Finite element analysis and structural design of pretensioned inverted T-beams with web openings

Abstract

This paper presents the results of a research project aimed at providing standard circular web openings to the popular precast pretensioned inverted T-beam. Opening size and placement and required materials strengths were investigated. In this paper the nonlinear analysis and design of simply supported pretensioned inverted T-beam with circular web openings are presented. Two design parameters are varied: opening location and number of openings. The results from nonlinear finite element analysis were substantiated by test results from five pretensioned inverted T-beams with web opening and one solid beam. Good agreement is shown between the theoretical and the experimental results. The test results obtained from this investigation show that the performance of the specimens with web openings is almost identical to that of the specimen without web openings. A simple design method for pretensioned inverted T-beam with circular web openings is proposed.

Keywords

prestressed concrete / inverted-T beam / web opening

Introduction

Utility requirement may make it desirable to use openings in pretensioned inverted T-beams. However, introducing an opening into the web of a prestressed concrete beam reduces stiffness and leads to more complicated behavior. Therefore, the effect of openings on strength and service ability must be considered in the design process. Numerous investigations such as Tan et al. [1] and Mansur [2] have been carried out on reinforced concrete beams with opening. The first published work on prestressed beam with web openings was conducted by Ragan et al. [3]. Since then, several other researchers [4-8] have investigated prestressed beams with web openings. Based on their researches, Thompson et al. [7], Barney et al. [9], and Abdalla et al. [10] developed design procedures for prestressed concrete beams with web openings.

They have proposed a rather involved procedure to design for the opening, but they have not considered simplifications. In this paper, the effect of openings on the structural response of simply supported pretensioned inverted T-beam with web openings is examined. Results from nonlinear analysis are substantiated by an experimental investigation on several pretensioned inverted T-beams with web openings. As a consequence, design procedures for pretensioned inverted T-beam with circular web openings are proposed.

Objectives and scopes

In this paper, the effect of openings on the structural response of simply supported pretensioned inverted T-beams is examined. The results of a nonlinear analysis are substantiated by experimental investigation on several pretensioned inverted T-beams with web openings. A simplified method to design pretensioned inverted T-beams with circular web openings is presented. The second objective of this paper is load-deflection modeling using finite element analysis. Deflection modeling of prestressed beam with web opening is, in part, dependent on empirical performance constants. The empirical component reflects the opening locations, number of openings along the beam, as well as the material specific composite behavior of prestressing steel and concrete. Therefore, wherever a web opening exists in lieu of prestressing steel, the measured response may have a significant difference if one compares to predicted value using finite element modeling. It is practically very useful to consultancy as well as research studies to investigate the suitability of using finite element modeling for predicting the deflection behavior of prestressed beam with web openings, as many time-consuming experimental works could be minimized later on because we could rely confidently on an accurate FEA model to predict load-deflection relationship on many related works in the future. The load-deflection results of six simply supported pretensioned inverted T-beams with web openings subject to four-point monotonic loadings are presented. Test results are compared with finite element modeling for predicting deflection under all load conditions.

Experimental analysis

Tests were carried out on six bonded prestensioned inverted T-beams. The cross-section geometries are shown in Figs. 1 to 6. Typical reinforcement details for all beams and the as-tested details are shown in Figs. 2 to 4. The concrete was proportioned to produce a seven-day compressive strength of 45 MPa. The percentage of fine aggregate to total aggregate was 50%. The water-cement ratio was 0.4 with an aggregate-cement ratio of 2.7, both by weight. High early strength Portland cement was used, and the maximum size of coarse aggregate was 10 mm. Straight, bonded 7-wired super high tensile wires, each 12.9 mm in diameter, with an ultimate strength of 1860 MPa, were used for prestressing. Mild steel bars of 10 mm diameter, with a minimum specified yield stress of 350 MPa, were used as longitudinal nonprestressing steel. Figure 5 shows the prestressed steel and rebar locations in the prestressed beams. Figure 6 shows the fabricated prestressed beams used in the testing program. The concrete cover was 50 mm thick. All mixing was done in a factory rotary drum mixer. Concrete was placed in steel molds and compacted by an immersion vibrator. Specimens were demolded 24 h after casting and the subsequent curing was performed by covering with wet burlap for at least two weeks.

