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Publication Number: FHWA-HRT-05-058
Date: October 2006

Optimized Sections for High-Strength Concrete Bridge Girders--Effect of Deck Concrete Strength

CHAPTER 3. TASK 2: ANALYSES OF FLEXURAL STRENGTH AND DUCTILITY

The general philosophy in prestressed concrete design is that concrete members shall be designed so that the steel is yielding as ultimate capacity is approached. This is generally achieved by specifying a maximum amount of reinforcement for a given cross section. When the reinforcement exceeds the specified amount, the design-moment strength is based on the compression portion of the moment couple. In an over-reinforced section, an increase in the compressive strength of the deck concrete results in an increase in the moment capacity. It should be noted that current design requirements have been developed based on lower strength concretes. It was, therefore, deemed appropriate to investigate the flexural strength and ductility that would result when higher-strength concretes are used in bridge decks.

RESEARCH APPROACH

Analyses for the effect of higher concrete strengths on the flexural strength and moment-curvature relationships were investigated using a computer program known as BEAM BUSTER. The program BEAM BUSTER performs a moment-curvature analysis of a reinforced or prestressed element using actual material properties. The program takes into account uncracked, cracked, and post-yield behavior of flexural members for a specified cross section and curvature. The program BEAM BUSTER calculates the strains and stresses that satisfy equilibrium of forces on the cross section and compatibility of strains. A cross section may be divided into many elements. A different stress–strain curve may be utilized for each element of the cross section. The stress–strain data are input as discrete data points along the complete stress–strain curve for each element specified in the cross section.

Analyses were performed for the following variables:

  • Girder concrete compressive strength: 41 and 83 MPa (6,000 and 12,000 psi, respectively).
  • Deck concrete compressive strength: 28, 41, 55, and 69 MPa (4,000, 6,000, 8,000 and 10,000 psi, respectively).
  • Girder cross section: PCI Bulb-Tee with a depth of 1.83 m (72 inches).
  • Span lengths: 24.3, 44.5, and 53.3 m (80, 146, and 175 ft, respectively).

The combination of variables selected for analysis is defined in tables 9 and 10. The combinations were selected to represent a range of girder strengths and spans.

Table 9. Task 2 variables (SI units).
Series Girder Strength (MPa) Deck Strength (MPa) Span (m) No. of Strands* Flange Width (m)
A 41 28 24.4 19 2.44
41 41 24.4 19 2.44
41 55 24.4 19 2.44
41 69 24.4 19 2.44
B 83 28 24.4 19 2.44
83 41 24.4 19 2.44
83 55 24.4 19 2.44
83 69 24.4 19 2.44
C 41 28 44.5 41 1.37
41 41 44.5 41 1.37
41 55 44.5 41 1.37
41 69 44.5 41 1.37
D 83 28 53.3 76 1.37
83 41 53.3 76 1.37
83 55 53.3 76 1.37
83 69 53.3 76 1.37

* For consistency between tasks, the odd number of strands calculated by program BRIDGE in task 1 was retained in task 2.

Table 10. Task 2 variables (English units).
Series Girder Strength (psi) Deck Strength (psi) Span (ft) No. of Strands* Flange Width (inches)
A 6,000 4,000 80 19 96
6,000 6,000 80 19 96
6,000 8,000 80 19 96
6,000 10,000 80 19 96
B 12,000 4,000 80 19 96
12,000 6,000 80 19 96
12,000 8,000 80 19 96
12,000 10,000 80 19 96
C 6,000 4,000 146 41 54
6,000 6,000 146 41 54
6,000 8,000 146 41 54
6,000 10,000 146 41 54
D 12,000 4,000 175 76 54
12,000 6,000 175 76 54
12,000 8,000 175 76 54
12,000 10,000 175 76 54

* For consistency between tasks, the odd number of strands calculated by program BRIDGE in task 1 were retained in task 2.

The cross sections used in the analyses are shown in figure 16. For purposes of simplicity, the non-prestressed reinforcement in the deck was not included in the analyses. The flange width of the composite section and number of strands were based on the calculations performed by the program BRIDGE in task 1.

Prior to performing the analyses, it was necessary to define the stress–strain curves for the individual constituent materials consisting of deck concrete, girder concrete, and prestressing strand.

