assumption is far from realistic, and therefore the shear stiffness transverse to web plate direction is overestimated. This is due to the fact that the laser weld in the T-joint deforms, as presented in Ref. [4] by extensive 3D FE-models and experiments. Therefore, in Ref. [5], the analytical response model was presented for the laser stake welded web-core sandwich beams, where the influence of
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laser welding could be taken into account by including the T-joint’s rotation stiffness when the shear stiffness formulation was derived. The purpose of this paper is to determine experimentally both the rotation stiffness of the T-joint and the influence of its geometrical parameters on this stiffness. Thus, the aim of this study is to gain a quantitative understanding of the stiffness, as compared to the qualitative understanding gained through the experiments presented in Ref. [6]. The physical phenomena associated with this stiffness are further identified with FEA. Finally, the measured joint stiffness values are applied to a case study, where the response of a web-core sandwich beam in four-point bending is calculated. These results are validated by the experimental results given in Ref. [4].
condition is assumed and taken into account by considering the effective Young’s modulus in the calculations as E ¼ E=ð1 n2 Þ. 3. Theoretical background of the stiffness tests 3.1. Shear behaviour of web-core sandwich structures The theory of the bending response of web-core sandwich beams with thick face plates has been presented in Ref. [5], where the shear stiffness transverse to web plate direction (see Fig. 2), is defined as DQy ¼
12Dw , s2 kQ ðDw =Db Þ þ 6ðd=sÞ þ 12ðDw =kby sÞ 2ðd=sÞ (1)
2. Definitions
where the bending stiffness of the face and web plates is
The web-core sandwich plate consists of web and face plates, which are connected by the laser welds. The web plates are parallel to the xz-plane and have a thickness tw and a height hc; see Fig. 1. The web plate spacing is s. The face plates are on the xy-plane and have a thickness t. Subscripts t, b and w are used for the top face, the bottom face and the web plates, respectively. The plate has length L, breadth B, total height h ¼ hc þ tt þ tb and the neutral axes of the face plates have distance d ¼ hc þ ðtt þ tb Þ=2. The T-joint consists of the laser-weld and the face and web plates connected to it. The laser-weld position, thickness, root gap and the radius of the web plate are denoted by e1, tweld, hrg and R, respectively. When the weld is not in the middle of the web plate, the additional dimension e2 ¼ tw tweld e1 is used. A unit cell is a representative cut of the sandwich plate in the yz-plane, whose borders are neighbouring web plates and the parts of the face plates between these; length is infinitesimal dx. The plate has two coordinate systems, namely the global xyz and the local xylzl; see Fig. 1. The origin of the global coordinate system xyz is located at the neutral axis of the beam or at the geometrical mid-plane of the plate. The origin of the local coordinate xylzl is located on the neutral axis of the face or the web plate under consideration. Notations q, F and M are used for the external distributed loads, point forces and point moments per unit breadth, respectively. The Young’s modulus and Poisson’s ratio are denoted with E and n, respectively. A plane-strain
E i t3i ; i ¼ t; b; w, (2) 12 and the stiffness parameter describing the ratio of shear force carried by the top and bottom plates is 1 þ 12ðDt =sÞ ð1=kty Þ ð1=kby Þ þ 6ðDt =Dw Þðd=sÞ kQ ¼ . (3) 1 þ 12ðDt =Dw Þðd=sÞ þ ðDt =Db Þ
Fig. 1. (A) Laser-welded all-metal sandwich panel and (B) laser-welds with their dimensions.
Di ¼
The T-joint rotation stiffness kiy in Eqs. (1) and (3) is defined as the ratio of the moment (M in ¼ QsQ ; see Ref. [5]) to rotation angle yic at the T-joint, and is written as kiy ¼
QQ yic
s;
i ¼ t; b,
(4)
where QQ is the shear force carried by the shear deformation of thin-faced sandwich structures, which considers equal slopes of the face plates at the ends of the unit cell. yic is the rotation induced by local deformation in the laser-weld in face plate i and in the web plate connection; see Fig. 2B. 3.2. Determination of the joint stiffness A web-core sandwich structure under shear force QQ is presented in Fig. 2. Assuming constant shear force and
Fig. 2. (A) The internal forces in a unit cell and (B) the slopes in the T-joint.
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symmetrical cross-section gives equal shear force in both faces, i.e., QtQ ¼ QbQ ¼ QQ =2 and dQtQ ¼ dQbQ ¼ 0. Further, the rotation in the laser-weld was assumed to be equal at both ends of the web plate, i.e., ytc ¼ ybc ¼ yc . Thus, the web-induced horizontal H in , vertical V in and moment M in reactions in face plate i can be written as
455
dN i ¼ H in ¼ H n ,
(5a)
Fig. 3. The deformations in the T-joint area were not restrained in order to prevent additional stresses there from supporting the specimen. Considering the ratio of the displacement v to the distance where it is measured dv gives the rotation angle as v yc ¼ . (11) dv
V in ¼ 0,
(5b)
Substitution of Eqs. (10) and (11) into Eq. (4) gives the rotation stiffness as
QQ s, (5c) 2 where dNi is the increase in the normal force Ni. From the equilibrium of the web plate one obtains
M in ¼
M tn þ M bn Mn ¼2 . d d So by substitution of Eq. (5c) to Eq. (6) one obtains s H n ¼ QQ . d Thus, the moment in the web plate can be written as zw H n d zw Mw 12 ¼ , n ðzw Þ ¼ M n 1 2 2 d d and the shear force as Hn ¼
(6)
(7)
(8)
M tn ¼ H n . (9) d This means that the shear force is constant and the bending moment is linearly distributed inside the web plate. Further, the inflection point M w n ¼ 0 of the bending moment is at zw ¼ d=2 and the maximum value is Mw n ðzw Þ ¼ H n d=2. This property can be used to design the test setup for the rotation stiffness of the T-joint, since the cantilever beam between 0pzw pd=2 has the same bending moment and shear force distribution along this span. In the test, the horizontal force Hn was applied on the specimen; see Fig. 3. The corresponding shear force was obtained from Eq. (7) by considering the force Hn to be applied at distance d M ¼ d=2 on the specimen. So Eq. (7) is then rewritten as s H n ¼ QQ . (10) 2d M Qw n ðzw Þ ¼ 2
The rotation at the T-joint was obtained by restraining bending deformations of the face and web plate with support plates, except right next to the laser-weld area; see
ky ¼ kty ¼ kby ¼ 2d M d v
Hn . v
(12)
4. Testing 4.1. Test specimens Two types of test specimens were considered, i.e., specimens with 2.5 and 3.0 mm face-plate thicknesses. The length of the face sheet was 120 mm and all the specimens had a breadth of 50 mm. The web plates had a thickness and height of 4 and 20 mm, respectively. The measured material properties for the test specimens are given in Table 1. In order to identify the geometrical properties of the welds, small pieces were cut from both ends of each test specimen; see Figs. 4 and 5. The results of the weld thickness, root gap and weld centricity are given in Figs. 6B, C, and D, respectively. Fig. 5A presents an ideal case where the root gap is zero and the weld is positioned in the middle of the web plate, while in Fig. 5B the weld is positioned at the other edge of the web plate. In Fig. 5C, the web plate is rounded and the root gap is therefore increasing when approaching the edge of the web plate from the weld. In Fig. 5D, the root gap is very large. Altogether, 80 specimens were analysed. In addition, 234 fracture surface measurements per plate thickness were carried out to obtain the weld thicknesses. As can be seen from Fig. 6B, the weld thicknesses are fairly concentrated and have average values of 1.36 and 1.41 mm for 2.5 and 3.0 mm specimens, respectively. The corresponding standard deviations are 0.17 and 0.23 mm. Fig. 6C indicates that the root gap is mainly between 0 and 10 mm. Also, the weld centricity varies quite a lot, as presented in Fig. 6D. It should be noted that the variation of the weld thickness, root gap and weld centricity can be significant inside one specimen. Table 1 The measured values for the materials used
Fig. 3. Schematic view of the test setup.