Testing setup

All tests were conducted with a closed-loop hydraulic servo-controlled MTS testing system. The 360 kN jack was capable of both displacement and load control for monotonic or cyclic loading. A four-point loading scheme, with an effective span of 4000 mm and a distance of 1200 mm between the load points was used to limit the presence of shear stress in the mid-span zone. Figure 7 shows the layout of four-points bending test of pretensioned inverted T-beam. LVDT was employed at the middle locations of the beam to provide recording of the beam deflection at mid-span.

Finite element analysis

The nonlinear finite element program ANSYS was used for theoretical study as shown in Figs. 8 to 13. Three-dimensional eight-noded isoparametric Solid65 elements were used to model the concrete portions of the inverted T-beam. The element is capable of plastic deformation, cracking in three orthogonal directions, and crushing. A Link8 element was used to model the prestressing steel reinforcement as well as non-prestressing steel. The element is also capable of plastic deformation. An eight-node solid element, Solid45, was used for the steel plates at the supports in the beam models. The thickness of the elements was varied to account for the difference in width between the flange and the web. In ANSYS, the load is applied gradually in small increments. The size of each increment depends on the convergence of the iteration process in the previous increments of loading. The concrete under compression is modeled by an elastic-plastic theory, using a simple form for the yield surface expressed in terms of the equivalent pressure stress. Isotropic hardening is accounted for. Cracking is assumed to occur when the stress reaches the failure surface represented by a simple Coulomb line in terms of the equivalent pressure, p, and the Von Mises equivalent deviatoric stress, q. The model is a smeared crack model, in the sense that it does not track individual macro cracks. Instead, constitutive calculations are performed independently at each integration point of the finite element model, and the presence of cracks is accounted for through the changes in the stresses and materials stiffness associated with the integration point. The concrete behavior is characterized by a stress-displacement response, in which the state of stress along the crack is a function of the opening strain across the crack. The cracking and compression responses of the concrete as well as the influence of tension stiffening due to the presence of the prestressing steel are investigated. The following design parameters were considered: 1) locations of the circular opening; and 2) the number of openings.

