4 DESIGN FOR FLEXURE
Class 1 members Class 2 members
4.4.6 Partial prestressing
Prestressed concrete members in which flexural tensile cracks are allowed to develop at service load levels are referred to as being partially prestressed. A partially prestressed member is reinforced by a combination of prestressed and non-prestressed reinforcement, which both contribute to the ultimate strength and serviceability behaviour of the member. Although the non-prestressed steel can be either ordinary reinforcing bars or non-prestressed prestressing steel, the former is usually used for this purpose.
When compared to ordinary reinforced concrete, partial prestress offers the advantage of improved deflection control as well as the advantages to be gained from the fact that the member is usually crack free under long-term loads, depending on the degree of prestress. Partial prestressing also offers some advantages over full prestressing (Ref. 4-10):
• Improved control of camber.
• Savings in the cost of prestressing. Since a smaller prestressing force is required, the use of partial prestressing usually leads to savings in the amount of prestressing steel required, the anchorages required and the cost of the work associated with tensioning and grouting (in the case of bonded post-tensioning) the tendons.
• Economical use of ordinary reinforcing steel.
• Possible improved ductility.
The most often quoted disadvantage of partial prestressing, when compared to full prestressing, is that such members can be cracked at service load levels. However, ample evidence exists that if appropriate steps are taken to control flexural cracks in terms of crack width and spacing, the presence of these cracks will not adversely effect the durability of a partially prestressed concrete member (Ref. 4-26).
In addition to providing adequate flexural capacity, together with the prestressed steel, the non-prestressed reinforcement performs the following functions (Ref. 4-10):
• Properly detailed non-prestressed steel can effectively control both crack width and spacing at service load levels.
• In an unbonded member, some non-prestressed bonded reinforcement should be provided to prevent the development of a single large crack at ultimate and, thereby, to increase the flexural capacity of the member.
• If, at transfer, large tensile stresses are induced in the compression flange, non-prestressed reinforcement can be provided to prevent possible fracture, e.g. in the top flange over the midspan region of a simply supported beam in which the live load is large in comparison to the self weight of the beam.
• In the case of precast beams, properly placed non-prestressed reinforcement will ensure that the beam is sufficiently robust with regard to unexpected stresses which may arise during handling and erection.
Numerous procedures have been developed for the flexural design of partially prestressed concrete beams and a comprehensive discussion of a number of these can be found in Ref. 4-26. The various methods can usually be grouped into one of the following three categories, depending on the limit state which the design procedure initially satisfies:
• Methods which initially satisfy the serviceability limit state. The British design codes BS 8110 (Ref. 4-7) and BS 5400 (Ref. 4-8) as well as the South African code SABS 0100 (Ref. 4-2) recommend a method which is based on a limiting hypothetical tensile stress. The hypothetical tensile stress in a cracked prestressed concrete beam section is defined as the flexural stress which would occur in the extreme tension fibre of the uncracked section. Leonhardt (Ref. 4-27) and Menn (Ref. 4-28) each outline procedures which use a crack width limitation as the point of departure for design.
• Methods which initially satisfy the ultimate limit state. The method proposed by Naaman (Ref. 4-29) makes use of the partial prestressing ratio, while the procedure developed by Bachmann (Refs. 4-30 and 4-31) employs the concept of the degree of prestress.
• Methods which simultaneously satisfy the ultimate and serviceability limit states. The procedure proposed by Huber (Ref. 4-32) is an example of such a method.
Over the years, a number of indices have been developed for quantifying the extent of prestressing in a partially prestressed concrete beam section (see Ref. 4-26). The partial prestressing ratio (PPR) and the degree of prestress k are two such indices: The partial prestressing ratio is defined as the ratio of the ultimate moment of resistance provided by the prestressing steel only to the ultimate moment of resistance provided by all the steel (i.e. the prestressed plus non-prestressed steel). The degree of prestress is defined as the ratio of the decompression moment (i.e. the moment which induces a zero stress in the extreme tension fibre of the section) to the total service load moment and therefore represents the fraction of the total service moment which is counteracted by prestressing effects. Consequently, a value of zero for the degree of prestress corresponds to reinforced concrete while a value of one applies to fully prestressed concrete.
