2D idealization of hollow partially pre-stressed concrete beams

Transcription

1 Computational Methods and Experimental Measurements XII 357 2D idealization of hollow partially pre-stressed concrete beams A. S. Alnuaimi Department of Civil and Architectural Engineering, Sultan Qaboos University, Sultanate of Oman Abstract This paper presents a finite element model for idealisation of partially prestressed concrete hollow beams using 2D plane elements. The method of ensuring compatibility between the plates using a two-dimensional model to analyze this type of structures is discussed. Cross-sectional distortion is minimised by incorporating end diaphragms in the finite element model. Experimental results from three partially pre-stressed concrete hollow beams are compared with the non-linear predictions produced by a 2D in-house finite element program. The beam dimensions were 3x3 mm cross section with 2x2 mm hollow core and 38 mm length. The beam ends were filled with concrete to form solid end diaphragms to prevent local distortion. The beams were subjected to combined bending, torsion and shear. It was found that the two-dimensional idealisation of hollow beams is adequate provided that compatibility of displacements between adjoining plates along the line of intersection is maintained and the cross-sectional distortion is reduced to a minimum. The results from the 2D in-house finite element program showed a good agreement with experimental results. Keywords: 2D analysis, pre-stressed concrete, finite element method, hollow beams, bending, torsion, shear, combined load, numerical model. Notations M d, T d, V d = design bending, torsion and shear force respectively M e, T e, V e = experimental (measured) failure bending, torsion and shear force respectively M c, T c, V c = Computed failure bending, torsion and shear force respectively

2 358 Computational Methods and Experimental Measurements XII = load factor = (M i /M d + T i /T d )/2 where i = measured incremental load L e /L c = measured to computed failure load ratio = (M e /M c + T e /T c )/2 τ tor = shear stress due torsion τ shr = shear stress due shear force ε/ε y = steel strain ratio = measured strain at load / measured yield strain f cu, f c = concrete cube and cylinder compressive strength respectively f y, f yv, f py = yield strength of longitudinal steel, transverse steel and pre-stressing wires respectively 2 Introduction Considering the complex behaviour of hollow beams, a detailed analysis would normally require a full three-dimensional finite element model especially with box-sections for large girder bridges. However, a study of the structural behaviour of typical thin-walled concrete beams indicates that the main stress conditions are those of direct stresses in the plates of the box beam. The forces involved in out-of-plane bending are very small and can be ignored. The distortion of cross-section is prevented by the use of reasonably thick plates and end diaphragms. This suggests that the main stresses are in-plane ones and, therefore, plane stress elements can be used to account for the major stresses. Zero stiffness is assumed for out-of-plane bending action of the component plates. Figure shows the state of stress in a typical box beam subjected to bending, torsion and shear. The wall thickness used for torsional resistance was 5 mm which is /6 of the beam depth Thurlimann [] and MacGregor and Ghoneim [2]. Bending Torsion Shear Figure : State of stress in a typical hollow beam. Bhatt and Beshara [3] studied the behaviour of bridge box girders using similar method with plane elements. In their work, out of plane bending was considered due to the large size of the box girder. The advantages of using 2D approach over a full 3D finite elements solution as studied by Abdel-Kader [4] is that it is easier and leads to cheaper computations while at the same time the main stresses are obtained with reasonable accuracy. In this study, a 2D in-house finite element program was used to analyse three partially pre-stressed concrete box beams. The actual beam ends were filled with concrete to form end diaphragms. The diaphragms were included in the

3 Computational Methods and Experimental Measurements XII 359 numerical analysis. The predicted behaviour was compared with the experimental results. In modelling the linear and non-linear responses of concrete as a continuum, the non-linear elasticity approach was adopted Kotsovos and Pavlovic [5] and Chen and Saleeb [6]. The above idealisation was implemented in a 2D finite element program originally developed for carrying out non-linear analysis of solid rectangular beams Bhatt and Abdel-Kader [7]. The effects of the material and numerical factors on the 2D finite element program response were presented in separate parametric studies Alnuaimi [8] and Alnuaimi [9]. Detailed description and derivation of equations used in the program as well as results from analysis of eight reinforced concrete hollow beams can be found in Alnuaimi and Bhatt []. Here, the program is used for the analysis of three pre-stressed hollow beams. The main variables in this series of beams were the ratio of torsion to bending Td/Md, which varied between.35 and 2.62 and the ratio of the shear stress due to torsion to the shear stress due to shear force τ τor /τ shr which varied between.9 and Geometrical relationship between displacements The two-dimensional idealisation of box girders is adequate provided compatibility of displacement between adjoining plates along the line of intersection is maintained and cross-sectional distortion is reduced to a minimum. To achieve these objectives, the following steps were adopted as shown in Figure 2: Top Flange Compatibility of Displacement Web Diaphragm Figure 2: Imposed displacement constraint.