Three load stages were investigated: transfer, service load, and ultimate load. The finite element models were nonlinear, so they were more exact for the analysis with ultimate loads because the model accounts for cracking. The analysis gave a good indication of overall inverted T-beam behavior, such as the deflection of the inverted T-beam under service loads, the locations of stress concentrations, and an idea of the magnitude of the stress concentrations. The opening depth was varied to see what impact that would have on the inverted T-beam stress. Based on required cover for the strands and any mild steel reinforcement required, the maximum opening depth could be 200 mm. With 200-mm-deep openings, there were high compressive stresses in the bottom chords at transfer and high tensile stresses at service. With 200-mm-deep opening, the tensile stresses were between \mathrm{6}\sqrt{{f^{\prime }}_{\mathrm{c}}} and \mathrm{12}\sqrt{{f^{\prime }}_{\mathrm{c}}} stress limits, which would indicate limited flexural cracking. Although cracking would significantly increase the deflection of the inverted T-beam, allowing the inverted T-beam to crack was preferable to reducing the opening depth any further. The clear cover to the prestressing strand was increased by 50 percent in accordance with ACI 318-99 Code [11], and the \mathrm{12}\sqrt{{f^{\prime }}_{\mathrm{c}}} limit was used for the allowable tensile stress limit at service. The final opening size used was 200 mm. The width of the post between openings was 200 mm, based on strut-and-tie theory and because Barney et al. [9] recommended that post be at least half of the width of the opening. This allowed nine openings along the web of the inverted T-beam. No openings were placed in the length required for strand development or in areas of high shear. One of the major variables investigated in this project was the effect of using straight prestressing strand versus two-point or one-point used in the previous research projects. Straight tendons allow placement of larger openings in the webs than one-point depression or two-point depression do. Application of ACI 318-99 Code [11] in design shows that one prestressing strand was required for the span and loading chosen. In the finite element analyses, one strand was investigated. Based on the discussion with producers and the finite element analysis, a prestressing strand was added in the compression chord. This strand was necessary because as the inverted T-beam is loaded, Vierendeel action of the inverted T-beam at the openings produces moments on the ends of the tension and compression chords. These moments cause tensile and compressive stress concentration at the ends of the chords. The top strand provides additional compression in the compression chord to counteract these tensile stress concentrations. It also makes placement of auxiliary steel in the inverted T-beam easier. Both 12.9 mm and 15 mm diameter, 1860 MPa and 2070 MPa prestressing steel were investigated using finite element analysis. The results of the analysis showed that 12.9 mm diameter, 1860 MPa low-relaxation strand was adequate. Higher steel strength and/or higher diameter strands may be desirable in other applications not considered in this study. Concrete strengths between 40Mpa and 70 MPa were considered. The finite element analysis showed that a higher concrete strength of {f^{\prime }}_{\mathrm{c}}=45 MPa and {f^{\prime }}_{\mathrm{ci}}=35 MPa was required. The higher concrete strength increased the stiffness of the inverted T-beam as well as the code allowable stresses. The increased stiffness of the inverted T-beam reduced localized stress concentrations and overall deflection.

There were tensile stress concentrations in the posts between openings at transfer. For both service and ultimate loads, the highest tensile and compressive stress concentrations occurred in the chords above and below the first full opening at each end of the inverted T-beam. Under service loads, there were also tensile and shear stress concentrations at the corners of the openings, tensile stress concentrations below the post on either side of the opening at midspan, and a tensile and shear stress concentration extending from the edge of the end openings towards the supports.

Concrete cracking and crushing

Concrete cracking and crushing failure sequences for all the six tested beams could be summarized as follows: 1) at an applied load in the range 125 kN to 135 kN (mid point load) with 2 mm to 2.5 mm displacement, the bending cracks occur vertically on the tension side of the prestressed beam base; 2) with increasing load, the cracking progresses toward the upper part of the prestressed beam; 3) at applied loading between 160 kN and 175 kN (mid point load) with displacement in the range of 12 mm to 18 mm, the cracks reach almost 75% of the prestressed beam height. The prestressing steel at this loading is about 75% to 80% of yield value. More cracks begin to form above the openings in the pure moment region towards the top fiber of the prestressed beam; 4) at 190 kN to 200 kN central loading with 30 mm to 40 mm displacement, the prestressing steels yield and cracks at the tension side grow relatively large in a short period of time; and 5) at the same time, yielding at the prestressing steel compression fiber starts followed by complete failure of the beam in a ductile manner. Load-deflection results are shown in Figs. 8 to 13. As expected, a ductile flexural failure occurred in all beams associated with yielding of prestressing steel.

The advantage of prestressing steel is demonstrated in the energy absorbed through plastic deformation in the prestressing steel. Although the specimens technically failed at yielding of prestressing steel, the applied load was sustained and moment capacity remained intact until secondary ductile failure occurred by concrete crushing. The load-deflection history for these five samples was characterized by two to three individual cracks in each sample. Cracks within and near the constant moment region were observed to grow in a vertical direction. The orientation became progressively more inclined as the distance from the beam centerline increased.