Figure 16 (part 1). Diagram. Cross sections of series A girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches). This figure shows a barbell shaped drawing, representing a cross section of the girder BT-72, standing straight up on end. On top of the girder is a long rectangular box representing the bridge deck. The top of the girder is wider and thinner, while the base is narrow and thick. The deck measures 2,440 millimeters long and 190 millimeters thick. An arrow pointing to the base indicates the presence of 19 12.7-millimeter-diameter strands. Figure 16 (part 2). Diagram. Cross sections of series B girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches). This figure shows a barbell shaped drawing, representing a cross section of the girder BT-72, standing straight up on end. On top of the girder is a long rectangular box representing the bridge deck. The top of the girder is wider and thinner, while the base is narrow and thick. The deck measures 2,440 millimeters long and 190 millimeters thick. An arrow pointing to the base indicates the presence of 19 12.7-millimeter-diameter strands.

Figure 16 (part 1). Cross section of series A girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches).

Figure 16 (part 2). Cross section of series B girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches).

Figure 16 (part 3). Diagram. Cross sections of series C girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches). This figure shows a barbell shaped drawing, representing a cross section of the girder BT-72, standing straight up on end. On top of the girder is a long rectangular box representing the bridge deck. The top of the girder is wider and thinner, while the base is narrow and thick. The deck measures 1,370 millimeters long and 190 millimeters thick. An arrow pointing to the base indicates the presence of 41 12.7-millimeter-diameter strands. Figure 16 (part 4). Diagram. Cross sections of series D girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches). This figure shows a barbell shaped drawing, representing a cross section of the girder BT-72, standing straight up on end. On top of the girder is a long rectangular box representing the bridge deck. The top of the girder is wider and thinner, while the base is narrow and thick. The deck measures 1,370 millimeters long and 190 millimeters thick. An arrow pointing to the base indicates the presence of 76 12.7-millimeter-diameter strands.

Figure 16 (part 3). Cross section of series C girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches).

Figure 16 (part 4). Cross section of series D girder (BT-72) analyzed in task 2. All dimensions are in millimeters (inches).

MATERIAL PROPERTIES

Required input for the program BEAM BUSTER consists of stress–strain curves for each element analyzed in the cross section. Therefore, it was necessary to define a family of stress–strain curves for different strength concretes and to select an appropriate curve for the prestressing strand.

Stress–Strain Curves for Concrete

Since the calculations for ductility are very sensitive to the assumed shape of the stress–strain curves, it was necessary to establish a family of curves that realistically represent the properties of high-strength concrete. Several researchers have experimentally determined the complete stress-strain curves of concrete for strengths up to about 100 MPa (14,500 psi).(18,19) The measurements have indicated that the following occur with increasing strength:

  • The initial slope of the stress–strain curve increases.
  • The ascending portion of the stress–strain curve is more linear.
  • The strain at peak stress increases.
  • The slope of the descending portion of the curve increases.

The slope of the ascending portion of the stress–strain curve is the modulus of elasticity of the concrete. Several equations have been proposed that relate the modulus of elasticity, Ec, to the concrete compressive strength, f'c, and the unit weight of the concrete, wc, including:

ACI 318: (20) Equation 1. The equation reads E subscript C equals 33 times W superscript 1.5, subscript C, times the square root of F prime subscript C. (1)

Martinez: (21) Equation 2. The equation reads E subscript C equals open parentheses, 40,000 times the square root of F prime subscript C, that sum plus 1,000,000 close parentheses, times open parentheses W subscript C divided by 145 close parentheses, all to the power of 1.5. (2)

Canadian Code: (22) Equation 3. The equation reads E subscript C is equal to open parentheses, 39,740 times the square root of F prime subscript C, that sum plus 1,000,000 close parentheses, times open parentheses W subscript C divided by 143.5 close parentheses, all to the power of 1.5. (3)

Ahmad: (19) Equation 4. The equation reads E subscript C is equal to open parentheses, W subscript C close parentheses, to the power of 2.5, open parentheses, F prime subscript C, close parentheses, to the power of 0.325. (4)

Values of modulus of elasticity calculated according to the above equations are tabulated in table 11.

