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Post-tensioning of glass beams: Analytical determination of the allowable pre-load

Post-tensioning of glass beams: Analytical determination of the allowable pre-load The effectiveness of post-tensioning in enhancing the fracture resistance of glass beams depends on the level of compressive pre-stress introduced at the glass edge surface that will in service be exposed to tensile stresses induced by bending. Maximum pre-load that can be applied in a post-tensioned glass beam system, yielding maximum compressive pre-stress, is limited by various failure mechanisms which might occur during post-tensioning. In this paper, failure mechanisms are identified for a post-tensioned glass beam system with a flat stainless steel tendon adhesively bonded at the bottom glass edge, including the rupture of the tendon, glass failure in tension and adhesive/glass failure in the load introduction zone. Special attention is given to the load introduction failure given that the transparent nature of glass limits the use of vertical confinement usually applied in concrete. An analytical model for determination of the allowable pre-load in post-tensioned glass beams is proposed, based on the model applied for externally post-tensioned concrete beams. The model is verified with the results of a numerical model, showing good correlation, and applied in a parametric study to determine the influence of various beam parameters on the effectiveness of post-tensioning glass beams. Keywords Post-tensioned glass beam · Pre-load introduction · Failure modes · Analytical model · Numerical model · Parametric study 1 Introduction terms of initial fracture resistance and redundancy in the post-fracture state (Bos et al. 2004; Schober et al. 2004; Post-tensioned glass beams are hybrid structural components Débonnaire 2013; Louter et al. 2013; Jordão et al. 2014; in which a ductile tendon is applied on a standard glass Louter et al. 2014; Engelmann and Weller 2019; Cupace ´ tal. section to enhance its in-plane bending behaviour. The ten- 2021). These studies have generally focused on the struc- don introduces compressive pre-stress into the glass and thus tural behaviour of post-tensioned beams in bending, which compensates for the rather low resistance of glass in tension. have been investigated experimentally and through numerical A number of studies have investigated various methodolo- modelling, where particular attention has been given to the gies of post-tensioning applied to glass beams, demonstrating modelling of the brittle fracture of glass (Bedon and Louter significantly enhanced structural performance in bending in 2016, 2017). Present study focuses on the effectiveness of post-tension- B Jagoda Cupac´ ing in enhancing the fracture resistance of glass beams which jagoda.cupac@tu-dresden.de depends on the level of compressive pre-stress introduced Christian Louter at the glass edge surface that will in service be exposed to christian.louter@tu-dresden.de tensile stresses induced by bending. The maximum pre-load Alain Nussbaumer that can be applied in a post-tensioned glass beam system, alain.nussbaumer@epfl.ch yielding maximum compressive pre-stress, is limited by a number of failure mechanisms which might occur during Institute of Building Construction, Technische Universität Dresden, August-Bebel-Straße 30, 01219 Dresden, Germany post-tensioning. This paper investigates the post-tensioning of laminated glass beams with an adhesively bonded flat Resilient Steel Structures Laboratory (RESSLab), School of Architecture, Civil and Environmental Engineering (ENAC), stainless steel tendon placed along the bottom glass edge École Polytechnique Fédérale de Lausanne (EPFL), GC B3 (Fig. 1). The tendon is first pre-tensioned by an external 495, Station 18, 1015 Lausanne, Switzerland 123 234 J. Cupace ´ tal. demonstrating by analogy that similar stress limitations may apply. Glass fracture at the top glass edge is avoided by limiting the tensile stresses induced by the eccentrically applied pre- load. Maximum tensile stress at mid-span, for the initial pre- load level P, can be assessed from the following expression, assuming full composite action in the steel-glass section P Pe σ = + z ≤ f (1) g,t,P g,t g,d A I eq c where A is the equivalent cross-sectional area of the beam, eq e is the eccentricity of the applied pre-load P from the neutral axis, I is the moment of inertia of the composite section, and z is the distance of the top glass edge from the neutral axis. Fig. 1 Schematic of the post-tensioned laminated glass beam cross- g,t section with nominal dimensions Equivalent cross-sectional area is defined as A = b h E /E (2) eq i i i g mechanism and subsequently adhesively bonded to the glass. The release of the pre-load set-up after the curing of the adhe- where b , h , E represent the width, height and Young’s i i i sive induces a compressive pre-stress and a hogging bending modulus of the considered component of the section, and E moment into the glass beam . Failure mechanisms which is the Young’s modulus of glass. The position of the neu- may occur at this stage are the following: (1) rupture of the tral axis, in reference to the top edge of the beam, can be tendon, (2) glass fracture in tension due to the eccentricity determined from the following expression of the pre-load, i.e. the hogging bending moment, (3) adhe- sive failure and (4) glass fracture caused by stress peaks in b h z E /E i i i,t i g z =  (3) the load introduction zone at beam ends. b h E /E i i i g The rupture of the steel tendon is prevented by limiting the where z is the distance from the centroid of the considered allowable stress induced by post-tensioning. In the related i,t component to the top beam edge. The inertia of the compos- field of conventional prestressing steels applied in concrete ite section is calculated according to Eq. (4), following the structures, the maximum allowable stress is restricted to 75% Steiner’s rule of the characteristic tensile strength, or 85% of 0.1% proof stress (EN 1992-1-1 2004), in order to limit the loss of pre- b h E E i i load due to stress relaxation of steel under constant strain. i 2 I = + b h z (4) c i i 12 E E g g Losses due to relaxation of prestressing steel are normally based on the value ρ , the percentage relaxation loss at where z determines the distance of the centroid of a compo- 1000 hours after tensioning at a mean temperature of 20 C, nent to the neutral axis. The contribution of the interlayer foils for an initial stress equal to 70% of the actual tensile strength in the calculation of the equivalent cross-sectional area and of the prestressing steel samples prEN (2000). Stainless steel, moment of inertia of a laminated glass beam can be neglected which is not commonly applied for prestressing, exhibits due to its several orders of magnitude lower Young’s modu- relaxation in the same order of magnitude as conventional lus, relative to the other components of the section. prestressing steels, with ρ < 8% (Alonso et al. 2010), At the release of the pre-load from the post-tensioning set-up, load introduction failure may occur in the adhesive, This method is referred to as post-tensioning (although the pre-load the glass or at the tendon-adhesive/adhesive-glass interface, is only introduced into the glass upon curing of the adhesive bond) in depending on the relative shear strength of the components analogy to post-tensioning in concrete, where post assumes an already of the load transfer. When the pre-load is too high, failure cured concrete element. Pre-tensioning generally applies to the appli- of the beam will occur at both beam ends due to high shear cation of pre-stress on a tendon, followed by curing of the concrete element and finally release of pre-stress. The laminated glass beam is stresses which develop as the load is introduced from the here considered an already formed structural element; the addition of tendon through the adhesive and into the glass. The transpar- the tendon therefore corresponds with post- rather than pre-processing. ent nature of glass limits the use of special anchorage which Given a fairly high compressive strength of glass, a failure of glass would provide vertical confinement in order to avoid this type in compression is unlikely to limit the allowable pre-load in practical of failure; thus, the design of the end zones requires special applications, as long as constructive measures are taken during fabrica- tion to avoid stability problems. attention. 123 Post-tensioning of glass beams 235 2 Analytical model of pre-load introduction The glass beam shown in Fig. 3 has a length L, height h and width b . Pre-stressed tendon is bonded at the bottom glass edge; the height and width of the tendon is h and b , t t respectively. The adhesive thickness is t . Young’s modulus of the glass beam is E , Young’s modulus of the tendon is E , and the shear modulus of the adhesive is G . The tendon t a is initially pre-stressed to a stress level of σ . Upon release of the tendon from the post-tensioning set-up, the stress at a distance x from the beam mid-length drops to σ (x ). The pre- stress is transferred into the glass through the adhesive layer, resulting in a shear stress τ(x ) at the interface, and a com- pressive stress σ (x ) at the bottom glass edge. The shear g,b stress distribution is considered uniform across the adhesive thickness; peeling stresses are assumed to be negligible for the investigated tendon thickness, i.e. not causing delami- nation. Given the relatively small thickness of the adhesive Fig. 2 Concrete beams with external pretensioned FRP sheets; fail- and the tendon, these simplifying assumptions are considered ure in the anchorage zone: a adhesive shear strength < beam shear acceptable for a derivation of a theoretical solution which strength, b adhesive shear strength > beam shear strength (Triantafil- lou and Deskovic 1991) aims to provide initial understanding of the mechanics of load-introduction in a post-tensioned glass beam system. The release of pre-stress is accompanied by a displacement in the An analytical model which describes the short-term beam components, shown in Fig. 3c (rotation, i.e. peeling, is mechanical behaviour of post-tensioning through bonded here neglected for simplicity). The initial state, just before the tendons is presented in Sect. 2. It is based on the model release, is marked with a dashed line; the solid lines indicate developed by Triantafillou and Deskovic (1991) for concrete the state of displacement just after the release. The initial beams externally post-tensioned through fiber-reinforced extension of the tendon at a distance x equals u (x ).The plastic (FRP) composite sheets bonded in the tensile zone of release of the pre-stress causes elastic shortening of the glass a structural element (Fig. 2). The model allows for determi- beam, which equals −u (x ) at the bottom glass edge, while nation of the maximum allowable pre-load, for two failure the deformation of the tendon drops to u (x ). scenarios: (1) cohesive failure of the adhesive (within the Assuming linear-elastic material behaviour, shear strain γ bulk material) in a system with superior glass shear strength, and shear stress τ can be defined as follows (2) glass fracture in a system with superior shear strength of the adhesive. Adhesive strength on both substrates is con- u − u + u t g γ = (5) sidered sufficiently high to avoid failure at the interface, assuming appropriate surface preparation prior to bonding τ = (u − u + u ) (6) t g (steel surface may be roughened with sand-paper, followed t by thorough cleaning, of both steel and glass, with iso- propyl alcohol; glass primer is applied on the glass surface Equation (6) differentiated with respect to x equals to improve adhesion). In Sect. 3, the analytical model is used for the calculation of the allowable pre-load for a beam spec- dτ G du du du a t = − + imen applied in a wider experimental study on the bending dx t dx dx dx (7) behaviour of post-tensioned glass beams (Cupace ´ tal. 2021); G σ σ a t g,b = − + the results of the analytical model are further verified with a t E E E a t t g numerical model of the investigated beam system. Finally, the model is applied in a parametric study, presented in Sect. 4,in Compressive pre-stress at the bottom glass edge, σ ,is g,b order to determine the influence of certain geometric beam assumed uniform across the width of the glass beam, for the parameters and adhesive properties on the effectiveness of the post-tensioning. The results are discussed in Sect. 5, with 3 The overall beam width includes the width of the glass