A reasonable basis for any procedure for the flexural design of a partially prestressed concrete beam section is to adopt a unified approach in the sense that the method must apply to the complete spectrum of possible levels of prestress, from fully prestressed concrete through to reinforced concrete. The method should also provide a smooth transition from fully prestressed concrete to reinforced concrete. The design procedure proposed by Bachmann (Refs. 4-30 and 4-31) satisfies these requirements and is presented in the following. The method assumes that all the dimensions of the concrete section are known, that all the material properties are known, and that the bending moments due to all dead and live loads can be determined. The section is initially designed to provide the required ultimate strength and is subsequently checked for serviceability as follows:
(a) Select a suitable value for the degree of prestressk or, alternatively, the decompression moment.
This choice is strongly dependent on engineering judgment to suit a given consideration such as durability, deflection, fatigue, crack control and cost. The decompression moment is commonly chosen at least equal to the dead load moment (Ref. 4-31) and it is suggested that in the case of bridges a value larger than the dead load moment is appropriate (Refs. 4-31 and 4-33). Wiessler (Ref. 4-33) recommends that the decompression moment should be taken equal to the dead load moment plus 33% of the live load moment for bridges in South Africa.
(b) Determine the required amount of prestressing steel. The prestressing force Pt required for developing the decompression moment selected in step (a) is determined by making use of the expression for calculating the flexural stress in an uncracked section. The following expression for calculating Pt is derived by setting fbot,s equal to zero and Mmax equal to the decompression moment Mdec in Eq. 4-43d, and solving the resulting expression for Pt:
(4-65)
The required amount of prestressing steel can subsequently be determined from the calculated value of Pt.
(c) Determine the required amount of non-prestressed steel. Non-prestressed reinforcement must be provided to ensure that the ultimate moment of resistance of the section exceeds the required value. The procedure outlined in Section 4.4.3 can be used to design this reinforcement.
(d) Detail the non-prestressed reinforcement carefully. In addition to its contribution to the ultimate strength, soundly detailed non-prestressed reinforcement can effectively control both crack width and spacing at service load levels. This is often the last step in the design procedure under normal circumstances.
(e) Check compliance with other limit states. Other limit states such as, for example, crack width, fatigue and deflection can be specifically examined, as required. These aspects are covered in later Chapters.
It should be noted that since design is essentially an iterative procedure, the section is usually determined on a trial and error basis, bearing in mind any specific design requirements. Generally, the number of iterations required for convergence to a solution rapidly reduces with experience.
P M
Z
A e
t
dec bot
=
-L
+NM O
QP
h
More expansive discussions on this design procedure can be found in Refs. 4-26 and 4-34. It should be noted that some of the practical advantages of the method are that it is code independent, that it treats uncertainties in a rational manner and that it does not preclude any serviceability check.
The design procedure is illustrated by example 4-16.
EXAMPLE 4-16
The midspan section of a pretensioned partially prestressed concrete beam, which is simply supported over a span of 14 m, is shown in Fig. 4-56. In addition to its self weight, the beam must support a uniformly distributed superimposed dead load of 4.9 kN/m and a live load of 8.8 kN/m. Make use of the provisions of SABS 0100 (Ref. 4-2) to design suitable prestressed and non-prestressed reinforcement for the midspan section so that the decompression moment Mdec is equal to the permanent load moment. Use 12.9 mm 7-wire super grade strand, for which fpu = 1860 MPa and Ep = 195 GPa, for the prestressed reinforcement and take fy = 450 MPa and Es = 200 GPa for the non-prestressed reinforcement. Take fcu = 50 MPa and Ec = 34 GPa for the concrete.
If the self weight of the concrete is taken as gc = 24 kN/m3, the self weight of the beam is given by wD = gcA = 3.96 kN/m. Therefore the total permanent load is wPerm = wD + wsdl = 3.96 + 4.9
= 8.86 kN/m. Using the load factors of SABS 0160, the design value of the permanent load moment appropriate to the serviceability limit state is given by
Since the decompression moment Mdec is taken to be equal to the permanent load moment, and the total design moment at the serviceability limit state is given by
the degree of prestress k corresponding to this choice of Mdec is
Assuming h = 0.84 (i.e. 16% losses), the prestressing force required for a decompression moment Mdec = 238.8 kN.m is subsequently obtained from Eq. 4-65
If 12.9 mm 7-wire super grade strand, jacked to 75% of characteristic strength (= 186 kN per strand), is used the jacking force per strand is 0.75 ´ 186 = 139.5 kN. Assuming the loss of prestress due
M w L
Figure 4-56: Midspan section for example 4-16.
to elastic shortening to be 5.0%, the initial force per strand at transfer is (1 - 0.05) ´ 139.5
= 132.5 kN (= Pt,strand). Therefore, |-639.1|/132.5 = 4.823, say 5 strands are required, for which Aps = 500 mm2.