Face-sheet thickness (mm)
Young’s modulus Yield strength (GPa) (MPa)
Tensile strength (MPa)
2.5 3.0 4.0
221 212 200
470–476 398–415 398–551
360–368 302–322 360–549
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4.2. Test setup and procedure The test system was built from INSTRON 1343 axial–torsional material testing equipment by adding a special gripping system to hold the specimen, which was fixed into INSTRON 1343 through a beam constructed from C200-profile and having length 1050 mm. A more detailed description of the test system is given in laboratory report [7]. The beam was checked to be stiff enough not to deform significantly under a maximum load of 5 kN. The
Fig. 4. Scheme of how weld dimensions are taken. Left: the specimens used to determine the root gap and weld centricity; right: the fracture surface measurement.
specimen was fixed to the beam through an aluminium block, support plates and screws. The test setup is presented in Fig. 7. An MTS TestStar II digital control unit controlled the operation of testing equipment and the tests were carried out as displacement controlled. The displacement of the specimen was measured with extensometer MTS632.27F-20 at distance dv ¼ 24.2 mm from the interface of the web and face plate. The force was measured with a Lebow 3173-104 load cell with maximum load capacity of 5 kN at distance dM ¼ 22.6 and 22.3 mm for specimens with 2.5 and 3.0 mm face-plate thicknesses, respectively. The specimens were loaded 3–4 times in one direction to a maximum force of 600 N (equal to 12 kN/m) in order to remove any material non-linear behaviour. Then the specimens were turned and the procedure was repeated. As a result, force–displacement curves were obtained from which the tangent stiffness could be determined. The maximum error, defined with total difference, in the tests was approximately 5%, mainly due to the inaccuracy in specimen dimensions. The tests were found to be repeatable.
Fig. 5. Different types of laser stake welds: (A) Zero root gap and weld close to middle of the web plate, (B) zero root gap and misaligned weld, (C) varying root gap, and (D) maximum root gap.
Fig. 6. Statistical properties of the welds: (A) Measured dimensions and distributions, (B) weld thickness, (C) root gap, and (D) weld centricity.
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Fig. 7. The test setup used in the rotation stiffness testing: (A) load frame, and (B) side view of the test setup.
4.3. Results Altogether 13 and 11 specimens were tested for 2.5 and 3.0 mm plate thicknesses, respectively. An example of a force–displacement curve is presented in Fig. 8A for four load cycles, where the microscopic pictures of the joint ends are also shown. The difference between the first and the rest of the closed loops during load cycles indicates that the plasticity disappears after the first cycle. This means that the force–displacement behaviour stabilises once the hardening of the material has occurred. In addition, the contact between the face and web plates at approximately 50 N can be seen as the slope increases in the force–displacement curve. Fig. 8B shows the stabilised force–displacement curves for 3-mm specimens. These stabilised curves were obtained after a small number of load cycles. Generally, the shape of the curve varies and three different types of curves exist: linear, bilinear and parabolic. However, when the average of all force–displacement curves is determined (see Fig. 8B), it can be seen that the force–displacement curve is quite linear up to the 600-N force level. In order to derive the rotation stiffness of the T-joint, the measured force was divided with specimen breadth. Fig. 8C shows how a tangent line was fitted for each test result between force values 0 N/m and 2 kN/m, which correspond to 0 and 100 N in the test, respectively. In this load range, the force–displacement curves were linear in almost all tests; see Table 2. The measured rotation stiffness values in two directions are given in Table 2, together with the resulting shear stiffness value calculated with Eq. (1). The corresponding statistical distributions are presented in Fig. 9. Table 2 shows that the rotation stiffness can be different in the same specimen in two directions. The rotation stiffness distributions for 2.5- and 3.0-mm specimens are presented in Fig. 9A. The average of the rotation stiffness is 113 and 107 kN for 2.5- and 3.0-mm specimens, respectively. The standard deviation for both types of specimens is 21 kN. The corresponding shear stiffness
distributions are presented in Fig. 9B. The corresponding average shear stiffness values are 299 and 376 kN/m with standard deviations of 19 and 34 kN/m, respectively. Comparison of the rotation stiffness in Fig. 9A with shear stiffness in Fig. 9B and their standard deviations reveals that the higher face plate thickness makes the shear stiffness more sensitive to the rotation stiffness of the T-joint. 5. Phenomena affecting stiffness 5.1. The effect of weld thickness and root gap In Ref. [4], an FE study into the influence of weld thickness and root gap on the shear stiffness when the contact effects are not considered is presented. The study is based on linear-elastic FE-analyses for the web-core sandwich beam in three-point bending. The study shows that shear stiffness decreases fast when the weld thickness decreases in the case of the sandwich beam thick face plates, while the decrease becomes smaller when the faceplate thickness is decreased. Regardless of the face-plate thickness, the influence of root gap on the shear stiffness is very small. The difference in the shear stiffness could vary easily over 200% in 3-mm specimens if a variation of between 0.5 and 2 mm in the measured weld thicknesses is assumed. 5.2. Contact effects In order to study the influence of contact on the rotation stiffness, detailed geometrically non-linear FE models were constructed, where the geometry of the weld in the T-joint was varied. Fig. 10 presents an example of FE-mesh, where eight-node plane strain elements (S8R in ABAQUS 6.5.1) were used. The FE models were restrained by fixing displacements of the nodes of the bottom surface of the face plate; see Fig. 10. The load was applied in the middle of the web plate at equal distance from the joint as in the tests. The material properties used are given in Table 1.
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Fig. 8. Force–displacement curve from test: (A) first four cycles for specimen S31t, where the ends of the joint are also shown, (B) test results for all 3-mm specimens, and (C) determination of the laser-weld rotation stiffness based on force values of 0 and 2 kN/m (equal to 100 N) for specimen A3t.
