Geometry Non-Linearity and Performance Base Design Procedure Relevant to Predict the Seismic Performance of Low-Rise Building and Structures in
a Developing Country (Indonesia)
G.A. SUSILA a, P. MANDAL b & T. SWAILES b
a School of Mechanical Aerospace and Civil Engineering, The University of Manchester, Manchester, M13 9PL, UK E-mail : [email protected]
b School of Mechanical Aerospace and Civil Engineering, The University of Manchester, Manchester, M13 9PL, UK
E-mail : [email protected] E-mail : [email protected]
Abstract: This paper reports numerical investigation of typical structures in Indonesia, either of non-engineered low-rise buildings or engineered structures by employing non-linear geometry analysis to determine the seismic performance and failures. Performance base design procedure utilized to predict behaviour of the structures: traditional timber frame, such masonry buildings (unconfined or confined by reinforcement concrete framing) and structural component of bridge. Results of simulation of non-linear Finite Element Model (FEM) are represented with validation between the numerical and experimental works. The results expected to confirm; modification procedure introduced, failure mode and carrying load capacity of the structures. A parallel study involving experimental and numerical work {(Adi S., 2012a); (Adi S., 2012b)} has indicated that performance of the structures may be predicted well. Some remarks delivered as consequences to the typical structures in Indonesia. Such an infrastructure, the collapsed of the longest bridge (Kertanegara-Indonesia) has also been examined with failure indicated of clamp connection may be the culprit of the disaster.
Key Words: Performance base design, geometric non-linearity, non-engineered & engineered structures, Indonesia.
1. INTRODUCTION
A study by using experimental and numerical work of material for non-engineering low-rise structure in Indonesia was committed. Two typical structural material used for those structures were investigated such as: masonry (Adi S., 2012b) and timber in order to determine the mechanical properties relevant for computational modelling. In numerical simulation, those mechanical material properties were needed and some material properties has also been added relevant for confined masonry structure such as local/tropical hardwood timber material, typical low-grade concrete frame and reinforcement steel bar to represent typical non- engineering structure in Indonesia.
Typical structures considered are shown in Fig.1. To measure the seismic performance of those structures is equally to find lateral load capacity and displacement format. In order to find the structural load carrying capacity, pushover analysis (Xu, 2011) was implemented as one of the prominence performance base design procedure relevant to predict the structural behaviour under seismic, dynamic wind load, so fort. In this paper, geometric non-linearity for solid material and structure (Crisfield, 2000) has also been adopted of the most rather then to use material model procedure.
(a) (b) (c) (d)
Figure 1. Schematic typical complex structures relevant for (a) confined masonry, (b &c) traditional timber frame pavilion “bale” and (d) barn “jineng”. (Adi S., 2012a)
(a) (b)
Figure 2. Shematic typical problem of (a). one (1) degree of freedom (DOF) (Crisfield, 2000) and (b). Six (6) DOF on traditional timber frame with spring stiffness, Ks described to build up typical
non-linear solution procedure.
2. DEVELOPMENT TYPICAL GEOMETRIC NON-LINEARITY AND ARC-LENGTH (RIKS METHOD).
2.1 Geometric Non-Linearity.
By adopting typical problem of one (1) DOF in Fig.1 (a) (Crisfield, 2000) that is relevant to a
“shallow truss theory”, numerical approximation developed by using typical Pythagoras’s theorem valid to take into account of strain, ε in the bar/frame. It has to be stiffness component of structure considered which are a structural member of area, A and Young’s modulus, E that is subject to a load P to produce movement to measure a displacement of w. By assuming of θ is so small, N is axial force in the member, lateral equilibrium of the structure developed as (Crisfield, 2000):
l w z N l
w z N N
P ( )
"
)
sin ( +
+ ≅
=
= θ (1)
2 2
2
2 2 2
2 1 '
' '
) (
+
= +
+
− +
= +
l w l
w l z l
z
l z l w
ε z (2)
2
2 1
+
=
= l
w l
w l EA z EA
N ε (3)
Eq.2 is relevant to non-linear problem. Relation between the load P and displacement w given:
2 ) 1 2
( 2 3 2 3
3 z w zw w
l
P= EA + + (4)
Tangent stiffness matrix take over the role once small change of load or displacement are taken into account, it can be defined as (Crisfield, 2000):
l N w N l
w z w
Kt P +
∂
∂
= +
∂
= ∂ ( )
l
N l
w zw l
EA l
z l
Kt EA +
+
+
= 2
2 2
2 (5)
2.2 Arc-Length / Riks Method
Non-linear geometry is affordable to find solutions on numerical practices and for structural analysis due to the increasing computer power today, therefore, analysis of non-linear finite element analysis is becoming potentially more advantages taken by engineer and researcher.
