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CHAPTER 2. WAFFLE DECK PANEL DESIGN

INTRODUCTION

The research team recognized that establishing efficient details for a full-depth waffle deck panel for a given bridge would be critical in ensuring successful implementation of this new technology in new and rehabilitation of highway bridges. This chapter presents a detailed design for the waffle deck panel components, including the reinforcement, typical deck panel dimensions, transverse and longitudinal rib configuration, and allowable dimensions for the different components including the connection regions.

UHPC WAFFLE DECK PANEL

A UHPC waffle deck panel consists of a slab cast integrally with concrete ribs spanning in the transverse and longitudinal directions (see figure 17).

Figure 17. Diagrams. Schematic view of UHPC deck panel.

Figure 17. Diagrams. Schematic view of UHPC deck panel.

This system is similar to the two-way joist system used by the building industry. A transverse strip along the deck panel acts as a T-beam, transferring wheel loads to the adjacent bridge girders. The longitudinal ribs help in distributing the wheel loads to the adjacent panels through the panel-to-panel connections. The reinforcement needed to resist the wheel loads is provided in the ribs running in both directions.

The important parameters that define the design of the full-depth UHPC waffle deck panel are as follows:

  • Length (dimension perpendicular to direction of the traffic), width (dimension in the direction of traffic), and thickness of the waffle deck panel.

  • Dimensions and spacing of the longitudinal and transverse ribs; note that the spacing of the ribs in both directions will be influenced largely by the girder-to-girder spacing, panel dimensions, and minimum detailing requirements of panel-to-panel connections.

  • Shear pocket dimensions, spacing, and size, type of shear connectors needed to achieve full composite action between the precast waffle panels and supporting systems (concrete girders, steel girders, stringers, etc.).

  • Design and details of mild steel reinforcement in the waffle panel ribs.

  • Configuration and reinforcement details of the panel-to-panel connections (transverse joint).

  • Configuration and reinforcement details of the panel-to-girder connections (longitudinal joint).

  • Overhang dimensions and details.

  • Parapets and parapet connection details at the deck level.

The major design and construction parameters that affect the details of the UHPC waffle deck panel system are described below.

Bridge Dimensions (Roadway Width): The waffle deck panel dimensions are dependent on the roadway width. For practical purposes, the panel dimensions are also limited by the allowable transportation and handling limits for deck panels at the precast plant and the job site. If the roadway width is more than 24 ft, it is recommended to use waffle panels with lengths equal to half of the roadway width. This will not only help in easy handling of the panels but also accelerate the construction of the bridge on site. This approach will also allow crowning requirements to be accommodated using steps in the beams and a constant haunch thickness.

Bridge Configuration: The focus of design in this chapter is on waffle deck panels applicable to straight bridges. For skewed bridges, the rectangular waffle deck panels may be used in conjunction with special end panels that can account for the effects of bridge skew. The end panels may be made of solid or waffle panels using UHPC or HPC. HPC panels may require prestressing to minimize cracking. The final choice for the end panel is left to the judgment of the designer with due consideration to costs of material and fabrication. In the case of curved bridges, similar deck panel details with necessary functional modifications may be used.

Bridge Type and Girder Spacing: There are currently more than 600,000 bridges in the US. Typical bridges use either prestressed precast concrete girders (see figure 18c) or steel girder (see figure 18a) or stringers at different center-to-center spacing for transferring loads from the deck to the substructure such as piers, foundations, and abutments. A small percentage of bridges use a superstructure with a deck supported by closely placed steel stringers, which in turn are supported by deep steel end girders or steel trusses, as shown in figure 18b and figure 18d. It is important to note that these types of bridges are fracture-critical and not used by several DOTs for new bridges. However, existing bridges with such configurations are considered candidates for replacement of deteriorated decks with UHPC waffle panels.

Figure 18. Diagrams. Illustrations of typical components of bridges. Diagram A steel girder bridge, diagram b steel stringer bridge, diagram c prestressed concrete girder bridge and diagram d truss bridge

Figure 18. Diagrams. Illustrations of typical components of bridges.

Based on the extensive review of current standard details used by several State DOTs (Alabama, Florida, Georgia, Illinois, Indiana, Iowa, Kentucky, Nebraska, New Jersey, New York, Ohio, Oklahoma, Virginia, and Wisconsin), the researchers found that the typical spacing used for the precast prestressed girders varies between 6 and 12 ft. The average girder spacing was found to be 8 ft for routine State and county bridges. In the case of steel girder bridges, the most often used maximum girder spacing was found to be 10 ft, although a girder spacing as much as 12 to 14 ft has been used occasionally.

Given the commonly used spacing and input received from the representatives of the Iowa DOT and the FHWA, the maximum girder spacing in this guide is limited to 10 ft to allow the waffle deck panels to be optimized for the most frequently used bridge designs.

Girder Type: The typical bridges in the US use standard prestressed concrete girders, steel beams, or stringers through the use of standard W sections, or built-up steel girders. The cross-section details of the girders vary depending on girder span, roadway width, and design practices specific to the local jurisdiction. The waffle deck panels in all cases are tied to the girders using appropriate connections (see chapter 3 for details) and designed to act compositely with the girders. Consequently, the top flange width of the girder will influence the spacing between the support ribs of the waffle deck (see figure 19).

Figure 19. Diagrams. Schematic of UHPC panels supported by girders.

Figure 19. Diagrams. Schematic of UHPC panels supported by girders.

In addition, the top flange width will dictate the location of the maximum negative bending moment in the deck panel. (27) Based on the review of the standard concrete girder details used by several DOTs, the minimum and maximum values for the top flange width were found to be 12 inches and 4 feet, respectively. To remain conservative, the top flange width of the girder is thus assumed to be 12 inches.

Dimensions of the Deck Panel

Panel Thickness: The AASHTO Load and Resistance Factor Design (LRFD) specification requires a minimum thickness of 7 inches for deck panels excluding any provision for grooving, grinding, or sacrificial surfaces.(27) However, the thickness of the deck panel is decided based on the minimum cover requirements and the overhang design. Following the preliminary analysis advantage of UHPC material properties, an 8-inch-thick panel was found to be structurally sufficient for most cases.

Panel Length and Width: As previously described, length and width of the panel depends on the handling requirements of the panels at the precast plant and at the job site along with the transportation constraints. The exact length and width of the panels will vary depending on the span and width of the bridge. Hence, it is left to the judgment of the designer to arrive at the desirable dimensions based on requirements of the bridge under consideration. However, as described below, this guide provides suggestions on the transverse and longitudinal rib spacing limits.