Table 11. Calculated values of modulus of elasticity.
Compressive Strength (MPa) Unit Weight (kg/m3) Equation
ACI (19) Martinez (20) CanadianCode (21) Ahmad (18)
(GPa) (GPa) (GPa) (GPa)
28 2320 25.1 24.3 24.5 25.8
41 2370 31.7 29.2 29.0 31.0
55 2420 37.8 33.5 33.9 35.9
69 2480 43.9 38.1 38.5 39.1
83 2500 48.5 41.4 41.8 44.3
(psi) (lb/ft3) (106 psi) (106 psi) (106 psi) (106 psi)
4,000 145 3.64 3.53 3.55 3.75
6,000 148 4.60 4.23 4.21 4.50
8,000 151 5.48 4.86 4.92 5.20
10,000 155 6.37 5.53 5.59 5.97
12,000 156 7.04 6.01 6.07 6.43

The ACI equation was based on an analysis for concrete strengths up to about 41 MPa (6,000 psi).(20, 23) Several investigators have indicated that the ACI equation tends to overestimate the modulus of elasticity for the higher-strength concretes.(1, 13, 19) The Martinez equation was developed as an alternative for the ACI equation and gives lower values of modulus of elasticity at the higher strength levels. (21) Some publications have indicated that the Martinez equation may underestimate the modulus of elasticity at the very high strength levels. (19) The Canadian Code equation was based on the Martinez equation with some rounding off for use in SI units and then converted back into English units. The rounding off results in slightly different calculated values. The Ahmad equation was based on a statistical analysis of data. Modulus values from the Ahmad equation lie between those of the ACI and the Martinez equations. For the present investigation, it was decided to use the ACI values for concrete strengths of 28 and 41 MPa (4,000 and 6,000 psi, respectively) and to use the values by the Ahmad equation for concrete strengths of 55, 69, and 83 MPa (8,000, 10,000, and 12,000 psi, respectively).

Collins et al. have indicated that the strain at peak stress, epsilon'c , can be calculated from the following equation: (24)

Equation 5. The equation reads epsilon prime subscript C equals F prime subscript C divided by E subscript C, that sum times the sum of N divided by N minus 1. (5)

where

Equation 6. The equation reads N equals 0.8 plus the sum of F prime subscript C divided by 17 in SI units, open parentheses, megapascals, close parentheses. in MPa units (6)

Equation 7. The equation reads N equals 0.8 plus the sum of F prime subscript C divided by 2500 in U.S. Customary units, open parentheses, pounds per square inch, close parentheses. in psi units (7)

These equations were used in the present analyses to calculate the strain at peak stress from the modulus of elasticity and concrete compressive strength.

Several investigators have published algebraic expressions to accurately describe the shape of the rising and descending branch of the stress-strain curve. The following expression was proposed by Popovics: (25)

Equation 8. The equation reads F subscript C divided by F prime subscript C is equal to epsilon subscript C divided by epsilon prime subscript C, that sum times N divided by the sum of open parentheses N minus 1 close parentheses plus open parentheses epsilon subscript C over epsilon prime subscript C close parentheses to the power of N. (8)

where fc = stress at a strain of epsilonc.

Thorenfeldt proposed a modification to Popovic's equation to increase the slope of the descending portion of the stress-strain curve. (26) He introduced a factor, k, to modify the equation as follows:

Equation 9. The equation reads F subscript C divided by F prime subscript C is equal to epsilon subscript C divided by epsilon prime subscript C, that sum times N divided by the sum of open parentheses N minus 1 close parentheses plus open parentheses epsilon subscript C over epsilon prime subscript C close parentheses to the power of NK. (9)

where k = 1 on the ascending portion of the curve, and k is greater than 1 for the descending portion of the curve. Collins has suggested the following for the descending portion of the curve.( 24)

Equation 10. The equation reads K equals 0.67 plus the sum of F prime subscript C divided by 62 in SI units open parentheses megapascals close parentheses. in MPa units (10)

Equation 11. The equation reads K equals 0.67 plus the sum of F prime subcript C divided by 9,000 in U.S. Customary units open parentheses pounds per square inch close parentheses. in psi units (11)

stress–strain curves generated using the above approach are shown in figure 17.

Figure 17. Graph. Stress-strain curves for concrete used in BEAM BUSTER analysis. This graph shows five distinct lines. The first line, solid with black squares running through it, is labeled 28 megapascals. The second line, solid with clear squares running through it, is labeled 41 megapascals. The third line, solid with black diamonds, is labeled 55 megapascals. The fourth line, solid with clear diamonds, is labeled 69 megapascals. The fifth line, solid with black triangles, is labeled 83 megapascals. The vertical axis of this graph ranges from -20 to 90 and represents the Stress in megapascals, while the horizontal axis of this graph ranges from -500 to 5,000 and represents Strain in millionths. All of the lines begin in the bottom left negative quadrant of the graph and come together at the point zero. At zero each line ascend upward until reaching point 1,800 on the vertical axis. The first line, solid black squares, reaches a peak of 25 on the vertical axis at which time it levels off and declines slightly ending at point 5,000 on the horizontal axis and 15 on the vertical axis. The second line, solid with clear squares, ascends to point 40 on the vertical axis at which time it enters into a downward slope and ending at point 5,000 on the horizontal axis and 10 on the vertical axis. The third line, solid with black diamonds, ascends to point 52 on the vertical axis at which time it slopes downward ending at point 5,000 on the horizontal axis and 3 on the vertical axis. The fourth line, solid with clear diamonds, ascends to point 68 on the vertical axis at which time it enters into a downward slope and ending at point 5,000 on the horizontal axis and 1 on the vertical axis. The fifth line, solid with black triangles, reaches the highest point at 82 on the vertical axis at which time it enters into a rapid downward slope and ending at point 5,000 on the horizontal axis and 0 on the vertical axis. All lines converge at point 3,000 on the horizontal axis and 21 on the vertical axis before reaching the end of the chart.