Proposed design method

The following proposed design procedure for pretensioned inverted T-beam with web openings is used to determine the prestressing requirements and to check the adequacy of the mild steel reinforcement. The flow chart of the design procedure is shown in Fig. 14 and briefly describes as follows: 1) Calculate the required strand development length. 2) Calculate the service and ultimate loads acting on the pretensioned inverted T-beam with web openings. 3) Determine a preliminary number of prestressing strands. An additional prestressing strand should be placed in each web at the same level as the bottom of the flange. 4) The bottom strands are placed concentric with the bottom chord. This requirement could slightly affect the capacity of the member compared with standard strand profile calculated using ACI 318-99 Code [11]. The final member capacity must be checked for the strand profile used. 5) Flexural stress at various critical sections are calculated at release of prestress and at final loading stages using the service load analysis, to ensure flexural cracking under full service load. Section properties and prestress eccentricity should be adjusted to account for the web opening where a section is through an opening. 6) Shear design in the solid part of the member is the same as for solid inverted T-beam. 7) Stirrups should be used at all vertical opening edges to control cracks extending from the corners of openings. The total factored shear force at the edge of the first opening should be used to compute the area of additional stirrups to be place adjacent to the opening using recommendation ACI 318-99 Code [11]. 8) Determine the required nominal shear strength value V_n=V_u/ϕ at a distance h/2 from the face of the support where ϕ=0.85. Alternatively, in the Architectural Institute of Japan (AIJ) Standard for Structural Calculation of Reinforced Concrete Structures (1988), a formula (designated as Hirosawa’s formula) has been incorporated to evaluate the shear capacity, V_n, of beams that contain a circular opening. This empirical formula considers that the total shear resistance is provided by both concrete and the steel crossing a 45° failure plane passing through the center of opening. The formula given is as follows:

(1)V_{\mathrm{n}}=\left[{\displaystyle \frac{\mathrm{0.092}{k}_{\mathrm{u}}{k}_{\mathrm{p}}({f^{\prime }}_{\mathrm{c}}+\mathrm{17.7})}{{\displaystyle \frac{M}{Vd}}+\mathrm{0.12}}}\left(\mathrm{1}-{\displaystyle \frac{\mathrm{1.61}{d}_{\mathrm{0}}}{h}}\right)+\mathrm{0.846}\sqrt{{\overline{p}}_{\mathrm{w}}{f}_{\mathrm{yv}}}\right]b{d}_{\mathrm{v}},

where k_u is a function of the effective depth dto account for the size effect in shear and has a value between 0.72 and 1.0; k_p=0.82(100A_s/bd)^0.23; d₀=diameter of the circular opening or diameter of the circumscribed circle in the case of a square opening which should be taken as less than or equal to h/3; h is the overall depth of the beam, and M/Vd is taken as less than or equal to 3.

The term {\overline{p}}_{\mathrm{w}} in Eq. (1) refers to the ratio of web reinforcement placed within a longitudinal distance d_v/2 from the center of the opening and is defined as

(2){\overline{p}}_{\mathrm{w}}={\displaystyle \frac{A_{\mathrm{v}}\mathrm{(sin}\alpha \mathrm{+cos}\alpha \mathrm{)}}{b{d}_{\mathrm{v}}}},

in which d_v=the distance between the top and bottom longitudinal bars; A_v=area of web reinforcement (vertical stirrup or diagonal bar); α=angle of inclination of web reinforcement; f_yv=yield strength of web reinforcement.

The first term in Eq. (1) gives the contribution of concrete to shear resistance, which is assumed to decrease in proportion to the opening depth. The second term gives the shear resistance due to the web reinforcement. 9) Deflection should be calculated using the reduced section properties and checked against the appropriate ACI 318-99 limits. 10) Calculate the required minimum web reinforcement. The spacing is s≤0.75h or 60 mm, whichever is smaller

(3)\mathrm{Minimum}{A}_{\mathrm{v}}={\displaystyle \frac{\mathrm{50}{b}_{\mathrm{w}}s}{f_{\mathrm{y}}}}.

where A_v=required area of stirrups; f_y=yield strength of stirrups; b_w=web width; s=spacing.