Flat Plate Thickness: The thickness of the flat plate connecting the ribs on top in the UHPC waffle deck panel is dictated by the punching shear capacity of the plate between the ribs, cover requirements of top transverse and longitudinal reinforcement, and any anticipated surface wearing over time. With the limited data available on the punching shear capacity of UHPC, a 2.5-inch-thick flat plate is recommended to prevent a punching shear failure between the ribs for the design truck load. Based on an experimental test completed at ISU, the punching shear capacity of UHPC was found to be 1 ksi, yielding a punching shear capacity of 200 kips for a 2.5-inch-thick UHPC plate under typical truck tire dimensions of 10 inches by 20 inches.

Rib Dimensions and Spacing: Based on the side cover requirements for the reinforcement, as well as the previous studies completed by the FHWA and ISU, the width of transverse and longitudinal ribs was chosen to be 3 inches at the bottom with a gradual increase to 4 inches at the top of the ribs at the rib-to-plate interface (see figure 19b). (17,40) The tapering of the rib helps with the easy removal of panel formwork during construction.

The spacing between the transverse and longitudinal ribs will depend on the girder-to-girder spacing, width of the panel, and minimum number of dowels required to establish sufficient panel-to-panel connections, as the dowels need to be accommodated within rib locations. Based on the limited available data on punching shear behavior, to limit the local damage to the flat plate and control cracking of the panel under service loads, the maximum allowable rib spacing is limited to 36 inches. Considering the wide range of girder-to-girder spacing used in current practice by the DOTs, it is expected that utilizing longitudinal and transverse rib spacing varying from 18 to 36 inches would accommodate the design of UHPC waffle deck for a variety of bridges.

The support longitudinal ribs that are located at the girder lines provide an enclosure for the girder-to-panel connection, which is referred to as the shear pocket connection, making the support rib spacing dependent on the top flange width of the girder. Consequently, it is recommended that the support rib spacing is limited to a value less than the beam top flange width, which uses a minimum value of 12 inches.

Shear Pockets: As noted above, shear pockets are provided along the precast panel width to facilitate a composite connection between the precast deck panels and the supporting girders using shear connectors and in situ UHPC. The spacing of the shear pockets depends on the arrangement of the shear connectors. If the shear studs/hooks are positioned uniformly along the girder length, the shear pocket spacing should be equal to the transverse rib spacing. However, if a group configuration is used for the shear studs, the shear pockets can be placed at a spacing of 2 to 4 ft apart, depending on the spacing between the shear stud groups. While the shear pockets are not required in every cell between the ribs, it is recommended that the shear pockets should be located in every other cell and close to the center of two adjacent transverse ribs.

The dimensions of the shear pocket should be enough to accommodate the easy pouring of in situ UHPC to create the composite action. An opening with minimum dimension of 4 inches by 8 inches is recommended. However, the designer can decide the dimensions and appropriate shape of the opening based on standard practice.

Panel Transverse Design

The waffle deck panel system may be designed using the strip method as described by the current AASHTO LRFD specifications. (27) A transverse strip of the deck is analyzed as a continuous beam supported by bridge girders, which are assumed to be considered as non-settling rigid supports. The transverse strip width depends on the regions along the deck panel cross-section (along the length of the panel) (+ve moment, -ve moment or overhang) is given by Article 4.6.2.1.3 in the AASHTO guidelines (see figure 20).

Figure 20. Equation. The transverse strip width according to article 4.6.2.1.3 of AASHTO LRFD guidelines. where S = girder-to-girder spacing in feet and X = distance of the critical location from the centerline of exterior girder (in feet).Equation W subscript ts is equal to the following: 26 plus 6.6 times Span or 48 plus 3 times Span or 45 plus 10 times X

Figure 20. Equation. The transverse strip width according to article 4.6.2.1.3 of AASHTO LRFD guidelines.

where S = girder-to-girder spacing in feet and X = distance of the critical location from the centerline of exterior girder (in feet).

The entire transverse strip is designed to resist the dead load and live load effects with appropriate load factors at different limit states. The equivalent transverse strip widths for different girder spacing are presented in table 4.

Table 4. Design strip width (W ts).

Span (ft-in.)

Transverse Strip Width (in.)

+ve moment

-ve moment

4’-0”

52.40

60.00

4’-3”

54.05

60.75

4’-6”

55.70

61.50

4’-9”

57.35

62.25

5’-0”

59.00

63.00

5’-3”

60.65

63.75

5’-6”

62.30

64.50

5’-9”

63.95

65.25

6’-0”

65.60

66.00

6’-3”

67.25

66.75

6’-6”

68.90

67.50

6’-9”

70.55

68.25

7’-0”

72.20

69.00

7’-3”

73.85

69.75

7’-6”

75.50

70.50

7’-9”

77.15

71.25

8’-0”

78.80

72.00

8’-3”

80.45

72.75

8’-6”

82.10

73.50

8’-9”

83.75

74.25

9’-0”

85.40

75.00

9’-3”

87.05

75.75

9’-6”

88.70

76.50

9’-9”

90.35

77.25

10’-0”

92.00

78.00

 

Design moments are determined at three different regions along the panel cross-section. These regions are for the span between girders, sections over interior girders, and the overhang section. As detailed in the AASHTO guidelines, the interior spans between girders are investigated for positive bending at the strength-I limit state. Sections over interior girders are examined for negative bending at the strength-I limit. The overhang region is investigated for different combinations of dead, live, and collision loads for the strength-I and extreme event II limit states. Load factors for the three different regions and accompanying limit states are presented in table 5 (adopted from Section 3.4 and Article 13.4 of the AASHTO specifications).(27)

Table 5. Load factors for different limit states.

Region

Limit State

DC

DW

LL

Between girders

Strength I

1.25

1.50

1.75

Over interior girders

Strength I

1.25

1.50

1.75

Overhang

Strength I

1.25

1.50

1.75

Overhang

Extreme Event II

1.00

1.00

1.00

DC = dead load of structural components; DW = dead load of wearing surface; LL = vehicular live load

The following subsections discuss the procedure used to determine live, dead, and collision loads on the UHPC deck for different limit states. The 2010 AASHTO LRFD Bridge Design Specifications have been used to determine the appropriate loads. (27) In addition, based on the previously established sections, the following design parameters are used while arriving at the loads on the deck panels.