Figure 17. stress–strain curves for concrete used in BEAM BUSTER analysis.

To provide some tensile capacity to the concrete prior to cracking, the stress–strain curves developed in compression were extrapolated backwards using the following assumptions:

  • The slope of the curve in tension is the same as that in compression near the origin.
  • A maximum tensile stress, ft, calculated by the following equation, was used: (19)
Equation 12. The equation reads F subscript T equals 2.3 times open parentheses F prime subscript C to the power of two thirds. (12)

The stress–strain curve in compression was calculated out to a strain of 0.005. The calculated stress–strain curves were then compared with the measured curves obtained by Kaar. (18) These comparisons were made to ensure that the calculated curves showed reasonable agreement with measured data. It should also be noted that in these curves, the peak stress corresponds to the designated concrete compressive strength. In concrete column testing, it has been observed that the maximum stress obtained in the column concrete can be less than the strength measured on a standard concrete cylinder. This difference is attributed to the difference in size and shape between the reinforced concrete column and the concrete cylinder; the differences in concrete casting, vibration, and curing; and to differences in the rate of loading. In his original column research, Hognestad chose a value of 0.85 for the ratio between maximum stress in the column and concrete cylinder strength. (27) He indicated that this factor may be systematically too high or too low and may not be a constant. In tests of plain concrete columns subjected to linearly varying strain, Hognestad and Kaar deduced (separately) that the factor was close to 1.0, varying from 0.96 to 1.12.(18, 28) In design, a value of 0.85 is normally used. However, for purposes of the present analyses, a value of 1.0 was selected.

stress–strain Properties of Strand

An assumed stress–strain curve for strand was used in computer program BRIDGE in task 1. This curve is shown in figure 18. A slight modification to the curve was made for use in the program BEAM BUSTER. This modification is shown in figure 18.

Figure 18. Graph. Stress-strain curve for prestressing strand used in BEAM BUSTER analysis. This graph shows two lines. The first line, solid with black squares, is labeled Bridge. The second line, solid with clear squares running through it, is labeled BEAM BUSTER. The vertical axis of this graph ranges from zero to 2,000 and represents the Stress in megapascals, while the horizontal axis of this graph ranges from zero to 6,000 and represents Strain in millionths. The first line, solid black squares, begins at point 25 on horizontal axis and point 2,700 on the vertical axis. The line slopes upward to point 51 on the horizontal axis and 4,100 on the vertical axis. The second line, solid with clear squares, begins at point 25 on horizontal axis and point 2,750 on the vertical axis. The line slopes upward to point 51 on the horizontal axis and 4,200 on the vertical axis. The third line, solid with black diamonds, begins at point 25 on horizontal axis and point 2,800 on the vertical axis. The line slopes upward to point 51 on the horizontal axis and 4,300 on the vertical axis. The fourth line, solid with clear diamonds, begins at point 25 on the horizontal axis and point 2,900 on the vertical axis. The line slopes upward to point 51 on the horizontal axis and 4,400 on the vertical axis.

Figure 18. stress–strain curve for prestressing strand used in BEAM BUSTER analysis.

MOMENT-CURVATURE RELATIONSHIPS

The moment-curvature relationships were calculated in two parts. In the first part, the moment was applied to the noncomposite section consisting of the prestressed concrete girder only. This part of the calculation represents the moments caused by girder dead load and deck dead load, and assumes that the dead load of the deck is carried entirely by the girder prior to development of composite action.