plies, b ,and g,i conclusions given in Sect. 6. the thickness of the interlayers, t , and normally matches the width of int the tendon, providing equal bonding surface on the glass beam and the tendon. For the calculation of beam resistance, only the thickness of the glass plies is taken into account, b = b . g g,i 123 236 J. Cupace ´ tal. Fig. 3 Components of the model of the post-tensioned beam system; a longitudinal and b cross-section, c axial deformations at the beam end upon release of the pre-load; adapted from Triantafillou and Deskovic (1991) sake of simplicity. It can be expressed in terms of the tensile Equation (11) then becomes a second order linear homoge- stress in the tendon, σ , through the following equation neous equation b h σ b h σ ez t t t t t t g,b d τ G 1 α σ =− − = + τ = ω τ (13) g,b A I g g dx h t E E t a t g b h b h ez t t t t g,b (8) =− + σ where A I g g =−ασ G 1 α ω = + (14) h t E E t a t g where A is the area and I the moment of inertia of the g g glass beam, z is the distance from the glass centroid to the g,b A general solution of Eq. (13)isofthe form bottom glass edge, e is the eccentricity of the force acting in the centroid of the tendon (b h σ ) from the glass centroid, t t t ωx −ωx τ = C e + C e (15) 1 2 thus e = z + t + h /2, and g,b a t The coefficients C and C can be determined from the 1 2 b h ez b h t t t t g,b α = + (9) boundary conditions, which depend on the considered fail- A I g g ure mechanism. The failure is governed by the shear strength of the glass or the adhesive, whichever is lower. The follow- By substituting Eq. (8)into(7), the following is obtained ing subsections provide the solution for the allowable level of initial pre-stress in the tendon for the two failure mecha- dτ G σ 1 α a nisms. = − + σ (10) dx t E E E a t t g 2.1 Allowable pre-load governed by the shear which, differentiated with respect to x, results in strength of the adhesive (model AF) The shear stress-shear strain relationship for a thermoset d τ G 1 α dσ a t =− + (11) 2 structural adhesive is schematically shown in Fig. 4.The dx t E E dx a t g dashed line shows the behaviour of a two component meth- acrylate adhesive Araldite 2047 in a single lap shear test, The equilibrium of the tendon under tensile stress, σ , and adopted from (Nhamoinesu 2015). In the current model, the shear stress, τ , at the interface with the adhesive, can be true behaviour is approximated by a bilinear curve (solid expressed as line), describing two characteristic behaviour modes: the initial linear-elastic response up to the strain level of γ , a,el dσ h =−τ (12) followed by the perfectly plastic path leading to failure once dx 123 Post-tensioning of glass beams 237 Fig. 4 Shear stress-shear strain curve for a thermoset structural adhesive; dashed line—true behaviour based on the tests on Araldite 2047 from Nhamoinesu (2015); solid line—bilinear approximation of the stress-strain curve the strain limit γ is reached. The shear strength equals By substituting (18)into(15), the following expression is a,max τ . obtained for the shear stress distribution in the elastic zone a,max The release of the pre-load induces high shear stresses γ G a,el a at beam ends. Figure 5a shows the stress distribution along τ = sinh(ωx ), 0 ≤ x ≤ L /2 (19) el sinh(ωL /2) the beam at the limit of the shear capacity of the adhesive el (note that x = 0 is located at beam mid-span): in the elastic In the plastic zone, L /2 ≤ 0 ≤ L/2, the shear stress is el range 0 ≤ x ≤ L /2, the stress distribution is described by el constant; however, the shear strain is assumed to follow the Eq. (15); for L /2 ≤ x ≤ L/2, the shear stress equals τ . el a,max same distribution as in the elastic zone, hence The corresponding shear strain equals γ at x = L /2, a,el el and γ at x = L/2. The coefficients C and C can be a,max 1 2 a,el determined from the following boundary conditions γ = sinh(ωx ), 0 ≤ x ≤ L/2(20) sinh(ωL /2) el The length of the elastic zone, L , follows from the condition el τ(x = 0) = 0 (16) γ(x = L/2) = γ a,max γ(x = L /2) = τ(x = L /2)/G = γ (17) el el a a,el β + β + 4 2ln L = , where (21) el resulting in 2γ a,el β = sinh(ωL/2) (22) a,max γ G γ G a,el a a,el a C = and C =− (18) 1 2 2sinh(ωL /2) 2sinh(ωL /2) el el Fig. 5 Stress distribution along the beam at the limit of the adhesive shear capacity; a shear stress at the interface; b tensile stress in the tendon; adapted from Triantafillou and Deskovic (1991) 123 238 J. Cupace ´ tal. Fig. 6 Stress distribution along the beam at the limit of the glass shear capacity; a shear stress at the interface; b tensile stress in the tendon; adapted from (Triantafillou and Deskovic 1991) The tensile stress distribution along the tendon, in the elas- dτ tic zone, can be obtained starting from Eq. (10); can be dx substituted with a derivative of (19) with respect to x E t ωγ t a a,el σ − cosh(ωx ) sinh(ωL /2) el σ = , 0 ≤ x ≤ L /2 t el 1 + α (23) In the plastic zone, the tensile stress linearly drops from σ (x = L /2) to zero at x = L/2 (Fig. 5b). The condition t el Fig. 7 Simplified model for shear stress-slip relationship in glass of slope continuity of σ at x = L /2 can be written as t el σ | dσ t t x =L /2 el at the lower glass edge, which results in a drop in shear stress = (24) L − L dx el x =L /2 el towards the beam end. The distribution of the shear stress is schematically shown in Fig. 6a. The elastic zone, 0 ≤ x ≤ L /2, is described by el Solving Eq. (24)for σ results in the expression for the ini- Eq. (15); in the non-linear zone, 0 ≤ x ≤ (L − L )/2, the el tial pre-stress level that will just cause failure in the adhesive fracturing behaviour is described by a softening law which upon release from the post-tensioning rig relates the shear stress at the interface (τ ) with a relative slip between the substrates (δ). In the lack of an existing model ω(L − L ) el σ = E t ωγ coth(ωL /2) + (25) for this type of failure in glass, an analogy with the softening t a a,el el of concrete in shear is assumed. A non-linear softening law is approximated with a simplified linearly descending τ − δ From σ , which can now be obtained from (23), the corre- model (Yuan et al. 2001), shown in Fig. 7. Once the fracture sponding compressive stress at the lower glass edge, σ , can g,b at the interface is initiated at τ , the stress linearly reduces g,max be calculated applying (8). with the increase of slip, reaching zero when the value of slip exceeds δ . The area below the curve presents the frac- max 2.2 Allowable pre-load governed by the shear ture energy in mode I, G , i.e. the energy dissipated in the Ic strength of glass (model GF) formation of new fracture surfaces , in case of brittle mate- rials. It should be noted that, unlike in concrete, where the When applying structural adhesives with high shear stiffness and shear strength, fracture in glass may occur at the release In crack mode I, the fracture energy (critical energy release rate) G Ic of the pre-load in the set-up, or with some delay. Once the is related to the critical stress intensity factor K by G = K /E , Ic Ic Ic shear stress at the beam end reaches the level of glass resis-   2 where E = E for plane stress state, and E = E /(1 − ν ) for plane tance in shear (< adhesive shear strength), a crack is initiated strain state (Haldimann et al. 2008). 123 Post-tensioning of glass beams 239 Linear approximation of the shear distribution in the non- linear zone can be written as ⎛ ⎞ ⎜ ⎟ τ = τ 1− , 0 ≤ x ≤ (L − L )/2(32) g,max ⎝ ⎠ el L − L el The equilibrium of the tendon under tensile stress and interface shear stress, taken as a triangle, at a distance x in the non-linear zone, equals 1 L − L el h σ = τ − x (33) Fig. 8 Shear crack at the release from the post-tensioning set-up t t 2 2 cracks propagate in mode II (sliding) in a layer above the ten- The combination of (32) and (33), solved for σ ,gives don, parallel with the interface (Triantafillou and Deskovic the expression for the distribution of the tensile stress in the 1991), pure mode I (opening) is assumed the governing mode tendon for crack propagation in glass. The observed cracking at the release of the pre-load (Fig. 8) shows an opening crack, which τ L − L g,max el propagates perpendicularly to the direction of the maximum σ = −x , 0 ≤ x ≤ (L − L )/2 t el h (L − L ) 2 principal stresses (Sect. 3.2). t el (34) Coefficients C and C in Eq. (15) can be determined from 1 2 the boundary conditions Substituting (34)into(31), after integration gives the fol- lowing τ(x = 0) = 0 (26) τ(x = L /2) = τ (27) el g,max σ τ (L − L ) τ g,max el g,max t    2 δ(x ) = x − x + (x ) E 4E h 2E h t t t t t resulting in the following expression for the shear stress dis- (35) g,max tribution in the elastic zone − (x ) + C 3E h (L − L ) t t el g,max τ = sinh(ωx ), 0 ≤ x ≤ L /2 (28) el sinh(ωL /2) For the condition δ(x = 0) = 0, (35) results in C = 0. For el δ(x = (L − L )/2) = δ ,Eq.(35) becomes el max Similarly to (23), the distribution of the tensile stress in dτ the tendon can be determined from (10), substituting with 0 2 τ (L − L ) dx σ (L − L ) el g,max el a derivative of (28) δ = − (36) max 2E 24E t t E t ωτ t a g,max σ − cosh(ωx ) The tensile stress in the tendon following from Eq. (29) G sinh(ωL /2) a el σ = , 0 ≤ x ≤ L /2 (29) t el for x = L /2 should be equal to that calculated from (34) el 1 + α for x = 0. This condition can be written as E t ωτ t a g,max The shear slip δ can be determined from the relative dis- σ − coth(ωL /2) el (L − L )τ el g,max placement of the substrates at the release of the pre-load. = (37) t 4h Assuming a fully rigid glass-adhesive system in the non- 1 + α linear zone, the slip at distance x results only from the straining of the tendon Knowing τ and δ , equations (36) and (37) can be g,max max 0 0 solved for the two remaining unknowns, the length of the d(u − u ) du du t t t t = − = ε − ε (30) t elastic zone, L , and the initial pre-stress in the tendon, σ , t el dx dx dx that will just initiate fracture in glass upon release. L can be el The shear slip follows from substituted in (29) and (34) to obtain the distribution of the tensile stress in the tendon in the linear and non-linear zone, x 0 x respectively (Fig. 6b). Finally, the compressive pre-stress at σ σ 0  t δ(x ) = (ε − ε )dx = x − dx (31) the bottom glass edge, σ , can be obtained from (8). g,b E E 0 t 0 t 123 240 J. Cupace ´ tal. Fig. 9 Shear and tensile stress distribution at the limit of the adhesive shear capacity; comparison of the analytical and numerical model (a) shear stress at the interface (b) tensile stress in the tendon 3 Numerical verification of the model 3.1 Model AF The analytical model was first applied for the calculation For the model AF, governed by the adhesive strength, Aral- of the allowable pre-load and stress distribution of a glass dite 2047-1 was selected as the reference adhesive. The beam post-tensioned through an adhesively bonded tendon parameters defining the bilinear shear stress-shear strain cur- placed along the bottom glass edge (Fig. 1). In order to verify ve were assessed based on the experimental results reported the analytical results, a numerical 2D model of the beam in Nhamoinesu (2015); the following values were adopted in was implemented in a finite element (FE) software Abaqus , the model: G = 211 MPa, γ = 2.69%, γ = 15%, a a,el a,max version 6.12-3. τ = 5.67 MPa. For the given beam properties, the initial a,max The beam comprises a triple-laminated annealed glass sec- pre-stress level that will just cause failure in the adhesive tion (6 +10 +6 mm) with a height of 122 mm and a length of upon release, σ , amounts to 363.24 MPa, i.e. the initial 1500 mm. The pre-stress is applied via stainless steel tendon pre-load P = 27.24 kN. The corresponding compressive 25 × 3 mm, grade EN 1.4301 (EN 10088-1 2005), and trans- pre-stress at the bottom glass edge, σ , equals −32.42 MPa. g,b ferred into the glass through 1.5 mm thick adhesive bond. Figure 9 shows the distribution of the shear stress in the The applied Young’s modulus of glass, E , and the tendon, adhesive, τ , and the tensile stress in the tendon, σ ,for g t E , equal 70 GPa (EN 572-1 2004) and 180 GPa [based on 0 ≤ x ≤ L/2, resulting from the release of the initial pre- uniaxial tensile tests reported in Cupac( ´ 2017)], respectively. load P (solid curves). The length of the elastic zone, L , el Two types of adhesives were considered, in order to simu- equals 1313 mm, i.e. the adhesive yielding at the beam ends late the two failure modes represented by the models AF and occurs over the initial 94 mm. GF. Adhesive properties are further detailed in the following In the numerical model, only half of the beam length L