Find the number of stirrups and place them at both ends of all openings, spaced at 25 mm. Note that if further experiments with smaller amounts of steel are conducted or if a more positive anchorage method of the 10-mm bar into the top flange is devised, this minimum requirement may be waived. Even with this conservative approach, the required shear reinforcement is a reasonable amount of reinforcement and does not significantly increase the cost of the inverted T-beam. 11) Check the shear capacity of the top chord at the first circular opening location, assuming that the top chord resist the total shear in the member at that section, combined with a compression force:

(4)C=T={\displaystyle \frac{M_{\mathrm{u}}}{z}},

where M_u=simultaneously applied moment with maximum factored shear at section; z=distance between centroid of tension reinforcement, i.e., bottom strands and compression area above opening.

Using ACI conservative method if f_pe>0.40f_pu,

(5)V_{\mathrm{c}}=\left(\mathrm{0.60}\lambda \sqrt{{f^{\prime }}_{\mathrm{c}}}+{\displaystyle \frac{\mathrm{700}{V}_{\mathrm{u}}{d}_{\mathrm{p}}}{M_{\mathrm{u}}}}\right)b_{\mathrm{w}}{d}_{\mathrm{p}},

where \mathrm{2}\lambda \sqrt{{f^{\prime }}_{\mathrm{c}}}{b}_{\mathrm{w}}{d}_{\mathrm{p}}\le V_{\mathrm{c}}\le \mathrm{5}\lambda \sqrt{{f^{\prime }}_{\mathrm{c}}}{b}_{\mathrm{w}}{d}_{\mathrm{p}},V_{\mathrm{u}}{d}_{\mathrm{p}}/M_{\mathrm{u}}\le \mathrm{1.0}, and λ=1.0 for normal-weight concrete; d_p= distance from the extreme compression fiber to the centroid of prestressed steel, or 0.8h, whichever is greater; V_u=factored shear force at section; M_u=factored moment at section; b_w=web width; {f^{\prime }}_{\mathrm{c}}=specified compressive strength concrete; f_pe=effective prestress; f_pu=specified tensile strength of prestressing tendon.

The total shear may be assumed to be resisted partly by the concrete and partly by the shear reinforcement crossing the failure plane. In designing for failure, a 45° inclined failure plane, similar to a solid beam may be assumed, the plane being transversed through the center of the opening. The V_c should then be calculated by Eq. (5). For the other component, V_s, it may be seen that the stirrups available to resist shear across the failure plane are those by the sides of the opening within a distance (d_v–d₀), where d_v is the distance between the top and bottom longitudinal rebars, and d₀ is the depth of opening. The contribution of diagonal reinforcement, if any, intercepted by the failure plane may also be taken into account in the calculation of shear resistance. Including the contribution of diagonal reinforcement, V_s is

(6)V_{\mathrm{s}}=V_{\mathrm{sv}}+V_{\mathrm{sd}}={\displaystyle \frac{A_{\mathrm{v}}{f}_{\mathrm{yv}}}{s}}(d_{\mathrm{v}}-d_{\mathrm{0}})+A_{\mathrm{d}}{f}_{\mathrm{yd}}\sin \alpha ,

in which V_sv and V_sd are the contribution of vertical and diagonal reinforcement, respectively; A_d is the total area of diagonal reinforcement through the failure surface; αis the inclination of diagonal reinforcement; and f_yd is the yield strength of diagonal reinforcement. The total amount of web reinforcement thus calculated should be contained within a distance (d_v-d₀)/2 on either side of the opening.

Crack control

The reinforcement designed as above would ensure adequate strength. However, due to the sudden reduction in beam cross section, stress concentration occurs at the edge of the opening. Adequate reinforcement with proper detailing should therefore be provided to prevent wide cracking under service load conditions.

