Professional Engineering Software Frequently Asked Questions and Solutions for RM2004 Technische Datenverarbeitung Dorian Janjic & Partner GmbH
Papers included in the series are detailed descriptions for RM2004 users to provide solutions for frequent modelling and analysis problems. Each document is structured into chapters relating the theoretical background, the method of implementation in RM2004 and application examples. Basic RM2004 Knowledge is presumed. Disclaimer Much time and effort have gone into the development and documentation of RM2004 and GP2004. The programs have been thoroughly tested and used. The user accepts and understands that no warranty is expressed or implied by the developers or the distributors on the accuracy or the reliability of the program. The user must understand the assumptions of the program and must apply engineering knowledge and skill to independently verify the results. Copyright The computer programs RM2004, GP2004 and all the associated documentation are proprietary and copyrighted products. Ownership of the program and the documentation remain with TDV Austria. Use of the program and the documentation is restricted to the licensed users. Unlicensed use of the program or reproduction of the documentation in any form, without prior written authorization from TDV is explicitly prohibited. RM2004 and GP2004 Copyright and support in Central Europe Tcl Copyright 1987-1994 The Regents of the University of California Tcl Copyright 1992-1995 Karl Lehenbauer and Mark Diekhans. Tcl Copyright 1993-1997 Bell Labs Innovations for Lucent Technologies Tcl Copyright 1994-1998 Sun Microsystems, Inc. Microsoft Windows Copyright Microsoft Corporation Dorian Janjic & Partner GmbH 2
1 Background External tendons are placed outside the concrete section after the concreting has been completed. They are unbonded from the concrete over long lengths and are anchored at diaphragms or on special anchor blocks along the span and deflected at deviators to give a draped profile. The principal difference between internal and external pre-stressing from a structural analysis point of view is the fact, that internal tendons after grouting are bonded with the concrete beam and full strain compatibility can be assumed in each cross-section. External tendons on the other hand elongate between their anchor or deviation points. Therefore the strain at any given point in an external tendon generally differs from the strain at the associated concrete cross-section. The strain in the tendon depends on the elongation due to the change in deviator geometry in the deflected beam (Figure 1a). Figure 1b illustrates another interesting effect. Due to the changes in tendon geometry in a deflected beam the eccentricity of the tendon changes and therefore influences the structural behaviour of the system. Consideration of this last effect requires the application of large displacements theory. stress-free beam deflected beam L (a) (b) e L elongation of tendon change of eccentricity e Figure 1. Structural implications in tendons due to deflected geometry of beams. 1.1 and Beam Theory Different modelling solutions for analysis of external pre-stressing in beam elements are proposed in the literature [3], [4]. In accordance with beam theory external tendons are commonly modelled as separate structural elements in tension between anchors and deviators. Over deviators the tendons are treated like internal tendons in this model. This piecewise model is generally consistent for all types of linear and non-linear calculations. For internal tendons, pre-stressing can be considered as a combination of primary and secondary effects. Primary forces result directly from application of tendon forces in each concrete section while secondary forces result from structural restraints that prevent the pre-stressed beam from freely deflecting. External tendons between their anchor or deviation points are structurally independent from the associated concrete beam elements. Tendon forces can be viewed as eccentric concentrated loads at anchor and deviator points. Pre-stressing of external tendons between their anchor or deviation points therefore only leads to total effects in the model and Dorian Janjic & Partner GmbH 3