The non-prestressed reinforcement is designed by considering the required ultimate moment, in exactly the same manner as in example 4-13. Following the requirements of SABS 0160, the design l o a d a p p r o p r i a t e t o t h e u l t i m a t e l i m i t s t a t e i s g i ve n b y wu = 1 . 2 wPerm + 1 . 6 wL = 1.2 ´ 8.86 + 1.6 ´ 8.8 = 24.71 kN/m, so that the required ultimate moment of resistance of the section is
For the equivalent rectangular stress block prescribed by SABS 0100, a = 0.45 and b = 0.9, while the design stress-strain curves for the prestressed and non-prestressed steel are as shown in Figs. 4- 17 and 4-23, respectively. Assuming fps = fpy = 1617 MPa, fs = fsy = 391.3 MPa and that the entire compression zone is contained in the flange, the depth to neutral axis is determined by taking moments about the position of the non-prestressed steel and solving the resulting expression for x.
Thus,
Solving for x yields x = 148.4 mm. Therefore s = bx = 0.9 ´ 148.4 = 133.6 mm is less than hf = 150 mm, which means that the entire compression zone is contained in the flange, as assumed.
Horizontal equilibrium requires that the following condition must be satisfied:
Solving this expression for As yields As = 621.8 mm2. This area of steel can be provided by 2 @ Y20 mm bars, for which As = 628 mm2.
The validity of the assumption that fps = fpy and that fs = fsy must be checked. This is done by calculating eps and es2 using the strain compatibility approach, as in example 4-13. Before this can be done, fse must be estimated so that its value is consistent with the assumptions made with regard to the various losses. Since the cross sectional area per strand Aps,strand = 100 mm2
If the effective prestress acting on the section is taken as P = -fseAps = -1113 ´ 500 ´ 10-3
= -556.6 kN, it can be shown that eps = 0.01752 and es2 = 0.01179. Since these values of eps and es2 are larger than epy = 0.01329 andesy = 0.00196, respectively, fps = fpy and fs = fsy (see Figs 4-17 and 4-23) as assumed.
Note that any other limit state can now be examined, as required. For example, consider the stresses in the concrete at transfer. For the losses assumed here, Pt = -5 Pt,strand = -5 ´ 132.5 = -662.6 kN,
while so that, at transfer, the stresses in the
top and bottom fibres of the section are given by M w Lu
To summarize: 5 @ 12.9 mm 7-wire super grade strands tensioned to 75% of their strength are required together with 2 @ Y20 mm bars.
If the concrete strength at transfer is taken as fci = -40 MPa then the permissible tensile and compressive stresses at transfer are and fct = 0.45 fci = 0.45
´ (-40) = -18 MPa, respectively. A comparison of the calculated concrete stresses with the permissible values clearly demonstrates that the stress limitations prescribed by SABS 0100 are satisfied at transfer.
4.5 REFERENCES
4-1 Park, R., and Paulay, T., Reinforced Concrete Structures, John Wiley & Sons, 1975.
4-2 South African Bureau of Standards, “The Structural Use of Concrete,” SABS 0100: 1992, Parts 1 and 2, SABS, Pretoria, 1992.
4-3 Khachaturian, N., Gurfinkel, G., Prestressed Concrete, McGraw-Hill Book Company, New York, 1969.
4-4 Hognestad, E., Hanson N. W. and McHenry D., “Concrete Stress Distribution in Ultimate Strength Design”. ACI Journal, Vol. 52, No. 6, December 1955.
4-5 Rüsch, H., “Researches Toward a General Flexural Theory for Structural Concrete”. ACI Journal, Vol. 57, No. 1, July 1960.
4-6 Committee of State Road Authorities, “Code of Practice for the Design of Highway Bridges and Culverts in South Africa,” TMH7 Part 3, CSRA, Pretoria, 1989.
4-7 British Standards Institution, “Structural Use of Concrete, Part 1, Code of Practice for Design and Construction,” BS 8110: Part 1: 1985, BSI, London, 1985.
4-8 British Standards Institution, “Steel, Concrete and Composite Bridges. Part 4: Code of Practice for Design of Concrete Bridges,” BS 5400: Part 4: 1984, BSI, London, 1984.
4-9 Warwaruk, J., Sozen, M. A., and Siess, C. P., “Strength Behaviour in Flexure of Prestressed Concrete Beams,” University of Illinois Engineering Experiment Station, Bulletin No. 464, 1962.
4-10 Lin, T. Y., and Burns, N. H., Design of Prestressed Concrete Structures, 3rd ed., John Wiley
& Sons, New York, 1981.