  • An 8-inch waffle deck panel with the transverse and longitudinal rib dimensions as shown in figure 19b.
  • The longitudinal and transverse rib spacing varying from 18 to 36 inches.
  • A girder spacing of 4 to 10 ft.
  • The assumed railing system will influence the design of the overhang in the deck system. Although the Iowa DOT standard F-shape concrete barrier is used as a railing system, here, the designer may choose appropriate railing/barrier details and make appropriate modifications to arrive at the overhang design of the waffle deck panel.

Dead Load

Dead load on the waffle panel includes the self-weight of the panel (DC) and the weight of any future wearing surface or overlays (if used by the DOT). The material properties shown in table 6 are recommended for determining the self-weight. Additional guidance on estimating different dead load components is given below.

Table 6. Recommended values for estimating dead load.

Component

Density

UHPC unit weight
(see Chapter 1)

157 pcf

Wearing surface unit weight ( bituminous)
AASHTO Table 3.5.1-1

140 pcf

 

Self-Weight of Waffle Deck Panel: Self-weight of the waffle deck panel depends on the dimensions such as overall deck thickness and rib spacing in both longitudinal and transverse directions. As described before, for wide usage of the waffle deck technology, this report presents several deck panel options with variable rib spacing to meet different bridge types and girder spacing requirements. The self-weight of the waffle deck can be estimated using the equation shown in figure 21, and values for different rib spacing are provided in table 7.

Figure 21. Equation. Self-weight of waffle deck for different rib spacing. Equation. w subscript waffle is equal to sum of h subscrit slab, b subscript w times h subcript w divided by S subscript tr,  b subscript w times h subcript w divided by S subscript lr, multiplied by gamma subsript uhpc  divided by 12.

Figure 21. Equation. Self-weight of waffle deck for different rib spacing.

where hslab = thickness of top slab in inches (= 2.5 in.), Str = transverse rib spacing in inches, Slr = longitudinal rib spacing in inches, hw = rib height in inches (= hdeck − hslab) (= 8 in. - 2.5 in. = 5.5 in.), and Yuhpc = unit density of UHPC in pcf (see table 6). Multiplying the tabulated values in table 7 by the design strip width (Wts) (in ft) will give the dead load value per linear foot.

Table 7 Self-weight of the deck panel (Waffle) in psf for different rib spacing.

 

Transverse Rib Spacing (in.)

Longitudinal Rib Spacing (in.)

36

33

30

27

24

21

18

36

46.70

47.34

48.10

49.03

50.20

51.70

53.70

33

47.34

47.97

48.74

49.67

50.83

52.33

54.33

30

48.10

48.74

49.50

50.43

51.60

53.10

55.10

27

49.03

49.67

50.43

51.36

52.53

54.03

56.03

24

50.20

50.83

51.60

52.53

53.70

55.20

57.19

21

51.70

52.33

53.10

54.03

55.20

56.69

58.69

18

53.70

54.33

55.10

56.03

57.19

58.69

60.69

 

Self-Weight of Wearing Surface: Note that the wearing surface may be eliminated if an appropriate riding surface is incorporated into the waffle panel during casting. However, if a designer chooses to include a wearing surface, its appropriate weight should be included. Given the thickness of the wearing surface varies depending on local design practices, a 2.0-inch-thick wearing surface is assumed and the dead load due to the wearing surface is estimated using relationship shown in figure 22. When a different thickness of tws is used for the wearing surface, the value in this equation may be modified by multiplying it by tws/2.

Figure 22. Equation. The dead load due to a 2-inch-thick wearing surface. Equation. the dead load due to 2 in. thick wearing surface, w subscript ws is equal to 23.3 psf

Figure 22. Equation. The dead load due to a 2-inch-thick wearing surface.

Dead Load from Barrier/Railing

Given the potential variations in dimensions of barriers, the Iowa DOT standard concrete barrier (F-section continuous barrier rail) is used to estimate the dead load on the bridge overhang section. This barrier is 44 inches tall, 17 inches wide at the base, and 8.5 inches wide at the top. It is reinforced with seven longitudinal #5 bars distributed throughout the cross section and #5 stirrups spaced at 12 inches. More information about this barrier is provided in table 8.

Table 8. Properties of F-section continuous barrier rail as used by the Iowa DOT

Property

Value

Weight per unit length

540 lb/ft

Barrier height

44 in.

Width at base of barrier

17 in.

Center of gravity location

5.73 in.

 

Live Load

The precast deck panel is designed for HL-93 truck loading. More details of the HL-93 truck loading can be found in section 3.6 of the AASHTO LRFD Bridge Design Specifications. (27)

Collision Load

The collision moment capacity of the Iowa DOT standard F-shape barrier (Mc) varies from 13 kip-ft/ft to 13.9 kip-ft/ft depending on the location and test level (TL). A detail of the standard F-type barrier is shown in figure 45. The collision design parameters for the barrier are taken from the Iowa DOT Bridge Design Manual and are summarized in table 9. (43) The values presented in table 9 may be replaced with appropriate information to meet the local design practice.

Table 9. Collision design parameters suggested for the Iowa standard F-type barrier.

Parameters

Rail Rating

Rw (kips)

Lc (ft)

Mc k-ft/ft (avg)

TL-4 interior

117

11.5

13.0

TL-4, end

74

8.0

13.0

TL-5, interior

138

16.7

13.9

TL-5, end

133.6

9.7

13.9

 

Interior Deck Design Moment Demand

As detailed in table 5, the moment demand for deck panel between the girders (+ve moment region) and at the interior girder locations (-ve moment region) is estimated using the strength-I limit state. The moment demands due to different loads are:

Equation. M subscript u superscript positive is equal to sum of M subscript DL,des superscript +ve and M subscript LL superscript positive and M subscript u superscript negative is equal to sum of M subscript DL,des superscript negativeve and M subscript LL superscript positive

Figure 25. Equation. The design moment demand equation for strength-I limit state.

The design moment values for different girder spacing and transverse rib spacing may be estimated using the values from tables 10 through 12. Tables 13 and 14 present the design moment demands for the positive and negative moment regions of the deck panel, respectively. The critical section location for the negative moment is taken as 3 inches from the centerline of the girder. This value is justified to be equal to one-fourth the minimum flange width for typical bridge girders, which was found to be 12 inches (based on the survey of the DOT standard details). From table 13, it is clear that the maximum +ve moment demand varies from 8.29 kip-ft/ft to 13.17 kip-ft/ft as the girder spacing varies from 4 ft to 10 ft. In addition, from table 14, it is noted that the maximum -ve moment demand varies from 3.81 kip-ft/ft to 13.41 kip-ft/ft as the girder spacing increases.

