Numerical Evaluation of the Thermo-Mechanical. Response of Shallow Reinforced Concrete Beams

Transcription

1 Adv. heor. Appl. Mech., ol. 6, 2013, no. 1, HIKARI Ltd, Numerical Evaluation of the hermo-mechanical Response of Shallow Reinforced Concrete Beams Mazen Musmar Al-ahliyya Amman University Civil Engineering Department, Amman, Jordan Copyright 2013 Mazen Musmar. his is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract A precise understanding of the thermo-mechanical response of shallow reinforced concrete beams is necessary to be able to design the proper sections for the shallow flexural elements, that could serve their intended purpose, in terms of safety and serviceability requirements, keeping in my mind the dominant use of ribbed slabs with concealed shallow beams in many countries. he study involves building a finite element structural model of a shallow reinforced concrete beam for the evaluation of the structural performance and thermal cracking at different temperatures. Material nonlinearity is taken into account because of the changes in material properties experienced in fire. he more complicated aspects of structural behaviour in fire conditions, such as thermal expansion, transient state strains in the concrete, cracking or crushing of concrete, yielding of steel are modelled. Keywords: Structural modeling, fire resistance, shallow reinforced concrete beams

2 34 Mazen Musmar 1. Introduction Fire causing high temperature is a serious potential risk to buildings. Improving fire resistance requires a proper knowledge of the responses of different construction materials including concrete and steel. It also demands understanding the material damage mechanisms, and investigation of structural responses for buildings subjected to fire attack. Zhaohui H, Ian W., and Roger J. [2006] stated that in terms of reinforced concrete construction, design is still based on simplistic methods which have been developed from standard fire tests that do not necessarily represent real building behaviour. his makes it very difficult, if not impossible, to determine the level of safety achieved in real concrete structures, or whether an appropriate level of safety could be achieved more efficiently. Alternative to existing methods of determining the fire resistance of building structures is their fire behaviour modelling by means of computer aided design. According to adim Kudryashov, Nguyen hanh Kien, and Aleksandr Lupandin, [2012], the use of specialized computer codes can significantly increase the profitability of the project works and increase their efficiency. he weakness in case of modelling is that the degradation of material properties is simplified. A number of researchers have developed structural modelling approaches such as Lie and Celikkod [1991], who developed a model for the high temperature analysis of circular reinforced concrete columns. Huang and Platten [1997] developed planar modelling software for reinforced concrete members in fire. According to Kasper [2009], the reinforced concrete beams possess high resistance to high temperature, high resistance to thermal shock, and strong resistance against fire action when compared with steel beams. he main disadvantage is the very low concrete tensile strength. his study involves the aspects connected with structural modelling and the numerical evaluation of thermal stresses and deformations induced by thermal gradient affecting a shallow reinforced concrete beam. he first law of thermodynamics states that thermal energy is conserved. Specializing this to a differential control volume, ANSYS [ 2009]: ρ c = + t + { }{} L {}{} L q q = &&& Where: ρ = Density c = Specific heat = emperature t = time (1)

3 Numerical evaluation of thermo-mechanical response 35 {} = Z Y X L = vector operator { } = z y x = velocity vector for mass transport of heat {}= q heat flux vector q&= & & heat generation rate per unit volume Next, Fourier s law is used to relate the heat flux vector to the thermal gradients: { } [ ]{ } L D q = (2) Where: [ ] = Kzz K K D yy xx = conductivity matrix = zz yy xx K K K,, conductivity in the element x, y, and Z directions respectively. Combining equation1 and equation 2 results in {}{} {} [ ]{} ( ) q L D L L t c +&&& = + ρ (3) Expanding Eqn. 3 to its more familiar form: = z K z y K y x K x q z y x t c y y x z y x & && ρ (4)

4 36 Mazen Musmar Assuming that no heat generation exists in the hardened concrete, thee term && q& may be neglected. 2. Finite element analysis ANSYS is used to model the beam having dimension as illustrated in Figure (1). In orderr to eliminate the effect of stirrups on stress distribution, no shear reinforcement is provided. his is in conformance with ACI [318-11] article d; that allows not to provide stirrups as minimumm shear reinforcement for f beams with depths not greater thann 250 mm. Longitudina al reinforcement of 4# #12 mm is provided as bottom steel reinforcement, giving a total steelel area of 452 mm 2. When performing thermal analysis, concrete is representedd by Solid70 element. Solid70 will then t be replaced by Solid65 when performing structural analysis. Figure 1: Detail of shallow reinforced concretee Beam detail Figure2: Finite element modelingg