4-11 ACI Committee 318,"Building Code Requirements for Reinforced Concrete (ACI 318-89) and Commentary - ACI 318 R-89," American Concrete Institute, Detroit, 1989.
4-12 Gamble, W. L., “Prestressed Concrete,” Lecture Notes for Prestressed Concrete: CE 368, Department of Civil Engineering, University of Illinois at Urbana-Champaign, Urbana, October 1991.
4-13 Libby, J. R., Modern Prestressed Concrete: Design Principles and Construction Methods, 4th ed., Van Nostrand Reinhold, New York, 1990.
4-14 Naaman, A. E., Prestressed Concrete Analysis and Design: Fundamentals, McGraw-Hill Book Company, New York, 1982.
4-15 Kajfasz, S., Somerville, G., and C., Rowe, R. E., “An investigation of the behaviour of composite concrete beams,” Cement and Concrete Association, Research Report 15, November 1963.
4-16 Clark, L. A., Concrete Bridge Design to BS 5400, Construction Press, London, 1983.
4-17 Kong, F. K., and Evans, R. H., Reinforced and Prestressed Concrete, 3rd ed., Van Nostrand Reinhold (UK), Workingham, 1987.
4-18 South African Bureau of Standards, “The General Procedures and Loadings to be Adopted in the Design of Buildings,” SABS 0160: 1989, SABS, Pretoria, as amended 1990.
4-19 British Standards Institution, “Dead and Imposed Loads,” CP 3: Chapter V: 1967. Loading.
Part I, BSI, London, 1967.
4-20 British Standards Institution, “Steel, Concrete and Composite Bridges. Part 2: Specification for Loads,” BS 5400: Part 2: 1978, BSI, London, 1978.
4-21 Guyon, Y., Prestressed Concrete, John Wiley & Sons, New York, Vol. 1, 1960.
4-22 Magnel, G., Prestressed Concrete, 3rd ed., revised and enlarged, Concrete Publications Ltd., London, 1954.
4-23 FIP “Shear at the interface of precast and in situ concrete,” Technical Report FIP/9/4, August 1978.
4-24 Committee of State Road Authorities, “Code of Practice for the Design of Highway Bridges and Culverts in South Africa,” TMH7 Parts 1 and 2, CSRA, Pretoria, 1989.
4-25 Handbook to British Standard BS 8110: 1985: Structural Use of Concrete, Palladian Publications Ltd., London, 1987.
4-26 Olivier, J. J., The Use of Partial Prestressing for Road Bridges in South Africa, MEng Thesis, Department of Civil Engineering, University of Pretoria, Pretoria, May 1993.
4-27 Leonhardt, F., “To New Frontiers for Prestressed Concrete Design and Construction,” PCI Journal, Vol. 19, No. 5, September 1974, pp. 54-69.
4-28 Menn, C., “Partial Prestressing from the Designer’s Point of View,” Concrete International, Vol. 5, No. 3, March 1983, pp. 52-59.
4-29 Naaman, A. E., “Partially Prestressed Concrete: Review and Recommendations,” PCI Journal, Vol. 30, No. 6, November/December 1985, pp. 30-71.
4-30 Bachmann, H., “Partial Prestressing of Concrete Structures,” IABSE Surveys S-11/79, International Association for Bridge an Structural Engineering, Zürich, 1979.
4-31 Bachmann, H., “Design of Partially Prestressed Concrete Structures based on Swiss Experi-ences,” PCI Journal, Vol. 29, No. 4, July/August 1984, pp. 84-105.
4-32 Huber, A., “Practical Design of Partially Prestressed Concrete Beams,” Concrete International, Vol. 5, No. 4, April 1983, pp. 49-54.
4-33 Wiessler, H. H., “Partial Prestress for Bridges,” International Concrete Symposium, Concrete Society of Southern Africa, Portland Park, Halfway House, May 1984.
4-34 Marshall, V., Wium, D. J. W., and Olivier, J. J., “Use of Partial Prestressing for Road Bridges,” Annual Transportation Convention, Session 5D: Structures, Pretoria, August 1991.
The authors gratefully acknowledge:
• The support of the Concrete Society of Southern Africa for publishing this text as a book.
• The support and encouragement of the committee of the Prestressed Concrete Division of the Society. The contributions made by various members of this committee in terms of planning the text and in terms of their review comments were particularly useful.
The authors are particularly indebted to Michael A. Vasarhelyi of the Prestressed Concrete Division for his careful review of the entire text. His comments and suggestions contributed significantly to the value of the book.
Vernon Marshall John M. Robberts