Civl010604.qxd The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 ______________________________ *Corresponding authors E-mail: alnuaimi@squ.edu.om 2D Idealisation of Hollow Reinforced Concrete Beams Subjected to Combined Torsion, Bending and Shear A.S. Al-Nuaimi*1 and P. Bhatt2 1Department of Civil and Architectural Engineering, College of Engineering, Sultan Qaboos University, P.O. Box 33, Al-Khodh 123, Muscat, Sultanate of Oman 2Honorary Senior Research Fellow, Department of Civil Engineering, Glasgow University, Glasgow, G12 8QQ, UK Received 1 June 2004; accepted 4 September 2004 Abstract : This paper presents a finite element model for idealisation of reinforced concrete hollow beams using 2D plane elements. The method of ensuring compatibility between the plates using two-dimensional model to analyze this type of structures is discussed. Cross-sectional distortion was minimised by incorporating end diaphragms in the FE model. Experimental results from eight reinforced concrete hollow beams are compared with the non-linear predictions produced by a 2D in-house FE program. The beam dimensions were 300x300 mm cross section with 200x200 mm hol- low core and 3800 mm long. The beam ends were filled with concrete to form solid end diaphragms to prevent local dis- tortion. The beams were subjected to combined bending, torsion and shear. It was found that the two-dimensional ideal- isation 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 minimum. The results from the 2D in-house finite element program showed a good agreement with experimental results. Keywords: 2D analysis, Finite element method, Hollow beams, Bending, Torsion, Shear, Combined load, Numerical model Notation B = The strain matrix composed of derivatives of shape functions D = The elasticity matrix {F} = The equivalent nodal forces for the continuum Ec = The initial modulus of elasticity of concrete for uniaxial loading. Es = The secant modulus of elasticity at peak stress Es = σp / εp fcu = Concrete cube compressive strength f'c = Concrete cylinder compressive strength ft’ = Tensile strength of concrete {Fi} = The total applied load vector ¢¢üü∏∏îîàà°°ùŸŸGGáfÉ°SôÿG øe áYƒæ°üŸG áaƒÛG Qƒ°ù÷G πªëàd iƒ°ü≤dG IQ~≤dGh ∑ƒ∏°ùdG π«ãªàd OÉ©H’G »FÉæK Oh~ÙG ô°üæ©dG ΩG~îà°SÉH »HÉ°ùM êPƒ‰ ábQƒdG Iòg ¢Vô©à°ùJ : íFÉØ°U ÚH •É≤æ∏d á«≤a’G áMGR’G ¿G .äÉ°ûf’G øe ´ƒædG Gòg π«∏ëàd OÉ©H’G »FÉæK êPƒªædG ΩG~îà°SÉH íFÉØ°üdG ÚH ≥aGƒàdG ¿Éª° d âe~îà°SG á≤jôW ábQƒdG ¢ûbÉæJh .áë∏°ùŸG èFÉàædG ™e OÉ©H’G »FÉæK Oh~ÙG ô°üæ©dG èeÉfôH ΩG~îà°SG øY áŒÉædG á«£NÓdG äGƒÄÑæàdG èFÉàf âfQƒb .√ÉŒ’G ¢ùØf ‘ äÉMGR’G §HGƒ°V ™°Vh ≥jôW øY É¡«∏Y ô£«°S Qƒ°ù÷G á≤jô£H Qƒ°ù÷G √òg ™«æ°üJ ”h .º∏e 200* 200 ™£≤à ÆQÉa ∞jƒŒ OƒLh ™e º∏e 3800 ¬dƒWh º∏e 300* 300 ô°ù÷G ™£≤e ¿Éc .áaƒ› Qƒ°ùL á«fɪK O~©d á«∏ª©ŸG »FÉæK ô°üæ©∏d »HÉ°ù◊G èeÉfÈdG èFÉàf äô¡XG ~≤d .»HÉ°ù◊G π«∏ëàdG ‘ Aõ÷G Gòg π«ã“ ” ~bh »∏ÙG AGƒàd’G ™æŸ ∂dPh áfÉ°SôÿÉH âÄ∏e å«M áરüe ô°ùL πc »àjÉ¡f π©Œ ‘ É¡«∏Y OɪàY’G øµÁ OÉ©H’G á«FÉæK IOh~fi ô°UÉæY ΩG~îà°SÉH π«ãªàdG á≤jôW ¿G ≈∏Y ∫~j ɇ …ÈıG πª©dG ‘ â°ù«b »àdG ∂∏J øe É¡ª«b ‘ áÑjôb á«©bƒJ º«b ,OÉ©H’G .Qƒ°ù÷G øe ´ƒædG Gò¡d iƒ°ü≤dG IQ~≤dGh ∑ƒ∏°ùdG äÉ©bƒJ ÜÉ°ùM ««MMÉÉààØØŸŸGG ääGGOOôôØØŸŸGG.»HÉ°ùM êPƒ‰ ,áë∏°ùe áfÉ°SôN ,Oh~fi ô°üæY ,OÉ©H’G »FÉæK π«ã“ ,Qƒ°ùL :á ¢ü≤dGh »æãdGh ≈∏dG iƒ≤d á°Vô©ŸGh áë∏°ùŸG ¬fÉ°SôÿG øe áYƒæ°üŸG áZôØŸG Qƒ°ùé∏d OÉ©HC’G »FÉæK π«ãªàdG 54 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 1. Introduction Considering the complex behaviour of hollow beams, a detailed analysis would normally require a full three- dimensional finite element model especially with box-sec- tions 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 bend- ing action of the component plates. Figure 1 shows the state of stress in a typical box beam subjected to bending, torsion and shear. The wall thickness used for torsional resistance was 50 mm which is 1/6 of the beam depth (Thurliman, 1979; MacGregor and Ghoneim, 1995). Bhatt and Beshara (1980) studied the behaviour of bridge box girders using similar method with plane ele- ments. 