In the second part of the calculation, the moment-curvature relationship was calculated for the composite section. This calculation represents any moment applied after the deck and girder act as a composite member. To ensure compatibility between the two parts of the analyses, it was necessary to match the moment-curvature relationships at the transition point between the two parts of the calculation. This was accomplished by making an artificial adjustment in the stress-strain curves for the concrete deck. The deck was assumed to consist of three layers of concrete and a separate adjustment was made for each layer. The girder concrete was assumed to consist of one layer. For a specified input curvature, the program BEAM BUSTER calculates the corresponding moment that results in equilibrium of forces and compatibility of strains. The output consists of curvature, moment, and selected strains and stresses. If compatibility of strains or equilibrium of forces cannot be calculated for a given curvature, the calculation for that curvature is terminated after a specified number of iterations. This generally occurs after the stress-strain curve for a concrete element or prestressing strand has returned to zero strain.

Moment-curvature relationships for the four series of analyses described in tables 9 and 10 are shown in figures 19 through 22. Figures 19 and 20 show the relationships for 41-MPa (6,000 psi) and 83-MPa (12,000-psi) girder strength concrete, respectively, at a span of 24.4 m (80 ft)—the minimum span length considered in this investigation. Figure 21 shows the relationship for a girder compressive strength of 41 MPa (6,000 psi) at a span of 44.5 m (146 ft)—the longest span for which a 41 MPa (6,000 psi) compressive girder can be designed. Figure 22 shows the relationship for a girder with concrete compressive strength of 83 MPa (12,000 psi) at a span of 53.3 m (175 ft)—the longest span length for which an 83 MPa (12,000 psi) girder can be designed.

All of the moment-curvature relationships show a similar shape that can be divided into four parts. The first part consists of the moment-curvature relationship for the noncomposite section. This begins at a negative curvature because of the prestressing force and continues until a moment equivalent to the dead load of the deck and girder have been applied. The second part of the moment-curvature relationship consists of the moment applied to the composite section prior to cracking of the concrete. This part has a slightly steeper slope than that for the noncomposite section because of the higher stiffness of the composite section. The third part of the curve consists of the moment-curvature relationship following cracking and prior to yielding of the prestressing strand. The slope of this portion of the curve is considerably less than the curve for the uncracked section. The slope is greatest for the sections that contain the largest number of strands. The final part of the curve consists of the moment-curvature relationship following yielding of the strand. This portion of the curve is largely horizontal and generally continues until the end point of one of the stress-strain curves is reached. The moments corresponding to girder and deck dead load, girder and deck dead load and live load plus impact, and required strength are also shown on the figures.

As shown in figures 19 and 20, the moment-curvature relationships for the girders containing 19 strands at a span of 24.4 m (80 ft) are not influenced by the deck concrete strength. Minor differences do occur, but at the scale shown in these figures, the differences are not discernible. The maximum moment for these sections is achieved when the prestressing strand reaches its maximum stress (which was assumed to be at a strain of 6 percent). This indicates that these cross sections will have a flexural capacity determined by fracture of the prestressing strand. These sections had a final curvature that was about 10 times the curvature when the first strands yielded.

The impact of utilizing higher-strength concrete in the deck is more evident in figures 21 and 22. In the cross sections with a larger number of strands, the breaking strength of the strand is not reached prior to the flexural strength of the section being achieved.

Figure 19. Graph. Moment-curvature relationships for BT-72, 41 megapascals at a span of 24.4 meters. This graph shows four lines representing desk strength. The first line, solid black, is labeled 28 megapascals. The second line, dashes, is labeled 41 megapascals. The third line, dotted, is labeled 55 megapascals. The fourth line dashes and dots, is labeled 69 megapascals. The vertical axis of this graph ranges from zero to 22,000 and represents the Moment, in kilonewton-meters, while the horizontal axis of this graph ranges from zero to 35 and represents curvature in microradians per millimeter. All of the lines follow the same general course beginning at zero on the horizontal axis and just under 2,000 on the vertical axis. The lines make a swift incline to 4,000. At approximately three on the horizontal axis, the lines have reached 6,000 on the vertical axis. At which time the lines level off stretching forward to approximately 30 on the horizontal axis before stopping.

Figure 19. Moment-curvature relationships for BT-72, 41 MPa at a span of 24.4 m.