Table 2 The measured rotation stiffness values and resulting shear stiffness Specimen Face plate, Normal direction t (mm) Curve Test, ktest (kN/mm) shape b1b b1t b2b b2t b3b b3t S11 S11top S12 S12top S14 S14top S15 A1t A2b A2t A3b A3t S14b S14t S31b S31t S91b S91t
2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0
4.45 6.51 3.38 6.26 4.11 6.03 5.93 4.00 5.04 4.60 6.11 4.73 6.13 4.29 5.35 5.17 3.62 6.48 6.74 5.86 5.78 4.45 4.71 2.92
Linear Linear Parabolic Linear Linear Parabolic Linear Parabolic Parabolic Parabolic Linear Linear Linear Linear Bilinear Bilinear Parabolic Linear Bilinear Bilinear Linear Parabolic Bilinear Parabolic
Reverse direction Linear Joint stiffness Shear stiffness, range (N) Ky (kN) DQy (kN/m)
100
300 400 500 400
200 350 200 300 200 200 350 50
97 143 74 137 90 132 130 88 110 101 134 104 134 93 115 112 78 140 146 126 125 96 102 63
286 322 258 319 278 316 314 276 299 290 317 293 317 356 392 387 328 422 428 406 404 362 372 293
Test, ktest (kN/mm)
Curve shape
Linear Joint stiffness Shear stiffness, range (N) ky (kN) DQy (kN/m)
4.64 6.18 3.51 7.04 4.28 5.20 5.42 4.72 5.32 4.87 4.54 4.77 6.08 5.29 4.99 5.25 3.64 5.74 5.36 3.70 5.75 4.61 4.88 4.19
Parabolic Linear Parabolic Linear Bilinear Linear Parabolic Parabolic Linear Parabolic Bilinear Bilinear Bilinear Linear Bilinear Bilinear Parabolic Bilinear Parabolic Parabolic Linear Bilinear Parabolic Parabolic
180 400 200 200 400 400 100 200 500 200 50 200 200 500 200 150 200 300
102 135 77 154 94 114 119 103 117 107 100 105 133 114 108 113 79 124 116 80 124 100 105 91
291 318 262 329 282 302 306 292 304 295 289 293 316 390 381 389 329 403 392 332 404 368 377 352
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Fig. 9. (A) The rotation stiffness distributions of the test specimens. (B) The resulting shear stiffness for specimens with s ¼ 120 mm, tw ¼ 4 mm, hc ¼ 40 mm. Materials are given in Table 1.
Fig. 10. Example of the FE model used to simulate the rotation test.
Fig. 11. The influence of contact on the force–displacement curve. Root gap (A) is constant at 11 mm, (B) varies and has minimum value of 1 mm at web plate edge, and (C) is zero at weld and 5 mm at the edge of web plate. Other dimensions are tt ¼ tb ¼ 3 mm, tw ¼ 4 mm, s ¼ 120 mm, h ¼ 46 mm, tweld ¼ 1.746 mm, e1 ¼ 0.89 mm and R ¼ 0.24 mm.
the contact area increases (Case C), giving a force–displacement curve similar to the parabolic curve from the tests. Case A occurs when the root gap is large; see Figs. 5D and 11A. Then the normal stress component sz exists only in the weld. Case B occurs when the root gap is small and is constant or has some surface roughness that can cause sudden contact; see Figs. 5A and B and 11B. The normal stress sz is distributed in two different locations. The first location is the weld, although there are normal stresses in the contact area also. The larger the distance of the contact from the weld, the higher the increase of slope in the force–displacement curve. If the contact exists already in the unloaded specimen, the slope of the force–displacement curve will be different in two directions for the whole load cycle. This can happen, for example, when there are some impurities in the root gap. Case C occurs when the root gap decreases towards the weld; see Figs. 5B and C and 11C. The extent of the contact area, represented by the normal stress distribution sz, increases during the whole load sequence. Comparison of the FE-analysis and the test results shows that the behaviour of the force–displacement curve is very sensitive to the contact and differences of up to 100% can occur whether or not the contact occurs. 6. Influence on sandwich beam 6.1. Analytical model for beam response
The influence of contact on the shape of the force–displacement curve is presented in Fig. 11. Fig. 11 clearly shows that the range of linear relationship between the force and displacement varies quite a lot. The FE analysis shows that the variation in shape of the force–displacement curve results from the contact. If there is no contact, the slope remains constant (Case A), which corresponds to the linear force–displacement curve obtained in the tests. If the contact occurs suddenly and very locally, there is a sudden increase in the slope (Case B), i.e., there is behaviour similar to that shown by the bilinear force–displacement curves from the tests. From when the contact starts right at the beginning of the loading, the slope increases during the whole load sequence, since
The beam in bending was solved with the theory of homogenous sandwich beams with thick face plates giving the deflection, bending moments and shear forces. The derivation of the formulations for stress is presented in Ref. [5] and the result is given in Appendix A. The general solution by Allen for homogenous sandwich beams with thick face plates in four-point bending is given in Ref. [8]. 6.2. Case study In order to show the influence of the T-joint rotation stiffness on the response of the beam, a case study presented in Ref. [4] was considered. A web-core sandwich
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Table 3 Stiffness values for the beam, together with the maximum deflection and the difference between the experiments and analytical prediction Rotation stiffness (kN)
Shear Stiffness (kN/m)
Maximum deflection, analytical (mm)
Maximum deflection, test (mm)
Difference (%)
Low 65 Avg. 107 High 148 N
283 355 399 590
2.453 2.024 1.835 1.352
2.026
+21.1 0.1 9.4 33.3
Fig. 12. Comparison of the deflection obtained from the test and with analytical methods. Load is 1 kN/m. The test results are marked with square dots.
beam in four-point bending was studied. The beam had length B ¼ 1.8 m, web plate spacing s ¼ 120 mm and face plate thickness tt ¼ tb ¼ 2.86 mm. The web plate thickness and height were tw ¼ 3.97 mm and hc ¼ 40 mm, respectively. The forces were located at y ¼ 600 and 1200 mm and the displacement was measured at y ¼ 200, 400, 600, 900 and 1200 mm. Four strain gauges were located on the top face plate at y ¼ 242.5, 357.5, 722.5 and 837.5 mm, respectively. The average joint stiffness of 107 kN/m from the tests with 3-mm specimens was used. To study the statistical influence of joint stiffness, the average plus or minus two standard deviations were also considered. The joint stiffness values used are presented in Table 3, together with resulting shear stiffness and maximum deflection values. The comparison of the calculated and measured deflection distributions is presented in Fig. 12. The corresponding top surface stress of the top face plate is presented in Fig. 13. As shown in Table 3 and Fig. 12, a serious error of 33% occurs in the maximum deflection if the rotation flexibility is not included in the analysis. However, if the average rotation stiffness is included, the accuracy of the analysis is very good. When the lower and higher rotation stiffness values are used, the deflection is overestimated by 21% and underestimated by 9%, respectively. Fig. 13 shows that normal stress in the top face plate is estimated with good accuracy; the result indicates that the
Fig. 13. Comparison of the surface stress with different rotation stiffness values. The test results are marked with square dots.
