Peer-reviewed & Open access journal www.academicpublishingplatorms.com Finite element modeling incorporating non-linearity o material behavior ATI - Applied Technologies & Innovations Volume 5 Issue November pp.5-6 Finite element modeling incorporating nonlinearity o material behavior based on the ib Model Code Han Ay Lie, Joko Purnomo Civil Engineering Department, Diponegoro University, Indonesia Researcher, Structural and Material Laboratory, Diponegoro University, Indonesia e-mails: ayliehan@indosat.net.id, a.joko.purnomo@gmail.com Non linearity is a prominent characteristic o most cement-based material. This nonlinear behavior is observed even at very low loading levels. When strain sotening is present, the increase in loading will result in a decrease o structural stiness. Most existing programs, including SAP, takes into account geometric nonlinearity, but assumes a constant stiness modulus throughout the loading process. This will result in a less accurate outcome, and can urther signiicantly inluence analysis o the overall behavior o the structure. A Finite Element Program written in the Visual Basic programming language was developed to take into account nonlinear behavior o the modulus o elasticity and the Poisson Ratio, as a unction o increasing principal stresses. The results o this program were validated by laboratory tested specimens to compare the load-deormation response and accuracy o the model. The Federal Institute o Technology, Europe Model Code was used to model the material behavior and ailure criterion. Keywords: Modulus, principal stresses, nonlinearity, FEM. Introduction Finite element modeling is a process o subdividing all systems into their individual components, whose behavior is readily understood, and then rebuilding the original system rom such components, to study its behavior (Cook et al., ; Bathe, ; Zienkiewicz et al., 6). In analysis, an idealization o the real system to a orm that can be analyzed based on equilibrium equations is constructed. From thereon, the obtained results are interpreted. The stiness matrix o the structural system is assembled as: T T [ K ] = [ k] = [ B] [ E][ B] dv = t [ B] [ E][ B] Γ rds () V The properties o material are included by the introduction o the material constitutive matrix [E] as: Where: [E] - is the material properties constitutive matrix E - is the modulus o elasticity, were: I 3 oct = () - 5 - Prague Development Center
Finite element modeling incorporating non-linearity o material behavior E c - is the material modulus o elasticity in compression in MPa E t - is the material modulus o elasticity in tension in MPa E ct - is the material modulus o elasticity in compression-tension in MPa υ - is the Poisson s ratio o material Material nonlinearity is incorporated through the [E] matrix. As the material stiness modulus decreases under increasing load, the corresponding structural stiness undergoes a sotening response. At low loading levels the stiness modulus tend to be linear, but at higher loading the behavior becomes signiicantly nonlinear. Assuming a constant stiness modulus will lead to substantial deviation rom the real behavior. Most inite element analysis models integrate geometric nonlinear behavior, but assume a constant stiness modulus and Poisson s ratio throughout the overall loading stage. The CEB-FIB Model Code deals with material nonlinearity in detail. The inite element model developed is based on the material constitutive model and ailure criteria as proposed by this code. The model is written in the Visual Basic language and constructed as a two dimensional plain stress model. Constitutive models or cementitious materials The most recent CEB-FIB Code is based on the CEB-FIB Report 8 (Bulletin 4, Task Group 8.) and research conducted by Ottosen (977, 979); Hillerborg (983); Vecchio and Collins (986) and Dahl (99). Modulus o elasticity The material behavior or concrete is expressed either in terms o the three stress invariants (,, 3) = in the Haigh-Westergaard system, or in the hydrostatic system as; ( ξ, ρ, θ ) =. Tensile stresses are signed positive and compression negative. Expressed in terms o their invariants, the unction becomes ( I, J,cos3 θ ) =. Ottosen (977) constructed a our parameter model or compression behavior that was validated by numerous experimental test results. The model is written as: J I = λ + λ 4 A( B ) cm A cm (3) Applied Innovations and Technologies Where, J - is the second invariant o the deviatoric stress tensor = S S ij I - is the irst invariant o stress tensors = ii λ - is a coeicient as a unction to K and K A, B, K and K - are Ottosen (977) parameters, as a unction o tm k = cm - is the uniaxial compressive strength in MPa rom experimental test results cm ji Prague Development Center - 53 -