sections where the two models are investigated separately. was considered, with symmetry restraint at the mid-section nodes reproducing the effective boundary conditions. The beam components - glass, tendon and adhesive—were rep- resented with 4-node monolithic shell elements with reduced integration (S4R). A regular mesh pattern was applied, with element size of 5 mm along the beam length. Glass height By Dassault Systémes Simulia Corp. 123 Post-tensioning of glass beams 241 Fig. 10 Distribution of principal stresses in the glass (half length) at the limit of the adhesive shear capacity (FEM results) was divided in 24 elements (element size ∼ 5 × 5 mm), three lated to the bottom glass edge, equals −32.16 MPa, which elements were applied across the thickness of the adhesive closely corresponds to the value of −32.42 MPa, obtained (0.5 × 5 mm), and one element over the height of the ten- analytically. don (3 × 5 mm), resulting in a total of 4200 elements. A rigid constraint (tie) was used at the tendon-adhesive and 3.2 Model GF adhesive-glass interface. Material properties equivalent to those applied in the analytical model AF were implemented For the model GF, epoxy adhesive 3M™ Scotch-Weld™ in the numerical simulation. The initial pre-stress level, DP490 was chosen as the reference adhesive due to its rel- σ = 363.24 MPa, obtained through the analytical solution, atively high shear modulus and shear strength, compared to was applied on the tendon as a pre-defined field (mechanical/ other adhesives, such as Araldite 2047. The shear modulus stress) in the initial step of the simulation. A geometrically G equals 239 MPa (Nhamoinesu 2015), the shear strength non-linear, static incremental computation was performed τ = 30.2 MPa at 23 C, according to the manufacturer’s a,max in Abaqus/Standard. The resulting stress distribution in the data sheet (3M 1996). adhesive and the tendon is plotted in Fig. 9 (dashed curves). DP490 was applied for the post-tensioning of glass beams The stress data represents the averaged nodal values extrapo- with the same nominal parameters (Fig. 1) in the scope of lated from the integration points of the connecting elements. a master’s thesis (Cokragan 2015); glass failure was con- The stress plots resulting from the analytical and numerical sistently observed at beam ends at the release of a 15 kN simulation demonstrate a good correlation in the computa- pre-load. In order to determine the maximum shear stress tion of both shear stresses in the adhesive layer and tensile at the interface which initiated glass fracture, τ , i.e. the g,max stresses in the tendon. The distribution of the principal shear resistance of glass, the release of the pre-load was sim- stresses in glass is shown in Fig. 10. The load-introduction ulated in a 2D numerical model in the present study. zone (Fig. 10b) is subjected to a complex stress state, which The results of the numerical model were further applied in tends to a linear stress-distribution over the beam height, as the calculation of the stress intensity factor (SIF), K , based the pre-stress is gradually introduced into the glass. In the on the approach proposed by Albrecht and Yamada (1977). mid-section, the stress varies linearly from tension at the top The procedure is based on the linear superposition princi- edge to compression at the bottom (Fig. 10c). The maximum ple (Broek 1986) used in linear elastic fracture mechanics value of compressive pre-stress in the FE model, extrapo- (LEFM) calculations to derive the SIF from an uncracked 123 242 J. Cupace ´ tal. Fig. 11 Surface crack in a semi-infinite body with a a uniform and b non-uniform stress distribution along the crack depth FE model, with the assumption that the cracking does not significantly influence the global stiffness of the component. The correction factor Y is divided in two parts, Y = Y Y , s g where Y accounts for the crack shape and the proximity of boundaries in a cracked body with a uniform stress dis- tribution, and Y is the correction factor for the local stress gradient due to the geometry of the modelled structural detail (Fig. 11). The expression for the SIF therefore equals K = Y (Y σ π a) = Y K (38) I s g s where Y = 1.12 for a shallow surface crack in a semi-infinite solid (Irwin 1962). The value of the SIF K , which contains Fig. 12 Stress gradient along the crack path perpendicular to the direc- the correction factor Y , can be determined in two steps by tion of the maximum principal stresses (1) computing the stresses in an uncracked model along a line where the anticipated crack will be inserted and (2) inte- soft double-sided adhesive pads were applied between the grating the normal stresses along the same line, for a given glass and the tendon to prevent the spread of adhesive to the crack depth, by applying the following expression glass beam corner edge). The mesh was refined in the zone of the load-introduction, to allow for the computation of K , 2 σ(x ) K = π a √ dx (39) I with 0.003 mm elements over an area of 0.4 × 0.2 mm. The 2 2 0 a − x element size was gradually increased towards the edges of the beam, to a maximum size of 5 mm, resulting in a total where a is the crack depth, σ(x ) is the stress distribution of 28648 elements. Initial pre-stress of 200 MPa was applied along the anticipated crack path, and x is the location along on the tendon elements, which corresponds to a 15 kN axial the crack path. For discrete values of stress obtained from the pre-load. FEM, (39) becomes The maximum shear stress of 8.28 MPa, obtained from the √ numerical model, was adopted as the glass shear resistance, 2 x x i +1 i K = π a σ arcsin − arcsin (40) τ , for this specific geometry and mechanism of load- g,max π a a i =1 introduction. For a known τ , the maximum shear slip, g,max δ , can be derived from the simplified τ − δ relationship max where σ is the discrete stress normal to the crack path, shown in Fig. 7 applied over the element width from x to x , and summed i i +1 over the total number of elements along the crack depth a. 2G Ic The numerical model applied for the simulation of pre- δ = (41) max g,max load introduction at the verge of adhesive failure was adapted 2 2 K (1 − ν ) by changing the material properties of the adhesive; DP490 g Ic G = (42) Ic was modelled as linear-elastic, with Young’s modulus E = 660 MPa and Poisson’s ratio ν = 0.38 (Nhamoinesu 2015). The adhesive was applied witha5mmoffsetfromthe where fracture toughness K = 0.75 MPa m and Pois- Ic beam end, corresponding to the bonding layout applied in son’s ratio ν = 0.23 (Haldimann et al. 2008). The resulting the experiments to avoid stressing the glass edge (5 mm long maximum shear slip equals δ = 1.8µm. max 123 Post-tensioning of glass beams 243 Fig. 13 Variation of the allowable initial pre-stress level, resulting compressive pre-stress on the bottom glass edge and length of the elastic zone with respect to the adhesive thickness and tendon area fraction (model AF) 123 244 J. Cupace ´ tal. The first path of the initial crack was assumed perpendic- height and beam length, and adhesive properties—strain ular to the glass edge surface, starting in the vicinity of the limit (model AF) and shear modulus (model GF). The study maximum shear stress at the interface; the SIF computed for was performed considering the nominal beam properties a crack length a = 0.2 mm equals 0.58 MPa m. The sec- described in Sect. 3, varying one of the parameters. The ond crack path followed a line perpendicular to the direction initial pre-stress applied on the tendon at the verge of the of the maximum principal stresses, as the most unfavourable adhesive/glass failure was calculated for each beam config- case for the effective glass resistance. In the observed refined uration. area of 0.4 × 0.2 mm, the angle equals 45 , measured coun- terclockwise from the glass edge surface. Further into the 4.1 Model AF global model of the beam, the direction of the principal stresses gradually changes (Fig. 10b), resulting in the angle of The results of the parametric study of the maximum allowable (visible) crack propagation of ∼ 30 (Fig. 8). The SIF com- pre-load governed by the adhesive failure are presented in puted along the 45 inclined crack path reached a value of Figs. 13, 14 and 15. 0.73 MPa mfor a = 0.28 mm, which closely corresponds Figure 13 shows the initial pre-stress applied on the ten- to the fracture toughness of glass (K = 0.75 MPa m), don, σ , resulting compressive pre-stress in glass, σ , and Ic g,b demonstrating that the applied 15 kN pre-load may initi- ratio of the length of the elastic zone over the total length, ate glass fracture. Although the assumed initial crack length L /L, with respect to the adhesive thickness and tendon el is rather large for a polished glass edge [Lindqvist (2013) area fraction, i.e. the cross-sectional area of the tendon, A , reported initial crack size in the range of 0.015 to 0.1 mm], a expressed as a percentage of the glass section, A . The limit result in the same order of magnitude is considered accept- of the initial pre-stress is set to 75% of the ultimate tensile able, given a large scatter of glass edge quality which depends strength of the tendon in order to avoid excessive stress relax- on the manufacturing process and varies among glass sup- ation. This amounts to 650 MPa for the stainless steel bars pliers. Stress gradient along the crack path at 45 is plotted employed in this research (based on uniaxial tensile tests in Fig. 12. reported in Cupac( ´ 2017)). It can be seen that the allowable For the nominal beam properties and τ = 8.28 MPa, initial pre-stress level increases with the adhesive thickness, g,max the analytical model yields a 7% lower maximum initial while it decreases with the increase in tendon area fraction pre-load P = 14 kN, at the verge of glass failure; the (Fig. 13a, b). However, a larger tendon area yields a higher corresponding compressive pre-stress at the bottom glass initial pre-load, P (Fig. 13b). Therefore, the compressive edge, σ , equals -16.64 MPa, compared to -17.69 MPa pre-stress at the bottom glass edge increases with both the g,b obtained numerically. This can be explained by the conser- adhesive thickness and tendon area fraction (Fig. 13c, d). An vative assumption of a fully rigid tendon-to-glass connection increase in both parameters results in a decrease in the ratio in the non-linear zone of the analytical model, while the of the elastic length, i.e. an increase in the yield zone in the FE model assumes linear-elastic adhesive behaviour over the adhesive (Fig. 13e, f). entire bond length. The variation of the compressive pre-stress in glass with For a better qualification of the shear resistance of glass, respect to the beam length is shown in Fig. 14. An increase release tests should be performed, in which the pre-load is in the pre-stress can be seen up to an effective bond length gradually released into the beam through a bonded tendon, at which the full pre-load is introduced into the glass; fur- while monitoring the relative shear displacement along the ther increase in beam length does not affect the resulting interface. The results in terms of shear-slip curve upon ini- compressive pre-stress. For the nominal dimensions of the tial glass failure (softening) could then be compared to the investigated beam specimen, 99% of the maximum compres- provided model, in order to validate the simplified linear soft- sive pre-stress at mid-length is achieved with a beam length ening law and the corresponding assumptions of the beam of L = 655 mm. behaviour. This was, however, not performed in the scope of Figure 15 shows the dependency of the post-tensioning the present study. system on the strain limit capacity of the applied adhesive. Similarly to the effect of the adhesive thickness, an increase 4 Application of the model in a parametric in the adhesive strain limit enhances the maximum level of study initial pre-stress in the tendon (Fig. 15a) and the achieved compressive pre-stress in glass (Fig. 15b), since the yielding The analytical models AF and GF were applied in a paramet- of the adhesive increases the overall flexibility of the joint, ric study in order to analyse the effectiveness of the inves- diminishing excessive stress peaks at load introduction. Con- tigated post-tensioned glass beam system, i.e. the achieved sequently, the ratio of the length of the elastic zone over the compressive pre-stress at the bottom glass edge, with vary- total beam length decreases with a higher yielding capacity ing geometric beam parameters—adhesive thickness, tendon of the adhesive (Fig. 15c). 