1. DESIGN CONCEPTS
UNIT II

2. Flexural Strength of Pre-stressed Concrete Sections
Types of Flexural Failure
When prestressed concrete members are subjected to bending loads, different types of flexural failures are possible at critical sections, depending upon the principal controlling parameters, such as
the percentage of reinforcement in the section,

3. degree of bond between tendons and concrete,
compressive strength of concrete and
the ultimate tensile strength of the tendons.
In the post cracking stage, the behaviour of a prestressed concrete member is same as that of a reinforced concrete member and the theories used for estimating the flexural strength of reinforced concrete section may as well be used for pre-stressed concrete sections.

4. The various types of flexural failures encountered in prestressed concrete members are examined in the light of recommendations of various codes of practice:
Fracture of steel in tension
Failure of under-reinforced sections
Failure of over-reinforced sections
Other modes of failure

5. Fracture of steel in tension
The sudden failure of a pre-stressed member without any warning is generally due to the fracture of steel in the tension zone.
This type of failure is imminent when the percentage of steel provided in the section is so low that when the concrete in the tension zone cracks, the steel is not in a position to bear up the additional tensile stress transferred to it by the cracked concrete.

6. This type of failure can be prevented by providing a certain minimum percentage of steel in the cross-section.
The Indian standard code IS: prescribes a minimum longitudinal reinforcement of 0.2 percent of the cross-sectional area in all cases except in the case of pre-tensioned units of small sections.
When a high-yield strength deformed reinforcement is used, the minimum steel percentage is reduced to 0.15 percent.

7. The percentage of steel provided, both tensioned and untensioned taken together, should be sufficient so that when concrete in the pre-compressed tensile zone cracks, the steel is in a position to bear the additional tensile stress transferred to it by the cracking of the adjacent fibres of the concrete, thereby preventing a sudden failure of the beam due to fracture of steel in tension.

8. In contrast, the British code BS: prescribes that the number of prestressing tendons should be such that cracking of the concrete precedes the failure of the beam.
This requirement will be satisfied if the ultimate moment of resistance of the section exceeds the moment necessary to produces a flexural tensile stress in the concrete at the extreme tension fibres of magnitude equal to 0.6 √fcu.

9. In these computations, the effective prestress in concrete should be considered after allowing for the various losses.
The American Concrete Institute code ACI: specifies that the minimum area of bonded reinforcement should be not less than times the area of that part of cross-section which is between flexural tension face and the centre of gravity of the gross concrete section.

10. Failure of under-reinforced sections
If the cross-section is provided with an amount of steel greater than the minimum prescribed, the failure is characterized by an excessive elongation of steel followed by the crushing of concrete.
As bending loads are increased, excessive elongation of the steel raises the neutral axis closer to the compression face at the critical section.
The member approaches failure due to the gradual reduction of the compression zone, exhibiting large deflections and cracks, which develop at the soffit and progress towards the compression face.

11. When the area of concrete in the compression zone is insufficient to resist the resultant internal compressive force, the ultimate flexural failure of the member takes place through the crushing of concrete.
Large deflections and wide cracks are the characteristic features of under-reinforced sections at failure.

12. This type of behaviour is generally desirable since there is considerable warning before the impending failure.
As such, it is common practice to design under-reinforced sections which become more important in the case of statically indeterminate structures.
An upper limit on the maximum area of steel is generally prescribed in various codes for under-reinforced sections.

13. Failure of over-reinforced sections
When the effective reinforcement index, which is expressed in terms of
the percentage of reinforcement
the compressive strength of concrete and
the tensile strength of steel
exceeds a certain range of values, the section is said to be over-reinforced.

14. Generally, over-reinforced members fail by the sudden crushing of concrete, the failure being characterised by small deflections and narrow cracks.

15. The area of steel being comparatively large, the stresses developed in steel at failure of the member may not reach the tensile strength and in many cases it may well be within the proof stress of the tendons.
In structural concrete members, it is undesirable to have sudden failures without any warning in the form of excessive deflections and widespread cracks, and consequently the use of over-reinforced sections are discouraged.
The amount of reinforcement used in practice should, preferably, not exceed that required for a balanced section.

16. In this connection, most of the codes follow a conservative approach in formulating the evaluation procedures for flexural strength calculations of over-reinforced sections.
The redistribution of moments in an indeterminate structure depends upon the rotation capacities of the critical sections of the member under a given system of loads.
The use of over-reinforced sections in such structures curtails the rotation capacity of the sections, consequently affecting the ultimate load on the structure.

17. Other modes of failure
Pre-stressed concrete members subjected to transverse loads may fail in shear before their full flexural strength is attained, if they are not adequately designed for shear.
Web shear cracks may develop if the principal stresses are excessive, and if thin webs are used, the failure may occur due to web crushing.

18. In the case of pre-tensioned members, the failure of the bond between the steel and the surrounding concrete is likely due to inadequate transmission lengths at the ends of members.
In post-tensioned members, anchorage failures may take place if the end block is not properly designed to resist the transverse tensile forces.

