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Recent Advances in Engineering Influence of Shear Forces on Deformation of Structural Glass Beams ONDREJ PESEK, JINDRICH MELCHER Institute of Metal and Timber Structures Brno University of Technology Veveri 331/95, 602 00 Brno CZECH REPUBLIC [email protected]; [email protected] http://www.kdk.fce.vutbr.cz Abstract: - This paper deals with actual behaviour of beams of structural glass. Simple beam was modeled in software based on finite element method. Due to the manufacturing process of structural glass components the cross-section is rectangular. Set of rectangular cross-section was modeled with variable aspect ratio ranging from 20 to 0,03. Numerical models were realized using space, planar and one dimensional finite elements and both linear and nonlinear solutions were performed. Results from numerical modeling were compared with analytical calculation results. In analytical computation deflections due to bending and shear were considered. Key-Words: - Structural glass beam, glass stairs, glass building envelope, bending deflection, shear deflection, nonlinear analysis, final element method Advanced structural glass structures are designed in today's, for example steel-glass composite structures or complex spatial structures. Fixing of glass members requires special attention because of local stresses which glass is not able to redistribute. There is difference from conventional materials as steel and concrete. 1 Introduction Currently glass structures are designed as load carrying, not as earlier, when they were used only for transparent building envelops. Glass structures are designed for active load bearing same as traditional concrete and steel structures. Beams are the simplest structures. Staircase steps (rectangular cross-section subjected on weak axis Fig. 2) and beams bearing glass roofs or floors or buildings envelops (cross-section subjected on rigid axis – Fig. 1) are the most frequently occurring cases. Fig. 1 Rigid axis subjected glass beam [2] ISBN: 978-1-61804-137-1 Fig. 2 Weak axis subjected glass beam [2] 240 Recent Advances in Engineering tensile strength of glass is not a material constant, but it depends on many aspects (on the condition of the surface, the size of glass element, the action history, the residual stress, the environmental conditions etc.) [1]. The compressive strength of glass is much larger than the tensile strength, because surface flaws do not grow of fail when element is in compression. The compressive strength is irrelevant for all structural applications, because an element’s tensile strength is exceeded long before it is loaded to its compressive strength [1]. Flaws behaviour in annealed and tempered glass is shown in Fig. 3. Base glass product is float glass (annealed glass, ANG). By secondary processing thermal treatment we get heat strengthened glass (partly toughened glass, HSG) and fully tempered glass (tempered glass, FTG). Laminated (safety) glass is consisting of two or more glass panes (it may be ANG, HSG or FTG) which are joined by PVB-foil. 2 Structural glass A glass is an inorganic product of fusion, which has been cooled to a rigid condition without crystallization. Most of the glass used in construction is soda lime silica glass (SLSG). For fire protecting glazing and heat resistant glazing is borosilicate glass (BSG) used. One of the most important properties of glass is its excellent chemical resistance to many aggressive substances which makes glass one of the most durable materials in construction [1]. 