123 Post-tensioning of glass beams 245 Fig. 14 Relationship between the compressive pre-stress at the bottom glass edge and beam length at the verge of the adhesive failure Fig. 15 Variation of the allowable initial pre-stress level, resulting compressive pre-stress at the bottom glass edge and length of the elastic zone with respect to the adhesive strain limit (model AF) 123 246 J. Cupace ´ tal. Fig. 16 Variation of the allowable initial pre-stress level and resulting compressive pre-stress at the bottom glass edge with respect to the adhesive thickness, tendon area fraction and adhesive shear modulus (model GF) 123 Post-tensioning of glass beams 247 4.2 Model GF The plastic deformation capacity of the adhesive has an important influence on the allowable pre-load level by reduc- The results of the parametric study of the maximum allow- ing the shear stress peaks. Even with a very high plastic strain, able pre-load governed by glass failure are shown in Fig. 16. the yield zone in the adhesive remains limited to a relatively It can be seen that the increasing adhesive thickness, t , small fraction of the total bond length (Fig. 15c). A complete positively influences the allowable initial pre-stress level, absence of the plastic zone in the adhesive is explored in the σ , resulting in a higher compressive pre-stress at the bot- model GF, governed by glass failure. The stress peaks are tom glass edge, σ , thus increasing the efficiency of the hence much higher; the assumed shear resistance of glass of g,b applied post-tensioning method (Fig. 16a, 16b). An increase 8.28 MPa is reached with a pre-load of 14 kN (analytically). in the tendon area relative to the cross-sectional area of glass, For comparison, in the adhesive failure (AF) model, a shear A / A , similarly provides a higher compressive pre-stress resistance limit of the adhesive of 5.67 MPa is reached with a t g in glass, achieved through an increasing initial pre-load, P, pre-load of 27.24 kN, considering a max. strain limit of 15%. which corresponds to a decrease in the initial pre-stress in the Plastic deformation is in this sense very beneficial for the tendon (Fig. 16c, d). A significant increase in the efficiency functioning of the system, while the limited yield zone does of the system can be achieved by applying adhesive with not pose a risk for the exploitation of the beam in bending, a lower shear modulus, G , assuming that sufficiently high as long as the added shear deformation in bending is consid- shear resistance of the adhesive is maintained (Fig. 16e, f). ered in the design. This, however, falls out of the scope of the present study which focuses on the pre-load introduction stage. 5 Discussion In order to determine the long-term behaviour of the pro- posed beam system, creep and relaxation behaviour of the Numerical verification of the proposed analytical model has constituent materials should be included in the model. In shown that the model can be applied with sufficient accu- particular, given the viscoelastic nature of the adhesive, load- racy for the prediction of short-term mechanical behaviour duration and temperature may affect the level of initially of post-tensioned glass beams, in terms of stress distribu- applied pre-load transferred into the glass, resulting in a lower tion in the tendon and the adhesive, and determination of the efficiency of the system in the long term. Chemical com- maximum compressive pre-stress that can be achieved in the patibility of the adhesive and the interlayer material should glass without causing premature failure at pre-load introduc- also be investigated; given that the pre-load introduction and tion. The maximum compressive pre-stress predicted by the composite action fully rely on the adhesive bond, a lack of analytical model governed by adhesive strength (model AF) compatibility in the bonding zone may potentially jeopardise corresponds very closely to the results of the FEM (99%); the entire system. in case of the model governed by the glass strength (model GF), the prediction is 6% lower than that obtained through 6 Conclusions numerical modelling. In the absence of an existing model for the shear failure of glass, an analogy with the softening of The effectiveness of post-tensioning in enhancing the in- concrete in shear has been assumed. The shear strength of plane bending behaviour of a laminated glass beam with an glass, τ , has been determined based on a 2D numerical adhesively bonded flat stainless steel tendon has been dis- g,max simulation of the release of pre-load. The obtained value has cussed taking into account the failure mechanisms that may been verified by means of LEFM calculations and applied in cause premature failure of the system during post-tensioning the assumed simplified model for shear stress-slip relation- or upon release of the applied pre-load from the post- ship in glass. In order to improve the understanding of the tensioning set-up. Certain failure mechanisms, such as the mechanism of glass failure in shear and enhance the proposed rupture of the tendon and glass failure at the top glass edge, analytical model, release tests should be performed by grad- can be easily avoided with adequate detailing and simple ually releasing the pre-stress applied on the tendon until the structural verifications (the complexity may increase taking first crack in the glass appears, while monitoring the relative into account the effects of load duration and temperature). slip between the glass and the tendon. Failure at load introduction has been investigated in more The parametric analysis of the effectiveness of the post- detail in order to determine a safe pre-load level that can tensioned glass beam system has shown that the maximum be applied on the tendon prior to bonding, without initiating level of compressive pre-stress in glass which can be attained adhesive failure or glass fracture upon release. Allowable through post-tensioning increases with tendon area fraction pre-load can be determined based on the provided analytical and increased flexibility of the bondline, achieved through models, which showed good correlation with the numerical increased adhesive thickness and strain capacity and lower model of the release of pre-load, in terms of stress distribu- adhesive stiffness. tion in the tendon and the adhesive. Glass shear resistance 123 248 J. Cupace ´ tal. has been verified by means of LEFM calculations; further Bedon, C., Louter, C.: Finite Element analysis of post-tensioned SG- laminated glass beams with adhesively bonded steel tendons. investigations into shear-slip behaviour of glass by means Compos. Struct. 167, 238–250 (2017). https://doi.org/10.1016/j. of release tests are advised for better understanding of this compstruct.2017.01.086 failure mechanism. Bos, F., Veer, F., Hobbelman, G., Louter, C.: Stainless steel reinforced Parametric study of the main beam parameters has shown and post-tensioned glass beams. In: ICEM12—12th International Conference on Experimental Mechanics, Bari (2004) that the effectiveness of the system, i.e. the level of the Broek, D.: Elementary Engineering Fracture Mechanics, 4th edn. Mar- attained compressive pre-stress in glass, increases with adhe- tinus Nijhoff Publishers, Dordrecht (1986) sive thickness and tendon area fraction (for a uniform shear Cokragan, M.: Etude expérimentale des poutres en verre précontraintes stress distribution across the adhesive thickness and negligi- par adhésifs. Master’s thesis, EPFL (2015) Cupac, ´ J.: Post-tensioned glass beams. Ph.D. thesis, EPFL (2017). ble peeling stresses). In terms of the choice of the applied https://doi.org/10.5075/epfl-thesis-7895 adhesive, high shear modulus and limited shear deformation Cupac, ´ J., Louter, C., Nussbaumer, A.: Flexural behaviour of post- capacity may lead to glass fracture at beam ends; therefore, tensioned glass beams: Experimental and analytical study of three increased flexibility of the joint should be sought through beam typologies. Compos. Struct. (2021). https://doi.org/10.1016/ j.compstruct.2020.112971 lower adhesive stiffness and plastic deformation of the adhe- Débonnaire, M.: Post-tensioned glass beams. Master’s thesis, EPFL sive in the load introduction region, as it will increase the (2013) efficiency of the system by distributing the stress peaks which EN 10088-1: Stainless steels—Part 1: List of stainless steels. CEN may initiate premature failure. (2005) EN 1992-1-1: Eurocode 2: Design of concrete structures—Part 1-1: General rules and rules for buildings. CEN (2004) Acknowledgements The authors would like to thank the Swiss National EN 572-1: Glass in building Basic soda lime silicate glass products Science Foundation for funding the present research through SNF Part 1: Definitions and general physical and mechanical properties. Grants 200021_143267 and 200020_159914. CEN (2004) Engelmann, M., Weller, B.: Residual load-bearing capacity of Funding Open Access funding enabled and organized by Projekt spannglass-beams: effect of post-tensioned reinforcement. Glass DEAL. Struct. Eng. 4(1), 83–97 (2019). https://doi.org/10.1007/s40940- 018-0079-4 Haldimann, M., Luible, A., Overend, M.: Structural Use of Glass. Compliance with ethical standards IABSE, Zürich (2008) Irwin, G.: Crack-extension force for a part-through crack in a plate. Conflict of interest On behalf of all authors, the corresponding author J. Appl. Mech. 29(4), 651–654 (1962). https://doi.org/10.1115/1. states that there is no conflict of interest. Jordão, S., Pinho, M., Martins, J.P., Santiago, A., Neves, L.C.: Open Access This article is licensed under a Creative Commons Behaviour of laminated glass beams reinforced with pre-stressed Attribution 4.0 International License, which permits use, sharing, adap- cables. Steel Constr. 7(3), 204–207 (2014). https://doi.org/10. tation, distribution and reproduction in any medium or format, as 1002/stco.201410027 long as you give appropriate credit to the original author(s) and the Lindqvist, M.: Structural Glass Strength Prediction Based on Edge Flaw source, provide a link to the Creative Commons licence, and indi- Characterization. Ph.D. thesis, EPFL (2013). https://doi.org/10. cate if changes were made. The images or other third party material 5075/epfl-thesis-5627 in this article are included in the article’s Creative Commons licence, Louter, C., Nielsen, J.H., Belis: Exploratory experimental investigations unless indicated otherwise in a credit line to the material. If material on post-tensioned structural glass beams. In: ICSA, Guimarães, pp. is not included in the article’s Creative Commons licence and your 358–365 (2013) intended use is not permitted by statutory regulation or exceeds the Louter, C., Cupac, ´ J., Lebet, J.P.: Exploratory experimental investigation permitted use, you will need to obtain permission directly from the copy- on post-tensioned structural glass beams. J. Facade Design Eng. right holder. To view a copy of this licence, visit http://creativecomm 2(1–2), 3–18 (2014). https://doi.org/10.3233/FDE-130012 ons.org/licenses/by/4.0/. Nhamoinesu, S.: Steel-Glass Composite Panels. Ph.D. thesis, Cam- bridge (2015) References prEN 10138: Prestressing steels. CEN (2000) Schober, H., Gerber, H., Schneider, J.: Ein Glashaus für die Therme in TM TM 3M: Scotch-Weld EPX Adhesive DP490 Product data sheet. Badenweiler. Stahlbau 73, 886–892 (2004) http://www.3m.com/ (1996) Triantafillou, T.C., Deskovic, N.: Innovative prestressing with FRP Albrecht, P., Yamada, K.: Rapid calculation of stress intensity factors. sheets: mechanics of short-term behavior. J. Eng. Mech. (ASCE) J. Struct. Div. 103(1–4), 377–389 (1977) 117(7), 1652–1672 (1991) Alonso MC, Sánchez M, Mazario E, Recio FJ, Mahmoud H, Hingo- Yuan, H., Wu, Z., Yoshizawa, H.: Theoretical solutions on interfa- rani R (2010) High strength stainless steel 14301 for prestressed cial stress transfer of externally bonded steel/composite laminates. concrete structures protection. In: 6th International Conference Doboku Gakkai Ronbunshu 18(1), 27–39 (2001). https://doi.org/ on Concrete Under Severe Conditions (CONSEC ’10) vol. 2, pp. 10.2208/jscej.2001.675_27 1047–1054. http://oa.upm.es/9342/2/INVE_MEM_2010_86612. pdf Publisher’s Note Springer Nature remains neutral with regard to juris- Bedon, C., Louter, C.: Finite-element numerical simulation of the bend- dictional claims in published maps and institutional affiliations. ing performance of post-tensioned structural glass beams with adhesively bonded CFRP tendons. Am. J. Eng. Appl. Sci. 9(3), 680–691 (2016). https://doi.org/10.3844/ajeassp.2016.680.691 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Glass Structures & Engineering Springer Journals