stress is not as sensitive to the joint stiffness as the deflection is. A minor difference in predicted stress is found only in the neighbourhood of the load application, where the use of infinite joint stiffness underestimates the stresses. Fig. 13 shows also that, when the joint stiffness decreases, the maximum stress increases and the amplitude of the zigzag pattern decreases. The rate of change is, however, very slow in the region of the stiffness values found in this paper. 7. Conclusions The purpose of this paper was to present the experimental investigation into T-joint rotation stiffness in laser stake welded steel sandwich panels. Test specimens with 2.5 and 3.0 mm face plate thicknesses were considered. First, microscopic analyses were carried out for altogether 80 specimens in order to identify the weld properties of the specimens. The weld thickness, root gap and weld centricity were found to vary quite a lot. It was also discovered that these properties vary inside the stiffness test specimens. In order to get a better insight into the properties of the welds inside test specimens, more sophisticated measurement techniques are required. In the T-joint stiffness testing, tangent stiffness of each specimen was determined in two directions. Altogether 13 and 11 specimens with face-sheet thicknesses of 2.5
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and 3.0 mm, respectively, were measured. It was found that the stiffness test results vary and that the slope is not necessarily constant for the whole applied load range. However, if the average of all tests was taken, the slope was very close to linear. The experimental results were explained using FE-analyses. The influence of weld thickness on the T-joint rotation stiffness is significant, especially when the plate thickness increases. The influence of contact was also identified through the FE-analyses. Depending on the contact mechanism, the slope of the force–displacement curve may change drastically. So the contact is significant on the joint stiffness. The influence of rotation stiffness on the response of a web-core sandwich beam was studied. A beam in fourpoint bending was considered as a case study. The use of the average T-joint stiffness gives exactly the same maximum deflection as the test; when this rotation stiffness is omitted, the modelling seriously underestimates the maximum deflection by 33%. To study the statistical influence of joint stiffness, the average plus or minus two standard deviations were also considered. Then the lower value of the joint stiffness results in a 21% higher maximum deflection than the measured value. Similarly, the use of a higher value results in a maximum deflection 9% smaller than the measured value. Comparison of stresses showed that the maximum stress is not as sensitive to T-joint stiffness as the maximum deflection is in the obtained rotation stiffness region. However, as the rotation stiffness was decreased outside this region, the maximum stress started to increase in the face plates. At the same time, the amplitude of the zigzag-pattern in the surface normal stress decreased.
QQ;avg . The total shear force Qg;avg is then
Acknowledgements
where
This work was initiated in the EU-funded research project Advanced Composite Sandwich Steel Structures SANDWICH in the 5th Framework Programme Growth KA III. Since then, the work has been ongoing in the Finnish Ministry of Education funded Graduate School of Engineering Mechanics, EU-funded Centre of Excellence for Laser Processing and Material Advanced Testing LAPROMAT and with grants given by Merenkulunsa¨a¨tio¨ in Finland. This financial support is gratefully acknowledged. The Meyer Werft shipyard in Germany and the personnel of the Ship Laboratory of Helsinki University of Technology are thanked for producing the test specimens. The personnel of the Institute of Fundamental Technological Research at the Polish Academy of Science are thanked for the preparation of the test setup. Appendix A. Analytical model for beam response The total averaged shear force is divided into two parts: averaged thick face plates Qtfg;avg and shear deformation
Qg;avg ¼ Qtfg;avg þ QQ;avg .
461
(13)
The total shear force Qg;avg is obtained as a result of equilibrium considerations of the beam. The shear force carried by shear deformation QQ;avg is then obtained from d2 Qg;avg k2 Qg;avg ¼ k2 Qtot;avg , dy2 where stiffness factor k is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DQ . k¼ Df ðD0 =Dg Þ
(14)
(15)
Bending stiffness of the face plates Df is Df ¼ Dt þ Db ,
(16)
and the global bending stiffness consists of the elongation D0 and face plate bending stiffness Dg ¼ D0 þ Dt þ Db ; ¼
E t tE E b tb d 2 E t t3t E b t3b þ . þ E t tt þ E b tb 12 12
(17)
Once Eq. (14) is solved for the shear force QQ;avg carried by the shear deformation, the thick face plate’s part of the shear force can be solved from Eq. (13). Further, the global bending and shear deflection components can be solved as shown in Ref. [8], together with shear forces and bending moments. Once these are known, the membrane stress can be calculated from siM ¼
M g D0 ; ti d D g
i ¼ t; b;
M g ¼ M g;avg M Q ¼ M g;avg ðM tQ þ M bQ Þ.
(18)
(19)
The moment components are M g;avg ¼ Dg
d2 wg;avg , dy2
(20)
and the shear-induced local bending moment at face plate i is Z Z i
Z Ql ðyl Þ QiQ ðyl Þ dyl Di dyl dyl , M iQ ðyl Þ ¼ sDi yl ¼s i ¼ t; b. Similarly the bending stress in the face plate is zi Di Di i i þ MQ þ Mg sb ¼ 12 3 M tf ; i ¼ t; b, Df D ti
ð21Þ
(22)
where zi is the local coordinate in the face plate running from the neutral axis in the direction of z and the moment due to the thick face plates effect is M tf ¼ M tot;avg M g;avg .
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References [1] Libove C, Hubka RE. Elastic constants for corrugated-core sandwich plates. NACA TN 2289. Langley Field, VA: Langley Aeronautical Laboratory; 1951. [2] Robertson JC, Pama RP, Cusens AR. Transverse shear deformation in multicell box beam bridges. IABSE Publ 1970;30(II):169–90. [3] Lok TS, Cheng QH. Elastic deflection of thin-walled sandwich panel. J Sandwich Struct Mater 1999;1:279–98. [4] Romanoff J, Kujala P. The effect of laser weld’s dimensions on transverse shear stiffness and stress state of steel sandwich panels. In: Sixth international conference on sandwich structures, Ft. Lauderdale; 31 March–2 April 2003. p. 895–910.
[5] Romanoff J, Varsta P, Klanac A. Stress analysis of homogenized webcore sandwich beams. Compos Struct 2007;79(3):411–22. [6] Kozak J. Strength tests of steel sandwich panel. In: Proceedings in ninth international symposium of practical design of ships and other floating structures—PRADS, Luebeck-Travemuende, Germany; 2004. p. 902–6. [7] Romanoff J, Remes H, Socha G, Jutila M. Stiffness and strength testing of laser stake welds in steel sandwich panels. Helsinki University of Technology, Ship Laboratory, Report M-291, Espoo; 2006 [8] Allen HG. Analysis and design of structural sandwich panels. Pergamon Press; 1969. p. 36.