3 Problem Formulation If member is sufficiently slender, their deformation state is given by shape of bending line. Bending line is curve which was originally straight axis of the beam before loading. Fig. 3 Structural glass behaviour [1] In Table 1 there are summarized the most important physical properties of soda lime silica glass. Glass shows an almost perfectly elastic, isotropic behaviour and exhibits brittle fracture. It does not yield plastically, which is why local stress concentrations are not reduced through stress redistribution as is the case for other construction materials like steel [1]. Quantity Density Young's modulus Poisson's ratio Symbol ρ E ν Table 1 Unit 3 kg/m MPa - Value 2500 70 000 0,23 Fig. 4 Bending theory [3] Rotation φ = φy is angle between the x-axis and the tangent to the bending line. Angle is very small (φ<<1) in small deflections theory, so that dw (1) ϕ ≈ tgϕ = = w′ dx One of the most important properties of any structural material is the strength. The theoretical tensile (and compressive) strength of glass is exceptionally high and may reach 32 GPa (this value is based on molecular forces). The relevant tensile strength for engineering is much lower. The reason is that glass is brittle material. The tensile strength of glass depends very much on mechanical flaws on the surface. But flaws are not necessarily visible to the naked eye (effective flaw depth may be between 10-6 and 10-1 mm). A glass element fails as soon as the stress intensity due to tensile stress at the tip of one flaw reaches its critical value. The ISBN: 978-1-61804-137-1 3.1 Bending deflection If influence of shear forces on shape of the bending line is neglected, so we can write w′′ = − My EI y (2) Second order differential equation of bending line. 241 Recent Advances in Engineering Designation and positive values Quantity Differential dependence for EI = const. Deflection w Rotation ϕ = w′ Bending mom. M = −EIw′′ Shear forces V = −EIw′′′ Transverse load q = EIw IV perpendicularity of section with respect to deformed axis is not valid here. The assumption about uniformly distributed shear stress in section of bending beam is contrary to the Grashof's equation. Actually section will collapse (warping) as you can see at Fig. 5. This incompatibility is possible to eliminate by substituting reduced cross-sectional area A*. Size of A* is derived by energetical methods. For rectangular cross-section is A* = 0,833A. Table 2 Bending moment at general section is 1 q M = M (x ) = Ax − qx 2 = 2 x − x 2 2 2 Substituting into (2) and successive integration q EIw′′ = − M = − 2 x − x 2 2 ( ( ) q x2 x3 EIw′ = − l − + C1 2 2 3 q x3 x4 EIw = − l − + C1 x + C 2 2 6 12 ) (3) Fig. 5 Shear deformation of element [3] (4) With this correction we can get rotation due to shear forces φV as shear strain γ (=γxz) from Hooke's law ϕV = γ xz = (5) τ xz G = VZ GA* (9) Because equation (1) is still valid, differential equation for deflection due to shear forces can be written as (6) wV′ = Integration constants are obtained from boundary conditions – deflections at both supports are zero. VZ GA* ⇒ GA* wV′ = VZ For our simple beam shear forces are l V = V z = q − x 2 l GA * wV′ = q − x 2 w(0) = 0 ⇒ C2 = 0 (10) (11) q l3 l4 w(l ) = 0 ⇒ − l − + C1l + C 2 = 0 2 6 12 1 3 ⇒ C1 = ql 24 By substituting constants into (6) we obtain q (7) wM (x ) = x l 3 − 2lx 2 + x 3 24 EI Problem is symmetrical, maximal deflection is in mid-span – substitute x = l/2 q l 3 l 2 l3 l l − 2l + = wM , max = w = 4 8 2 24 EI 2 5 ql 4 l (8) wM , max = w = 2 384 EI Integration constant is obtained from boundary conditions – deflections at both supports are zero. 3.2 Shear deflection q l l 1 l2 l = wV ,max = w = − * 2 GA 2 2 2 4 2 l 1 ql wV ,max = w = * 2 8 GA ( l x2 GA* wV = q x − + C1 2 2 ) (13) wV (0 ) = 0 ⇒ C1 = 0 Substituting constant into (13) we obtain wV ( x ) = q l x2 x − GA* 2 2 (14) Problem is symmetrical, maximal deflection is in mid-span – substitute x = l/2 For determination of deflections due to shear