tm Finite element modeling incorporating non-linearity o material behavior - is the uniaxial tensile strength in MPa rom experimental test results Further, Ottosen (979) developed an algorithm to incorporate nonlinearity by introducing the nonlinearity index β, relating the actual minor principal stress in compression to the stress at ailure state in compression. At ailure the value o β =. β = (4) Where: - is the minor principal compression stress in MPa, < - is the compression ailure stress o concrete in MPa assuming a constant For this research work, the ailure surace is urther transormed to the octahedral plane ( oct, τ oct). The transormation o the meridian system to the octahedral stresses is as ollowing: I 3 oct = and oct J τ = (5) 3 The ailure suraces based on experimental data are presented in Figure. The value o having a constant is derived rom the octahedral -τ relationship based on the Mohr s circle theorem. FIGURE. FAILURE SURFACE TRANSFORMATION FROM THE MERIDIAN TO THE OCTAHEDRAL PLANE τ oct cm oct cm - 54 - Prague Development Center
Finite element modeling incorporating non-linearity o material behavior Octahedral ailure curve τ = D + E τ oct = (I ) ( i, τ i ) α E oct = (J ) + The expression o the secant modulus under multi-axial loading is generated rom the stress-stain curves developed by Sarin and urther elaborated by Ottosen (979). E Jc Ec = = εc E J + Ec cm The modulus o elasticity or concrete in compression E c at any given stress level can be calculated using the ollowing equation. 3 c = β ( ) ± β ( ) β E E E E E E E E The two equations (6) and (7) are used to generate the stiness modulus at any loading stage in the Finite Element model. For tensile behavior, the code proposed a bilinear unction. The irst branch holds till 9% to the tension strength tm, than micro cracks signiicantly reduce the stiness o the material. Stresses and deormations in the acture process are described using a stress-crack opening diagram (Hillerborg, 983). The equations incorporated into the FEM are: (6) (7) Applied Innovations and Technologies E t Et. tme =.5E.9 (8) = E or.9 tm tm E or.9 tm < tm (9) Where: E - is the initial Young s Modulus in compression MPa E c - is the uniaxial secant modulus at ailure in compression in MPa E t - is modulus in tension as a unction o strain cm - is the uniaxial compressive strength in MPa Prague Development Center - 55 -
Finite element modeling incorporating non-linearity o material behavior tm - is the uniaxial tensile strength in MPa β - is the nonlinearity index rom (4) J - is the second invariant o the deviatoric stress tensor - is the major principal tensile stress in MPa rom When non uniorms stresses (tension and compression) are present, the work o Vecchio and Collins (986) is used to modiy the stiness modulus (Figure ). For elements in combined tension and compression, the relationship is expressed as: max = ε / ε cm cm.8.34 ε ε = max ( ) ( ) and = ε cm ε cm () () FIGURE. STRESS-STRAIN BEHAVIOR FOR BIAXIAL STRESSES (VECCHIO AND COLLINS, 986) cm max ε ε max cm max = ε / ε cm cm.8.34-56 - Prague Development Center
Finite element modeling incorporating non-linearity o material behavior From the equations, the principal stresses taking into account the biaxial stress eect, can be calculated. The secant modulus o elasticity is calculated. E ct ( ε ε ) cm = Ecm.8ε cm.34ε and = () Where: - is the major principal stress in tension in MPa - is the minor principal stress in compression in MPa cm - is the concrete compressions strength in uniaxial compression in MPa E cm - is the modulus o elasticity at peak stress in MPa ε cm - is the strain at peak stress in uniaxial compression ε - is the principal tensile strain at the i th iteration ε - is the principal compression strain at the i th iteration Poisson s ratio The Poisson s ratios or concrete based on tests results vary rom.4 to.6 which all within the elastic range (FIB Bulletin Nr. 55, ). To account or non-linearity, the ollowing equations are introduced (Ottosen, 979; CEB-FIB Bulletin Nr. 4, 8). The non-linear Poisson s ratio υ c is a unction o β (Figure 3). ϑ ϑ c = or i c = ( i ) β ϑ ϑ ϑ ϑ β β (3) β β Where: υ i - is the initial Poisson s ratio taken as. β o - is the initial non-linearity index taken as.8 υ - is the secant Poisson s ratio rom experimental test results υ c - is the Poisson s ratio taking into account non-linearity or β > β (4) FIGURE 3. POISSON S RATIO BEHAVIOR Applied Innovations and Technologies Non-linearity index β..8 υ i β o υ. υ c Prague Development Center - 57 -
Finite element modeling incorporating non-linearity o material behavior As or tensile and tensile-compression behavior, the stresses exhibited are low, and the Poisson s ratio is taken as to be the initial Poisson s ratio υ i. Failure criteria The ailure criterion is evaluated based on the principal stresses at the Gauss points. Failure criterion is distinguished as either crushing, or racture o the matrix. Based on the Kuper-Hilsdor-Rusch s (969) ailure envelope crushing will occur in the third quadrant, when all principal stresses are in compression. The irst quadrant is racture due to tension, while the remaining quadrants characterize the tension-compression ailure. Fracture o a Gauss point under a certain loading condition, will inluence the stiness o its element, and a reduction in the element stiness matrix will be resulted. Progressive incremental loading will lead to ailure o one or more Gauss point up till collapse o the element as a whole. Compression behavior The