splitting into primary and secondary components is, strictly speaking, not appropriate. At the deviators a connection between the tendon and the concrete cross-section is established due to friction. Therefore, the behaviour of an external tendon over a deviator is akin to the behaviour of an internal tendon at this location. 1.2 and Ultimate Limit States For external tendons between their anchor or deviation points additional strains at ultimate state depend on the change in length due to relative movement of the end points (Figure 1a). Conservatively, these relative deflections may be assumed to be small and are usually neglected for ULS checks. For example, this is stated explicitly in [1] (Clause 6.3.3.1f): the strain in unbonded tendons shall be assumed not to increase above the initial value... Some codes of practice allow a small additional strain for special cases. In a general case, external tendons therefore do not actively contribute to the load capacity of a structure except for the increase of capacity due to the induced compressive normal force in the concrete. For external tendons at deviators, at ultimate state loss of friction between saddle and tendon has to be assumed leading to slightly conservative results [2] for the section capacities. Expr 1 gives the equation governing the ULS check for internally and externally pre-stressed concrete cross-sections. Forces in this equation are given as vectors {F} with components N x, M y and M z. the following indices are used in Expr 1: p, ult Forces due to additional strain in pre-stressing. s Forces due to reinforcement steel. c Forces due to compression in concrete. Load Design forces. p,secondary Secondary components of (internal) pre-stressing. p0 Primary components of (internal) pre-stressing. p Total forces due to external pre-stressing. The assumption of slip between tendon and deviators implies, that initial strain of external tendons does not contribute to the computation of ultimate states (ie. Resistance in Expr 1). Resistance { F} p, ult + { F} s + { F} c { F} Load + { F} p,secondary { F} p0 (internal) { F} + { F} { F} + { F} (external) 144442 { F} s c 44443 ULT Load Load Expr 1. Design equation for pre-stressed concrete as used in RM2004 p Dorian Janjic & Partner GmbH 4
deviator (a) pre-stressing Strain ε c,top Stress M z,0 ε p,0 f p,0 N x,0 (b) loaded state ε c M z ε p =0 f p =0 N x (c) moment capacity ε c,ult M Ult N x,coexisting M C ε p,ult =0 f p,ult =0 TOTAL Figure 2. Strain and stress distribution in externally pre-stressed beam. Dorian Janjic & Partner GmbH 5
2 Implementation in RM2004 2.1 Geometry Definition In RM2004 deviator blocks are defined by referencing certain structural elements within a series of beam elements forming an externally pre-stressed girder. Deviator block limits have to coincide with element ends. To obtain results at (or near) the saddle peak it is recommended to use two elements for each deviator block definition. These element divisions are to be considered already when the overall structural model is set up. Wobble does not occur in external tendons, the wobble factor should therefore always be set to 0 (β=0). The friction coefficient depends on the pre-stressing system and is usually less than in internal tendons. The exact geometry definition of external tendons on saddles in space is complex and has to be distinguished from simplified geometry definitions for general structural behaviour calculations. In RM2004 the saddle geometry is calculated in the plane spanned by the two straight lines of the external tendons running on and off the deviator block. The geometry is simply defined by three points (tangent intersection points) and the longitudinal limits of the deviator block (Figure 3). The starting point of line 1 (P 0 ), an intersection point (P) of the two lines and an end point of line 2 (P 2 ) define a plane. With a given radius of the circular saddle, the position of the tendon tangent points are defined and automatically calculated by RM2004. Internally the saddle geometry is approximated by a 3D-spline with tangent points assumed to be approximately at the saddle limits. Alternatively, the actual tangent points (T 1 and T 2 ) on the saddle can be calculated more precisely by inserting so-called free points during the tendon definition (Figure 4). Independent from tendon definition, the lengths of individual tendon sections are always calculated between the limits of structural elements as defined in RM2004. the tangent points of the tendon do not exactly match the element limits which may lead to negligible inaccuracies in calculated tendon lengths as may be seen in Figure 4. A detailed input description for the geometry and the definition of stressing actions of external tendons will not be given here. Loading the attached examples and studying the included data should be self-explanatory. Dorian Janjic & Partner GmbH 6
P 0 P 2 P 0 P 2 RM2004 B112 B113 B114 B115 P line F 1 F 2 line Deviatior Block Figure 3: minimum definitions: P 0, P, P 2 span the tendon plane; F 1, F 2 are defined to be calculated by RM2004 at the deviator block limits. B112 B113 B114 B115 line T 1 P T 2 F 1 F 2 line Deviatior Block Figure 4: additional definition of tangent points T to be calculated by RM2004. 2.2 Tendon Tensioning Tendon tensioning is defined by three individual action commands in the RM2004 construction schedule. Action Stress is used to calculate the normal force diagram including all friction losses along one tendon. Action Calc is used to calculated the pre-stressing effects on the structure and finally action Grout is used to finalise the tendon stressing and calculation. External tendons between their anchor or deviation points are herewith activated as additional structural elements. In RM2004, the term grout is used for all types of external tendons although only some are really cement grouted while others have ducts that are grease filled or are embedded in PE plastics. Dorian Janjic & Partner GmbH 7