Figure 20. Graph. Moment-curvature relationships for BT-72, 83 megapascals at a span of 24.4 meters. This graph shows four lines representing deck strength. The first line, solid black, is labeled 28 megapascals. The second line, dashes, is labeled 41 megapascals. The third line, dotted, is labeled 55 megapascals. The fourth line dashes and dots, is labeled 69 megapascals. The vertical axis of this graph ranges from zero to 22,000 and represents the Moment in kilonewton-meters, while the horizontal axis of this graph ranges from zero to 35 and represents curvature in microradians per millimeter. All of the lines follow the same general course beginning at zero on the horizontal axis and just under 2,000 on the vertical axis. The lines make a swift incline to 4,000. At approximately three on the horizontal axis, the lines have reached 6,000 on the vertical axis. At which time the lines level off stretching forward to approximately 30 on the horizontal axis before stopping.

Figure 20. Moment-curvature relationships for BT-72, 83 MPa at a span of 24.4 m.

Figure 21. Graph. Moment-curvature relationships for BT-72, 41 megapascals at a span of 44.5 meters. This graph shows four lines representing deck strength. The first line, solid black, is labeled 28 megapascals. The second line, dashes, is labeled 41 megapascals. The third line, dotted, is labeled 55 megapascals. The fourth line dashes and dots, is labeled 69 megapascals. The vertical axis of this graph ranges from zero to 22,000 and represents the Moment in kilonewton-meters, while the horizontal axis of this graph ranges from zero to 30 and represents curvature in microradians millimeter. All of the lines follow the same general course beginning at zero on the horizontal axis and just under 4,000 on the vertical axis. The lines make a swift incline to 7,000. At approximately three on the horizontal axis, the lines have reached 12,000 on the vertical axis. At which time the lines level off stretching forward to approximately 20 on the horizontal axis. Separately, each line declines ever so slightly until ending. The solid line stops at 21 on the horizontal axis and just above 12,000 on the vertical axis. The dashed line stops at 23 on the horizontal axis and just above the solid line on the vertical axis. The dotted line stops at 25 on the horizontal axis and just above the dashed line on the vertical axis. The dashed and dotted line stops at 27 on the horizontal axis and just above the dotted line on the vertical axis.

Figure 21. Moment-curvature relationships for BT-72, 41 MPa at a span of 44.5 m.

Figure 22. Graph. Moment-curvature relationships for BT-72, 83 megapascals at a span of 53.3 meters. This graph shows four lines representing deck strength. The first line, solid black, is labeled 28 megapascals. The second line, dashes, is labeled 41 megapascals. The third line, dotted, is labeled 55 megapascals. The fourth line dashes and dots, is labeled 69 megapascals. The vertical axis of this graph ranges from zero to 22,000 and represents the Moment in kilonewton-meters, while the horizontal axis of this graph ranges from zero to 30 and represents curvature in microradians per millimeter. All of the lines follow the same general course beginning at zero on the horizontal axis and approximately 5,000 on the vertical axis. The lines make a swift incline to 10,000. At approximately five on the horizontal axis, the lines have reached 18,000 on the vertical axis. At which time the lines level off individually make a slight incline before making their descent and final stop. The solid line reaches a peak just over 18,000 and stops at 15 on the horizontal axis and just above 18,000 on the vertical axis. The dashed line stops at 15 on the horizontal axis and just above the solid line on the vertical axis. The dotted line stops at 16 on the horizontal axis and just above the dashed line on the vertical axis, it reaches a peak of approximately 19,000. The dashed and dotted line stops at 17 on the horizontal axis and just above the dotted line on the vertical axis but reaches a peak of 20,000 on the vertical axis.

Figure 22. Moment-curvature relationships for BT-72, 83 MPa at a span of 53.3 m.

For the 44.5-m (146-ft) and 53.3-m (175-ft) spans, the use of higher-strength concrete in the deck resulted in a slightly higher flexural capacity and a small increase in the final curvature. For the 44.5-m (146-ft) span girders, the final curvature was six to eight times the curvature at yield of the lower layer of strands. This indicated that the sections achieve adequate ductility even though fracture of the strand was not achieved at maximum moment. For the 53.3-m (175-ft) spans, the final curvature was about four times the curvature at yield of the lower layer of strands. This ductility is less than was obtained with the shorter span lengths but one that still corresponds to a very large deflection which would give adequate visual warning of impending failure.

The strand stresses, deck strains, and girder strains at maximum moment are tabulated in tables 12 and 13. The strand stress is the stress in the lower layer of strands. The deck strain is the strain at the top surface of the deck. The girder strain is the strain at the top surface of the girder. This table shows that for series A and B, the strand stress reached the breaking strength of the strand while the maximum deck strain was 2,160 millionths, which is only slightly greater than the assumed strain at peak stress for the 28 MPa (4,000 psi) concrete. For the other strength concretes, the deck strains are less than the strains at peak stress. Also for series A and B sections, the depth of the neutral axis was less than the deck thickness so that the strain at the top of the girder was tensile and sufficient to cause cracking. This is denoted by the letter "T" in the last column of the table. Series A and B sections had a flange width of 2.44 m (96 inches), so there was sufficient concrete area to allow development of the compression force for the flexural resistance.

