1.5 Finite-element Numerical Modeling
1.5.1 FE Solving Approach and Parametric Study
FE models were developed with the computer software ABAQUS/
Standard [64], to verify first the accuracy of Eq. (1.25). Specifically, eigenvalue buckling (lba) analyses were carried out on a wide range of beam geometrical properties, with continuous lateral restraints char- acterized by a sufficiently extended set of shear stiffnesses ky. For each FE model, the first 30 eigenvalues and eigen shapes were numerically predicted.
Throughout this FE study, two main aspects were in fact assessed. The critical buckling moment Mcr,R of each beam subjected to a constant, posi- tive bending moment My was numerically predicted and compared to the corresponding analytical estimation (Eq. 1.25). At the same time, the cor- respondence between the FE numerical and analytical critical numbers of half-sine waves nR was also properly assessed.
In doing so, the typical FE model consisted of S4R 4-node, quadrilateral, stress/displacement shell elements with reduced integration and large- strain formulation (type S4R of ABAQUS element library), see the detailed view of Figure 1.16. To ensure the accuracy of FE results, a refined and regular mesh pattern was used, with lmesh the characteristic size of quadri- lateral shell elements typically comprised between 3 and 15 mm, depend- ing on the h × L0 dimensions of the studied beams.
The positive bending moments My acting on each simply supported, fork-end restrained beam in LTB were introduced in the FE models in the form of point bending moments applied at the middle node of the end sections. Similarly, the simply supports and fork-end boundaries were described in the form of translational and rotational restraints for the same beam cross-sectional nodes.
For lba purposes only, glass was described in the form of a fully isotro- pic, indefinitely linear elastic material (E = 70 GPa, = 0.23 [25]). Careful attention was paid to the description of the adhesive joints applied along the top edge of the beams, e.g. to save the modeling and computational cost of parametric FE simulations but at the same time to preserve the accuracy of the same FE models. A series of indefinitely linear elastic
springs (“axial” connectors available in the ABAQUS library) directly connected to the ground and characterized each one in terms of elas- tic stiffness Ky, was in fact introduced along the top edge of each beam (zM = h/2, Figure 1.16), according to the adopted mesh pattern. The elas- tic stiffness Ky of each spring, being dependent on the reference length lmesh, was estimated as Ky = ky × lmesh (with the exception of the axial springs at the corners of each beam, where the value Ky = ky × 0.5 lmesh was taken into account). The same approach was carried out for several beam geom- etries as well as for various adhesive joint stiffnesses, with 10–4 N/mm2
≤ ky ≤ 104 N/mm2.
In terms of analytical and FE numerically predicted amplification fac- tors RM, a rather close agreement was generally found between the so- collected data, at least for a wide range of geometrical and mechanical configurations of practical interest for structural glass applications with adhesive joints.
Some examples are proposed in Figure 1.17, where data are associated to several joint shear stiffness ky. An average percentage discrepancy RM equal to ≈ ±0.5% was found between the collected data, for joint shear stiffnesses per unit of-length ky of practical interest for continuous sealant restraints (e.g. ≈0.184 N/mm2 ≤ ky ≤ ≈0.6136 N/mm2), with:
R
M Analytical M ABAQUS M ABAQUS
M
R R
= R −
100( ) ( )
( )
(1.30)
Figure 1.16 FE numerical model for a laterally restrained glass beam with continuous adhesive joints, under the action of a constant positive bending moment My (ABAQUS/
Standard).
Middle axis Ky
My L0
Imesh
h x’ z’
y’
X Y Z
Fork-support
Spring local axis
Simply support
t
As expected, FE lba analyses also confirmed that the presence of flexible adhesive joints provides a substantial modification of their reference buckling shape (e.g. critical nR value), thus a variation of their global LTB response. In this sense, a detailed incremental buck- ling investigation should be performed by taking into account the actual critical buckling shape for each geometrical configuration, with appropriate amplitude. As a first estimation, in absence of more detailed experi mental measurements and analyses on LR beams, this maximum amplitude could be taken equal to u0,max = L0/400, as experimentally obtained in [65].
Parametric studies also highlighted an almost exact correlation between the analytical and numerical number of half-sine waves nR associated to comparative data collected in Figure 1.17, especially in presence of lateral restraints able to provide a buckling strength increase up to ≈10 times the unrestrained beams (e.g. 1 ≤ RM ≤ 10 with 1 ≤ nR ≤ 8–10 depending on the beam geometry). Some examples are proposed in Figure 1.18 for two configurations with identical geometrical properties but different joint stiffnesses. A minor lack of correlation between analytical and numerical critical numbers of half-sine waves nR was found, through the full FE study, only in the presence of very stiff joints typically associated to a large number of half-sine waves (e.g. 10 N/mm2 ≤ ky ≤ 102 N/mm2 and ≈ 10 ≤ nR ≤ ≈ 18, in this investigation), but not of primary interest for structural glass and adhesive applications.
10–4 10–2 100 102
ky [N/mm2] –20
–16 –12 –8 –4 0 4
RM [%] Avg.Exp.
Figure 1.17 Comparison between analytical and FE magnifying coefficients RM (Eq. 1.30) calculated for various glass beam geometries, by changing the lateral restraint stiffness ky.
1.5.1.2 Incremental Nonlinear Analyses (inl)
Following the preliminary lba assessments, additional geometrical non- linear, static incremental simulations were successively carried out on the same geometrical/mechanical configurations.