2.3 Ultimate State In RM2004 during ultimate state computations, the initial strain of all tendons is properly included on the load side of the governing equation (Expr 1). No primary components are generated by RM2004 for the external tendons between their anchor or deviation points. As explained in 1.1 and 1.2 initial strain of external tendons does not contribute to the ultimate capacity of a beam. Therefore, for comparing ultimate moments and cross-section capacity the results stored as Total Results must be used. For internal tendons the story is different since primary results must be subtracted from both sides of Expr 1. Therefore, for comparing ultimate moments and cross-section capacity of internal tendons the results stored as Secondary Results must be used. This leads to complications in structures where internal and external pre-stressing tendons are used at the same time. In this case, the user of RM2004 must take care that the correct result components are compared in the ULS check (construction schedule action UltSup- Mz). In this case, primary components of all grouted internal tendons have to be collected and subtracted from total components manually. For the automatic calculation of the required reinforcement (construction schedule action UltSup-Rein) the Total Results can be used in both cases because initial tendon strain is included on both sides of Expr 1. Dorian Janjic & Partner GmbH 8
3 Examples 3.1 General The following three examples have been set up for demonstration purpose only. Tendon definitions in these examples do not represent typical engineering applications. Three examples are provided. They all use one common structural model as explained below. Internal and external tendons are defined with roughly matching geometry and normal force distribution after stressing including friction losses. For comparison of cross-section capacity in ultimate limit states, the proportion of external and internal tendons is varied keeping the sum of pre-stressing steel area constant (Table 1). Reinforcement calculations for ultimate limit states follow the same procedure for all of the three given cases. To perform an ultimate moment check, moment diagrams of design loads and cross-section resistance diagrams are compared. These diagrams have to include the correct primary components to account for initial tendon strains. As discussed, this is different for the three given cases and the procedures are described below. Table 1 Variation of tendon types in the given RM2004 examples. Example No. tcl file name Tendons 1 external_only.tcl 2 external tendons no internal tendon 2 external_internal.tcl 1 external tendon 1 internal tendon 3 internal_only.tcl no external tendon 2 internal tendons 3.2 Structural Model 3.2.1 Numbering Scheme Node numbers: 101-121 Element numbers: 101-120 Supports: A1: 1100-1102, A2: 1200-1202, A3: 1300-1302 Tendon numbers: 1 for internal tendons, 2 for external tendons. For the for the external tendons between their anchor or deviation points separate cable elements are created automatically in the model. These elements have the element numbers 501-506 (Figure 5). Dorian Janjic & Partner GmbH 9
A1 A2 A3 30.0m 30.0m 10 x 3.0m 10 x 3.0m El 101-110 El 111-120 deviator region for external tendons 501 503 504 506 502 505 Figure 5. Structural system and tendon geometry for the given examples. 3.2.2 Loading For all three examples the same loading was assumed: self weight (lc0101), post-tensioning (lc0501 for internal tendons and lc0502 for external tendons) and a simple traffic loading for which the results are stored in traffic.sup. Design loads have been combined into a result file called combination.sup. 3.2.3 Construction Stages For the given examples it is assumed that the whole structure is built at once and prestressing is applied at the same time. No time-dependent effects are considered. In all three examples Stage1 contains load case and superposition file initialisation and calculation of the loading as defined above. The resulting design loads are then available in superposition file combination.sup. Stage2 includes the operations necessary for the automated reinforcement design and the ultimate check calculations. 3.3 Analysis Results 3.3.1 Example 1 - External Tendons Only In the following, typical results of structures with external tendon are described without going into detail on standard procedures for load case definitions and result calculations. The action command Stress in Stage1 of the RM2004 construction schedule is calculated by application of external tendon stress actions defined in stress label CS2. The resulting normal force distribution after friction losses of the external tendons is shown in Figure 6. Typically, all friction losses take place over the deviators. Action command Calc for load case lc0502 (pre-stressing of external tendon) produces the following results. The total bending moments for this load case are indicated by the green line with dot symbols in Figure 7. Over the deviators the tendon is treated in a similar way Dorian Janjic & Partner GmbH 10