Thin-Walled Structures 45 (2007) 453–462 www.elsevier.com/locate/tws
The stiffness of laser stake welded T-joints in web-core sandwich structures Jani Romanoffa,, Heikki Remesa, Grzegorz Sochab, Mikko Jutilaa, Petri Varstaa a
Ship Laboratory, Department of Mechanical Engineering, Helsinki University of Technology, P.O. Box 5300, 02015 TKK, Finland b Institute of Fundamental Technological Research PAS, Swietokrzyska 21, 00-049 Warsaw, Poland Received 20 July 2006; accepted 16 March 2007 Available online 23 May 2007
Abstract The purpose of this paper is to describe experiments carried out on laser stake welded T-joints of web-core steel sandwich structures. A special test setup was developed to measure the shear-induced rotation at the T-joint. The ratio of the shear force to rotation angle gave the joint stiffness. This stiffness was measured for specimens with two different face-plate thicknesses. The influence of weld thickness, root gap and occurrence of contact were further investigated with finite element simulations. Finally, the shear stiffness of the sandwich structure transverse to the web plate direction was determined using the experimentally obtained average joint stiffness value. The validation of the shear stiffness was carried out by considering a beam in four-point bending. The agreement between calculated deflection and stress and experimental results was found to be good. r 2007 Elsevier Ltd. All rights reserved. Keywords: T-joint stiffness; Laser stake weld; Web-core sandwich panel
1. Introduction The demand for lighter and safer structures has stimulated the need to study new materials and new structural configurations in ship structures. All-metal sandwich panels offer an option that can fulfill these requirements. These panels are composed of face plates separated by a core. The simplest core geometry of allmetal sandwich structures is built from flat web plates perpendicular to the face plates. In this web-core structure, the web plates run only in one direction and are usually spaced 10–100 times the thickness of the face plate. The core produces continuous support to the face plates in the direction of the web plate, together with discrete support in the transverse direction. Thus, the web-core panel is highly orthotropic. The structural analysis of these panels can be carried out using either the finite element method (FEM) or analytical Corresponding author. Tel.: +358 9 451 4171; fax: +358 9 451 4173.
E-mail address: jani.romanoff@tkk.fi (J. Romanoff). 0263-8231/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2007.03.008
methods. The FEM is very flexible and well known, but it suffers from long analysis time and fails to offer deeper understanding of the problem. Therefore, analytical methods are more promising. In order to model the periodic core of the sandwich panel analytically, the stiffness properties of the panel are homogenized. In this homogenisation, the initial periodic structure is transformed into a homogenous continuum, where, for example, the beam- or plate-bending problem is solved, see Refs. [1,2]. The stiffness models for web-core sandwich structures have been developed mainly by assuming that the T-joint between the web and face plates remains undeformed when loaded. Thus, the joint stiffness is assumed to be infinite, see, as an example, Refs. [2,3]. Due to the laser stake welding, this assumption is far from realistic, and therefore the shear stiffness transverse to web plate direction is overestimated. This is due to the fact that the laser weld in the T-joint deforms, as presented in Ref. [4] by extensive 3D FE-models and experiments. Therefore, in Ref. [5], the analytical response model was presented for the laser stake welded web-core sandwich beams, where the influence of
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laser welding could be taken into account by including the T-joint’s rotation stiffness when the shear stiffness formulation was derived. The purpose of this paper is to determine experimentally both the rotation stiffness of the T-joint and the influence of its geometrical parameters on this stiffness. Thus, the aim of this study is to gain a quantitative understanding of the stiffness, as compared to the qualitative understanding gained through the experiments presented in Ref. [6]. The physical phenomena associated with this stiffness are further identified with FEA. Finally, the measured joint stiffness values are applied to a case study, where the response of a web-core sandwich beam in four-point bending is calculated. These results are validated by the experimental results given in Ref. [4].
condition is assumed and taken into account by considering the effective Young’s modulus in the calculations as E ¼ E=ð1 n2 Þ. 3. Theoretical background of the stiffness tests 3.1. Shear behaviour of web-core sandwich structures The theory of the bending response of web-core sandwich beams with thick face plates has been presented in Ref. [5], where the shear stiffness transverse to web plate direction (see Fig. 2), is defined as DQy ¼
12Dw , s2 kQ ðDw =Db Þ þ 6ðd=sÞ þ 12ðDw =kby sÞ 2ðd=sÞ (1)
2. Definitions
where the bending stiffness of the face and web plates is
The web-core sandwich plate consists of web and face plates, which are connected by the laser welds. The web plates are parallel to the xz-plane and have a thickness tw and a height hc; see Fig. 1. The web plate spacing is s. The face plates are on the xy-plane and have a thickness t. Subscripts t, b and w are used for the top face, the bottom face and the web plates, respectively. The plate has length L, breadth B, total height h ¼ hc þ tt þ tb and the neutral axes of the face plates have distance d ¼ hc þ ðtt þ tb Þ=2. The T-joint consists of the laser-weld and the face and web plates connected to it. The laser-weld position, thickness, root gap and the radius of the web plate are denoted by e1, tweld, hrg and R, respectively. When the weld is not in the middle of the web plate, the additional dimension e2 ¼ tw tweld e1 is used. A unit cell is a representative cut of the sandwich plate in the yz-plane, whose borders are neighbouring web plates and the parts of the face plates between these; length is infinitesimal dx. The plate has two coordinate systems, namely the global xyz and the local xylzl; see Fig. 1. The origin of the global coordinate system xyz is located at the neutral axis of the beam or at the geometrical mid-plane of the plate. The origin of the local coordinate xylzl is located on the neutral axis of the face or the web plate under consideration. Notations q, F and M are used for the external distributed loads, point forces and point moments per unit breadth, respectively. The Young’s modulus and Poisson’s ratio are denoted with E and n, respectively. A plane-strain
E i t3i ; i ¼ t; b; w, (2) 12 and the stiffness parameter describing the ratio of shear force carried by the top and bottom plates is 1 þ 12ðDt =sÞ ð1=kty Þ ð1=kby Þ þ 6ðDt =Dw Þðd=sÞ kQ ¼ . (3) 1 þ 12ðDt =Dw Þðd=sÞ þ ðDt =Db Þ
Fig. 1. (A) Laser-welded all-metal sandwich panel and (B) laser-welds with their dimensions.
Di ¼
The T-joint rotation stiffness kiy in Eqs. (1) and (3) is defined as the ratio of the moment (M in ¼ QsQ ; see Ref. [5]) to rotation angle yic at the T-joint, and is written as kiy ¼
QQ yic
s;
i ¼ t; b,
(4)
where QQ is the shear force carried by the shear deformation of thin-faced sandwich structures, which considers equal slopes of the face plates at the ends of the unit cell. yic is the rotation induced by local deformation in the laser-weld in face plate i and in the web plate connection; see Fig. 2B. 3.2. Determination of the joint stiffness A web-core sandwich structure under shear force QQ is presented in Fig. 2. Assuming constant shear force and
Fig. 2. (A) The internal forces in a unit cell and (B) the slopes in the T-joint.