The benefits are also potentially reduced cost for experimental works. The physical experiment can be slightly avoided by using it. The ability of solutions provided is complemented to such direct simulation of collapse behaviour into a structure with no extra cost on physical test (Crisfield, 2000). This method is typically ‘incremental’ procedures in which assembling of
“geometric stiffness matrix” is associated with an updating of coordinate system on structural geometry and in conjunction of initial displacement matrix. It is adopted by ABAQUS (Abaqus, 2012) which is known as an extended Riks-Method with modified Newton Rapson iteration apart from “material non-linearity” such as: elasto-plastic, plastic material model (Hill-Model, Draker-Prager Model, etc). Fig. 3 shows typical expected numerical iteration, predictor and corrector (iteration) method to predict the structural behaviour This method is intended to predict non-linear behaviour of structure.
Figure 3. Reproduced typical arch length (a) Orthogonality method and (b) (Riks, 1972; 1979;
Wempner, 1971) method for non-linear FEM analysis. (Bruce WR, 1987, Feng, 1998, Ahmed, 2003) The aim of arc-length procedures is to control iteration in simulation for numerical solution of such complex non-linear problems. Abaqus adopted Riks method (Abaqus, 2012) to trace the complex path of force and displacement development into the range of elasto-plastic or into the post critical range. (Bruce WR, 1987, Feng, 1998, Ahmed, 2003). Standard equilibrium equation for proportional loading is written as follow (Bruce WR, 1987):
0 )
( =
−F x
λP (6)
Initial solution was introduced traditionally by using Newton incremental iteration in (Bruce WR, 1987) as follow:
) ( ()
)
(i u P F xi
K ∆ =λ − ; x(i+1) = x(i) +∆u (7-8)
( )
() ())
(i u P m i P F(xi ) Gi
K ∆ −∆λ = λ+λ − =− (9)
2.3 The Heat Equation and Modified Arc-Length Method
Such numerical solution of 2nd-order partial differential equation (PDE) has been overlooked relevant to the general heat equation with the used of the finite difference method (FDM). In the heat equation case, one term on left hand side (LHS) is derivatives with respect to time, and the right hand side (RHS) is derivatives with respect to space (eq. 10), it was pursuing heat transfer scheme in time dependent. The one-dimensional heat equation is classified as a parabolic in PDE relevant to B2-AC=0 (zero) in the following:
2 2
x
t ∂
= ∂
∂
∂ φ
φ α
or 2
2
dy u k d dt
du = ; 0≤x≤L ; t≥0 where φ=φ(x,t) (10)
on the interval of ∂Ω=
[
0,L]
with the used of boundary conditions (BC) take the form of:x c b
a =
∂ + ∂φ
φ ;
( )
= = = =
0)
= (c Robin
0)
= (a Newmann
) ( 0 ,
; )
, (
; )
, 0 ( );
0
(b t (0)t L t ( )t x f0 x
Dirichlet φ φ φ φ L φ
(11)
where φ or u is a function depending two variables of (x/space and t/time, α or k is thermal conductivity /diffusion coefficient, Ω is domain of BC; say
[
0,L] [ ]
= 0,1 . (Recktenwald, 2011) Solution to ordinary differential equations (ODE) of arch-length is taken to be relevant of the called “similarity solution” to the PDE. (Bergara, 2011)
(a) (b)
Figure 4. Typical grid discretization of Euler explicit (FLCD) and Euler implicit scheme of BLCD.
In this study, the similarity solution of the arc-length procedure has been modified by using solution of heat equation in which derivatives developed as grid discretization shown in Fig.4.
Such the case; the forward time, centered space (FTCS) has been changed into forward load centered displacement (FLCD) and the backward time, centered space (BTCS) into backward load, centered displacement (BLCD) to develop solution of load and displacement relationship under typical grid discretization of FDM in one dimensional. The arc-length method relevant to derivatives with respect to displacement and derivatives with respect to loads can be described as follows:
λ λ + ∆
=
∆
+ i i i i
i K u P P
Fu (12)
Forward Load Centered Displacement (FLCD); the approximation known as an explicit scheme for which stability of iteration is taken into account with results considerably of unstable for advection equation (hyperbolic). Set of finite difference method on a discrete grid with points of (ui, λi) where the discretization point of load and displacement described of the form: ui=ihu;λn=nhλand hu =du; hλ =dλ
[ ] [
i i]
i i i i i u i
i h
P P u
h u
Fu K λ λ λ
λ
− +
=
−
+ +1 −1 +1
2 ;
( )
[
1]
2 2
1 1
1 = + −
−
+ − +
+ λ
λ
λ
λ h
K h
P u h
K F u h
u i i
i i u i
i u i
i (13)
Backward Load Centered Displacement (BLCD); the approximation known as an implicit scheme for which stability of iteration is taken into account with results considerably of unconditionally stable for advection and even more, it shows stable in iteration for which amplification factor is involved.