In the case of circular openings, since full-depth stirrups are already provided by the sides of the opening to ensure adequate strength, the provision of diagonal reinforcement may be considered to restrict the growth of cracks along the failure plane. An amount of diagonal reinforcement that is sufficient to carry the total shear along the 45° failure plane is proposed here for pretensioned beam with web openings. It is proposed that the total area of diagonal reinforcement, A_d, through the failure surface is

(7)A_{\mathrm{d}}={\displaystyle \frac{V_{\mathrm{u}}}{\varphi f_{\mathrm{yd}}\sin \alpha }},

in which α is the inclination of diagonal reinforcement and f_yd is the yield strength of diagonal reinforcement. This amount should be distributed equally on either side of the circular opening and be placed normal to the potential failure surface. An equal amount should be placed perpendicular to this reinforcement to avoid confusion during construction and to take care of any possible load reversal.

Restrictions on the proposed design procedure are as follows:

1) The openings should be placed horizontally in the flexural region and vertically below the concrete compression stress block.

2) The beams are subjected to four-point loading conditions.

3) The distance between circular openings is at least the opening diameter.

4) At least 20 mm concrete clearance is required on top and bottom of the circular opening around the web region.

5) The top edge of the opening should be at least 100 mm below the top surface of the flange.

6) No openings should be allowed within the strand development length at member ends.

Conclusions

Based on the theoretical and experimental results of this investigation, the following conclusions may be drawn:

1) The agreement between the results of the experiments and of the finite element analysis is quite favorable.

2) The results of the analysis and testing were based on the assumption that the member was subjected primarily to four-point loading case. They must be validated for cases involving significant uniform loadings.

3) When properly reinforced, the deflection of inverted T-beams with web openings is comparable with inverted T-beam without web openings. Reinforcing the sides of the opening can substantially increase the cracking load of prestressed beams with web openings.

4) Longitudinal reinforcement should be provided near the top and bottom of the two chords of the circular opening to resist cracking resulting from the secondary moments, and to prevent tension cracking resulting from the axial tensile force in the tension chord.

5) Cracking at the opening corners may cause splitting along the interface between the flange and the web. Such splitting should be resisted by vertical stirrup reinforcing or shear studs.

6) As many web openings as needed can be placed in the middle two thirds of four-points loaded inverted T-beam as long as the dimensional and reinforcement requirements given in this paper are satisfied.

References

[1] Tan K H, Mansur M A. Design procedure for reinforced concrete beams with large web openings. ACI Structural Journal, 1996, 93(4): 404-411

[2] Mansur M. Ultimate strength design of beam with large openings. International Journal of Structure, 1988, 8(2): 107-125

[3] Ragan H S, Warwaruk J. Tee Members with large web openings. PCI Journal, 1967, 12(4): 52-65

[4] Pessiki S, Prior R, Sause R, Slaughter S. Review of existing precast concrete gravity load floor framing systems. PCI Journal, 1995, 40(2): 52-68

[5] Savage J M, Tadros M R, Arumugasaamy P, Fischer L G. Behavior and design of double tees with web openings. PCI Journal, 1996, 41(1): 46-62

[6] Pessiki S, van Zyverden W, Sause R, Slaughter S. Proposed concepts for floor framing systems for precast/prestressed concrete office Buildings. PCI Journal, 1997, 42(5): 66-76

[7] Thompson J M, Pessiki S. Experimental investigation of precast,prestressed inverted-tee girders with large web openings. PCI Journal, 2006, 51(6): 2-17

[8] Dinakaran V, Sastry M K. Behavior of post-tensioned prestressed concrete T-beam with large web openings. Indian Concrete Journal, 1984, 58(2): 34-38

[9] Barney G B, Corley W G, Hanson J M, Parmelee R A. Behavior and design of prestressed concrete beams with large web openings. PCI Journal, 1977, 22(6): 32-61

[10] Abdalla H, Kennedy J B. Design of prestressed concrete beams with openings. Journal of Structural Engineering, 1995, 121(5): 890-898

[11] ACI Committee 318. Building Code Requirements for Structural Concrete (ACI 318-99) and Commentary(ACI 318R-99). Farmington Hills, MI: American Concrete Institute, 1999

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