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symmetrical cross-section gives equal shear force in both faces, i.e., QtQ ¼ QbQ ¼ QQ =2 and dQtQ ¼ dQbQ ¼ 0. Further, the rotation in the laser-weld was assumed to be equal at both ends of the web plate, i.e., ytc ¼ ybc ¼ yc . Thus, the web-induced horizontal H in , vertical V in and moment M in reactions in face plate i can be written as
455
dN i ¼ H in ¼ H n ,
(5a)
Fig. 3. The deformations in the T-joint area were not restrained in order to prevent additional stresses there from supporting the specimen. Considering the ratio of the displacement v to the distance where it is measured dv gives the rotation angle as v yc ¼ . (11) dv
V in ¼ 0,
(5b)
Substitution of Eqs. (10) and (11) into Eq. (4) gives the rotation stiffness as
QQ s, (5c) 2 where dNi is the increase in the normal force Ni. From the equilibrium of the web plate one obtains
M in ¼
M tn þ M bn Mn ¼2 . d d So by substitution of Eq. (5c) to Eq. (6) one obtains s H n ¼ QQ . d Thus, the moment in the web plate can be written as zw H n d zw Mw 12 ¼ , n ðzw Þ ¼ M n 1 2 2 d d and the shear force as Hn ¼
(6)
(7)
(8)
M tn ¼ H n . (9) d This means that the shear force is constant and the bending moment is linearly distributed inside the web plate. Further, the inflection point M w n ¼ 0 of the bending moment is at zw ¼ d=2 and the maximum value is Mw n ðzw Þ ¼ H n d=2. This property can be used to design the test setup for the rotation stiffness of the T-joint, since the cantilever beam between 0pzw pd=2 has the same bending moment and shear force distribution along this span. In the test, the horizontal force Hn was applied on the specimen; see Fig. 3. The corresponding shear force was obtained from Eq. (7) by considering the force Hn to be applied at distance d M ¼ d=2 on the specimen. So Eq. (7) is then rewritten as s H n ¼ QQ . (10) 2d M Qw n ðzw Þ ¼ 2
The rotation at the T-joint was obtained by restraining bending deformations of the face and web plate with support plates, except right next to the laser-weld area; see
ky ¼ kty ¼ kby ¼ 2d M d v
Hn . v
(12)
4. Testing 4.1. Test specimens Two types of test specimens were considered, i.e., specimens with 2.5 and 3.0 mm face-plate thicknesses. The length of the face sheet was 120 mm and all the specimens had a breadth of 50 mm. The web plates had a thickness and height of 4 and 20 mm, respectively. The measured material properties for the test specimens are given in Table 1. In order to identify the geometrical properties of the welds, small pieces were cut from both ends of each test specimen; see Figs. 4 and 5. The results of the weld thickness, root gap and weld centricity are given in Figs. 6B, C, and D, respectively. Fig. 5A presents an ideal case where the root gap is zero and the weld is positioned in the middle of the web plate, while in Fig. 5B the weld is positioned at the other edge of the web plate. In Fig. 5C, the web plate is rounded and the root gap is therefore increasing when approaching the edge of the web plate from the weld. In Fig. 5D, the root gap is very large. Altogether, 80 specimens were analysed. In addition, 234 fracture surface measurements per plate thickness were carried out to obtain the weld thicknesses. As can be seen from Fig. 6B, the weld thicknesses are fairly concentrated and have average values of 1.36 and 1.41 mm for 2.5 and 3.0 mm specimens, respectively. The corresponding standard deviations are 0.17 and 0.23 mm. Fig. 6C indicates that the root gap is mainly between 0 and 10 mm. Also, the weld centricity varies quite a lot, as presented in Fig. 6D. It should be noted that the variation of the weld thickness, root gap and weld centricity can be significant inside one specimen. Table 1 The measured values for the materials used
Fig. 3. Schematic view of the test setup.
Face-sheet thickness (mm)
Young’s modulus Yield strength (GPa) (MPa)
Tensile strength (MPa)
2.5 3.0 4.0
221 212 200
470–476 398–415 398–551
360–368 302–322 360–549
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4.2. Test setup and procedure The test system was built from INSTRON 1343 axial–torsional material testing equipment by adding a special gripping system to hold the specimen, which was fixed into INSTRON 1343 through a beam constructed from C200-profile and having length 1050 mm. A more detailed description of the test system is given in laboratory report [7]. The beam was checked to be stiff enough not to deform significantly under a maximum load of 5 kN. The
Fig. 4. Scheme of how weld dimensions are taken. Left: the specimens used to determine the root gap and weld centricity; right: the fracture surface measurement.
specimen was fixed to the beam through an aluminium block, support plates and screws. The test setup is presented in Fig. 7. An MTS TestStar II digital control unit controlled the operation of testing equipment and the tests were carried out as displacement controlled. The displacement of the specimen was measured with extensometer MTS632.27F-20 at distance dv ¼ 24.2 mm from the interface of the web and face plate. The force was measured with a Lebow 3173-104 load cell with maximum load capacity of 5 kN at distance dM ¼ 22.6 and 22.3 mm for specimens with 2.5 and 3.0 mm face-plate thicknesses, respectively. The specimens were loaded 3–4 times in one direction to a maximum force of 600 N (equal to 12 kN/m) in order to remove any material non-linear behaviour. Then the specimens were turned and the procedure was repeated. As a result, force–displacement curves were obtained from which the tangent stiffness could be determined. The maximum error, defined with total difference, in the tests was approximately 5%, mainly due to the inaccuracy in specimen dimensions. The tests were found to be repeatable.
Fig. 5. Different types of laser stake welds: (A) Zero root gap and weld close to middle of the web plate, (B) zero root gap and misaligned weld, (C) varying root gap, and (D) maximum root gap.
Fig. 6. Statistical properties of the welds: (A) Measured dimensions and distributions, (B) weld thickness, (C) root gap, and (D) weld centricity.
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Fig. 7. The test setup used in the rotation stiffness testing: (A) load frame, and (B) side view of the test setup.
4.3. Results Altogether 13 and 11 specimens were tested for 2.5 and 3.0 mm plate thicknesses, respectively. An example of a force–displacement curve is presented in Fig. 8A for four load cycles, where the microscopic pictures of the joint ends are also shown. The difference between the first and the rest of the closed loops during load cycles indicates that the plasticity disappears after the first cycle. This means that the force–displacement behaviour stabilises once the hardening of the material has occurred. In addition, the contact between the face and web plates at approximately 50 N can be seen as the slope increases in the force–displacement curve. Fig. 8B shows the stabilised force–displacement curves for 3-mm specimens. These stabilised curves were obtained after a small number of load cycles. Generally, the shape of the curve varies and three different types of curves exist: linear, bilinear and parabolic. However, when the average of all force–displacement curves is determined (see Fig. 8B), it can be seen that the force–displacement curve is quite linear up to the 600-N force level. In order to derive the rotation stiffness of the T-joint, the measured force was divided with specimen breadth. Fig. 8C shows how a tangent line was fitted for each test result between force values 0 N/m and 2 kN/m, which correspond to 0 and 100 N in the test, respectively. In this load range, the force–displacement curves were linear in almost all tests; see Table 2. The measured rotation stiffness values in two directions are given in Table 2, together with the resulting shear stiffness value calculated with Eq. (1). The corresponding statistical distributions are presented in Fig. 9. Table 2 shows that the rotation stiffness can be different in the same specimen in two directions. The rotation stiffness distributions for 2.5- and 3.0-mm specimens are presented in Fig. 9A. The average of the rotation stiffness is 113 and 107 kN for 2.5- and 3.0-mm specimens, respectively. The standard deviation for both types of specimens is 21 kN. The corresponding shear stiffness
distributions are presented in Fig. 9B. The corresponding average shear stiffness values are 299 and 376 kN/m with standard deviations of 19 and 34 kN/m, respectively. Comparison of the rotation stiffness in Fig. 9A with shear stiffness in Fig. 9B and their standard deviations reveals that the higher face plate thickness makes the shear stiffness more sensitive to the rotation stiffness of the T-joint. 5. Phenomena affecting stiffness 5.1. The effect of weld thickness and root gap In Ref. [4], an FE study into the influence of weld thickness and root gap on the shear stiffness when the contact effects are not considered is presented. The study is based on linear-elastic FE-analyses for the web-core sandwich beam in three-point bending. The study shows that shear stiffness decreases fast when the weld thickness decreases in the case of the sandwich beam thick face plates, while the decrease becomes smaller when the faceplate thickness is decreased. Regardless of the face-plate thickness, the influence of root gap on the shear stiffness is very small. The difference in the shear stiffness could vary easily over 200% in 3-mm specimens if a variation of between 0.5 and 2 mm in the measured weld thicknesses is assumed. 5.2. Contact effects In order to study the influence of contact on the rotation stiffness, detailed geometrically non-linear FE models were constructed, where the geometry of the weld in the T-joint was varied. Fig. 10 presents an example of FE-mesh, where eight-node plane strain elements (S8R in ABAQUS 6.5.1) were used. The FE models were restrained by fixing displacements of the nodes of the bottom surface of the face plate; see Fig. 10. The load was applied in the middle of the web plate at equal distance from the joint as in the tests. The material properties used are given in Table 1.