 

Table 10. Suitable values for estimating the design moments for waffle deck panels due to dead load (self weight and wearing surface)

Girder Spacing right pointing arrow

Design positive and negative moment values due to dead load (kip-ft/ft) (M+ve DL, des  and M-ve DL, des)

4’-0” 4’-3” 4’-6” 4’-9” 5’-0” 5’-3” 5’-6” 5’-9” 6’-0” 6’-3” 6’-6” 6’-9” 7’-0” 7’-3” 7’-6” 7’-9” 8’-0” 8’-3” 8’-6” 8’-9” 9’-0” 9’-3” 9’-6” 9’-9” 10’
Transverse rib spacing (in.) Longitudinal rib spacing = 36”
36” 0.15 0.17 0.19 0.21 0.23 0.26 0.28 0.31 0.34 0.36 0.39 0.43 0.46 0.49 0.53 0.56 0.60 0.64 0.67 0.71 0.76 0.80 0.84 0.89 0.93
33” 0.15 0.17 0.19 0.21 0.24 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.46 0.49 0.53 0.57 0.60 0.64 0.68 0.72 0.76 0.81 0.85 0.90 0.94
30” 0.15 0.17 0.19 0.21 0.24 0.26 0.29 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.57 0.61 0.65 0.69 0.73 0.77 0.81 0.86 0.90 0.95
27” 0.15 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.51 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.82 0.87 0.92 0.96
24” 0.16 0.18 0.20 0.22 0.24 0.27 0.30 0.32 0.35 0.38 0.41 0.45 0.48 0.51 0.55 0.59 0.63 0.67 0.71 0.75 0.79 0.84 0.88 0.93 0.98
21” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.33 0.36 0.39 0.42 0.45 0.49 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.81 0.85 0.90 0.95 1.00
18” 0.16 0.18 0.21 0.23 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.57 0.61 0.65 0.70 0.74 0.78 0.83 0.87 0.92 0.97 1.02
Transverse rib spacing (in.)

Longitudinal rib spacing = 33”

36” 0.15 0.17 0.19 0.21 0.24 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.46 0.49 0.53 0.57 0.60 0.64 0.68 0.72 0.76 0.81 0.85 0.90 0.94
33” 0.15 0.17 0.19 0.21 0.24 0.26 0.29 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.53 0.57 0.61 0.65 0.69 0.73 0.77 0.81 0.86 0.90 0.95
30” 0.15 0.17 0.19 0.22 0.24 0.26 0.29 0.32 0.35 0.37 0.41 0.44 0.47 0.50 0.54 0.58 0.61 0.65 0.69 0.73 0.78 0.82 0.87 0.91 0.96
27” 0.16 0.18 0.20 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.41 0.44 0.48 0.51 0.55 0.58 0.62 0.66 0.70 0.74 0.79 0.83 0.88 0.92 0.97
24” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.33 0.35 0.38 0.42 0.45 0.48 0.52 0.55 0.59 0.63 0.67 0.71 0.75 0.80 0.84 0.89 0.94 0.99
21” 0.16 0.18 0.20 0.23 0.25 0.28 0.30 0.33 0.36 0.39 0.42 0.46 0.49 0.53 0.56 0.60 0.64 0.68 0.73 0.77 0.81 0.86 0.91 0.95 1.00
18” 0.16 0.19 0.21 0.23 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.79 0.83 0.88 0.93 0.98 1.03
Transverse rib spacing (in.)

Longitudinal rib spacing = 30”

36” 0.15 0.17 0.19 0.21 0.24 0.26 0.29 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.57 0.61 0.65 0.69 0.73 0.77 0.81 0.86 0.90 0.95
33” 0.15 0.17 0.19 0.22 0.24 0.26 0.29 0.32 0.35 0.37 0.41 0.44 0.47 0.50 0.54 0.58 0.61 0.65 0.69 0.73 0.78 0.82 0.87 0.91 0.96
30” 0.15 0.17 0.20 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.51 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.83 0.87 0.92 0.97
27” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.32 0.35 0.38 0.41 0.45 0.48 0.52 0.55 0.59 0.63 0.67 0.71 0.75 0.79 0.84 0.88 0.93 0.98
24” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.33 0.36 0.39 0.42 0.45 0.49 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.81 0.85 0.90 0.95 0.99
21” 0.16 0.18 0.21 0.23 0.25 0.28 0.31 0.34 0.36 0.40 0.43 0.46 0.50 0.53 0.57 0.61 0.65 0.69 0.73 0.78 0.82 0.87 0.91 0.96 1.01
18” 0.17 0.19 0.21 0.23 0.26 0.29 0.31 0.34 0.37 0.41 0.44 0.47 0.51 0.55 0.58 0.62 0.66 0.71 0.75 0.80 0.84 0.89 0.94 0.99 1.04
Transverse rib spacing (in.) Longitudinal rib spacing = 27”
36” 0.15 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.41 0.44 0.47 0.51 0.54 0.58 0.62 0.66 0.70 0.74 0.78 0.82 0.87 0.92 0.96
33” 0.16 0.18 0.20 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.41 0.44 0.48 0.51 0.55 0.58 0.62 0.66 0.70 0.74 0.79 0.83 0.88 0.92 0.97
30” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.32 0.35 0.38 0.41 0.45 0.48 0.52 0.55 0.59 0.63 0.67 0.71 0.75 0.79 0.84 0.88 0.93 0.98
27” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.33 0.36 0.39 0.42 0.45 0.49 0.52 0.56 0.60 0.63 0.68 0.72 0.76 0.80 0.85 0.90 0.94 0.99
24” 0.16 0.18 0.20 0.23 0.25 0.28 0.30 0.33 0.36 0.39 0.43 0.46 0.49 0.53 0.57 0.60 0.64 0.69 0.73 0.77 0.82 0.86 0.91 0.96 1.01
21” 0.16 0.19 0.21 0.23 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.79 0.83 0.88 0.93 0.97 1.03
18” 0.17 0.19 0.21 0.24 0.26 0.29 0.32 0.35 0.38 0.41 0.44 0.48 0.51 0.55 0.59 0.63 0.67 0.71 0.76 0.80 0.85 0.90 0.95 1.00 1.05
Transverse rib spacing (in.)