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 (1993) 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 eight reinforced concrete box beams. The actu- al beam ends were filled with concrete to form end diaphragms. The diaphragms were included in the numer- ical analysis. The predicted behaviour was compared with the experimental results. Lc = Computational failure load = (Mc+Tc)/2 Ld = Design ultimate load = (Md+Td)/2 Le = Experimental failure load = (Me+Te)/2 L.F. = Load factor L.F. = (Me/Md + Te/Td)/2 Md,Td,Vd = Design ultimate bending moment, twisting moment and shear force, respectively Me,Te,Ve = Experimental failure bending, torsion and shear force, respectively Mc,Tc,Vc = Computed failure bending, torsion and shear force, respectively {Ri} = The residual force vector at ith iteration α = The ratio of the principal stresses σ2 / σ1 {δ } = The nodal displacements of the continuum εcc = Concrete strain at peak stress εcr = The tensile crack strain (corresponds to the peak tensile stress) εmax = Maximum compressive strain εn = A fictitious strain normal to the crack plane εp = The strain at the peak (maximum) compressive stress of the concrete σp ε/εy = Steel strain ratio = Measured strain at L.F. load / Measured yield strain v = The Poisson's ratio σ and ε = The current stress and strain σp = The ultimate strength of concrete in compression, equal fc’ τtor = Shear stress due torsion τshr = Shear stress due shear force. TorsionBending Shear Figure 1. State of stress in a typical beam 55 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 2. Research Significance A method using 2D plane elements to model 3D hol- low beams is introduced. The compatibility of displace- ment between adjoining plates was maintained by intro- ducing constraints on the displacement. The cross-section- al distortion was minimized by incorporating the end diaphragms into the in-house FE program. This work is restricted to the analysis of hollow cross-sections with in- plane stresses. 3. Geometrical Relationship between Displacements The two-dimensional idealisation of box girders is ade- quate provided compatibility of displacement between adjoining plates along the line of intersection is main- tained and cross-sectional distortion is reduced to a mini- mum. To achieve these objectives, the following steps were adopted as shown in Fig. 2: 1. 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 Fig. 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 dis- placements perpendicular to the joining line are independ- ent 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 com- patibility along the lines of intersections. This is done by giving the same freedom number for those equal displace- ments. Other freedoms which are independent are given different numbers. In other words, every pair of displace- ments in the x-direction (direction of the beam axis) of the joining line between the flange and the web will be hav- ing the same freedom number and, therefore, the same dis- placement value for that pair. The displacements perpen- dicular 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. 4. A 2-D Finite Element Program In modelling the linear and non-linear responses of concrete as a continuum, the non-linear elasticity approach was adopted. In this approach, the bulk modu- lus, shear modulus, Poisson's ratio and Young's modulus of concrete are expressed in terms of stress/strain vari- ables, such as deviatoric stresses or strains, stress or strain invariants, normal and octahedral strain ... etc. The mod- uli were used to formulate an isotropic matrix to represent the behaviour of concrete at a certain load level (Alnuaimi, 1999; Kotsovos and Pavlovic, 1995; Chen and Saleeb, 1994). No bar dual action or bar kinking were included in the analysis. The above idealisation was implemented in a 2D finite element program originally developed for carrying out non-linear analysis of solid rectangular beams Abdel- Kader, (1993). In this program, Eq. 1 was used to repre- sent the ascending portion of the uniaxial compressive stress-strain curve. This equation was tested by Darwin and Pecknold, (1977) on many experimental results and used