Post-tensioning of glass beams: Analytical determination of the allowable pre-load

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Abstract

The effectiveness of post-tensioning in enhancing the fracture resistance of glass beams depends on the level of compressive pre-stress introduced at the glass edge surface that will in service be exposed to tensile stresses induced by bending. Maximum pre-load that can be applied in a post-tensioned glass beam system, yielding maximum compressive pre-stress, is limited by various failure mechanisms which might occur during post-tensioning. In this paper, failure mechanisms are identified for a post-tensioned glass beam system with a flat stainless steel tendon adhesively bonded at the bottom glass edge, including the rupture of the tendon, glass failure in tension and adhesive/glass failure in the load introduction zone. Special attention is given to the load introduction failure given that the transparent nature of glass limits the use of vertical confinement usually applied in concrete. An analytical model for determination of the allowable pre-load in post-tensioned glass beams is proposed, based on the model applied for externally post-tensioned concrete beams. The model is verified with the results of a numerical model, showing good correlation, and applied in a parametric study to determine the influence of various beam parameters on the effectiveness of post-tensioning glass beams. Keywords Post-tensioned glass beam · Pre-load introduction · Failure modes · Analytical model · Numerical model · Parametric study 1 Introduction terms of initial fracture resistance and redundancy in the post-fracture state (Bos et al. 2004; Schober et al. 2004; Post-tensioned glass beams are hybrid structural components Débonnaire 2013; Louter et al. 2013; Jordão et al. 2014; in which a ductile tendon is applied on a standard glass Louter et al. 2014; Engelmann and Weller 2019; Cupace ´ tal. section to enhance its in-plane bending behaviour. The ten- 2021). These studies have generally focused on the struc- don introduces compressive pre-stress into the glass and thus tural behaviour of post-tensioned beams in bending, which compensates for the rather low resistance of glass in tension. have been investigated experimentally and through numerical A number of studies have investigated various methodolo- modelling, where particular attention has been given to the gies of post-tensioning applied to glass beams, demonstrating modelling of the brittle fracture of glass (Bedon and Louter significantly enhanced structural performance in bending in 2016, 2017). Present study focuses on the effectiveness of post-tension- B Jagoda Cupac´ ing in enhancing the fracture resistance of glass beams which jagoda.cupac@tu-dresden.de depends on the level of compressive pre-stress introduced Christian Louter at the glass edge surface that will in service be exposed to christian.louter@tu-dresden.de tensile stresses induced by bending. The maximum pre-load Alain Nussbaumer that can be applied in a post-tensioned glass beam system, alain.nussbaumer@epfl.ch yielding maximum compressive pre-stress, is limited by a number of failure mechanisms which might occur during Institute of Building Construction, Technische Universität Dresden, August-Bebel-Straße 30, 01219 Dresden, Germany post-tensioning. This paper investigates the post-tensioning of laminated glass beams with an adhesively bonded flat Resilient Steel Structures Laboratory (RESSLab), School of Architecture, Civil and Environmental Engineering (ENAC), stainless steel tendon placed along the bottom glass edge École Polytechnique Fédérale de Lausanne (EPFL), GC B3 (Fig. 1). The tendon is first pre-tensioned by an external 495, Station 18, 1015 Lausanne, Switzerland 123 234 J. Cupace ´ tal. demonstrating by analogy that similar stress limitations may apply. Glass fracture at the top glass edge is avoided by limiting the tensile stresses induced by the eccentrically applied pre- load. Maximum tensile stress at mid-span, for the initial pre- load level P, can be assessed from the following expression, assuming full composite action in the steel-glass section P Pe σ = + z ≤ f (1) g,t,P g,t g,d A I eq c where A is the equivalent cross-sectional area of the beam, eq e is the eccentricity of the applied pre-load P from the neutral axis, I is the moment of inertia of the composite section, and z is the distance of the top glass edge from the neutral axis. Fig. 1 Schematic of the post-tensioned laminated glass beam cross- g,t section with nominal dimensions Equivalent cross-sectional area is defined as A = b h E /E (2) eq i i i g mechanism and subsequently adhesively bonded to the glass. The release of the pre-load set-up after the curing of the adhe- where b , h , E represent the width, height and Young’s i i i sive induces a compressive pre-stress and a hogging bending modulus of the considered component of the section, and E moment into the glass beam . Failure mechanisms which is the Young’s modulus of glass. The position of the neu- may occur at this stage are the following: (1) rupture of the tral axis, in reference to the top edge of the beam, can be tendon, (2) glass fracture in tension due to the eccentricity determined from the following expression of the pre-load, i.e. the hogging bending moment, (3) adhe- sive failure and (4) glass fracture caused by stress peaks in b h z E /E i i i,t i g z =  (3) the load introduction zone at beam ends. b h E /E i i i g The rupture of the steel tendon is prevented by limiting the where z is the distance from the centroid of the considered allowable stress induced by post-tensioning. In the related i,t component to the top beam edge. The inertia of the compos- field of conventional prestressing steels applied in concrete ite section is calculated according to Eq. (4), following the structures, the maximum allowable stress is restricted to 75% Steiner’s rule of the characteristic tensile strength, or 85% of 0.1% proof stress (EN 1992-1-1 2004), in order to limit the loss of pre- b h E E i i load due to stress relaxation of steel under constant strain. i 2 I = + b h z (4) c i i 12 E E g g Losses due to relaxation of prestressing steel are normally based on the value ρ , the percentage relaxation loss at where z determines the distance of the centroid of a compo- 1000 hours after tensioning at a mean temperature of 20 C, nent to the neutral axis. The contribution of the interlayer foils for an initial stress equal to 70% of the actual tensile strength in the calculation of the equivalent cross-sectional area and of the prestressing steel samples prEN (2000). Stainless steel, moment of inertia of a laminated glass beam can be neglected which is not commonly applied for prestressing, exhibits due to its several orders of magnitude lower Young’s modu- relaxation in the same order of magnitude as conventional lus, relative to the other components of the section. prestressing steels, with ρ < 8% (Alonso et al. 2010), At the release of the pre-load from the post-tensioning set-up, load introduction failure may occur in the adhesive, This method is referred to as post-tensioning (although the pre-load the glass or at the tendon-adhesive/adhesive-glass interface, is only introduced into the glass upon curing of the adhesive bond) in depending on the relative shear strength of the components analogy to post-tensioning in concrete, where post assumes an already of the load transfer. When the pre-load is too high, failure cured concrete element. Pre-tensioning generally applies to the appli- of the beam will occur at both beam ends due to high shear cation of pre-stress on a tendon, followed by curing of the concrete element and finally release of pre-stress. The laminated glass beam is stresses which develop as the load is introduced from the here considered an already formed structural element; the addition of tendon through the adhesive and into the glass. The transpar- the tendon therefore corresponds with post- rather than pre-processing. ent nature of glass limits the use of special anchorage which Given a fairly high compressive strength of glass, a failure of glass would provide vertical confinement in order to avoid this type in compression is unlikely to limit the allowable pre-load in practical of failure; thus, the design of the end zones requires special applications, as long as constructive measures are taken during fabrica- tion to avoid stability problems. attention. 123 Post-tensioning of glass beams 235 2 Analytical model of pre-load introduction The glass beam shown in Fig. 3 has a length L, height h and width b . Pre-stressed tendon is bonded at the bottom glass edge; the height and width of the tendon is h and b , t t respectively. The adhesive thickness is t . Young’s modulus of the glass beam is E , Young’s modulus of the tendon is E , and the shear modulus of the adhesive is G . The tendon t a is initially pre-stressed to a stress level of σ . Upon release of the tendon from the post-tensioning set-up, the stress at a distance x from the beam mid-length drops to σ (x ). The pre- stress is transferred into the glass through the adhesive layer, resulting in a shear stress τ(x ) at the interface, and a com- pressive stress σ (x ) at the bottom glass edge. The shear g,b stress distribution is considered uniform across the adhesive thickness; peeling stresses are assumed to be negligible for the investigated tendon thickness, i.e. not causing delami- nation. Given the relatively small thickness of the adhesive Fig. 2 Concrete beams with external pretensioned FRP sheets; fail- and the tendon, these simplifying assumptions are considered ure in the anchorage zone: a adhesive shear strength < beam shear acceptable for a derivation of a theoretical solution which strength, b adhesive shear strength > beam shear strength (Triantafil- lou and Deskovic 1991) aims to provide initial understanding of the mechanics of load-introduction in a post-tensioned glass beam system. The release of pre-stress is accompanied by a displacement in the An analytical model which describes the short-term beam components, shown in Fig. 3c (rotation, i.e. peeling, is mechanical behaviour of post-tensioning through bonded here neglected for simplicity). The initial state, just before the tendons is presented in Sect. 2. It is based on the model release, is marked with a dashed line; the solid lines indicate developed by Triantafillou and Deskovic (1991) for concrete the state of displacement just after the release. The initial beams externally post-tensioned through fiber-reinforced extension of the tendon at a distance x equals u (x ).The plastic (FRP) composite sheets bonded in the tensile zone of release of the pre-stress causes elastic shortening of the glass a structural element (Fig. 2). The model allows for determi- beam, which equals −u (x ) at the bottom glass edge, while nation of the maximum allowable pre-load, for two failure the deformation of the tendon drops to u (x ). scenarios: (1) cohesive failure of the adhesive (within the Assuming linear-elastic material behaviour, shear strain γ bulk material) in a system with superior glass shear strength, and shear stress τ can be defined as follows (2) glass fracture in a system with superior shear strength of the adhesive. Adhesive strength on both substrates is con- u − u + u t g γ = (5) sidered sufficiently high to avoid failure at the interface, assuming appropriate surface preparation prior to bonding τ = (u − u + u ) (6) t g (steel surface may be roughened with sand-paper, followed t by thorough cleaning, of both steel and glass, with iso- propyl alcohol; glass primer is applied on the glass surface Equation (6) differentiated with respect to x equals to improve adhesion). In Sect. 3, the analytical model is used for the calculation of the allowable pre-load for a beam spec- dτ G du du du a t = − + imen applied in a wider experimental study on the bending dx t dx dx dx (7) behaviour of post-tensioned glass beams (Cupace ´ tal. 2021); G σ σ a t g,b = − + the results of the analytical model are further verified with a t E E E a t t g numerical model of the investigated beam system. Finally, the model is applied in a parametric study, presented in Sect. 4,in Compressive pre-stress at the bottom glass edge, σ ,is g,b order to determine the influence of certain geometric beam assumed uniform across the width of the glass beam, for the parameters and adhesive properties on the effectiveness of the post-tensioning. The results are discussed in Sect. 5, with 3 The overall beam width includes the width of the glass plies, b ,and g,i conclusions given in Sect. 6. the thickness of the interlayers, t , and