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Fig. 8. Force–displacement curve from test: (A) first four cycles for specimen S31t, where the ends of the joint are also shown, (B) test results for all 3-mm specimens, and (C) determination of the laser-weld rotation stiffness based on force values of 0 and 2 kN/m (equal to 100 N) for specimen A3t.
Table 2 The measured rotation stiffness values and resulting shear stiffness Specimen Face plate, Normal direction t (mm) Curve Test, ktest (kN/mm) shape b1b b1t b2b b2t b3b b3t S11 S11top S12 S12top S14 S14top S15 A1t A2b A2t A3b A3t S14b S14t S31b S31t S91b S91t
2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 2.5 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0
4.45 6.51 3.38 6.26 4.11 6.03 5.93 4.00 5.04 4.60 6.11 4.73 6.13 4.29 5.35 5.17 3.62 6.48 6.74 5.86 5.78 4.45 4.71 2.92
Linear Linear Parabolic Linear Linear Parabolic Linear Parabolic Parabolic Parabolic Linear Linear Linear Linear Bilinear Bilinear Parabolic Linear Bilinear Bilinear Linear Parabolic Bilinear Parabolic
Reverse direction Linear Joint stiffness Shear stiffness, range (N) Ky (kN) DQy (kN/m)
100
300 400 500 400
200 350 200 300 200 200 350 50
97 143 74 137 90 132 130 88 110 101 134 104 134 93 115 112 78 140 146 126 125 96 102 63
286 322 258 319 278 316 314 276 299 290 317 293 317 356 392 387 328 422 428 406 404 362 372 293
Test, ktest (kN/mm)
Curve shape
Linear Joint stiffness Shear stiffness, range (N) ky (kN) DQy (kN/m)
4.64 6.18 3.51 7.04 4.28 5.20 5.42 4.72 5.32 4.87 4.54 4.77 6.08 5.29 4.99 5.25 3.64 5.74 5.36 3.70 5.75 4.61 4.88 4.19
Parabolic Linear Parabolic Linear Bilinear Linear Parabolic Parabolic Linear Parabolic Bilinear Bilinear Bilinear Linear Bilinear Bilinear Parabolic Bilinear Parabolic Parabolic Linear Bilinear Parabolic Parabolic
180 400 200 200 400 400 100 200 500 200 50 200 200 500 200 150 200 300
102 135 77 154 94 114 119 103 117 107 100 105 133 114 108 113 79 124 116 80 124 100 105 91
291 318 262 329 282 302 306 292 304 295 289 293 316 390 381 389 329 403 392 332 404 368 377 352
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Fig. 9. (A) The rotation stiffness distributions of the test specimens. (B) The resulting shear stiffness for specimens with s ¼ 120 mm, tw ¼ 4 mm, hc ¼ 40 mm. Materials are given in Table 1.
Fig. 10. Example of the FE model used to simulate the rotation test.
Fig. 11. The influence of contact on the force–displacement curve. Root gap (A) is constant at 11 mm, (B) varies and has minimum value of 1 mm at web plate edge, and (C) is zero at weld and 5 mm at the edge of web plate. Other dimensions are tt ¼ tb ¼ 3 mm, tw ¼ 4 mm, s ¼ 120 mm, h ¼ 46 mm, tweld ¼ 1.746 mm, e1 ¼ 0.89 mm and R ¼ 0.24 mm.
the contact area increases (Case C), giving a force–displacement curve similar to the parabolic curve from the tests. Case A occurs when the root gap is large; see Figs. 5D and 11A. Then the normal stress component sz exists only in the weld. Case B occurs when the root gap is small and is constant or has some surface roughness that can cause sudden contact; see Figs. 5A and B and 11B. The normal stress sz is distributed in two different locations. The first location is the weld, although there are normal stresses in the contact area also. The larger the distance of the contact from the weld, the higher the increase of slope in the force–displacement curve. If the contact exists already in the unloaded specimen, the slope of the force–displacement curve will be different in two directions for the whole load cycle. This can happen, for example, when there are some impurities in the root gap. Case C occurs when the root gap decreases towards the weld; see Figs. 5B and C and 11C. The extent of the contact area, represented by the normal stress distribution sz, increases during the whole load sequence. Comparison of the FE-analysis and the test results shows that the behaviour of the force–displacement curve is very sensitive to the contact and differences of up to 100% can occur whether or not the contact occurs. 6. Influence on sandwich beam 6.1. Analytical model for beam response
The influence of contact on the shape of the force–displacement curve is presented in Fig. 11. Fig. 11 clearly shows that the range of linear relationship between the force and displacement varies quite a lot. The FE analysis shows that the variation in shape of the force–displacement curve results from the contact. If there is no contact, the slope remains constant (Case A), which corresponds to the linear force–displacement curve obtained in the tests. If the contact occurs suddenly and very locally, there is a sudden increase in the slope (Case B), i.e., there is behaviour similar to that shown by the bilinear force–displacement curves from the tests. From when the contact starts right at the beginning of the loading, the slope increases during the whole load sequence, since
The beam in bending was solved with the theory of homogenous sandwich beams with thick face plates giving the deflection, bending moments and shear forces. The derivation of the formulations for stress is presented in Ref. [5] and the result is given in Appendix A. The general solution by Allen for homogenous sandwich beams with thick face plates in four-point bending is given in Ref. [8]. 6.2. Case study In order to show the influence of the T-joint rotation stiffness on the response of the beam, a case study presented in Ref. [4] was considered. A web-core sandwich
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Table 3 Stiffness values for the beam, together with the maximum deflection and the difference between the experiments and analytical prediction Rotation stiffness (kN)
Shear Stiffness (kN/m)
Maximum deflection, analytical (mm)
Maximum deflection, test (mm)
Difference (%)
Low 65 Avg. 107 High 148 N
283 355 399 590
2.453 2.024 1.835 1.352
2.026
+21.1 0.1 9.4 33.3
Fig. 12. Comparison of the deflection obtained from the test and with analytical methods. Load is 1 kN/m. The test results are marked with square dots.