Table 12. Calculated stresses and strains at maximum moment (SI units).
Series Girder Strength (MPa) Deck Strength (MPa) Span (m) Strand Stress (GPa) Deck Strain (millionths) Girder Strain* (millionths)
A 41 28 24.4 1.90 2,160 T
41 41 24.4 1.90 1,720 T
41 55 24.4 1.90 1,540 T
41 69 24.4 1.90 1,430 T
B 83 28 24.4 1.90 2,160 T
83 41 24.4 1.90 1,720 T
83 55 24.4 1.90 1,550 T
83 69 24.4 1.90 1,430 T
C 41 28 44.5 1.84 4,250 1,340
41 41 44.5 1.83 2,900 360
41 55 44.5 1.86 3,020 T
41 69 44.5 1.87 2,740 T
D 83 28 53.3 1.80 3,640 2,025
83 41 53.3 1.79 2,560 1,320
83 55 53.3 1.81 2,710 1,090
83 69 53.3 1.80 2,410 800

* T denotes tensile strain sufficient to cause cracking.

Table 13. Calculated stresses and strains at maximum moment (English units).
Series Girder Strength (psi) Deck Strength (psi) Span (ft) Strand Stress (ksi) Deck Strain (millionths) Girder Strain* (millionths)
A 6,000 4,000 80 275 2,160 T
6,000 6,000 80 275 1,720 T
6,000 8,000 80 275 1,540 T
6,000 10,000 80 275 1,430 T
B 12,000 4,000 80 275 2,160 T
12,000 6,000 80 275 1,720 T
12,000 8,000 80 275 1,550 T
12,000 10,000 80 275 1,430 T
C 6,000 4,000 146 267 4,250 1,340
6,000 6,000 146 266 2,900 360
6,000 8,000 146 270 3,020 T
6,000 10,000 146 271 2,740 T
D 12,000 4,000 175 261 3,640 2,025
12,000 6,000 175 260 2,560 1,320
12,000 8,000 175 262 2,710 1,090
12,000 10,000 175 262 2,410 800

* T denotes tensile strain sufficient to cause cracking.

Series C and D cross sections contained 41 and 76 strands, respectively, and had a flange width of 1.38 m (54 inches). Consequently, the demands on the deck for the development of the compressive force were much higher than in series A and B. Flexural strength of the section was limited by the strain capacity of the deck concrete for all deck concrete strength levels. For all strengths of deck concrete, the deck concrete strains at maximum moment exceeded the strains at peak stress as shown in figure 17. For the sections with 28-MPa (4,000-psi) deck concrete, the strains exceeded the normally assumed limit of 3,000 millionths (0.003 strain). To maintain the deck strains below 3,000 millionths, a concrete deck strength of at least 41 MPa (6,000 psi) is needed. It should also be noted that with one exception, all of the cross sections in series C and D have reinforcement indices that permit the flexural design to be based on yielding of the steel. The exception is the 83/28-MPa (12,000/4,000-psi) combination of girder and deck strengths. For this section, the strain in the girder is close to the strain at peak stress for the 83-MPa (12,000‑psi) girder concrete. This is a combination that should be avoided. Based on the analyses, it appears that there should be a limit on the difference between the girder concrete strengths and the deck concrete strength. However, the limited scope of this investigation does not permit the development of a rationale analysis. As an interim measure, it is proposed that the specified deck concrete strength should be at least 60 percent of the specified girder concrete strength at 28 days when the specified girder concrete strength exceeds 41 MPa (6,000 psi).

FLEXURAL STRENGTH

The flexural strengths for each of the sections analyzed in task 2 are tabulated in tables 14 and 15. The required strength and design strength were calculated using the computer program BRIDGE in task 1. The required strength was based on the following equation:

Equation 13. The equation reads M is equal to 1.3 times open parentheses Dead Load Moment plus 1.67 times open parentheses Live Load plus Impact moment, close parentheses, close parentheses. (13)

The design strength was calculated through an iteration process. (17) The number in the first column of nominal strengths was based on the design strength divided by a strength reduction factor of 0.9. The second column of nominal strengths were calculated using the AASHTO provisions. (16) The calculated strengths represent the maximum moments determined by the program BEAM BUSTER and are assumed to represent the real strengths of the sections. For all sections analyzed, the design strength exceeded the required strength and the calculated strength was greater than the nominal strength calculated by two different methods. It should also be noted that the nominal strengths provided an excellent prediction of the calculated strengths. Comparisons of required strengths with calculated strengths are shown in figures 19 through 22.