Longitudinal rib spacing = 24”

36” 0.16 0.18 0.20 0.22 0.24 0.27 0.30 0.32 0.35 0.38 0.41 0.45 0.48 0.51 0.55 0.59 0.63 0.67 0.71 0.75 0.79 0.84 0.88 0.93 0.98
33” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.33 0.35 0.38 0.42 0.45 0.48 0.52 0.55 0.59 0.63 0.67 0.71 0.75 0.80 0.84 0.89 0.94 0.99
30” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.33 0.36 0.39 0.42 0.45 0.49 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.81 0.85 0.90 0.95 0.99
27” 0.16 0.18 0.20 0.23 0.25 0.28 0.30 0.33 0.36 0.39 0.43 0.46 0.49 0.53 0.57 0.60 0.64 0.69 0.73 0.77 0.82 0.86 0.91 0.96 1.01
24” 0.16 0.18 0.21 0.23 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.57 0.61 0.65 0.70 0.74 0.78 0.83 0.87 0.92 0.97 1.02
21” 0.17 0.19 0.21 0.23 0.26 0.29 0.31 0.34 0.37 0.41 0.44 0.47 0.51 0.55 0.58 0.62 0.67 0.71 0.75 0.80 0.84 0.89 0.94 0.99 1.04
18” 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.42 0.45 0.49 0.52 0.56 0.60 0.64 0.68 0.72 0.77 0.82 0.86 0.91 0.96 1.01 1.06
Transverse rib spacing (in.)

Longitudinal rib spacing = 21”

36” 0.16 0.18 0.20 0.22 0.25 0.27 0.30 0.33 0.36 0.39 0.42 0.45 0.49 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.81 0.85 0.90 0.95 1.00
33” 0.16 0.18 0.20 0.23 0.25 0.28 0.30 0.33 0.36 0.39 0.42 0.46 0.49 0.53 0.56 0.60 0.64 0.68 0.73 0.77 0.81 0.86 0.91 0.95 1.00
30” 0.16 0.18 0.21 0.23 0.25 0.28 0.31 0.34 0.36 0.40 0.43 0.46 0.50 0.53 0.57 0.61 0.65 0.69 0.73 0.78 0.82 0.87 0.91 0.96 1.01
27” 0.16 0.19 0.21 0.23 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.79 0.83 0.88 0.93 0.97 1.03
24” 0.17 0.19 0.21 0.23 0.26 0.29 0.31 0.34 0.37 0.41 0.44 0.47 0.51 0.55 0.58 0.62 0.67 0.71 0.75 0.80 0.84 0.89 0.94 0.99 1.04
21” 0.17 0.19 0.21 0.24 0.26 0.29 0.32 0.35 0.38 0.41 0.45 0.48 0.52 0.56 0.60 0.64 0.68 0.72 0.76 0.81 0.86 0.91 0.96 1.01 1.06
18” 0.17 0.20 0.22 0.24 0.27 0.30 0.33 0.36 0.39 0.42 0.46 0.49 0.53 0.57 0.61 0.65 0.69 0.74 0.78 0.83 0.88 0.93 0.98 1.03 1.08
Transverse rib spacing (in.)

Longitudinal rib spacing = 18”

36” 0.16 0.18 0.21 0.23 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.57 0.61 0.65 0.70 0.74 0.78 0.83 0.87 0.92 0.97 1.02
33” 0.16 0.19 0.21 0.23 0.26 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.50 0.54 0.58 0.62 0.66 0.70 0.74 0.79 0.83 0.88 0.93 0.98 1.03
30” 0.17 0.19 0.21 0.23 0.26 0.29 0.31 0.34 0.37 0.41 0.44 0.47 0.51 0.55 0.58 0.62 0.66 0.71 0.75 0.80 0.84 0.89 0.94 0.99 1.04
27” 0.17 0.19 0.21 0.24 0.26 0.29 0.32 0.35 0.38 0.41 0.44 0.48 0.51 0.55 0.59 0.63 0.67 0.71 0.76 0.80 0.85 0.90 0.95 1.00 1.05
24” 0.17 0.19 0.22 0.24 0.27 0.29 0.32 0.35 0.38 0.42 0.45 0.49 0.52 0.56 0.60 0.64 0.68 0.72 0.77 0.82 0.86 0.91 0.96 1.01 1.06
21” 0.17 0.20 0.22 0.24 0.27 0.30 0.33 0.36 0.39 0.42 0.46 0.49 0.53 0.57 0.61 0.65 0.69 0.74 0.78 0.83 0.88 0.93 0.98 1.03 1.08
18” 0.18 0.20 0.22 0.25 0.28 0.31 0.34 0.37 0.40 0.43 0.47 0.51 0.54 0.58 0.62 0.67 0.71 0.75 0.80 0.85 0.90 0.95 1.00 1.05 1.11

 

Table 11. Table A4-1 from AASHTO LRFD specifications with a load factor of 1.75 as the LL positive moment effect in strength-I limit state.(27)

Girder Spacing (ft-in.)

Live Load Positive Moment (MLL+ve)
kip-ft/ft

Design Positive Moment (MLL,des+ve) = 1.75a (MLL+ve)
kip-ft/ft

4’-0”

4.68

8.19

4’-3”

4.66

8.16

4’-6”

4.63

8.10

4’-9”

4.64

8.12

5’-0”

4.65

8.14

5’-3”

4.67

8.17

5’-6”

4.71

8.24

5’-9”

4.77

8.35

6’-0”

4.83

8.45

6’-3”

4.91

8.59

6’-6”

5.00

8.75

6’-9”

5.10

8.93

7’-0”

5.21

9.12

7’-3”

5.32

9.31

7’-6”

5.44

9.52

7’-9”

5.56

9.73

8’-0”

5.69

9.96

8’-3”

5.83

10.20

8’-6”

5.99

10.48

8’-9”

6.14

10.75

9’-0”

6.29

11.01

9’-3”

6.44

11.27

9’-6”

6.59

11.53

9’-9”

6.74

11.80

10’-0”

6.89

12.06

 

Table 12. Table A4-1 from AASHTO LRFD specifications with a load factor of 1.75 as the LL negative moment effect in strength-I limit state). (27)

Girder Spacing
(ft-in)

Live Load Negative Moment (MLL-ve)
kip-ft/ft

Design Negative Moment (MLL,des-ve) = 1.75a (MLL+ve) kip-ft/ft

 

0”

3”

6”

0”

3”

6”

4’-0”

2.68

2.07

1.74

4.69

3.62

3.05

4’-3”

2.73

2.25

1.95

4.78

3.94

3.41

4’-6”

3.00

2.58

2.19

5.25

4.52

3.83

4’-9”