as an internal tendon and therefore primary results are produced by RM2004 in these places as shown by the other (red) line in Figure 7. Figure 6. Normal force diagram for external tendons in Example 1. Dorian Janjic & Partner GmbH 11
Figure 7. Primary and total results of the pre-stressing load case lc0502. 3.4 Reinforcement Design Automated reinforcement design is performed using the action command UltSup with the option Rein in Stage2 for all three given examples. As described in 1.1, in case of external tendons no strain compatibility between tendon and adjacent concrete can be assumed and therefore external tendons cannot contribute to the resistance with additional strain at ultimate state. It may be expected, that the cross-section capacity gradually increases as the proportion of external tendons in the sum all tendons decreases. These expectations are met as shown in Figure 8, Figure 10 and Figure 12. 3.5 Ultimate Moment Check 3.5.1 Example 1 - External Tendons Only No primary components are generated for the external tendons between their anchor or deviation points as indicated in Figure 7. Because initial strain of external tendons does not contribute to the ultimate capacity of the corresponding beams, the results stored as Total Results were used for comparing ultimate moments and cross-section capacity as given in Figure 9. Dorian Janjic & Partner GmbH 12
Figure 8. Reinforcement for Example 1, external tendons only. Figure 9. Ultimate moment check for Example 1, external tendons only. Dorian Janjic & Partner GmbH 13
3.5.2 Example 2 - External and Internal Tendons In order to achieve meaningful results according to Expr 1 it must be distinguished between primary results from grouted internal tendons and (total) results from external tendons. While for ultimate state calculations the initial strain of internal grouted tendons must be subtracted from the load and the resistance side of the design equation (Expr 1) this is not the case for results of external tendons. Therefore, in the given example, primary components of all grouted internal tendons were collected in a summation load case called lc_intpt using the load manager label intpt (Table 2a,b). Subsequently, the primary components stored in lc_intpt were subtracted from the total components of the design loads (combination.sup) and also from the result file of the ultimate capacity calculation (umz_tot.sub). The results of these subtractions were stored in umz_ult.sup and comb_ult.sup (Table 2c) respectively. The total results of these two files umz_ult.sup and comb_ult.sup can then be used for comparison in a checking operation. The corresponding diagram is stored in plotfile ultmz.pl and shown in Figure 11. Table 2 TCL-Syntax of definitions a) LMANAGE "intpt" LCASE "LC1000" TOTAL LCASE "lc_intpt" PRIMARY LMANAGE END b) LCASE "LC0501" PERMANENT INFO "stress Tendon 1" LMANAGE "intpt" LSET "LS0501" 1 VAR LCASE END c) MODULE SupAddLc "umz_tot.sup" "lc_intpt" "-1,-1" "umz_ult.sup" MODULE SupAddLc "combination.sup" "lc_intpt" "-1,-1" "comb_ult.sup" MODULE DgmSet "dgm_ultcheck_tot" "" "" "ultmz.pl" 3.5.3 Example 3 - Internal Tendons Only For internal tendons the primary results must be shifted from design loads to the section capacity side of Expr 1. Therefore, for comparing ultimate moments and cross-section capacity of internal tendons the results stored as Secondary Results are used as given in Figure 13. Dorian Janjic & Partner GmbH 14
Figure 10. Reinforcement for Example 2, external and internal tendons. Figure 11. Ultimate moment check for Example 2, external and internal tendons. Dorian Janjic & Partner GmbH 15
Figure 12. Reinforcement for Example 3, internal tendons only. Figure 13. Ultimate moment check for example 3, internal tendons only. Dorian Janjic & Partner GmbH 16
4 References [1] British Highways Agency (1994) The Design of Concrete Highway Bridges and Structures with External and Unbonded Prestressing in Design Manual for Roads and Bridges, Volume 1, Section 4, Part 9 BD 58/94 [2] Tandler, J. (2001) Collapse analysis of external prestressed structures, Dissertation at the University of Surrey, Department of Civil Engineering, UK [3] Ariyawardena N., Ghali A. (2002) Prestressing with Unbonded Internal or External Tendons: Analysis and Computer Model, Journal of Structural Engineering, ASCE, v128, n12, pp 1493-1501 [4] Diep B.K., Umehara H. (2002) Non-Linear Analysis of Externally Prestressed Concrete Beams, Electronic Journal of Structural Engineering, v2, pp 85-96 Dorian Janjic & Partner GmbH 17