by Bhatt and Abdel-Kader, (1995) to numerically analyse many reinforced and pre-stressed concrete beams from different experimental investigations. (1) In modelling the non-linear stress-strain relationship of the concrete in the principle stress direction, Eq. (2) was adopted as proposed by Liu et al. (1972) and used by Bhatt and Abdel-Kader, (1995). This equation accounts for non-linear behaviour of concrete in biaxial compres- sion and takes the form: (2) If α = 0, i.e. for uniaxial state of stress, Eqs. (1) and (2) become identical. Equation (2) was used to generate the stress-strain behaviour of concrete in biaxial compression up to peak strain εp, after which this equation ceases to be valid due to softening deformation. A linear descending curve was adopted by Bhatt and Abdel Kader (1995) as given by Eq. (3) after the maximum stress was reached until a maxi- mum strain of 0.0035 after which the stress drops to zero. (3) Figure 4 shows typical stress-strain curve for combined ascending and descending parts plotted using Eqs. (1) and (3). Smeared crack model with orthogonal cracks was adopted. In this model, it is assumed that before cracking the concrete is a homogeneous, isotropic and linear elastic 2 21 . ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ −+ = pps c c E E E ε ε ε ε ε σ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ +− − +− = 2 )2 1 1 (1)1( . pps c c E E v v E ε ε ε ε α α ε σ 0.0035< )1.0( )1.0( ' ε ε ε σ c cc f − − = 56 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 Web Compatibility of Displacement Top Flange Diaphragm Figure 2. Imposed displacement constraint A B C 9 3 6 15 18 1 2 4 5 7 8 10 11 14 13 16 17 12 A B C A 5 7 9 23 11 13 15 17 2119 20 22 24 25 2627 3940 2 4 656 53 54 192740 55 Top flange Front web Left diaphragm Figure 3. Freedom constraints 57 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 material and after cracking the concrete remains a contin- uum but becomes orthotropic with one of the material axis orientated along the direction of the crack. The idea of orthogonal cracks was adopted, where cracks are allowed to open only in directions orthogonal to the existing cracks. These fixed orthogonal cracks are governed by the direction of the first principal (tensile) stress that exceeds the cracking strength. When a first crack occurs, it is assumed that direct tensile stresses cannot be supported in the direction normal to the crack and therefore, the modu- lus of elasticity corresponding to this direction (normal to crack) is reduced to zero unless tension stiffening is con- sidered. The reduced shear stiffness due to aggregate interlock is accounted for by a reduced shear modulus of elasticity. On further loading, orthogonal cracks occur when the tensile stress parallel to the first cracks becomes greater than the concrete tensile strength f’t (Alnuaimi, 1999; Kotsovos and Pavlovic, 1995; Chen and Saleeb, 1994; Abdel-Kader, 1993). The effect of the shear reten- tion factor was considered as given by Eq. (4) while Eq. (5) was used for modelling the tension stiffening: (4) (5) Figure 5 shows typical shear retention curve and Fig. 6 shows tension stiffening curve used in this model. The program was based on the finite element displacement method, load control, where the displacements are the prime unknowns, with stresses being determined from the calculated displacement field. The non-linear problems were solved satisfying the basic laws of continuum mechanics: equilibrium, compatibility and constitutive relationship of materials. In this program, the continuum (structure) is divided into a series of distinct non-overlap- ping eight nodded 2D isoparametric elements with nine Gauss points for integration. Element stiffness matrix is given by Eq. (6). (6) The summation of terms in Eq. (6) over all elements leads to the continuum stiffness matrix [K]. This is used in a system of equilibrium equations for complete continuum given by (7) This system of equations is solved using direct Gaussian elimination algorithms in conjunction with frontal method of equation assembly and