forces we are assumption that all sections will remain planar after loading and they will deformation by uniform shear. This model is called Timoshenko's beam. Bernoulli’s assumption about ISBN: 978-1-61804-137-1 (12) 242 (15) Recent Advances in Engineering 4.1 Analytical solution Simple beam with span L = 1,8m was modeled with set of various rectangular cross section. Everyone cross section has the same moment of inertia Iy = 675.103 mm4. Aspect ratio b/h is ranging from 20 to 0,03. Totally were modeled sixteen variable cross sections. In Fig. 7 are shown some shapes of crosssection. Fig. 7 Cross-sections shapes In Table 3 are listed every cross-sections, which were used in analytical solution and numerical models. Reduced area A* in equation (15) is 0,833A. Fig. 6 Bending and shear deflection [3] 3.3 Superposition Provided that there are small deformations, we can use the principle of superposition. Than total deflection at mid-span due to bending moment and shear forces is wmax = wM ,max + wV ,max = 5 ql 4 1 ql 2 + 384 EI 8 GA* (16) 4 Problem solution Influence of beam stability (lateral flexural buckling) is not considered. The structure is designed so that lateral buckling cannot occur. Glass units are modeled as monolithic. Actually deflections achieve little higher values, because of no perfect shear connection – only for laminated glass. ISBN: 978-1-61804-137-1 Width Thickness b [mm] 500,00 400,00 300,00 200,00 121,61 100,00 53,35 50,00 30,00 23,40 20,00 10,00 9,49 8,00 6,00 4,00 h [mm] 25,30 27,26 30,00 34,34 40,54 43,27 53,35 54,51 64,63 70,21 73,99 93,22 94,87 100,41 110,52 126,51 Area A [mm2] 12651 10903 9000 6868 4930 4327 2846 2726 1939 1643 1480 932 900 803 663 506 Table 3 Section modulus W [mm3] 53353 49529 45000 39311 33304 31201 25305 24764 20887 19228 18247 14482 14230 13444 12215 10671 Aspect ratio b/h [-] 19,761 14,675 10,000 5,824 3,000 2,311 1,000 0,917 0,464 0,333 0,270 0,107 0,100 0,080 0,054 0,032 Curve of cross-section area depending on aspect ratio b/h is shown in Fig. 8. Is obvious that rectangular cross-section subjected on weak axis have much more larger area than cross-section subjected on rigid axis. 243 Recent Advances in Engineering Cross-sectio n area [mm2] 14000 elements (SHELL181) (SOLID45). 12000 and 3D elements 10000 8000 6000 4000 2000 0 0 5 10 15 Aspect ratio b/h [mm] 20 Fig. 8 Shape’s cross-section area The results of analytical solution are in Table 4. Deflection wM due to bending is the same for all shapes because it is depend on moment of inertia (equation 8). Deflection wV due to shear increases with reduced area A* decreasing (equation 15). Total deflection wtotal from equation (16) are in next column in the table. In the last column are values of quotient of total deflection and shear deflection in percents. Aspect ratio b/h [-] 19,7605 14,6752 10,0000 5,8239 3,0000 2,3112 1,0000 0,9172 0,4642 0,3333 0,2703 0,1073 0,1000 0,0797 0,0543 0,0316 Deflections wM [mm] 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 4,339 wV wtotal [mm] [mm] 0,002 4,341 0,002 4,342 0,003 4,342 0,004 4,343 0,005 4,344 0,006 4,345 0,009 4,348 0,009 4,349 0,013 4,353 0,016 4,355 0,017 4,357 0,027 4,367 0,028 4,368 0,032 4,371 0,039 4,378 0,051 4,390 Table 4 Percentage of shear deformation wV/wtotal [%] 0,047 0,054 0,066 0,086 0,120 0,136 0,207 0,216 0,304 0,358 0,398 0,630 0,652 0,730 0,883 1,154 Fig. 9 Numerical models meshing In Fig. 9 and Table 5 meshing of beam into finite elements is shown. Number of elements along the x, y and z axis is designated nx, ny and nz respectively. Each beam was divided on 1000 elements. Load was modeled as uniform surface stress load at 2D and 3D problem. At 1D problem it was modeled as uniform line load. Dimens ionality 1D 2D 3D nx ny nz Σni 1000 100 10 10 10 10 1000 1000 1000 Table 5 Each beam was computed by linear and nonlinear analysis. Deflections at mid-span, normal stresses in the extreme fibers and displacements of movable support of simple beam were observed. Numerical modeling solutions were compared with analytical solutions and they are shown in Fig. 10 and 11. 