normally matches the width of int the tendon, providing equal bonding surface on the glass beam and the tendon. For the calculation of beam resistance, only the thickness of the glass plies is taken into account, b = b . g g,i 123 236 J. Cupace ´ tal. Fig. 3 Components of the model of the post-tensioned beam system; a longitudinal and b cross-section, c axial deformations at the beam end upon release of the pre-load; adapted from Triantafillou and Deskovic (1991) sake of simplicity. It can be expressed in terms of the tensile Equation (11) then becomes a second order linear homoge- stress in the tendon, σ , through the following equation neous equation b h σ b h σ ez t t t t t t g,b d τ G 1 α σ =− − = + τ = ω τ (13) g,b A I g g dx h t E E t a t g b h b h ez t t t t g,b (8) =− + σ where A I g g =−ασ G 1 α ω = + (14) h t E E t a t g where A is the area and I the moment of inertia of the g g glass beam, z is the distance from the glass centroid to the g,b A general solution of Eq. (13)isofthe form bottom glass edge, e is the eccentricity of the force acting in the centroid of the tendon (b h σ ) from the glass centroid, t t t ωx −ωx τ = C e + C e (15) 1 2 thus e = z + t + h /2, and g,b a t The coefficients C and C can be determined from the 1 2 b h ez b h t t t t g,b α = + (9) boundary conditions, which depend on the considered fail- A I g g ure mechanism. The failure is governed by the shear strength of the glass or the adhesive, whichever is lower. The follow- By substituting Eq. (8)into(7), the following is obtained ing subsections provide the solution for the allowable level of initial pre-stress in the tendon for the two failure mecha- dτ G σ 1 α a nisms. = − + σ (10) dx t E E E a t t g 2.1 Allowable pre-load governed by the shear which, differentiated with respect to x, results in strength of the adhesive (model AF) The shear stress-shear strain relationship for a thermoset d τ G 1 α dσ a t =− + (11) 2 structural adhesive is schematically shown in Fig. 4.The dx t E E dx a t g dashed line shows the behaviour of a two component meth- acrylate adhesive Araldite 2047 in a single lap shear test, The equilibrium of the tendon under tensile stress, σ , and adopted from (Nhamoinesu 2015). In the current model, the shear stress, τ , at the interface with the adhesive, can be true behaviour is approximated by a bilinear curve (solid expressed as line), describing two characteristic behaviour modes: the initial linear-elastic response up to the strain level of γ , a,el dσ h =−τ (12) followed by the perfectly plastic path leading to failure once dx 123 Post-tensioning of glass beams 237 Fig. 4 Shear stress-shear strain curve for a thermoset structural adhesive; dashed line—true behaviour based on the tests on Araldite 2047 from Nhamoinesu (2015); solid line—bilinear approximation of the stress-strain curve the strain limit γ is reached. The shear strength equals By substituting (18)into(15), the following expression is a,max τ . obtained for the shear stress distribution in the elastic zone a,max The release of the pre-load induces high shear stresses γ G a,el a at beam ends. Figure 5a shows the stress distribution along τ = sinh(ωx ), 0 ≤ x ≤ L /2 (19) el sinh(ωL /2) the beam at the limit of the shear capacity of the adhesive el (note that x = 0 is located at beam mid-span): in the elastic In the plastic zone, L /2 ≤ 0 ≤ L/2, the shear stress is el range 0 ≤ x ≤ L /2, the stress distribution is described by el constant; however, the shear strain is assumed to follow the Eq. (15); for L /2 ≤ x ≤ L/2, the shear stress equals τ . el a,max same distribution as in the elastic zone, hence The corresponding shear strain equals γ at x = L /2, a,el el and γ at x = L/2. The coefficients C and C can be a,max 1 2 a,el determined from the following boundary conditions γ = sinh(ωx ), 0 ≤ x ≤ L/2(20) sinh(ωL /2) el The length of the elastic zone, L , follows from the condition el τ(x = 0) = 0 (16) γ(x = L/2) = γ a,max γ(x = L /2) = τ(x = L /2)/G = γ (17) el el a a,el β + β + 4 2ln L = , where (21) el resulting in 2γ a,el β = sinh(ωL/2) (22) a,max γ G γ G a,el a a,el a C = and C =− (18) 1 2 2sinh(ωL /2) 2sinh(ωL /2) el el Fig. 5 Stress distribution along the beam at the limit of the adhesive shear capacity; a shear stress at the interface; b tensile stress in the tendon; adapted from Triantafillou and Deskovic (1991) 123 238 J. Cupace ´ tal. Fig. 6 Stress distribution along the beam at the limit of the glass shear capacity; a shear stress at the interface; b tensile stress in the tendon; adapted from (Triantafillou and Deskovic 1991) The tensile stress distribution along the tendon, in the elas- dτ tic zone, can be obtained starting from Eq. (10); can be dx substituted with a derivative of (19) with respect to x E t ωγ t a a,el σ − cosh(ωx ) sinh(ωL /2) el σ = , 0 ≤ x ≤ L /2 t el 1 + α (23) In the plastic zone, the tensile stress linearly drops from σ (x = L /2) to zero at x = L/2 (Fig. 5b). The condition t el Fig. 7 Simplified model for shear stress-slip relationship in glass of slope continuity of σ at x = L /2 can be written as t el σ | dσ t t x =L /2 el at the lower glass edge, which results in a drop in shear stress = (24) L − L dx el x =L /2 el towards the beam end. The distribution of the shear stress is schematically shown in Fig. 6a. The elastic zone, 0 ≤ x ≤ L /2, is described by el Solving Eq. (24)for σ results in the expression for the ini- Eq. (15); in the non-linear zone, 0 ≤ x ≤ (L − L )/2, the el tial pre-stress level that will just cause failure in the adhesive fracturing behaviour is described by a softening law which upon release from the post-tensioning rig relates the shear stress at the interface (τ ) with a relative slip between the substrates (δ). In the lack of an existing model ω(L − L ) el σ = E t ωγ coth(ωL /2) + (25) for this type of failure in glass, an analogy with the softening t a a,el el of concrete in shear is assumed. A non-linear softening law is approximated with a simplified linearly descending τ − δ From σ , which can now be obtained from (23), the corre- model (Yuan et al. 2001), shown in Fig. 7. Once the fracture sponding compressive stress at the lower glass edge, σ , can g,b at the interface is initiated at τ , the stress linearly reduces g,max be calculated applying (8). with the increase of slip, reaching zero when the value of slip exceeds δ . The area below the curve presents the frac- max 2.2 Allowable pre-load governed by the shear ture energy in mode I, G , i.e. the energy dissipated in the Ic strength of glass (model GF) formation of new fracture surfaces , in case of brittle mate- rials. It should be noted that, unlike in concrete, where the When applying structural adhesives with high shear stiffness and shear strength, fracture in glass may occur at the release In crack mode I, the fracture energy (critical energy release rate) G Ic of the pre-load in the set-up, or with some delay. Once the is related to the critical stress intensity factor K by G = K /E , Ic Ic Ic shear stress at the beam end reaches the level of glass resis-   2 where E = E for plane stress state, and E = E /(1 − ν ) for plane tance in shear (< adhesive shear strength), a crack is initiated strain state (Haldimann et al. 2008). 123 Post-tensioning of glass beams 239 Linear approximation of the shear distribution in the non- linear zone can be written as ⎛ ⎞ ⎜ ⎟ τ = τ 1− , 0 ≤ x ≤ (L − L )/2(32) g,max ⎝ ⎠ el L − L el The equilibrium of the tendon under tensile stress and interface shear stress, taken as a triangle, at a distance x in the non-linear zone, equals 1 L − L el h σ = τ − x (33) Fig. 8 Shear crack at the release from the post-tensioning set-up t t 2 2 cracks propagate in mode II (sliding) in a layer above the ten- The combination of (32) and (33), solved for σ ,gives don, parallel with the interface (Triantafillou and Deskovic the expression for the distribution of the tensile stress in the 1991), pure mode I (opening) is assumed the governing mode tendon for crack propagation in glass. The observed cracking at the release of the pre-load (Fig. 8) shows an opening crack, which τ L − L g,max el propagates perpendicularly to the direction of the maximum σ = −x , 0 ≤ x ≤ (L − L )/2 t el h (L − L ) 2 principal stresses (Sect. 3.2). t el (34) Coefficients C and C in Eq. (15) can be determined from 1 2 the boundary conditions Substituting (34)into(31), after integration gives the fol- lowing τ(x = 0) = 0 (26) τ(x = L /2) = τ (27) el g,max σ τ (L − L ) τ g,max el g,max t    2 δ(x ) = x − x + (x ) E 4E h 2E h t t t t t resulting in the following expression for the shear stress dis- (35) g,max tribution in the elastic zone − (x ) + C 3E h (L − L ) t t el g,max τ = sinh(ωx ), 0 ≤ x ≤ L /2 (28) el sinh(ωL /2) For the condition δ(x = 0) = 0, (35) results in C = 0. For el δ(x = (L − L )/2) = δ ,Eq.(35) becomes el max Similarly to (23), the distribution of the tensile stress in dτ the tendon can be determined from (10), substituting with 0 2 τ (L − L ) dx σ (L − L ) el g,max el a derivative of (28) δ = − (36) max 2E 24E t t E t ωτ t a g,max σ − cosh(ωx ) The tensile stress in the tendon following from Eq. (29) G sinh(ωL /2) a el σ = , 0 ≤ x ≤ L /2 (29) t el for x = L /2 should be equal to that calculated from (34) el 1 + α for x = 0. This condition can be written as E t ωτ t a g,max The shear slip δ can be determined from the relative dis- σ − coth(ωL /2) el (L − L )τ el g,max placement of the substrates at the release of the pre-load. = (37) t 4h Assuming a fully rigid glass-adhesive system in the non- 1 + α linear zone, the slip at distance x results only from the straining of the tendon Knowing τ and δ , equations (36) and (37) can be g,max max 0 0 solved for the two remaining unknowns, the length of the d(u − u ) du du t t t t = − = ε − ε (30) t elastic zone, L , and the initial pre-stress in the tendon, σ , t el dx dx dx that will just initiate fracture in glass upon release. L can be el The shear slip follows from substituted in (29) and (34) to obtain the distribution of the tensile stress in the tendon in the linear and non-linear zone, x 0 x respectively (Fig. 6b). Finally, the compressive pre-stress at σ σ 0  t δ(x ) = (ε − ε )dx = x − dx (31) the bottom glass edge, σ , can be obtained from (8). g,b E E 0 t 0 t 123 240 J. Cupace ´ tal. Fig. 9 Shear and tensile stress distribution at the limit of the adhesive shear capacity; comparison of the analytical and numerical model (a) shear stress at the interface (b) tensile stress in the tendon 3 Numerical verification of the model 3.1 Model AF The analytical model was first applied for the calculation For the model AF, governed by the adhesive strength, Aral- of the allowable pre-load and stress distribution of a glass dite 2047-1 was selected as the reference adhesive. The beam post-tensioned through an adhesively bonded tendon parameters defining the bilinear shear stress-shear strain cur- placed along the bottom glass edge (Fig. 1). In order to verify ve were assessed based on the experimental results reported the analytical results, a numerical 2D model of the beam in Nhamoinesu (2015); the following values were adopted in was implemented in a finite element (FE) software Abaqus , the model: G = 211 MPa, γ = 2.69%, γ = 15%, a a,el a,max version 6.12-3. τ = 5.67 MPa. For the given beam properties, the initial a,max The beam comprises a triple-laminated annealed glass sec- pre-stress level that will just cause failure in the adhesive tion (6 +10 +6 mm) with a height of 122 mm and a length of upon release, σ , amounts to 363.24 MPa, i.e. the initial 1500 mm. The pre-stress is applied via stainless steel tendon pre-load P = 27.24 kN. The corresponding compressive 25 × 3 mm, grade EN 1.4301 (EN 10088-1 2005), and trans- pre-stress at the bottom glass edge, σ , equals −32.42 MPa. g,b ferred into the glass through 1.5 mm thick adhesive bond. Figure 9 shows the distribution of the shear stress in the The applied Young’s modulus of glass, E , and the tendon, adhesive, τ , and the tensile stress in the tendon, σ ,for g t E , equal 70 GPa (EN 572-1 2004) and 180 GPa [based on 0 ≤ x ≤ L/2, resulting from the release of the initial pre- uniaxial tensile tests reported in Cupac( ´ 2017)], respectively. load P (solid curves). The length of the elastic zone, L , el Two types of adhesives were considered, in order to simu- equals 1313 mm, i.e. the adhesive yielding at the beam ends late the two failure modes represented by the models AF and occurs over the initial 94 mm. GF. Adhesive properties are further detailed in the following In the numerical model, only half of the beam length L sections where the two models are investigated separately. was considered, with symmetry restraint