− +
=
−
+ + − −
λ
λ λ λ
P h h P
u K u
Fu i i i i i
u i i i i
1 1
1
2 ; 1 2 1 2
[ (
1)
1]
−
−
+ = + −
−
+ i i
i i u i
i u i
i h
K h
P u h
K F u h
u λ λ λ
λ
(14)
2. RESULTS OF STUDY
Result of modified PDE schame on arc-length procedure may be brought appropriate approximation. From in-house simulation by using fortran or matlab, the result shown that adopted numerical procedure on heat transfer was providing relatively close to the expected solution. Development of tangent stiffness of Ks =∆λ ∆u is given and variation of Ks expected to be taken from the original of “strutural geometric stiffness matrix” for the structural evaluated, initial predictor started from the given first ratio of Ks and corrector was developed during the iteration related to the modified arc-length procedure in Thomas algorithm (diagonal matrix) iteration. Typical solution was developed by initial value of sinusoidal function as an amplitude. The results can be seen in Fig.5.(a).
(a)
(b)
Figure 5. (a) Results of in-house simulation and (b) Result of Abaqus simulation results for idealized confined masonry ( brittle collapse behaviour) relevan to Fig. 1.(a) (Adi S., 2012a)
Un-confined masonry (UMR) modeling was developed for two storey masonry building relevant to Fig.8. The UMR structure is 7.5m x 7.5 m in plan with height of 7.1m and masonry wall thickness of 110mm. Continuum model (Lourenco, 1996) is adopted by using shell element. Such the most vulnerable structure, the horizontal racking load (catastrophic seismic force) produced unstable condition either in-plane and out-plane into the wall. The wall behave is in a brittle manner. Therefore, commonly, reinforcement concrete framing shown in Fig.1 (a) is attached to improve strength and provide ductility. Under the given initial deformation, load capacity of UMR (Bali clay brick) has been found up to 3000 kN and the displacement predicted of 200mm at 1st floor level and at the 2nd floor level, the load was up to 2000 kN with displacement predicted maximum of 400mm. However, under the given of displacement control either for both of small or large deformation, UMR (Bali-Indonesia clay brick) has been found to be weaker at least of 10 times (Adi S., 2012b) compare to the minimum
requirement of clay brick (UMR) in seismic active area of British/Eropean Standard Masonry.
(British Standard, 2005)
(a) (b)
Figure 6. (a) Results of simulation for Shank’s experimental works on traditional Oak timber frame (J. D. Shanks, 2005). The result suggests with good agreement between numerical and experimental
works to evaluate load and difflection values.
(a) (b)
Figure 7. Results of simulation for typical traditional (Bali) timber frame of Bingkarai timber, the structure measured of 3m height and 3m bay with typical diagonal brace and mortice-tenon joint. (a)
Ductile collapse behaviour (b) Spliting failure on the beam and European Yield Model on the peg.
(a) (b)
Figure 8. Result of simulation for (a) Lateral load capacity of two storey UMR relevant to {(Yi et al., 2006); (Gerd-Jan Schreppers, 2011)}, and (b) stresses on the masonry wall (Adi S., 2012b).
3.1. Cable (Kutanegara-Indonesia) Bridge to be Vulnerable Structures to Collapse In this particular case, failure of cable bridge component was examined in which linear elastic material for iron cast taken into account in order to evaluate initial crack growth and the propagation as per Abaqus requirement. Typical displacement-controlled pushover procedure has been employed by starting with small deformation (d=10-20mm) initially and increased afterward. It has been found that stresses concentrated at the hole of the clamp and at the corner of the pinned bolt. Crack tip and directive propagation have been put on the element of the highest stress concentration. By fining mesh generation, stress and crack path was more clearly concentrated. Detail result of simulation relevant to the crack growth per (spectrum/cyclic) load rate related to range of stress intensity factor will be reported for the next publications. However, these results may be indicated that failure on the clamp and bolt suggested to be the cause of total collapse of the cable bridge.
(a)
(c)
(b)
(d)
Figure 9. Result of Abaqus simulation for failure clamp of collapsed cable bridge Kutanegara- Indoneisa (2011); (a) Photo of failures of iron cast bolt, (b) modelling development, (c-d) stresses
concentration on the clamp and at the corner of the pinned bolt.
4. CONCLUSION AND DISCUSSION
Modification on numerical procedure introduced by adopting typical parabolic PDE (heat equation), ODE of arch-length has been taken into account with “similarity solution” to the PDE. The method is limited to examine only on iteration procedure. The result of modified scheme may be confirmed as an alternative procedure during the iteration process as part of computational strategy. By enforcing control iteration, this may be remarked that displacement control is most relevant solution strategy for obtaining the trace of complex path for load- displacement problem (Crisfield, 2000), relevant to performance base design procedure.
It can be reported that most simulations were using Abaqus with geometric non-linearity under typical pushover procedure. It has been found that those procedure were simultaneously giving reasonable results, such validation, verification of numerical work may have been satisfactory to confirm the experimental result and to predict the performance of structures.
Further works will be plastic material model, time history procedure or ground peak acceleration control to G-force over displacement failures. Significant works in progress will be reported for the next publications.
ACKNOWLEDGMENT
Financial assistance under grant (Batch 2 Scholarship DIKTI) from the National Educational Council of Indonesian Republic and University of Udayana-Bali.
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