Table 14. Calculated flexural strengths (SI units).
Series Girder Strength (MPa) Deck Strength (MPa) Span (m) Moments (kN·m)
Required Strength Design Strength* Nominal Strength* Nominal Strength** Calculated Strength***
A 41 28 24.4 6,060 6,100 6,780 6,640 6,820
41 41 24.4 6,060 6,100 6,780 6,700 6,860
41 55 24.4 6,060 6,100 6,780 6,720 6,860
41 69 24.4 6,060 6,100 6,780 6,740 6,960
B 83 28 24.4 6,060 6,100 6,780 6,640 6,820
83 41 24.4 6,060 6,100 6,780 6,700 6,860
83 55 24.4 6,060 6,100 6,780 6,720 6,860
83 69 24.4 6,060 6,100 6,780 6,740 6,870
C 41 28 44.5 9,720 11,730 13,030 12,950 13,150
41 41 44.5 9,720 12,120 13,463 13,330 13,440
41 55 44.5 9,720 12,370 13,734 13,520 13,750
41 69 44.5 9,720 12,550 13,951 13,670 13,920
D 83 28 53.3 13,600 16,970 18,660 18,240 19,040
83 41 53.3 13,600 17,490 19,430 19,160 19,520
83 55 53.3 13,600 17,960 19,950 19,770 19,980
83 69 53.3 13,600 18,330 20,360 20,240 20,310

* Calculated by the program BRIDGE.
** Calculated per AASHTO.
*** Calculated by the program BEAM BUSTER.

Table 15. Calculated flexural strengths (English units).
Series Girder Strength (psi) Deck Strength (psi) Span(ft) Moments (ft-kip)
Required Strength Design Strength* Nominal Strength* Nominal Strength** Calculated Strength***
A 6,000 4,000 80 4,470 4,500 5,000 4,900 5,030
6,000 6,000 80 4,470 4,500 5,000 4,940 5,060
6,000 8,000 80 4,470 4,500 5,000 4,960 5,060
6,000 10,000 80 4,470 4,500 5,000 4,970 5,060
B 12,000 4,000 80 4,470 4,500 5,000 4,900 5,030
12,000 6,000 80 4,470 4,500 5,000 4,940 5,060
12,000 8,000 80 4,470 4,500 5,000 4,960 5,060
12,000 10,000 80 4,470 4,500 5,000 4,970 5,070
C 6,000 4,000 175 7,170 8,650 9,610 9,550 9,700
6,000 6,000 175 7,170 8,940 9,930 9,830 9,910
6,000 8,000 175 7,170 9,120 10,130 9,970 10,140
6,000 10,000 175 7,170 9,260 10,290 10,080 10,270
D 12,000 4,000 175 10,030 12,520 13,910 13,450 14,040
12,000 6,000 175 10,030 12,900 14,330 14,130 14,400
12,000 8,000 175 10,030 13,250 14,720 14,580 14,740
12,000 10,000 175 10,030 13,520 15,020 14,930 14,980

* Calculated by the program BRIDGE.
** Calculated per AASHTO.
*** Calculated by the program BEAM BUSTER.

TASK 2 CONCLUSIONS

Based on the task 2 analyses, the following conclusions are made:

  • For span lengths of 24.4 m (80 ft) with lower amounts of prestressing strands, the use of high-strength concrete did not affect the flexural strengths.
  • At the maximum span lengths for each girder concrete strength, the use of high-strength concrete in the decks had a slight effect in increasing the design and nominal strengths. The use of high-strength concrete also increased the flexural ductility of the cross sections.
  • To ensure that the flexural strength is based on yielding of the reinforcement and to limit the strains in the deck, a minimum deck concrete strength of 41 MPa (6,000 psi) is recommended for span lengths in excess of 24.4 m (80 ft) when concrete girder strength exceeds 41 MPa (6,000 psi). Until further analyses can be performed, the specified deck concrete strength should be at least 60 percent of the specified girder concrete strength at 28 days when the specified girder concrete strength exceeds 41 MPa (6,000 psi).
  • The applicability of the AASHTO specification for flexural design should be examined with respect to the use of higher-strength concretes and larger differential strengths between the deck and girder concretes. (16)
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