19. Strain Compatibility Method
The rigorous method of estimating the flexural strength of prestressed concrete section is based on the compatibility of strains and equilibrium of forces acting on the section at the stage of failure.
The basic theory is applicable to all structural concrete sections, whether reinforced or prestressed, and generally the following assumptions are made:

20. The stress distribution in the compression zone of concrete can be defined by means of coefficients applied to the characteristic compressive strength and the average compressive stress and the position of the centre of compression can be assessed.
The distribution of concrete strain is linear (plane sections normal to axis remain plane after bending).
The resistance of concrete in tension is neglected.
The maximum compressive strain in concrete at failure reaches a particular value.

21. The flexural compressive stress in the compressive zone closely follows the stress strain curve of concrete.
The properties of the concrete stress block can be expressed in terms of the characteristic ratios k1 and k2.
Figure below shows the stress and strain distribution at the limit state of collapse for a rectangular section with steel in the tension zone.

23. The parameters k1 and k2 are not constant but depend upon the compressive strength of concrete.
Investigations by Hognestad et.al. have shown that k1 varies between 0.5 and 0.7 and k2 between 0.42 and 0.47 for the compressive strength varying from 60 to 20 N/mm2.
The figure below summarizes the characteristics of Hognestad et. al’s stress block.

25. In which the concrete cube strengths fck have been obtained from the cylinder strength fc’ using conversion factor of 0.8.
The figure shows that the ultimate strain ecu varies with the concrete strength.
However, the current British, American and Indian standard codes assume, for the sake of simplicity, a constant value for ultimate compressive strain in concrete irrespective of the strength of concrete.

26. Based on the stress block, Total compressive force Cu = k1. fck. b
Based on the stress block, Total compressive force Cu = k1.fck. b. x Total tensile force Tu = Aps.fpb The ultimate flexural strength of the section is expressed as

27. The major steps to be followed in the strain compatibility method are summarised below:
Compute the effective strain Ɛse in steel due to prestress after allowing for a losses from the stress-strain curve for steel.
Assume a trial value for the neutral axis depth x and evaluate (Ɛsu - Ɛse) from the strain diagram (assuming Ɛcu = , compute the value of Ɛsu)

28. 3. Using the stress-strain curve for steel, determine the value of stress in steel at failure fpb corresponding to Ɛsu. 4. Compute the total compression Cu and tension Tu. 5. If the compressive and tensile forces are equal, then the assumed value of x is correct.

29. 6. If the tension is less than compression, decrease the value of x and if tension exceeds compression, increase x and repeat steps 2 to 4 until a reasonable agreement is achieved.
7. Evaluate the ultimate moment Mu using the expression,
Mu = Aps.fpb(d-k2x)

30. Generally, it is possible to achieve force equilibrium within two or three trials.
The strain compatibility method is useful for estimating the ultimate flexural strength of over-reinforced sections in which the stresses in steel at failure do not reach the ultimate strength values.

31. A pretensioned concrete beam with a rectangular section, 100 mm wide by 160 mm deep, is prestressed by 10 high-tensile wires of 2.5 mm diameter located at an eccentricity of 40 mm. The initial force in each wire is 6.8kN. The strain loss in wires due to elastic shortening, creep and shrinkage of concrete is estimated to be units. The characteristic cube strength of concrete is 40 N/mm2. Given the load-strain curve of 2.5 mm diameter steel wire, estimate the ultimate flexural strength of the section using the strain compatibility method.

33. For fck = 40 N/mm2, From Figure Hognestad’s stress block, Ɛcu = 0
For fck = 40 N/mm2, From Figure Hognestad’s stress block, Ɛcu = , k1 = 0.57 and k2 = 0.45 Strain due to load of 6.8 kN in wire is (figure given in the problem) Effective strain in steel after all losses is given by

34. (Ɛsu – Ɛse)= Ɛcu

38. Simplified Code Procedures
Indian Code Provisions
The Indian standard code method (IS: ) for computing the flexural strength of rectangular sections or T-sections in which neutral axis lies within the flange is based on the rectangular and parabolic stress block as shown in the Figure.

41. For pretensioned and post tensioned members with an effective bond between concrete and tendons, the value of fpu and xu are given in Table 11, Appendix B of IS:
The effective prestress fpe after all losses should be not less than 0.451fp.

43. For post-tensioned rectangular beams with unbonded tendons, the values of fpu and xu are influenced by the effective span to depth ratios, and their values for different span/depth ratios are given in Table 12, Appendix B of IS: 1343 – 1980.

45. The ultimate moment of resistance of flanged sections in which the neutral axis falls outside the flange is computed by combining the moment of resistance of the web and flange portions and considering the stress blocks shown in the figure below.

48. For the effective reinforcement ratio of (Apw. fp/bw. d
For the effective reinforcement ratio of (Apw.fp/bw.d.fck), the corresponding values of (fpu/0.87fp) and (xu/d) are obtained from Table 11, Appendix B of IS:
The ultimate moment of resistance of the flanged section is obtained from the expression,

49. A pretensioned prestressed concrete beam having a rectangular section, 150 mm wide and 350 mm deep, has an effective cover of 50 mm. If fck = 40 N/mm2, fp = 1600 N/mm2, and the area of prestressing steel Ap = 461 mm2, calculate the ultimate flexural strength of the section using IS : 1343 code provisions.