reduction to yield the nodal dis- placements. The main feature of this method is that it assembles the equations and eliminates the variable at the same time. Hence, the complete structural stiffness is never formed. This reduces computer storage significant- ly. Once this is done, the strains {ε} and thereafter the Figure 4. Typical combined stress-strain curve 05.0 /4.0 ≥= ncr εεβ ∫= eV Te dVBDBK ]][[][][ }]{[}{ δKF = 2/')4( mmNtfn cr t ε εσ = Eq. (1) Eq. (3) 58 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 εn åcr 05.04.0 ≥⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ = n cr ε ε β Figure 5. Shear retention curve εεn ◊◊ t f′′cr εcr ' 4 t n cr t f⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = ε ε σ Figure 6. Tension stiffening curve σt εn εn β εcr βmin (11) And the norm of the applied loads Fi* is given by Eq. (12) (12) When a reinforced concrete member is subjected to a tensile stress exceeding its tensile strength, the concrete cracks at discrete locations. The total force is then trans- ferred across the crack by the tensile steel. The reinforcing bars were modelled as embedded in orthogonal directions. Perfect bond (full compatibility) between steel and con- crete was assumed. The displacement of any point on the bar was obtained from the displacement field of the isoparametric element in the bar direction. The stiffness matrix of the bar Ks was calculated separately and then added to the stiffness matrix of the concrete Kc to form the element stiffness matrix Eq. (13) (13) In addition, the following parameters were used as con- stants: 5. Validation of the Proposed Model The program was used in a parametric study for the analysis of eight hollow reinforced concrete beams tested at the University of Glasgow, UK (Alnuaimi, 1999; Alnuaimi, 2002; Alnuaimi, 2003) under combined bend- 59 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 stresses {σ} in each element are evaluated using Eqs. (8) and (9) respectively (8) (9) Incremental iterative procedure with Modified Newton- Raphson method was used for the numerical solution of the non-linear problems. In this approach, the total load is divided into a number of load increments and the solutions are obtained iteratively until equilibrium is achieved to an acceptable level of accuracy. For every increment the stiffness matrix was updated at the first iteration. To ensure adequate convergence to the exact solution, conti- nuity of displacement between adjoining elements, mate- rial constitutive relation as well as equilibrium are main- tained by constraints on nodal points and ensuring that at any load level, stresses are consistent with displacement field and material constitutive relationship. This is obtained by successive linear solutions until specified accuracy is reached. If the material is within the elastic region, the relationship between the nodal forces and the displacement is linear and the stiffness matrix is un- changed. However, if the material yields then the relation- ship is non-linear and the stiffness matrix has to be updat- ed. This is done by a succession of linear approximations considering the new material law. This piece-wise lin- earization is used to form a global non-linear solution for the problem. A yield criterion for biaxial stress conditions based on non-dimensionalised values of the principal stresses σ1 and σ2 with respect to the uniaxial cylinder strength of concrete f’c’ was adopted. This criterion was originally proposed by Kupfer et al. (1969) and tested and used by many researchers. Detailed derivations of compression- compression, compression-tension and tension-tension criteria can be found in many references (i.e. Alnuaimi, 1999; Kotsovos and Pavlovic, 1995; Chen and Saleeb, 1994; Kupfer et al. 1969). To ensure gradual elimination of out-of-balance (resid- ual) forces and to terminate the iterative process when the desired accuracy has been achieved, a convergence crite- rion based on the out-of-balance load norms was used. Since it is difficult and expensive to check the decay of residual forces for every