5 Numerical evaluation of thermo-mechanical response 37 Figure 3: Structural Solid65 and thermal Solid70 elements, representing concrete. A schematic of structural solid65 and thermal solid700 representing concretee is shown in figure (3). hee structural solid65 element models the nonlinear response of reinforced concrete, it models the concrete material based on a constitutive model for the triaxial behavior of concrete after Williams and Warnke [1975]. Itt is capable of plastic deformation, cracking in three orthogonal directions at each integration point. he criterion for failure of concrete due to a multi axial stress iss expressed in the form: F S 0 (5) f c Where: F: a function of principal stress state S: failure surface f : uniaxial crushing strength. c If equation (5) is satisfied, the material will crack or o crush.

6 38 Mazen Musmar Figure 4: Structural LINK180 and thermal LINK33 element geometry Bottom steel reinforcement of 4#12mm bars with total steel area of 452 mm 2 is provided. Steel reinforcement bars are represented by link33 thermal finite elements during thermal analysis, switchable to link180 structural elements on conducting structural analysis. LINK33 is a uniaxial element with the ability to conduct heat between its nodes. he element has a single degree of freedom, temperature, at each node. On performing structural analysis LINK33 is replaced by LINK180. It is 3-D spar uniaxial tension-compression element with three degrees of freedom at each node; translations in the nodal x, y, and Z directions. 3. Boundary conditions A temperature of 600 C is applied by convection with a film coefficient of 50 W/m 2 /C to the bottom face of the reinforced concrete beam, and also 25 C is applied at the upper face. Figure 5 illustrates the thermal boundary conditions.

7 Numerical evaluation of thermo-mechanical response 39 Figure5 : hermal boundary conditionss Figure 6: Structural boundary conditions According to Moaveni [2003], the numerical calculation forr temperature distribution is carried out by Ansys utilizing Galarkin finite element technique that is capable to perform heatt exchanger calculations where the thermal conductivity of the structure is taken into account. he shallow reinforced concrete beam is pin supported at the left support and roller supported at the right, as shown in figure 6. he total beam length is 1.90m, and the distance between supports is 1.80m, beam widthh is 0.3m and total depth equals 0.15m. 4. Analysiss and discussion off results A nonlinear transient thermal structural analysis is carried out taking into account the thermal dependant properties of the concretee as thermall conductivity and specific heat. he analysis is performed on the result of the solution of two types of problems. First time transient analysis is carried out to determine the t temperature distribution within the beam as a function of time. he field temperature distribution for the transient thermal analysis is then applied as a load to perform structural analysis. he obtained results of thermal analysis are plotted

8 40 Mazen Musmar in figures 7,8,9 regarding the temperature profiles, the thermal flux vector, and the t temperature distribution in steel reinforcement. Figure 7: emperature profile Figure 8: hermal Flux

9 Numerical evaluation of thermo-mechanical response 41 Figure 9: emperature distribution in reinforcement bars. able 1 lists the thermal and structural finite elements that represent concrete and steel reinforcementt in thermal and structural analysis. Nodal temperatures from the transient thermal analysis are applied at a specified time in the t subsequent steady state stress analysis. he change from thermal l to structural analysis is performed by switchingg the thermal SOLID70 elementss to structural SOLID65 elements, and the thermal LINK33 elements too the structural LINK180 elements. able 1 lists the aforementioned switchable elements. Material properties illustrated in table t 2 are used in the analysis. he relationship between the temperature variation andd the associated thermal strains is as follows: Where: ε thermal = αδ ε therma al: the thermal deformation α: the thermal coefficientt of expansion o o Δ: the thermal gradient c c 1 (6)