3.38

2.90

2.43

5.92

5.08

4.25

5’-0”

3.74

3.20

2.66

6.55

5.60

4.66

5’-3”

4.06

3.47

2.89

7.11

6.07

5.06

5’-6”

4.36

3.73

3.11

7.63

6.53

5.44

5’-9”

4.63

3.97

3.31

8.10

6.95

5.79

6’-0”

4.88

4.19

3.50

8.54

7.33

6.13

6’-3”

5.10

4.39

3.68

8.93

7.68

6.44

6’-6”

5.31

4.57

3.84

9.29

8.00

6.72

6’-9”

5.50

4.74

3.99

9.63

8.30

6.98

7’-0”

5.98

5.17

4.36

10.47

9.05

7.63

7’-3”

6.13

5.31

4.49

10.73

9.29

7.86

7’-6”

6.26

5.43

4.61

10.96

9.50

8.07

7’-9”

6.38

5.54

4.71

11.17

9.70

8.24

8’-0”

6.48

5.65

4.81

11.34

9.89

8.42

8’-3”

6.58

5.74

4.90

11.52

10.05

8.58

8’-6”

6.66

5.82

4.98

11.66

10.19

8.72

8’-9”

6.74

5.90

5.06

11.80

10.33

8.86

9’-0”

6.81

5.97

5.13

11.92

10.45

8.98

9’-3”

6.87

6.03

5.19

12.02

10.55

9.08

9’-6”

7.15

6.31

5.46

12.51

11.04

9.56

9’-9”

7.51

6.65

5.80

13.14

11.64

10.15

10’-0”

7.85

6.99

6.13

13.74

12.23

10.73

 

Table 13. The positive movement demand for waffle deck panel at strength-I limit state.

Girder Spacing right pointing arrow

Design positive moment values for UHPC waffle deck panel (kip-ft/ft)

4’-0”

4’-3”

4’-6”

4’-9”

5’-0”

5’-3”

5’-6”

5’-9”

6’-0”

6’-3”

6’-6”

6’-9”

7’-0”

7’-3”

7’-6”

7’-9”

8’-0”

8’-3”

8’-6”

8’-9”

9’-0”

9’-3”

9’-6”

9’-9”

10’

Transverse rib spacing (in.)

Longitudinal rib spacing = 36”

36”

8.34

8.32

8.29

8.33

8.37

8.43

8.52

8.66

8.79

8.96

9.14

9.35

9.58

9.80

10.05

10.29

10.56

10.84

11.16

11.46

11.76

12.07

12.38

12.68

12.99

33”

8.34

8.33

8.29

8.33

8.37

8.43

8.53

8.66

8.79

8.96

9.15

9.35

9.58

9.80

10.05

10.30

10.56

10.84

11.16

11.47

11.77

12.08

12.38

12.69

13.00

30”

8.34

8.33

8.30

8.33

8.38

8.43

8.53

8.66

8.79

8.96

9.15

9.36

9.58

9.81

10.06

10.30

10.57

10.85

11.17

11.47

11.78

12.08

12.39

12.70

13.01

27”

8.34

8.33

8.30

8.34

8.38

8.44

8.53

8.67

8.80

8.97

9.16

9.36

9.59

9.82

10.06

10.31

10.57

10.86

11.18

11.48

11.79

12.09

12.40

12.71

13.02

24”

8.35

8.33

8.30

8.34

8.38

8.44

8.54

8.67

8.80

8.97

9.16

9.37

9.60

9.82

10.07

10.32

10.58

10.87

11.19

11.49

11.80

12.11

12.41

12.72

13.03

21”

8.35

8.33

8.30

8.34

8.39

8.45

8.54

8.68

8.81

8.98

9.17

9.38

9.61

9.83

10.08

10.33

10.60

10.88

11.20

11.51

11.81

12.12

12.43

12.74

13.05

18”

8.35

8.34

8.31

8.35

8.39

8.45

8.55

8.69

8.82

8.99

9.18

9.39

9.62

9.85

10.09

10.34

10.61

10.90

11.22

11.53

11.83

12.14

12.45

12.77

13.08

 

Longitudinal rib spacing = 33”

Transverse rib spacing (in.)

36”

8.34

8.33

8.29

8.33

8.37

8.43

8.53

8.66

8.79

8.96

9.15

9.35

9.58

9.80

10.05

10.30

10.56

10.84

11.16

11.47

11.77

12.08

12.38

12.69

13.00

33”

8.34

8.33

8.29

8.33

8.37

8.43

8.53

8.66

8.79

8.96

9.15

9.36

9.58

9.81

10.05

10.30

10.57

10.85

11.17

11.47

11.78

12.08

12.39

12.70

13.01

30”

8.34

8.33

8.30

8.34

8.38

8.44

8.53

8.66

8.80

8.97

9.16

9.36

9.59

9.81

10.06

10.31

10.57

10.86

11.18

11.48

11.78

12.09

12.40

12.71

13.02

27”

8.35

8.33

8.30

8.34

8.38

8.44

8.54

8.67

8.80

8.97

9.16

9.37

9.59

9.82

10.07

10.31

10.58

10.86

11.18

11.49

11.79

12.10

12.41

12.72

13.03

24”

8.35

8.33

8.30

8.34

8.38

8.44

8.54

8.67

8.81

8.98

9.17

9.37

9.60

9.83

10.07

10.32

10.59

10.87

11.19

11.50

11.81

12.11

12.42

12.73

13.04

21”

8.35

8.34

8.31

8.35

8.39

8.45

8.55

8.68

8.81

8.98

9.17

9.38

9.61

9.84

10.08

10.33

10.60

10.89

11.21

11.51

11.82

12.13

12.44

12.75

13.06

18”

8.35

8.34

8.31

8.35

8.39

8.46

8.55

8.69

8.82

8.99

9.18

9.39

9.62

9.85

10.10

10.35

10.62

10.90

11.23

11.53

11.84

12.15

12.46

12.77

13.09

 

Longitudinal rib spacing = 30”

Transverse rib spacing (in.)