degree of freedom, an overall evaluation of convergence was used. This was done by using the residual force norm in each iteration i. This cri- terion assumes that convergence is achieved if: (10) { } { }eB δε ][= { } { }εσ ][D= 04.0* * ≤ ∆ i i F R Where the norm of residuals *iR∆ at each iteration i, is given by E q. (11) }{}{* i T ii RRR =∆ }{}{* i T ii FFF = sce KKK += • Concrete cube compressive strength: cuf as measured for each beam. • Concrete cylinder compressive strength: cuc ff 8.0 ' = N/mm2. • Young’s modulus: 2' /5000 mmNfE cc = . • Compressive strain at peak stress: 2500 ' c cc f =ε . • Maximum compressive strain: 0035.0max =ε . • Tensile strength of concrete: '' 54.0 ct ff = N/mm2. • Mesh size: 84x84mm. • Maximum number of load increments: 30. • Maximum number of iteration s per increment: 100. • Load increment: 10% for the first five increments and 5% thereafter. • Steel strain points for comparison with experimental results were located as in the actual beam. 60 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 ing, shear and torsion. The main variables studied were the ratios of twisting moment to bending moment which varied between 0.19 to 2.62 and the ratio of shear stress due torsion to shear stress due shear which varied from 0.59 to 6.84 as shown in Table 2. 5.1 Test Beams The tested beams were 300x300 mm cross section with 200x200 mm hollow core and 3800 mm length. The outer 400 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. The middle 1200 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 7 shows loading and support arrangement and Fig. 8 shows close look of the torsional arm and concrete filled end of beam diaphragm. 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 Fig. 9. Bending and shear were direct result of mid-span point down-ward 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, Ld, 10% for the first three increments, in anticipation of crack initiation, and then 5% until fail- ure. The beam was considered to have collapsed when it could resist no more loads. Table 1 shows average meas- ured material properties. Figure 10 shows the reinforce- ment provided along the test span of each beam. The solid circles refer to bars on which strain gauges were stuck while the hollow ones refer to bars without strain gauges installed. 5.2. Comparison between Experimental and the 2D Computational Results Here some experimental and computational results are compared. The comparison was carried out using the fol- lowing criteria: * Load displacement relationship, * Longitudinal steel strain, * Transverse steel strain, * Failure load, * Crack pattern and mode of failure. 5.2.1 Load-Displacement Relationship Figure 11 shows displacements at mid-span of the beam. It is clear from this figure that, in general, an acceptable agreement between experimental and compu- tational results was achieved in most cases. It should be noted that in few cases, due to technical problems, it was not possible to record displacement readings just before failure (i.e. BTV6). 5.2.2 Longitudinal Steel Strain Figure 12 shows that a very 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. 5.2.3 Transverse Steel Strain Figure 13 shows that a very good 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. 5.2.4 Failure Load The ratios of experimental to computational failure fcu f'c fy fyv Age Beam No Unit N/mm2 N/mm2 N/mm2 N/mm2 days BTV1 39 33 495 516 10 BTV2 37 24 490 472 7 BTV3 38 27 490 472 7 BTV4 42 33 480 472 7 BTV5 35 27 490 472 8 BTV6 35 28 490 472 7 BTV7 54 34 500 472 7 BTV8 53 36 500 472 8 Average 41.6 30.2 492 477.5 7.6 Table 1. Average measured material properties 61 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 Test span 1000mm 900mm 900mm 1000mm 1200mm Figure 7. Loading and support arrangement Figure 8. Torsion arm with a jack and load cell installed (see the solid end of the beam) 1200 mm 1000 mm 900 mm 900 mm 1000 mm Test span 62 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 Figure 9. Set of perpendicular rollers BTV1 BTV2 BTV3 BTV4 BTV5 BTV6 BTV7 BTV8 Y8 @66 mm c /c 4Y12 2Y8 2Y10 2Y8 Y8 @60 mm c /c 4Y12 2Y8 2Y10 2Y10 Y8 @93 mm c /c 5Y10 2Y8 2Y10 2Y8 Y8 @84 mm c /c 5Y12+1Y8 2Y8 2Y10 2Y8 3Y10 2Y8 2Y8 2Y8 Y8 @173 mm c /c Y8 @120 mm c /c 4Y10 2Y8 2Y10 3Y10 2Y8 2Y8 2Y10 Y8 @92 mm c /c Y8 @80 mm c /c 4Y10+1Y8 2Y8 2Y10 2Y8 Figure 10. Reinforcement provided in the test span 63 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Disp.