10 42 Mazen Musmar able 1: hermal and structural elements Switchable elements concrete Switchable elements Steel reinforcement Element hermal Structural hermal Structural ype Solid 70 Solid65 Link33 Link180 Number of nodes Number of DOF per node Nature emperature ranslations in nodal x,y,z emperature ranslations in nodal x,y,z he solid65 element models the nonlinear response of reinforced concrete, it models the concrete material based on a constitutive model for the triaxial behavior of concrete after Williams and Warnke. able 2: Material properties for concrete and steel Material Concrete Steel reinforcement Compressive strength 30 MPa ensile strength 3.78 MPa 420 MPa Elastic modulus MPa MPa Poisson s ratio Density 2400 Kg/m Kg/m 3 hermal conductivity (k) 1.2W/m o c 60W/m o c Specific heat capacity ( c) 1000J/kg o c 500J/kg o c hermal expansion coefficient α 1.2x10-5 / o c 1.08x10-5 / o c Solid65 is capable of plastic deformation; cracking in three orthogonal directions at each integration point. he cracking is modeled through an adjustment of the material properties that is done by changing the element stiffness matrices.

11 Numerical evaluation of thermo-mechanical response 43 able 3: Concrete parameters beyond initial yield surface Open shear transfer coefficient, β t Closed shear transfer coefficient, β c Uniaxial cracking stress Uniaxial crushing stresss f'c Mpa 30 Mpa If the concrete at an integration point fails in uniaxial, biaxial, or triaxial compression, the concrete is assumed crushed at that point. Crushingg is defined as the completee deterioration of the structural integrityy of the concrete. he determined displacements, deformations, internal stresses, andd strains are illustrated in figures 10, 11,12. Figure 10: Deflected shape

12 44 Mazen Musmar Figure 11: Nodal displacement for bars at midspan. Figure 12. Max nodal displacements, Uy Figure 13. Longitudinal normal stresses s

13 Numerical evaluation of thermo-mechanical response 45 Figure 14. on Mises stresses Figure 15. hermal cracking able 4: he stress and strain statee emperature at beam bottom face 600 Longitudinal tensile t normal stress MPa 2.08 on Mises MPa 16.4 Max vertical nodal displacement (mm)

14 46 Mazen Musmar he stress state in the shallow beam is determined by a number of indicators such as on Mises stresses and the longitudinal normal stresses. he strain state is determined by a number of factors such as full deformation (mechanical and thermal), and temperature deformation. able (4) summarizes number of results obtained from the coupled analysis. 5. Conclusions In a shallow beam the thermal gradient is steeper than in the case of an ordinary beam, the beam has a pronounced deflected shape and curvature. he high temperature gradient between the beam bottom face and beam top results in high values of tension stresses that would provoke cracking and undermine the beam structural integrity. he temperatures within bottom reinforcement are about 70% of the temperature at the beam base. Cracks spread and propagate close to the supports. he amount of cracks and their propagation increases as the temperature increases. he high temperature gradient between the base and the top of the beam enhances tension cracks and undermines the strength of the shallow beam. References [1] Ansys (Release 12), heory Reference for the Mechanical APDL and Mechanical applications,(2009). [2] Building Code Requirements for Structural Concrete ACI (318M-11) and Commentary, American Concrete Institute. [3] Huang, Z., and Platten, A., Non-linear finite element analysis of planar reinforced concrete members subjected to fire, ACI Structural Journal, 94(3), pp , 1997 [4] Kaspar, W, Yunping Xi, Keun Lee and Byunhum, K. hermal Response of Reinforced Concrete Structures in Nuclear Power Plants, Report no. SESM [5] Lie,.., and Celikkod, B., Method to calculate the fire resistance of circular reinforced concrete columns, ACI Material Journal, 88(1),pp.84-91, 1991

15 Numerical evaluation of thermo-mechanical response 47 [6] Moaveni, S, Finite element Analysis: heory and Application with ANSYS, Pearson Education Inc., 2003, New Jersey. [7] adim, K, Nguyen hank, K, Aleksandr Lupandin, Fire Resistance Evaluation of Reinforced Concrete Structures, Safety of echnogenic Environment, 2012 [8] William, K.J. and Warnke, E.P., Constitutive Model for the riaxial Behavior of Concrete. Proceedings of the International Association for Bridge and Structural Engineering, ol. 19, ISMES, Bergamo, Itali, 1975,pp. 174 [9] Zhaohui H, Ian W, and Roger J., Nonlinear Analysis of Reinforced Concrete Slabs Subjected to Fire, ACI Structural Journal, ol 96, No.1, 1999 Received: May 17, 2013