at the mid-section nodes reproducing the effective boundary conditions. The beam components - glass, tendon and adhesive—were rep- resented with 4-node monolithic shell elements with reduced integration (S4R). A regular mesh pattern was applied, with element size of 5 mm along the beam length. Glass height By Dassault Systémes Simulia Corp. 123 Post-tensioning of glass beams 241 Fig. 10 Distribution of principal stresses in the glass (half length) at the limit of the adhesive shear capacity (FEM results) was divided in 24 elements (element size ∼ 5 × 5 mm), three lated to the bottom glass edge, equals −32.16 MPa, which elements were applied across the thickness of the adhesive closely corresponds to the value of −32.42 MPa, obtained (0.5 × 5 mm), and one element over the height of the ten- analytically. don (3 × 5 mm), resulting in a total of 4200 elements. A rigid constraint (tie) was used at the tendon-adhesive and 3.2 Model GF adhesive-glass interface. Material properties equivalent to those applied in the analytical model AF were implemented For the model GF, epoxy adhesive 3M™ Scotch-Weld™ in the numerical simulation. The initial pre-stress level, DP490 was chosen as the reference adhesive due to its rel- σ = 363.24 MPa, obtained through the analytical solution, atively high shear modulus and shear strength, compared to was applied on the tendon as a pre-defined field (mechanical/ other adhesives, such as Araldite 2047. The shear modulus stress) in the initial step of the simulation. A geometrically G equals 239 MPa (Nhamoinesu 2015), the shear strength non-linear, static incremental computation was performed τ = 30.2 MPa at 23 C, according to the manufacturer’s a,max in Abaqus/Standard. The resulting stress distribution in the data sheet (3M 1996). adhesive and the tendon is plotted in Fig. 9 (dashed curves). DP490 was applied for the post-tensioning of glass beams The stress data represents the averaged nodal values extrapo- with the same nominal parameters (Fig. 1) in the scope of lated from the integration points of the connecting elements. a master’s thesis (Cokragan 2015); glass failure was con- The stress plots resulting from the analytical and numerical sistently observed at beam ends at the release of a 15 kN simulation demonstrate a good correlation in the computa- pre-load. In order to determine the maximum shear stress tion of both shear stresses in the adhesive layer and tensile at the interface which initiated glass fracture, τ , i.e. the g,max stresses in the tendon. The distribution of the principal shear resistance of glass, the release of the pre-load was sim- stresses in glass is shown in Fig. 10. The load-introduction ulated in a 2D numerical model in the present study. zone (Fig. 10b) is subjected to a complex stress state, which The results of the numerical model were further applied in tends to a linear stress-distribution over the beam height, as the calculation of the stress intensity factor (SIF), K , based the pre-stress is gradually introduced into the glass. In the on the approach proposed by Albrecht and Yamada (1977). mid-section, the stress varies linearly from tension at the top The procedure is based on the linear superposition princi- edge to compression at the bottom (Fig. 10c). The maximum ple (Broek 1986) used in linear elastic fracture mechanics value of compressive pre-stress in the FE model, extrapo- (LEFM) calculations to derive the SIF from an uncracked 123 242 J. Cupace ´ tal. Fig. 11 Surface crack in a semi-infinite body with a a uniform and b non-uniform stress distribution along the crack depth FE model, with the assumption that the cracking does not significantly influence the global stiffness of the component. The correction factor Y is divided in two parts, Y = Y Y , s g where Y accounts for the crack shape and the proximity of boundaries in a cracked body with a uniform stress dis- tribution, and Y is the correction factor for the local stress gradient due to the geometry of the modelled structural detail (Fig. 11). The expression for the SIF therefore equals K = Y (Y σ π a) = Y K (38) I s g s where Y = 1.12 for a shallow surface crack in a semi-infinite solid (Irwin 1962). The value of the SIF K , which contains Fig. 12 Stress gradient along the crack path perpendicular to the direc- the correction factor Y , can be determined in two steps by tion of the maximum principal stresses (1) computing the stresses in an uncracked model along a line where the anticipated crack will be inserted and (2) inte- soft double-sided adhesive pads were applied between the grating the normal stresses along the same line, for a given glass and the tendon to prevent the spread of adhesive to the crack depth, by applying the following expression glass beam corner edge). The mesh was refined in the zone of the load-introduction, to allow for the computation of K , 2 σ(x ) K = π a √ dx (39) I with 0.003 mm elements over an area of 0.4 × 0.2 mm. The 2 2 0 a − x element size was gradually increased towards the edges of the beam, to a maximum size of 5 mm, resulting in a total where a is the crack depth, σ(x ) is the stress distribution of 28648 elements. Initial pre-stress of 200 MPa was applied along the anticipated crack path, and x is the location along on the tendon elements, which corresponds to a 15 kN axial the crack path. For discrete values of stress obtained from the pre-load. FEM, (39) becomes The maximum shear stress of 8.28 MPa, obtained from the √ numerical model, was adopted as the glass shear resistance, 2 x x i +1 i K = π a σ arcsin − arcsin (40) τ , for this specific geometry and mechanism of load- g,max π a a i =1 introduction. For a known τ , the maximum shear slip, g,max δ , can be derived from the simplified τ − δ relationship max where σ is the discrete stress normal to the crack path, shown in Fig. 7 applied over the element width from x to x , and summed i i +1 over the total number of elements along the crack depth a. 2G Ic The numerical model applied for the simulation of pre- δ = (41) max g,max load introduction at the verge of adhesive failure was adapted 2 2 K (1 − ν ) by changing the material properties of the adhesive; DP490 g Ic G = (42) Ic was modelled as linear-elastic, with Young’s modulus E = 660 MPa and Poisson’s ratio ν = 0.38 (Nhamoinesu 2015). The adhesive was applied witha5mmoffsetfromthe where fracture toughness K = 0.75 MPa m and Pois- Ic beam end, corresponding to the bonding layout applied in son’s ratio ν = 0.23 (Haldimann et al. 2008). The resulting the experiments to avoid stressing the glass edge (5 mm long maximum shear slip equals δ = 1.8µm. max 123 Post-tensioning of glass beams 243 Fig. 13 Variation of the allowable initial pre-stress level, resulting compressive pre-stress on the bottom glass edge and length of the elastic zone with respect to the adhesive thickness and tendon area fraction (model AF) 123 244 J. Cupace ´ tal. The first path of the initial crack was assumed perpendic- height and beam length, and adhesive properties—strain ular to the glass edge surface, starting in the vicinity of the limit (model AF) and shear modulus (model GF). The study maximum shear stress at the interface; the SIF computed for was performed considering the nominal beam properties a crack length a = 0.2 mm equals 0.58 MPa m. The sec- described in Sect. 3, varying one of the parameters. The ond crack path followed a line perpendicular to the direction initial pre-stress applied on the tendon at the verge of the of the maximum principal stresses, as the most unfavourable adhesive/glass failure was calculated for each beam config- case for the effective glass resistance. In the observed refined uration. area of 0.4 × 0.2 mm, the angle equals 45 , measured coun- terclockwise from the glass edge surface. Further into the 4.1 Model AF global model of the beam, the direction of the principal stresses gradually changes (Fig. 10b), resulting in the angle of The results of the parametric study of the maximum allowable (visible) crack propagation of ∼ 30 (Fig. 8). The SIF com- pre-load governed by the adhesive failure are presented in puted along the 45 inclined crack path reached a value of Figs. 13, 14 and 15. 0.73 MPa mfor a = 0.28 mm, which closely corresponds Figure 13 shows the initial pre-stress applied on the ten- to the fracture toughness of glass (K = 0.75 MPa m), don, σ , resulting compressive pre-stress in glass, σ , and Ic g,b demonstrating that the applied 15 kN pre-load may initi- ratio of the length of the elastic zone over the total length, ate glass fracture. Although the assumed initial crack length L /L, with respect to the adhesive thickness and tendon el is rather large for a polished glass edge [Lindqvist (2013) area fraction, i.e. the cross-sectional area of the tendon, A , reported initial crack size in the range of 0.015 to 0.1 mm], a expressed as a percentage of the glass section, A . The limit result in the same order of magnitude is considered accept- of the initial pre-stress is set to 75% of the ultimate tensile able, given a large scatter of glass edge quality which depends strength of the tendon in order to avoid excessive stress relax- on the manufacturing process and varies among glass sup- ation. This amounts to 650 MPa for the stainless steel bars pliers. Stress gradient along the crack path at 45 is plotted employed in this research (based on uniaxial tensile tests in Fig. 12. reported in Cupac( ´ 2017)). It can be seen that the allowable For the nominal beam properties and τ = 8.28 MPa, initial pre-stress level increases with the adhesive thickness, g,max the analytical model yields a 7% lower maximum initial while it decreases with the increase in tendon area fraction pre-load P = 14 kN, at the verge of glass failure; the (Fig. 13a, b). However, a larger tendon area yields a higher corresponding compressive pre-stress at the bottom glass initial pre-load, P (Fig. 13b). Therefore, the compressive edge, σ , equals -16.64 MPa, compared to -17.69 MPa pre-stress at the bottom glass edge increases with both the g,b obtained numerically. This can be explained by the conser- adhesive thickness and tendon area fraction (Fig. 13c, d). An vative assumption of a fully rigid tendon-to-glass connection increase in both parameters results in a decrease in the ratio in the non-linear zone of the analytical model, while the of the elastic length, i.e. an increase in the yield zone in the FE model assumes linear-elastic adhesive behaviour over the adhesive (Fig. 13e, f). entire bond length. The variation of the compressive pre-stress in glass with For a better qualification of the shear resistance of glass, respect to the beam length is shown in Fig. 14. An increase release tests should be performed, in which the pre-load is in the pre-stress can be seen up to an effective bond length gradually released into the beam through a bonded tendon, at which the full pre-load is introduced into the glass; fur- while monitoring the relative shear displacement along the ther increase in beam length does not affect the resulting interface. The results in terms of shear-slip curve upon ini- compressive pre-stress. For the nominal dimensions of the tial glass failure (softening) could then be compared to the investigated beam specimen, 99% of the maximum compres- provided model, in order to validate the simplified linear soft- sive pre-stress at mid-length is achieved with a beam length ening law and the corresponding assumptions of the beam of L = 655 mm. behaviour. This was, however, not performed in the scope of Figure 15 shows the dependency of the post-tensioning the present study. system on the strain limit capacity of the applied adhesive. Similarly to the effect of the adhesive thickness, an increase 4 Application of the model in a parametric in the adhesive strain limit enhances the maximum level of study initial pre-stress in the tendon (Fig. 15a) and the achieved compressive pre-stress in glass (Fig. 15b), since the yielding The analytical models AF and GF were applied in a paramet- of the adhesive increases the overall flexibility of the joint, ric study in order to analyse the effectiveness of the inves- diminishing excessive stress peaks at load introduction. Con- tigated post-tensioned glass beam system, i.e. the achieved sequently, the ratio of the length of the elastic zone over the compressive pre-stress at the bottom glass edge, with vary- total beam length decreases with a higher yielding capacity ing geometric beam parameters—adhesive thickness, tendon of the adhesive (Fig. 15c). 