(m m ) L. F. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Disp(mm) L .F . -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Disp.(m m ) L. F. 0 0.2 0.4 0.6 0.8 1 1.2 Disp.(m m ) L. F. Comput. EXP 10mm 10mm BTV 3 BTV 4 10mm 10mm BTV 1 BTV 2 20mm 20mm BTV 5 BTV 6 20mm 20mm BTV 7 BTV 8 Figure 11. Vertical displacement at mid-span 64 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ε/εy L. F. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ε/εy L. F. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 ε/εy L. F. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 ε/εy L. F. Comput EXP 1 1 BTV1 BTV2 1 1 BTV3 BTV4 1 1 BTV5 BTV6 1 1 BTV7 BTV8 Figure 12. Longitudinal steel strain ratios 65 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ε/εy L. F. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 ε/εy L. F. 0 0.2 0.4 0.6 0.8 1 1.2 ε/εy L. F. -0.2 0 0.2 0.4 0.6 0.8 1 1.2 ε/εy L. F. Comput EXP 1 1 BTV 1 BTV 2 1 1 BTV 3 BTV 4 1 1 BTV 5 BTV 6 1 1 BTV 7 BTV 8 Figure 13. Transverse steel strain ratios 66 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 loads Le/Lc are given in Table 2. A very good agreement was attained in failure loads. In most cases, the experi- mental results were slightly larger than the computed ones. Wide range of T/M and τtor/τshr ratios did not result in large differences between experimental and computa- tional results. 5.2.5 Crack pattern and Mode of Failure When the principle tensile stress at any point in the structure exceeds the maximum tensile strength of the concrete a crack is initiated perpendicular to the direction of the principle tensile stress. Figure 14 shows typical computed and observed crack patterns of the tested beams. The first graph in each pair represents the predict- ed location of cracks and their orientations in the test span just before failure load. It can be seen that a good agree- ment was achieved between the experimental and compu- tational results on the crack concentration and orientation. All beams failed in a ductile manner as can be seen from figures on displacements and steel ratios (Figs. 11 - 13). Steel yielded in most cases or reached near yield before the concrete crushed. The load was transferred from the concrete to steel at about 35% of failure load. Enough fine cracks have developed in each case long before major cracks development near the failure load. Beam Td Md Vd Td/Md τtor/τshr Le/Lc No. kNm kNm kN Ratio Ratio Ratio BTV1 13 14.89 21.08 0.87 2.28 0.99 BTV2 13 32.89 41.08 0.4 1.17 1.1 BTV3 13 50.89 61.08 0.26 0.79 1.14 BTV4 13 68.89 81.08 0.19 0.59 1.12 BTV5 26 14.89 21.08 1.75 4.56 0.97 BTV6 26 32.89 41.08 0.79 2.34 0.88 BTV7 39 14.89 21.08 2.62 6.84 1.06 BTV8 39 32.89 41.08 1.19 3.51 1.01 Table 2. Design, experimental and computed failure loads Figure 14a. Computed and observed crack development in the back web (BTV1) 67 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 Figure 14b. Computed and observed crack development in the front web (BTV3) Figure 14c. Computed and observed crack development in the front web (BTV6) Figure 14d. Computed and observed crack development in the front web (BTV8) 68 The Journal of Engineering Research Vol. 2, No. 1 (2005) 53-68 6. Conclusions From the results presented in this paper, it can be con- cluded 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, T/M, and shear stress due to torsion to shear stress due to shear force, τtor/ τshr, did not result in large differences between experimental and computation- al failure loads. References Abdel-Kader, M.M.A., 1993, “Prediction of Shear Strength of Reinforced and Prestressed Concrete Beams by Finite Element Method,” Ph.D. Thesis, University of Glasgow. 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