beam in four-point bending was studied. The beam had length B ¼ 1.8 m, web plate spacing s ¼ 120 mm and face plate thickness tt ¼ tb ¼ 2.86 mm. The web plate thickness and height were tw ¼ 3.97 mm and hc ¼ 40 mm, respectively. The forces were located at y ¼ 600 and 1200 mm and the displacement was measured at y ¼ 200, 400, 600, 900 and 1200 mm. Four strain gauges were located on the top face plate at y ¼ 242.5, 357.5, 722.5 and 837.5 mm, respectively. The average joint stiffness of 107 kN/m from the tests with 3-mm specimens was used. To study the statistical influence of joint stiffness, the average plus or minus two standard deviations were also considered. The joint stiffness values used are presented in Table 3, together with resulting shear stiffness and maximum deflection values. The comparison of the calculated and measured deflection distributions is presented in Fig. 12. The corresponding top surface stress of the top face plate is presented in Fig. 13. As shown in Table 3 and Fig. 12, a serious error of 33% occurs in the maximum deflection if the rotation flexibility is not included in the analysis. However, if the average rotation stiffness is included, the accuracy of the analysis is very good. When the lower and higher rotation stiffness values are used, the deflection is overestimated by 21% and underestimated by 9%, respectively. Fig. 13 shows that normal stress in the top face plate is estimated with good accuracy; the result indicates that the
Fig. 13. Comparison of the surface stress with different rotation stiffness values. The test results are marked with square dots.
stress is not as sensitive to the joint stiffness as the deflection is. A minor difference in predicted stress is found only in the neighbourhood of the load application, where the use of infinite joint stiffness underestimates the stresses. Fig. 13 shows also that, when the joint stiffness decreases, the maximum stress increases and the amplitude of the zigzag pattern decreases. The rate of change is, however, very slow in the region of the stiffness values found in this paper. 7. Conclusions The purpose of this paper was to present the experimental investigation into T-joint rotation stiffness in laser stake welded steel sandwich panels. Test specimens with 2.5 and 3.0 mm face plate thicknesses were considered. First, microscopic analyses were carried out for altogether 80 specimens in order to identify the weld properties of the specimens. The weld thickness, root gap and weld centricity were found to vary quite a lot. It was also discovered that these properties vary inside the stiffness test specimens. In order to get a better insight into the properties of the welds inside test specimens, more sophisticated measurement techniques are required. In the T-joint stiffness testing, tangent stiffness of each specimen was determined in two directions. Altogether 13 and 11 specimens with face-sheet thicknesses of 2.5
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and 3.0 mm, respectively, were measured. It was found that the stiffness test results vary and that the slope is not necessarily constant for the whole applied load range. However, if the average of all tests was taken, the slope was very close to linear. The experimental results were explained using FE-analyses. The influence of weld thickness on the T-joint rotation stiffness is significant, especially when the plate thickness increases. The influence of contact was also identified through the FE-analyses. Depending on the contact mechanism, the slope of the force–displacement curve may change drastically. So the contact is significant on the joint stiffness. The influence of rotation stiffness on the response of a web-core sandwich beam was studied. A beam in fourpoint bending was considered as a case study. The use of the average T-joint stiffness gives exactly the same maximum deflection as the test; when this rotation stiffness is omitted, the modelling seriously underestimates the maximum deflection by 33%. To study the statistical influence of joint stiffness, the average plus or minus two standard deviations were also considered. Then the lower value of the joint stiffness results in a 21% higher maximum deflection than the measured value. Similarly, the use of a higher value results in a maximum deflection 9% smaller than the measured value. Comparison of stresses showed that the maximum stress is not as sensitive to T-joint stiffness as the maximum deflection is in the obtained rotation stiffness region. However, as the rotation stiffness was decreased outside this region, the maximum stress started to increase in the face plates. At the same time, the amplitude of the zigzag-pattern in the surface normal stress decreased.
QQ;avg . The total shear force Qg;avg is then
Acknowledgements
where
This work was initiated in the EU-funded research project Advanced Composite Sandwich Steel Structures SANDWICH in the 5th Framework Programme Growth KA III. Since then, the work has been ongoing in the Finnish Ministry of Education funded Graduate School of Engineering Mechanics, EU-funded Centre of Excellence for Laser Processing and Material Advanced Testing LAPROMAT and with grants given by Merenkulunsa¨a¨tio¨ in Finland. This financial support is gratefully acknowledged. The Meyer Werft shipyard in Germany and the personnel of the Ship Laboratory of Helsinki University of Technology are thanked for producing the test specimens. The personnel of the Institute of Fundamental Technological Research at the Polish Academy of Science are thanked for the preparation of the test setup. Appendix A. Analytical model for beam response The total averaged shear force is divided into two parts: averaged thick face plates Qtfg;avg and shear deformation
Qg;avg ¼ Qtfg;avg þ QQ;avg .
461
(13)
The total shear force Qg;avg is obtained as a result of equilibrium considerations of the beam. The shear force carried by shear deformation QQ;avg is then obtained from d2 Qg;avg k2 Qg;avg ¼ k2 Qtot;avg , dy2 where stiffness factor k is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi DQ . k¼ Df ðD0 =Dg Þ
(14)
(15)
Bending stiffness of the face plates Df is Df ¼ Dt þ Db ,
(16)
and the global bending stiffness consists of the elongation D0 and face plate bending stiffness Dg ¼ D0 þ Dt þ Db ; ¼
E t tE E b tb d 2 E t t3t E b t3b þ . þ E t tt þ E b tb 12 12
(17)
Once Eq. (14) is solved for the shear force QQ;avg carried by the shear deformation, the thick face plate’s part of the shear force can be solved from Eq. (13). Further, the global bending and shear deflection components can be solved as shown in Ref. [8], together with shear forces and bending moments. Once these are known, the membrane stress can be calculated from siM ¼
M g D0 ; ti d D g
i ¼ t; b;
M g ¼ M g;avg M Q ¼ M g;avg ðM tQ þ M bQ Þ.
(18)
(19)
The moment components are M g;avg ¼ Dg
d2 wg;avg , dy2
(20)
and the shear-induced local bending moment at face plate i is Z Z i
Z Ql ðyl Þ QiQ ðyl Þ dyl Di dyl dyl , M iQ ðyl Þ ¼ sDi yl ¼s i ¼ t; b. Similarly the bending stress in the face plate is zi Di Di i i þ MQ þ Mg sb ¼ 12 3 M tf ; i ¼ t; b, Df D ti
ð21Þ
(22)
where zi is the local coordinate in the face plate running from the neutral axis in the direction of z and the moment due to the thick face plates effect is M tf ¼ M tot;avg M g;avg .
ARTICLE IN PRESS 462
J. Romanoff et al. / Thin-Walled Structures 45 (2007) 453–462
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