123 Post-tensioning of glass beams 245 Fig. 14 Relationship between the compressive pre-stress at the bottom glass edge and beam length at the verge of the adhesive failure Fig. 15 Variation of the allowable initial pre-stress level, resulting compressive pre-stress at the bottom glass edge and length of the elastic zone with respect to the adhesive strain limit (model AF) 123 246 J. Cupace ´ tal. Fig. 16 Variation of the allowable initial pre-stress level and resulting compressive pre-stress at the bottom glass edge with respect to the adhesive thickness, tendon area fraction and adhesive shear modulus (model GF) 123 Post-tensioning of glass beams 247 4.2 Model GF The plastic deformation capacity of the adhesive has an important influence on the allowable pre-load level by reduc- The results of the parametric study of the maximum allow- ing the shear stress peaks. Even with a very high plastic strain, able pre-load governed by glass failure are shown in Fig. 16. the yield zone in the adhesive remains limited to a relatively It can be seen that the increasing adhesive thickness, t , small fraction of the total bond length (Fig. 15c). A complete positively influences the allowable initial pre-stress level, absence of the plastic zone in the adhesive is explored in the σ , resulting in a higher compressive pre-stress at the bot- model GF, governed by glass failure. The stress peaks are tom glass edge, σ , thus increasing the efficiency of the hence much higher; the assumed shear resistance of glass of g,b applied post-tensioning method (Fig. 16a, 16b). An increase 8.28 MPa is reached with a pre-load of 14 kN (analytically). in the tendon area relative to the cross-sectional area of glass, For comparison, in the adhesive failure (AF) model, a shear A / A , similarly provides a higher compressive pre-stress resistance limit of the adhesive of 5.67 MPa is reached with a t g in glass, achieved through an increasing initial pre-load, P, pre-load of 27.24 kN, considering a max. strain limit of 15%. which corresponds to a decrease in the initial pre-stress in the Plastic deformation is in this sense very beneficial for the tendon (Fig. 16c, d). A significant increase in the efficiency functioning of the system, while the limited yield zone does of the system can be achieved by applying adhesive with not pose a risk for the exploitation of the beam in bending, a lower shear modulus, G , assuming that sufficiently high as long as the added shear deformation in bending is consid- shear resistance of the adhesive is maintained (Fig. 16e, f). ered in the design. This, however, falls out of the scope of the present study which focuses on the pre-load introduction stage. 5 Discussion In order to determine the long-term behaviour of the pro- posed beam system, creep and relaxation behaviour of the Numerical verification of the proposed analytical model has constituent materials should be included in the model. In shown that the model can be applied with sufficient accu- particular, given the viscoelastic nature of the adhesive, load- racy for the prediction of short-term mechanical behaviour duration and temperature may affect the level of initially of post-tensioned glass beams, in terms of stress distribu- applied pre-load transferred into the glass, resulting in a lower tion in the tendon and the adhesive, and determination of the efficiency of the system in the long term. Chemical com- maximum compressive pre-stress that can be achieved in the patibility of the adhesive and the interlayer material should glass without causing premature failure at pre-load introduc- also be investigated; given that the pre-load introduction and tion. The maximum compressive pre-stress predicted by the composite action fully rely on the adhesive bond, a lack of analytical model governed by adhesive strength (model AF) compatibility in the bonding zone may potentially jeopardise corresponds very closely to the results of the FEM (99%); the entire system. in case of the model governed by the glass strength (model GF), the prediction is 6% lower than that obtained through 6 Conclusions numerical modelling. In the absence of an existing model for the shear failure of glass, an analogy with the softening of The effectiveness of post-tensioning in enhancing the in- concrete in shear has been assumed. The shear strength of plane bending behaviour of a laminated glass beam with an glass, τ , has been determined based on a 2D numerical adhesively bonded flat stainless steel tendon has been dis- g,max simulation of the release of pre-load. The obtained value has cussed taking into account the failure mechanisms that may been verified by means of LEFM calculations and applied in cause premature failure of the system during post-tensioning the assumed simplified model for shear stress-slip relation- or upon release of the applied pre-load from the post- ship in glass. In order to improve the understanding of the tensioning set-up. Certain failure mechanisms, such as the mechanism of glass failure in shear and enhance the proposed rupture of the tendon and glass failure at the top glass edge, analytical model, release tests should be performed by grad- can be easily avoided with adequate detailing and simple ually releasing the pre-stress applied on the tendon until the structural verifications (the complexity may increase taking first crack in the glass appears, while monitoring the relative into account the effects of load duration and temperature). slip between the glass and the tendon. Failure at load introduction has been investigated in more The parametric analysis of the effectiveness of the post- detail in order to determine a safe pre-load level that can tensioned glass beam system has shown that the maximum be applied on the tendon prior to bonding, without initiating level of compressive pre-stress in glass which can be attained adhesive failure or glass fracture upon release. Allowable through post-tensioning increases with tendon area fraction pre-load can be determined based on the provided analytical and increased flexibility of the bondline, achieved through models, which showed good correlation with the numerical increased adhesive thickness and strain capacity and lower model of the release of pre-load, in terms of stress distribu- adhesive stiffness. tion in the tendon and the adhesive. Glass shear resistance 123 248 J. Cupace ´ tal. has been verified by means of LEFM calculations; further Bedon, C., Louter, C.: Finite Element analysis of post-tensioned SG- laminated glass beams with adhesively bonded steel tendons. investigations into shear-slip behaviour of glass by means Compos. Struct. 167, 238–250 (2017). https://doi.org/10.1016/j. of release tests are advised for better understanding of this compstruct.2017.01.086 failure mechanism. Bos, F., Veer, F., Hobbelman, G., Louter, C.: Stainless steel reinforced Parametric study of the main beam parameters has shown and post-tensioned glass beams. In: ICEM12—12th International Conference on Experimental Mechanics, Bari (2004) that the effectiveness of the system, i.e. the level of the Broek, D.: Elementary Engineering Fracture Mechanics, 4th edn. Mar- attained compressive pre-stress in glass, increases with adhe- tinus Nijhoff Publishers, Dordrecht (1986) sive thickness and tendon area fraction (for a uniform shear Cokragan, M.: Etude expérimentale des poutres en verre précontraintes stress distribution across the adhesive thickness and negligi- par adhésifs. Master’s thesis, EPFL (2015) Cupac, ´ J.: Post-tensioned glass beams. Ph.D. thesis, EPFL (2017). ble peeling stresses). In terms of the choice of the applied https://doi.org/10.5075/epfl-thesis-7895 adhesive, high shear modulus and limited shear deformation Cupac, ´ J., Louter, C., Nussbaumer, A.: Flexural behaviour of post- capacity may lead to glass fracture at beam ends; therefore, tensioned glass beams: Experimental and analytical study of three increased flexibility of the joint should be sought through beam typologies. Compos. Struct. (2021). https://doi.org/10.1016/ j.compstruct.2020.112971 lower adhesive stiffness and plastic deformation of the adhe- Débonnaire, M.: Post-tensioned glass beams. Master’s thesis, EPFL sive in the load introduction region, as it will increase the (2013) efficiency of the system by distributing the stress peaks which EN 10088-1: Stainless steels—Part 1: List of stainless steels. CEN may initiate premature failure. (2005) EN 1992-1-1: Eurocode 2: Design of concrete structures—Part 1-1: General rules and rules for buildings. CEN (2004) Acknowledgements The authors would like to thank the Swiss National EN 572-1: Glass in building Basic soda lime silicate glass products Science Foundation for funding the present research through SNF Part 1: Definitions and general physical and mechanical properties. Grants 200021_143267 and 200020_159914. CEN (2004) Engelmann, M., Weller, B.: Residual load-bearing capacity of Funding Open Access funding enabled and organized by Projekt spannglass-beams: effect of post-tensioned reinforcement. Glass DEAL. Struct. Eng. 4(1), 83–97 (2019). https://doi.org/10.1007/s40940- 018-0079-4 Haldimann, M., Luible, A., Overend, M.: Structural Use of Glass. Compliance with ethical standards IABSE, Zürich (2008) Irwin, G.: Crack-extension force for a part-through crack in a plate. Conflict of interest On behalf of all authors, the corresponding author J. Appl. Mech. 29(4), 651–654 (1962). https://doi.org/10.1115/1. states that there is no conflict of interest. Jordão, S., Pinho, M., Martins, J.P., Santiago, A., Neves, L.C.: Open Access This article is licensed under a Creative Commons Behaviour of laminated glass beams reinforced with pre-stressed Attribution 4.0 International License, which permits use, sharing, adap- cables. Steel Constr. 7(3), 204–207 (2014). https://doi.org/10. tation, distribution and reproduction in any medium or format, as 1002/stco.201410027 long as you give appropriate credit to the original author(s) and the Lindqvist, M.: Structural Glass Strength Prediction Based on Edge Flaw source, provide a link to the Creative Commons licence, and indi- Characterization. Ph.D. thesis, EPFL (2013). https://doi.org/10. cate if changes were made. The images or other third party material 5075/epfl-thesis-5627 in this article are included in the article’s Creative Commons licence, Louter, C., Nielsen, J.H., Belis: Exploratory experimental investigations unless indicated otherwise in a credit line to the material. If material on post-tensioned structural glass beams. In: ICSA, Guimarães, pp. is not included in the article’s Creative Commons licence and your 358–365 (2013) intended use is not permitted by statutory regulation or exceeds the Louter, C., Cupac, ´ J., Lebet, J.P.: Exploratory experimental investigation permitted use, you will need to obtain permission directly from the copy- on post-tensioned structural glass beams. J. Facade Design Eng. right holder. To view a copy of this licence, visit http://creativecomm 2(1–2), 3–18 (2014). https://doi.org/10.3233/FDE-130012 ons.org/licenses/by/4.0/. Nhamoinesu, S.: Steel-Glass Composite Panels. Ph.D. thesis, Cam- bridge (2015) References prEN 10138: Prestressing steels. CEN (2000) Schober, H., Gerber, H., Schneider, J.: Ein Glashaus für die Therme in TM TM 3M: Scotch-Weld EPX Adhesive DP490 Product data sheet. Badenweiler. Stahlbau 73, 886–892 (2004) http://www.3m.com/ (1996) Triantafillou, T.C., Deskovic, N.: Innovative prestressing with FRP Albrecht, P., Yamada, K.: Rapid calculation of stress intensity factors. sheets: mechanics of short-term behavior. J. Eng. Mech. (ASCE) J. Struct. Div. 103(1–4), 377–389 (1977) 117(7), 1652–1672 (1991) Alonso MC, Sánchez M, Mazario E, Recio FJ, Mahmoud H, Hingo- Yuan, H., Wu, Z., Yoshizawa, H.: Theoretical solutions on interfa- rani R (2010) High strength stainless steel 14301 for prestressed cial stress transfer of externally bonded steel/composite laminates. concrete structures protection. In: 6th International Conference Doboku Gakkai Ronbunshu 18(1), 27–39 (2001). https://doi.org/ on Concrete Under Severe Conditions (CONSEC ’10) vol. 2, pp. 10.2208/jscej.2001.675_27 1047–1054. http://oa.upm.es/9342/2/INVE_MEM_2010_86612. pdf Publisher’s Note Springer Nature remains neutral with regard to juris- Bedon, C., Louter, C.: Finite-element numerical simulation of the bend- dictional claims in published maps and institutional affiliations. ing performance of post-tensioned structural glass beams with adhesively bonded CFRP tendons. Am. J. Eng. Appl. Sci. 9(3), 680–691 (2016). https://doi.org/10.3844/ajeassp.2016.680.691

Journal

Glass Structures & EngineeringSpringer Journals

Published: Mar 30, 2021

Keywords: Post-tensioned glass beam; Pre-load introduction; Failure modes; Analytical model; Numerical model; Parametric study

References