4 36 Computational Methods and Experimental Measurements XII. To ensure shear transfer between adjoining plates of the beam, compatibility of displacement along the line of intersection at the common edge of adjoining plates is maintained by introducing geometrical constraints. 2. To reduce cross-sectional distortion, end diaphragms are introduced into the analysis. To illustrate this technique, consider top flange, front web and left diaphragm of a typical beam as shown in Figure 3. For ease, only corner nodes of some elements are shown in this illustration. If the out-of-plane bending is ignored, then the web and the flange can be considered as thin plates in a state of plane stress. However, the displacements of both plates along the joining line are equal to each other. The displacements perpendicular to the joining line are independent of each other. The same applies to the lines joining the plates with the diaphragm. It is therefore, necessary in this analysis to enforce geometrical constraints to ensure compatibility along the lines of intersections. Giving the same freedom number for those equal displacements does this. Other freedoms, which are independent, are given different numbers. In other words, every pair of displacements in the x-direction (direction of the beam axis) of the joining line between the flange and the web will have the same freedom number and, therefore, the same displacement value for that pair. The displacements perpendicular to this line will have different numbers. In addition to the freedom numbering, attention should be given to plate orientation when the whole structure is assembled, to prevent contradicting directions of displacements. The rigid body movement is prevented by proper restraints, which are dependent on the load conditions and support locations. A Top flange B Front web Left diaphragm C A 4 56 A B C Figure 3: Freedom constraints.

5 Computational Methods and Experimental Measurements XII 36 4 Test beams The tested beams were 3x3 mm cross section with 2x2 mm hollow core and 38 mm length. The outer 6 mm of each end was filled with concrete to make it solid to form a diaphragm to prevent distortion. The beams were subjected to bending moment, twisting moment and shear force as shown in Table. The middle 2 mm of the beam was considered as test span. This is the region where both maximum moment and shear occur and with least effect of concentrated stresses near the ends. Figure 4 shows loading and support arrangement. The beam was simply supported by a set of two perpendicular rollers at each support to allow axial displacement and rotation about a horizontal axis at the soffit of the beam. Bending and shear were direct result of mid-span point downward load while constant torsion was applied by means of the torsion arms. During testing the load was applied in increments as a percentage of the design load, % for the first three increments, in anticipation of crack initiation, and then 5% until failure. The beam was considered to have collapsed when it could resist no more loads. Table 2 shows average measured material properties. Figure 5 shows the reinforcement provided along the test span of each beam. The solid circles refer to bars or pre-stressing wires on which strain gauges were stuck while the hollow ones refer to bars or pre-stressing wires without strain gauges installed. Table : Design loads. Beam T d M d V d T d /M d τ tor /τ shr No. knm knm kn Ratio Ratio BTV BTV BTV mm Test span mm 9mm 9mm mm Figure 4: Loading and support arrangement.

6 362 Computational Methods and Experimental Measurements XII Table 2: Average measured material properties. Beam No f cu f' c f y f yv f py Unit N/mm 2 N/mm 2 N/mm 2 N/mm 2 N/mm 2 BTV BTV BTV BTV9 2 wires +3 wires Y8@2 mm BTV 2 wires +2 wires mm BTV 2 wires +2 wires mm Figure 5: Provided reinforcement in the test span. 5 Comparison between measured and computed results Here some experimental and computational results are compared. The comparison was carried out using the following criteria: Failure load and mode of failure Load displacement relationship Longitudinal steel strain Transverse steel strain 5. Failure load and mode of failure Table 3 shows the ratio of experimental to computational failure load L e /L c. A very good agreement was attained in failure loads. In all cases, the experimental results were slightly larger than the computed ones. Wide range of Td/Md and τ tor /τ shr ratios did not result in large differences between experimental and computational results. Steel yielded in most cases or reached near yield before the concrete crushed. Enough fine cracks have developed in each case long before major cracks development near the failure load. Table 3: Measured to computed failure load ratios. Beam T e (kn.m) M e (kn.m) V e (kn) L e /L c (Ratio) BTV BTV BTV