36”

8.34

8.33

8.30

8.33

8.38

8.43

8.53

8.66

8.79

8.96

9.15

9.36

9.58

9.81

10.06

10.30

10.57

10.85

11.17

11.47

11.78

12.08

12.39

12.70

13.01

33”

8.34

8.33

8.30

8.34

8.38

8.44

8.53

8.66

8.80

8.97

9.16

9.36

9.59

9.81

10.06

10.31

10.57

10.86

11.18

11.48

11.78

12.09

12.40

12.71

13.02

30”

8.34

8.33

8.30

8.34

8.38

8.44

8.54

8.67

8.80

8.97

9.16

9.37

9.59

9.82

10.06

10.31

10.58

10.86

11.18

11.49

11.79

12.10

12.41

12.72

13.03

27”

8.35

8.33

8.30

8.34

8.38

8.44

8.54

8.67

8.81

8.98

9.16

9.37

9.60

9.83

10.07

10.32

10.58

10.87

11.19

11.50

11.80

12.11

12.42

12.73

13.04

24”

8.35

8.33

8.30

8.34

8.39

8.45

8.54

8.68

8.81

8.98

9.17

9.38

9.61

9.83

10.08

10.33

10.59

10.88

11.20

11.51

11.81

12.12

12.43

12.74

13.05

21”

8.35

8.34

8.31

8.35

8.39

8.45

8.55

8.68

8.82

8.99

9.18

9.39

9.61

9.84

10.09

10.34

10.61

10.89

11.21

11.52

11.83

12.14

12.45

12.76

13.07

18”

8.36

8.34

8.31

8.35

8.40

8.46

8.56

8.69

8.83

9.00

9.19

9.40

9.63

9.86

10.10

10.35

10.62

10.91

11.23

11.54

11.85

12.16

12.47

12.78

13.10

 

Longitudinal rib spacing = 27”

Transverse rib spacing (in.)

36”

8.34

8.33

8.30

8.34

8.38

8.44

8.53

8.67

8.80

8.97

9.16

9.36

9.59

9.82

10.06

10.31

10.57

10.86

11.18

11.48

11.79

12.09

12.40

12.71

13.02

33”

8.35

8.33

8.30

8.34

8.38

8.44

8.54

8.67

8.80

8.97

9.16

9.37

9.59

9.82

10.07

10.31

10.58

10.86

11.18

11.49

11.79

12.10

12.41

12.72

13.03

30”

8.35

8.33

8.30

8.34

8.38

8.44

8.54

8.67

8.81

8.98

9.16

9.37

9.60

9.83

10.07

10.32

10.58

10.87

11.19

11.50

11.80

12.11

12.42

12.73

13.04

27”

8.35

8.33

8.30

8.34

8.39

8.45

8.54

8.68

8.81

8.98

9.17

9.38

9.60

9.83

10.08

10.33

10.59

10.88

11.20

11.50

11.81

12.12

12.43

12.74

13.05

24”

8.35

8.34

8.31

8.35

8.39

8.45

8.55

8.68

8.81

8.99

9.18

9.38

9.61

9.84

10.09

10.33

10.60

10.89

11.21

11.52

11.82

12.13

12.44

12.75

13.06

21”

8.35

8.34

8.31

8.35

8.39

8.46

8.55

8.69

8.82

8.99

9.18

9.39

9.62

9.85

10.10

10.35

10.61

10.90

11.22

11.53

11.84

12.15

12.46

12.77

13.08

18”

8.36

8.34

8.32

8.36

8.40

8.46

8.56

8.69

8.83

9.00

9.19

9.40

9.63

9.86

10.11

10.36

10.63

10.92

11.24

11.55

11.86

12.17

12.48

12.79

13.11

 

Table 14. The negative moment demand for waffle deck panel at strength-I limit state.

Girder Spacing right pointing arrow

Design negative moment values for UHPC waffle deck panel (kip-ft/ft)

4’-0”

4’-3”

4’-6”

4’-9”

5’-0”

5’-3”

5’-6”

5’-9”

6’-0”

6’-3”

6’-6”

6’-9”

7’-0”

7’-3”

7’-6”

7’-9”

8’-0”

8’-3”

8’-6”

8’-9”

9’-0”

9’-3”

9’-6”

9’-9”

10’

Transverse rib spacing (in.)

Longitudinal rib spacing = 36”

36”

3.77

4.11

4.70

5.29

5.83

6.33

6.81

7.26

7.67

8.05

8.39

8.72

9.51

9.78

10.03

10.26

10.49

10.68

10.86

11.04

11.20

11.35

11.89

12.53

13.17

33”

3.77

4.11

4.71

5.29

5.84

6.33

6.81

7.26

7.67

8.05

8.40

8.72

9.51

9.79

10.03

10.26

10.49

10.69

10.87

11.05

11.21

11.36

11.89

12.53

13.17

30”

3.77

4.11

4.71

5.29

5.84

6.33

6.82

7.26

7.67

8.05

8.40

8.73

9.51

9.79

10.04

10.27

10.50

10.69

10.87

11.05

11.22

11.37

11.90

12.54

13.18

27”

3.78

4.11

4.71

5.29

5.84

6.34

6.82

7.27

7.68

8.06

8.40

8.73

9.52

9.80

10.04

10.27

10.50

10.70

10.88

11.06

11.23

11.38

11.91

12.55

13.20

24”

3.78

4.11

4.71

5.30

5.84

6.34

6.82

7.27

7.68

8.06

8.41

8.74

9.53

9.81

10.05

10.28

10.51

10.71

10.89

11.07

11.24

11.39

11.92

12.57

13.21

21”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.07

8.42

8.75

9.54

9.82

10.06

10.29

10.53

10.72

10.90

11.09

11.25

11.40

11.94

12.58

13.23

18”

3.79

4.12

4.72

5.31

5.86

6.35

6.84

7.29

7.70

8.08

8.43

8.76

9.55

9.83

10.08

10.31

10.54

10.74

10.92

11.11

11.27

11.43

11.96

12.61

13.25

 

Longitudinal rib spacing = 33”

Transverse rib spacing (in.)