ultimate strength under biaxial compression is higher than the strength under uniaxial compression, resulting in a 5% to % increase o the uniaxial compression strength. The relationship is expressed by an ellipse in the third quadrant (Kuper et al., 969; Dahl, 99; Hampel et al., ; 9). This ellipse depends on the two parameters a and b that characterize the radii o this ellipse. The center o the ellipse is expressed as c. All these parameters are a unction o the uniaxial compression strength cm. The unction is written as: ( + c) ( ) cm cm cm cm + = a b (5) With: a = i ( cm); b = ( cm); c = 3( cm) i ( cm)=a i ( cm) +B i cm+c i with i =,,3 A i, B i, C i are coeicients (Hampel et. al., ) When stress combinations at any loading stage exceed the value in equation (5), the Gauss point under consideration has ailed in compression. Tensile behavior The ailure envelope o the biaxial tensile stresses lies in the irst quadrant o the Kuper- Hilsdor-Rusch s curve. Since the envelope is a symmetrical square, the ailure criteria can be analyzed based on the uniaxial tensile strength. A crack will orm when the principal major tensile strain exceeds the ultimate tensile strain o the material. Compression-tension behavior The biaxial compression-tension ailure envelope can be approached by a linear relationship. The intersection point between the compression-compression and the tensioncompression area can be considered as a biurcation point. A boundary o 5% is assumed to the compression margin, to identiy the ailure criteria to the compression-compression ailure. - 58 - Prague Development Center
Finite Element Modeling Finite element modeling incorporating non-linearity o material behavior In the inite element analysis, the model chosen is a our node quadrilateral, having x Gauss Points. The numbering o elements ollows the designations as shown in Figure 4. FIGURE 4. FOUR NODE QUADRILATERAL ELEMENT Y Node point s Gi s=+ s=+ r s=s=- Gauss point 4 r=+ X 3 r=- r=- r=+ The notation X and Y reer to the global coordinate system, while r and s reer to the natural coordinate system. The notation G i or i = to 4 are denoting the Gauss points in each element. In order to incorporate the nonlinear material behavior, the program should update the stiness modulus o the material or ever loading increment. At primary stages, linear behavior is assumed, and the constitutive material matrix [E] is set as constant. The initial tangent stiness and Poisson s ratio are used. For the next increments, the stiness matrix is adjusted to the actual secant stiness, generated rom the material behavior under biaxial stress condition. The resulting load-displacement curves are validated against the outcome o an identical structure ran through the SAP program. As the two resulting curves coincide, the program in considered valid or a constant [E]. Further, nonlinearity is introduced by reconstructing the [E] matrix to adjust the secant stiness modulus o the material, at each loading stage. This stiness modulus modiication is based on the principal stresses, acting on the Gauss point. The combinations o principal stresses are distinguished as tension, compression or tension-compression. Upon reaching convergence, the stress ailure criteria based on the Kuper-Hilsdor- Rusch s curve is evaluated. I the particular Gauss point has ailed, the stiness matrix or this point is set to zero. When this zero matrix is assemblage into the structural matrix, a reduction in stiness is resulted. The procedure is repeated, and at every loading increment, the stresses in the Gauss points will increase, while the material stiness will decrease as a result o the sotening o the concrete stress-strain curves. As soon as all our Gauss points in an element have ailed, the element is taken out o the structure. The remaining nodes are re-arranged and numbered. The process is repeated, consequentially in the reduction o the size o the structural stiness matrix. The program is developed to show the stages o ailure at Gauss points, and all resulting data such as stresses and strains, both in the global system as well as in their principal direction, are recorded. For the model discussed in this paper (Davies and Nath, 967), Prague Development Center - 59 - Applied Innovations and Technologies
Finite element modeling incorporating non-linearity o material behavior only hal o the structure was generated, since the model was symmetrical. This approach will provide the opportunity or smaller elements. Meshing was perormed by the mesh generator QUAD beta-version developed in Australia by Dr. Alexander Tsvelikh. The program consist o two base programs: the QB and QPRO4, the later is to observe the mesh structure and to evaluate the outputs. The generator is based on the Lagrangian (Joseph Louis Lagrange, 83) analysis. Results and evaluation The FEM program was validated with the experimental data o Davies and Nath (967) who perormed lexure tests on plain-concrete beams size 4x4x. The supports were placed at a distance o 8 apart, and the two point loadings were positioned at 6 rom the supports. The concrete compression strength was measured to be 54 lb/inch, and the ratio o tensile-to-compression strength was 9.