7 Computational Methods and Experimental Measurements XII Load-displacement relationship Figure 6 shows vertical displacements at mid-span of the tested beams. It is clear from this figure that, in general, an acceptable agreement between experimental and computational results was achieved BTV Disp. (mm) BTV Disp. (mm) BTV.2 EXP COMP Disp. (mm) Figure 6: Vertical displacement at mid-span.

8 364 Computational Methods and Experimental Measurements XII 5.3 Longitudinal steel strain Figure 7 shows that a good agreement between experimental and computational results was obtained for longitudinal steel strains. The reported strain ratios were closest to the mid-span of the beam L.F BTV ε/ε y BTV ε/ε y BTV.2 EXP COMP ε/εy Figure 7: Longitudinal steel strain ratios.

9 Computational Methods and Experimental Measurements XII Transverse steel strain Figure 8 shows that an acceptable agreement between experimental and computational results was obtained for transverse steel strains. The reported strain ratios were at the mid-depth of the beam section, closest to mid-span BTV ε/ε y BTV ε/ε y BTV.2 EXP COMP ε/ε y Figure 8: Transverse steel strain ratios.

10 366 Computational Methods and Experimental Measurements XII 6 Conclusion From the results presented in this paper it can be said that the 2-D idealisation of hollow beams using plane elements is adequate for cross-sections with in-plane stresses. The 2-D in-house finite element program used for the non-linear analysis gave good results when compared with experimental ones. Wide range of ratios of torsion to bending, Td/Md, and shear stress due to torsion to shear stress due to shear force, τ tor / τ shr, did not result in large differences between experimental and computational failure loads. References [] Thurlimann B. Torsional Strength of Reinforced and Prestressed Concrete Beams-CEB Approach, Institut fur Baustatik und konstruktion, ETH. Zurich. No. 92, pp. 7-43, June 979. [2] Mac Gregor J.G. and Ghoneim M.G. Design for Torsion, ACI Structural Journal, No. 92-S2, pp. 2-28, March- April 995. [3] Bhatt, P.; Beshara, A.W. In-plane Stresses in End Diaphragms of Box Girder Bridges, Journal of the Institution of Engineers (India), Part CV, Civil Engineering Division, v6, pp. 3-9, September 98. [4] Abdel-Kader M.M.A. Prediction of Shear Strength of Reinforced and Prestressed Concrete Beams by Finite Element Method., Ph.D. Thesis, University of Glasgow, 993. [5] Kotsovos M.D. and Pavlovic M.N. Structural Concrete, Thomas Telford Publications. Heron Quay, London, E4 4JD, UK, 995. [6] Chen W-F. and Saleeb A.F. Constitutive Equations for Engineering Materials, Elsevier Science B.V., Sara Burgerhartstraat 25, P.O. Box 2, AE Amsterdam, The Netherlands, 994. [7] Bhatt P. and Abdel Kader M. Prediction of Shear Strength of Reinforced Concrete Beams by Nonlinear Finite Element Analysis, Developments in Computational Techniques for Civil Engineering. CIVIL-COMP Press, Edinburgh, Scotland, UK. pp.9-24, 995. [8] Alnuaimi A.S. Parametric Study on the Computational Behaviour of Hollow Beams Designed Using the Direct Design Method Material Factors, Proceedings of High Performance Structures and Composites, Seville, Spain, WIT Press Publishers, pp , March 22. [9] Alnuaimi A.S. Parametric Study on the Computational Behaviour of Hollow Beams Designed Using the Direct Design Method Numerical Factors, Proceedings of International Conference on Advances in Structures, ASSCCA 3, Sydney, Australia, A.A., Balkema publishers, Volume 2, pp. 7 2, June 23. [] Alnuaimi A.S. and Bhatt P. 2D Idealization of Hollow Reinforced Concrete Beams Subjected to Combined Torsion, Bending and Shear, The Journal of Engineering Research, V2, No., pp , January 25.