4.2 Numerical modeling This problem was modeled in ANSYS software based on finite element method. Each beam was modeled by 1D elements (BEAM3), 2D ISBN: 978-1-61804-137-1 Finite element BEAM3 SHELL181 SOLID45 244 Recent Advances in Engineering 5 Conclusion -60 In figure 10 is shown relationship between aspect ratio b/h and defection at mid-span. Black line indicates analytical solution, Blue line indicates 2D FEM solution and red line indicates 3D FEM analysis. The difference between linear and nonlinear solution is negligible. The shape of the blue curve is very similar to the black line (analytical solution) for aspect ratio ranging from 0,03 to 1,0 (for cross-sections subjected on rigid axis). Difference between curves is significant for higher values of aspect ratio (cross-sections subjected on weak axis). Analytically computed deflection is still decreasing but deflection computed numerically is increasing. 2D and 3D curves have the same shape but 3D curve is placed below a certain constant. This constant is approximately 0,08 mm. The deflections computed by finite element method using 1D elements (green line) are the same as for analytical computed deflections due to bending wM. -50 20,00 -20 -10 2,00 -30 0,20 Normal stress σx [MPa] 0,02 -40 0 10 20 30 40 50 60 Aspect ratio b/h [mm] Analytical sigma x + Analytical sigma x - 3D linear FEM sigma x + 3D nonlinear FEM sigma x + 3D linear FEM sigma x 3D nonlinear FEM sigma x - 2D linear FEM sigma x + 2D nonlinear FEM sigma x + 2D linear FEM sigma x 2D nonlinear FEM sigma x - Fig. 11 Results – normal stresses Acknowledgements: This paper has been supported by the Czech Ministry of Education, Youth and Sports in specific research FAST-J-12-24/1699 and within the frame of the projects OPVK CZ.1.07/2.2.00/15.0428. 4,40 4,38 4,36 4,34 4,32 4,30 References: [1] Haldimann, M., Luible, E., Overend, M.: Structural Use of Glass, Zurich, ETH Zurich, 2008, ISBN 3-85748-119-2. [2] Glassdesignandbuild: glass floor. GLASS DESIGN AND BUILD: INSPIRATIONAL GLASS STRUCTURES [online]. [cit. 2012-0913]. Accessible from: http://www.glassdesign andbuild.co.uk/Glass-floor.html. [3] Smirak, S.: Elasticity and Plasticity I. CERM Brno, 2006, ISBN 80-7204-468-0. [4] Pesek, O., Melcher, J.: Study of Behaviour of Beams and Panels Based on Influence of Rigidity. In Proceedings of Steel Structures and Bridges 2012, Podbanske, Slovakia, 2012, ISBN 978-80-89619-00-9. [5] Melcher, J., Karmazinova, M.: The Experimental Verification of Actual Behaviour of the Glass Roofing Structure under Uniform Loading. In Proceedings of EUROSTEEL 2005 4th European Conference on Steel and Composite Structures, volume B, Maastricht, 2005, ISBN 3-86130-812-6. 4,28 25,00 2,50 0,25 0,03 4,26 Aspect ratio b/h [-] analytical wTOT analytical wM 1D linear FEM 1D nonlinear FEM 2D linear FEM 2D nonlinear FEM 3D linear FEM 3D nonlinear FEM Fig. 10 Results- deflections Normal stresses due to bending are shown in Fig. 11. The waveform of all curves is similar. The normal stress computed 2D analysis is similar as for analytical computed stress for aspect ratio b/h = 0,03 – 1,0. For other cross-sections difference is higher (0,5 percent). The difference between analytical solution and 3D numerical solution is ranging from 1,2 % to 2,0 %. Stresses from mathematical models are lower than from analytical computation. The shear forces have more influence at short and high beams and they are usually neglected because of their low values. ISBN: 978-1-61804-137-1 245