50. From Table 12,IS:

52. A post-tensioned bridge girder with unbonded tendons is of box section of overall dimensions 1200 mm wide by 1800 mm deep, with wall thickness of 150 mm. The high-tensile steel has an area of 4000 mm2 and is located at an effective depth of 1600 mm. The effective prestress in steel after all losses is 1000 N/mm2 and the effective span of the girder is 24 m. If fck = 40 N/mm2 and fp = 1600 N/mm2, estimate the ultimate flexural strength of the section.

56. British Code Provisions
The British code BS: provides that for prestressed concrete members, the stress distribution in concrete at failure may be assumed to be rectangular with an average stress value of 0.45fcu and the depth of the stress block is assumed to be equal to 0.9 times the depth of the compression zone as shown in the Figure below.

58. The ultimate moment of resistance of a beam containing bonded or unbonded tendons, all of which are located in the tension zone, may be obtained from the equation:
fpb

59. For a rectangular or flanged beam in which the flange thickness is not less than 0.9x, dn may be taken as 0.45x.
For bonded tendons, the values of fpb and x may be obtained from the Table below.

61. For unbonded tendons, values of fpb and x may be obtained from the following equations:

62. The maximum value of fpb is limited to 0.7fpu.
In the absence of any rigorous analysis, any additional untensioned reinforcement As provided in the tension zone is replaced by an equivalent area of prestressing tendons (As.fy/fpu).
In contrast to the Indian Standard Code, in the British Code, the stress in tendons at failure fpb is influenced by the effective reinforcement ratio(fpu.Aps/fcu.b.d.) and the ratio (fpe/fpu).

63. The ultimate flexural strength of tee sections in which the neutral axis falls outside the flange is computed by combining the flexural strengths of the web and flange portions having the stress blocks as shown in the Figure below.

65. The ultimate moment of resistance is expressed as

66. A pretensioned beam of rectangular section 400 mm wide and 600 mm overall depth is stressed by 1700 mm2 of high-tensile wires located 100 mm from the soffit of the section. If the characteristic cube strength of concrete is 50 N/mm2 and the tensile strength of prestressing steel is 1600 N/mm2, estimate the flexural strength of the section using the British code recommendations. Assume the effective prestress after all losses as 960 N/mm2.

69. American Code Recommendations
The building code requirements of the American Concrete Institute ACI: 318 – 1989 recommends separate expressions for estimating the ultimate moment capacity of under reinforced and over reinforced rectangular and flanged sections with or without compression reinforcement.
The expressions are based on the assumption that the maximum strain in concrete, Ɛcu = 0.003, and the average concrete compressive stress in the rectangular stress block is 0.85 fc’ at the limit state of flexural failure.

70. The following notations are applicable in the various strength equations:

74. (a) Limitation of effective reinforcement index
The reinforcements in under-reinforced sections, which fail by yielding of reinforcement, which fail by yielding of reinforcement, should satisfy the following conditions:
For sections satisfying the above conditions, the ultimate flexural strength is computed by using the following relations

75. (b) Sections with tension reinforcement only (bonded tendons)
From the Figure given below, for rectangular or flanged sections in which the depth of the stress block (a) does not exceed the thickness of the flange (hf), the ultimate moment Mu is computed by:

77. When the compression flange thickness (hf) is less than the depth of the stress block (a), as shown in the Figure below, the ultimate moment may be computed by:

79. (c) Rectangular sections with compression reinforcement
For rectangular sections, the ultimate moment Mu is computed as follows:

80. When the value of ((Aps. fps + As. fy – As’
When the value of ((Aps.fps + As.fy – As’.fy)/bd) is less than the value specified above, the stress in the compression reinforcement is less than the yield strength fy.
In such cases, the effect of compression reinforcement may be neglected and a conservative estimate of the moment capacity may be obtained by equations given for rectangular sections.
However, a rigorous analysis based on stress-strain compatibility will yield an accurate estimate of the ultimate moment capacity of the section.

81. (d) Sections with unbonded tendons
For members with unbonded tendons and with a span/depth ratio not exceeding 35, the stress in tendons at failure is computed by the relation,

83. (e) Moment Capacity of Over-Reinforced Sections
Rectangular or flanged sections having the effective reinforcement index exceeding 0.36 β1 are considered as over-reinforced, and when the neutral axis lies within the flange, the ultimate moment is computed by the relation,

84. For flanged sections in which the neutral axis is located outside the flange, the ultimate moment is computed by the relation,

85. A pretensioned, prestressed concrete beam of rectangular section 150 mm wide and 350 mm deep, has an effective cover of 50 mm. If fc’ = 40 N/mm2, fpu = 1600 N/mm2, (fpy/fpu) = 0.90, determine, using ACI recommendations:
the minimum area of prestressing steel to avoid failure of section by fracture of steel;
The maximum area of prestressing steel which just ensures failure by yielding of steel;
Ultimate flexural strength corresponding to case (b); and
The ultimate flexural strength of the section if the area of prestressing steel in case (b) is doubled in the section.