In each FE analysis, the shape of the initial geometrical imperfection was derived from the buckling mode belonging to the lowest lba eigenvalue (e.g. Figure 1.18). The maximum amplitude of the so-scaled fundamental buckling shapes were assumed – in the absence of more detailed experi- mental measurements and investigations for LR glass beams in LTB – equal to 1/400 of the total span L0, as this approach is generally accepted for the analysis of structural glass members and for the calibration of design buckling curves [65].
The mechanical characterization of the adhesive joints was also properly modified, compared to lba studies. At this stage, a brittle elastic constitu- tive law was in fact taken into account, so that a possible progressive fail- ure mechanism in the adhesive layers could be properly simulated. This mechanical calibration was carried out by taking into account, for the equivalent axial springs depicted in Figure 1.16, the ultimate shear stress and elongation values derived from shear experiments performed on small adhesive specimens (Section 1.3.4).
Some examples of the parametric FE results derived from this further inl study are proposed in Figures 1.19 and 1.20 for a L0 = 3000 mm × h = 300 mm × t = 10 mm beam subjected to positive constant bending moment My. In Figure 1.19, specifically, the maximum envelope of out-of- plane displacements umax is proposed as a function of the RM amplification factor for the reference LU beam, as well as for the same beam geometry laterally restrained by means of continuous adhesive joints (LR). For both the LU and LR beams, the maximum amplitude of the initial geometrical
Figure 1.18 Critical buckling shapes of (a) LU glass beams, compared to (b) LR glass beams. L0 = 3000 mm, h = 300 mm, t = 10 mm; ABAQUS/Standard, white-to-black contour plot.
x y
nR = 1 (ky= 0) nR = 4 (ky= 0.814 N/mm2)
z x y
z
(a) (b)
imperfection is set equal to u0,max = L0/400, being the corresponding lba buckling shapes obtained performed on both the FE models (with nR = 1 and nR = 4 for the LU and LR beams, respectively). As shown, preliminary neglecting possible cracking mechanisms in glass, the LU beam would ide- ally carry on a maximum bending moment asymptotically tending toward the theoretical critical buckling moment Mcr( )E given by Eq. (1.27) (e.g.
RM → 1 for the LU beam).
The laterally restrained beam (LR), otherwise, would be theoretically able to offer (when neglecting possible cracking in glass and damage in the adhesive joint) a significantly higher buckling resistance, e.g. up to ≈5.7 times the LU geometry, almost comparable to the corresponding theoretical critical buckling moment (e.g. Mcr R( )E, ≅5 70. Mcr( )E ).
In the same figure, it is also possible to notice that for the LR beam both possible failure mechanisms occurring in glass or in the adhesive joint would result in marked decrease of its ideal LTB resistance. By assuming in the same beam an indefinitely linear elastic mechanical behavior for glass, for example, the LTB failure mechanism would be governed by the progressive collapse of few axial connectors, representative of the adhesive joint, hence typically resulting in an ultimate failure load significantly lower than the the- oretical Mcr R( )E, value (point A of Figure 1.19, RM = 4.22). Due to the progres- sive failure of these equivalent axial connectors along the beam buckling length L0, the post-cracked LTB response would also be characterized by an unsymmetrical deformed configuration (e.g. path AB of Figure 1.19).
Figure 1.19 LTB response of a monolithic glass beam laterally unrestrained (LU, nR = 1) or restrained (LR, nR = 4). Effects of continuous lateral restraints (ABAQUS-inl).
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 (umax - u0, max) [m]
0 1 2 3 4 5 6 7 8
RM
L0 = 3000mm, h = 300mm, t = 10mm ky = 0.184N/mm2
u0,max = L0/400 LU, nR = 1 LR, nR = 4 A
B Mcr,R(E)
Mcr(E) 5.70Mcr(E)
, , , AN FT
HS
My
&
This latter effect can be noticed in Figure 1.20, where the R stress co efficient – denoting the ratio between the measured stress max in each axial connector and the corresponding ultimate resistance u – is proposed for the adhesive joint belonging to the LR beam of Figure 1.19. The so- defined R values are shown, along the beam buckling length x/L0 (with 0 ≤ x ≤ L0), as a function of the applied bending moment (e.g. the RM loading configurations derived from Figure 1.19). As shown, due to the assumed geometrical configuration for the examined LR beam (nR = 4), the damage in the axial connectors (e.g. R = ± 1) first occurs where the beam undergoes the maximum out-of-plane deflections. In the same figure, it is also possible to notice – according to Figure 1.19 – that the ulti- mate LTB resistance of the examined LR beam would be clearly affected by the limited tensile resistance of glass. Depending on the type of glass and the corresponding characteristic tensile strength Rk, the LTB collapse would occur due to premature glass failure (with AN, HS, FT in Figures 1.19 and 1.20 denoting the attainment of the tensile resistance for AN, HS, and FT glass types, respectively).
It is interesting to notice, in this context, that almost the same buckling collapse mechanism was found for all the beam geometries taken into account in this parametric investigation, and the failure of the adhesive joints, accordingly, typically occurred for higher bending loads only.
However, a detailed LTB investigation for a general beam and adhesive joint geometrical/mechanical configuration should necessarily take into
Figure 1.20 Stress ratio R evolution in the silicone joint, as a function of the applied bending moment (ABAQUS-inl).
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x / L0 –1.0
–0.8 –0.6 –0.4 –0.2 0.0 0.2 0.4 0.6 0.8 1.0
Rσ
L0 = 3000 mm, h = 300 mm, t = 10 mm
ky = 0.184 N/mm2, nR = 4, u0,max = L0/400 RM = 4.22 (joint) RM = 3.78 (FT)
RM = 2.81 (HS) RM = 1.94 (AN) x/L0 = 0
x/L0 = 1 My
account both these possible collapse mechanisms, since strictly related to a combination of several influencing parameters.