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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]
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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.
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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.
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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
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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
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