%. The material behavior model was constructed using the CEB-FIB code. The compression strength, initial stiness modulus, the modulus at peak stress and ailure were inputted into the FEM. The Poisson s ratio was taken as.3. To validate the model in the elastic range, an identical model was run by the SAP program. At this linear stage, the program s load-displacement curve coincides perectly with both the SAP outcome as well as the experimental test results. Then material non linearity was introduced and the resulting curves, compared. The size, geometrical coniguration and number o elements in a structure will inluence the outcome o the inite element analysis. Using non uniorm meshing patterns have the advantage that areas in tension, which are more susceptible to early ailure, can be meshed closely so that this area is observed in depth. To analyze the inluence o the meshing pattern to the load-displacement curves predicted by the Finite Element Model program, a range o meshing types were evaluated (Table ). TABLE. INFLUENCE OF MESHING ANALYSIS MODEL LIST Notation Element-to-structure Geometric element size area ratio and area A :3.7x.7/ 6.3 Coniguration A :8 6.3x6.3/ 4.3 A3 :64.7x.7/ 6.3.7x6.35/ 8.7 A4 :8 5.4x5.4/ 645. The loading increment is set as a constant and is / o the ultimate experimental load. For all our meshing types the angles are taken 9 degrees and the area dierence between adjacent elements are within the guideline boundaries o 5 to % to avoid poor elements. The resulting curves are presented in Figure 5. - 6 - Prague Development Center
Finite element modeling incorporating non-linearity o material behavior FIGURE 5. LOAD-DISPLACEMENT CURVE COMPARISON 6 inch 6 inch 8 inch Based on the results, it can be concluded that the ideal element size or this FEM program lies between 65 to 6 mm, thus an element-to-structure area ratio o /8 to /64, and with an element dimension o.5 to 5 mm is ideal. It is thereore suggested that the element meshing or the developed FEM program ollows the ollowing guidelines:. An element area o 5 to 5 mm, with an element-to-structure area ratio o / to /5 to avoid instability o the structure at high stress stages.. Square elements or trapezoids with a ratio o length-to-width ratio close to one, and interior angles approaching 9 to avoid poor elements 3. A progressive loading increment o % to the ultimate load at loading or the initial loading up till 3% o the ultimate load; a % loading increment or stages o loading till 7% o the ultimate load; and a 5% loading increment or stages above these levels, up till ailure. 4. The dierence between adjacent elements should be within the guideline boundaries o 5 to %. The model showed a remarkably close prediction to the actual behavior. It is thereore concluded that the CEB-FIB code in conjunction with the developed FEM program is accurate and versatile to predict material non linearity. Applied Innovations and Technologies Reerences Bathe, K-J., 6. Finite element procedures, First edition, Prentice-Hall Cook, R., Malkus, D., Plesha, M., and Witt, R.,, Concepts and applications o inite element analysis, Fourth Edition, John Wiley and Sons Dahl, K., 99. Rapport 7.6, Project 7, Uniaxial stress-strain curves or normal and high strength concretes, Department o Structural Engineering, Technical University Denmark Prague Development Center - 6 -
Finite element modeling incorporating non-linearity o material behavior Davies, J. and Nath, P., 967. Complete load-deormation curves or plain concrete beams, Building Science, Vol., pp. 5-, Pergamon Press Hampel, T., Scheerer, S., Speck, K. and Curbach, M.,. High strength concrete under biaxial and triaxial loading, Proceeding o the 6th International Symposium on Utilization o High Strength/High Perormance Concrete, Vol., Leipzig, Germany, pp.7-36 Hillerborg, A, 983. Analysis o one single crack, in: Wittman, F. (Ed.), Facture mechanics o concrete, development in civil engineering 7, Elsevier Science Publisher, Amsterdam, pp.3-49 Kuper, H., Hilsdor, H., and Rusch, H., 969. Behavior o concrete under biaxial stresses, American Concrete Institute Journal, Proceedings Vol.66, No.8, August, pp.656-66 Ottosen, N. S., 977. A ailure criterion or concrete, Journal o the Engineering Mechanics Division, ASCE, Vol.3, No.EM4, pp.57-35 Ottosen, N., 979. Constitutive model or short-time loading o concrete, Journal o the Engineering Mechanics Division, ASCE, Vol.5, No.EM, pp.7-4 Task Group 8., CEB-FIB, 8. State-o-art report on Constitutive modeling o high strength/high perormance concrete, International Federation or Structural Concrete, Switzerland Vecchio, F. and Collins, M., 986. The modiied compression-ield theory or reinorced concrete elements subjected to shear, ACI Journal Proceedings, Vol.83, No., pp.9-3 Zienkiewicz, O., Taylor, R., and Zhu, J., 6. The inite element method or solid and structural mechanics, Sixth Edition, Elsevier Butterworth-Heinemann, Burlington, UK FIB Bulletin, Nr.55 and 56,. Model Code, First Complete Drat, Vol. and, Federal Institute o Technology, Lausanne, Switzerland - 6 - Prague Development Center