36”

3.77

4.11

4.71

5.29

5.84

6.33

6.81

7.26

7.67

8.05

8.40

8.72

9.51

9.79

10.03

10.26

10.49

10.69

10.87

11.05

11.21

11.36

11.89

12.53

13.17

33”

3.77

4.11

4.71

5.29

5.84

6.33

6.81

7.26

7.67

8.05

8.40

8.73

9.51

9.79

10.04

10.27

10.50

10.69

10.87

11.05

11.22

11.37

11.90

12.54

13.18

30”

3.78

4.11

4.71

5.29

5.84

6.34

6.82

7.26

7.68

8.06

8.40

8.73

9.52

9.80

10.04

10.27

10.50

10.70

10.88

11.06

11.22

11.37

11.91

12.55

13.19

27”

3.78

4.11

4.71

5.29

5.84

6.34

6.82

7.27

7.68

8.06

8.41

8.74

9.52

9.80

10.05

10.28

10.51

10.71

10.89

11.07

11.23

11.38

11.92

12.56

13.20

24”

3.78

4.12

4.71

5.30

5.85

6.34

6.83

7.27

7.69

8.07

8.41

8.74

9.53

9.81

10.06

10.29

10.52

10.72

10.90

11.08

11.25

11.40

11.93

12.57

13.22

21”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.07

8.42

8.75

9.54

9.82

10.07

10.30

10.53

10.73

10.91

11.09

11.26

11.41

11.95

12.59

13.24

18”

3.79

4.12

4.72

5.31

5.86

6.36

6.84

7.29

7.70

8.08

8.43

8.76

9.55

9.83

10.08

10.31

10.55

10.75

10.93

11.11

11.28

11.43

11.97

12.62

13.26

 

Longitudinal rib spacing = 30”

Transverse rib spacing (in.)

36”

3.77

4.11

4.71

5.29

5.84

6.33

6.82

7.26

7.67

8.05

8.40

8.73

9.51

9.79

10.04

10.27

10.50

10.69

10.87

11.05

11.22

11.37

11.90

12.54

13.18

33”

3.78

4.11

4.71

5.29

5.84

6.34

6.82

7.26

7.68

8.06

8.40

8.73

9.52

9.80

10.04

10.27

10.50

10.70

10.88

11.06

11.22

11.37

11.91

12.55

13.19

30”

3.78

4.11

4.71

5.29

5.84

6.34

6.82

7.27

7.68

8.06

8.41

8.74

9.52

9.80

10.05

10.28

10.51

10.70

10.88

11.07

11.23

11.38

11.92

12.56

13.20

27”

3.78

4.11

4.71

5.30

5.85

6.34

6.82

7.27

7.69

8.07

8.41

8.74

9.53

9.81

10.05

10.28

10.51

10.71

10.89

11.08

11.24

11.39

11.93

12.57

13.21

24”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.07

8.42

8.75

9.54

9.82

10.06

10.29

10.52

10.72

10.90

11.09

11.25

11.40

11.94

12.58

13.23

21”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.70

8.08

8.43

8.76

9.54

9.83

10.07

10.30

10.54

10.73

10.92

11.10

11.27

11.42

11.96

12.60

13.25

18”

3.79

4.13

4.73

5.31

5.86

6.36

6.84

7.29

7.71

8.09

8.44

8.77

9.56

9.84

10.09

10.32

10.55

10.75

10.94

11.12

11.29

11.44

11.98

12.62

13.27

 

Longitudinal rib spacing = 27”

Transverse rib spacing (in.)

36”

3.78

4.11

4.71

5.29

5.84

6.34

6.82

7.27

7.68

8.06

8.40

8.73

9.52

9.80

10.04

10.27

10.50

10.70

10.88

11.06

11.23

11.38

11.91

12.55

13.20

33”

3.78

4.11

4.71

5.29

5.84

6.34

6.82

7.27

7.68

8.06

8.41

8.74

9.52

9.80

10.05

10.28

10.51

10.71

10.89

11.07

11.23

11.38

11.92

12.56

13.20

30”

3.78

4.11

4.71

5.30

5.85

6.34

6.82

7.27

7.69

8.07

8.41

8.74

9.53

9.81

10.05

10.28

10.51

10.71

10.89

11.08

11.24

11.39

11.93

12.57

13.21

27”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.07

8.42

8.75

9.53

9.81

10.06

10.29

10.52

10.72

10.90

11.08

11.25

11.40

11.94

12.58

13.22

24”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.08

8.42

8.75

9.54

9.82

10.07

10.30

10.53

10.73

10.91

11.10

11.26

11.41

11.95

12.59

13.24

21”

3.79

4.12

4.72

5.31

5.86

6.36

6.84

7.29

7.70

8.08

8.43

8.76

9.55

9.83

10.08

10.31

10.54

10.74

10.93

11.11

11.28

11.43

11.97

12.61

13.26

18”

3.79

4.13

4.73

5.31

5.86

6.36

6.85

7.29

7.71

8.09

8.44

8.77

9.56

9.84

10.09

10.33

10.56

10.76

10.94

11.13

11.30

11.45

11.99

12.64

13.28

 

Longitudinal rib spacing = 24”

Transverse rib spacing (in.)

36”

3.78

4.11

4.71

5.30

5.84

6.34

6.82

7.27

7.68

8.06

8.41

8.74

9.53

9.81

10.05

10.28

10.51

10.71

10.89

11.07

11.24

11.39

11.92

12.57

13.21

33”

3.78

4.12

4.71

5.30

5.85

6.34

6.83

7.27

7.69

8.07

8.41

8.74

9.53

9.81

10.06

10.29

10.52

10.72

10.90

11.08

11.25

11.40

11.93

12.57

13.22

30”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.07

8.42

8.75

9.54

9.82

10.06

10.29

10.52

10.72

10.90

11.09

11.25

11.40

11.94

12.58

13.23

27”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.08

8.42

8.75

9.54

9.82

10.07

10.30

10.53

10.73

10.91

11.10

11.26

11.41

11.95

12.59

13.24

24”

3.79

4.12

4.72

5.31

5.86

6.35

6.84

7.29

7.70

8.08

8.43

8.76

9.55

9.83

10.08

10.31

10.54

10.74

10.92

11.11

11.27

11.43

11.96

12.61

13.25

21”

3.79

4.13

4.73

5.31

5.86

6.36

6.84

7.29

7.71

8.09

8.44

8.77

9.56

9.84

10.09

10.32

10.55

10.75

10.94

11.12

11.29

11.44

11.98

12.63

13.27

18”

3.79

4.13

4.73

5.32

5.87

6.37

6.85

7.30

7.72

8.10

8.45

8.78

9.57

9.85

10.10

10.33

10.57

10.77

10.95

11.14

11.31

11.46

12.00

12.65

13.30

 

Longitudinal rib spacing = 21”

Transverse rib spacing (in.)

36”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.07

8.42

8.75

9.54

9.82

10.06

10.29

10.53

10.72

10.90

11.09

11.25

11.40

11.94

12.58

13.23

33”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.69

8.07

8.42

8.75

9.54

9.82

10.07

10.30

10.53

10.73

10.91

11.09

11.26

11.41

11.95

12.59

13.24

30”

3.78

4.12

4.72

5.30

5.85

6.35

6.83

7.28

7.70

8.08

8.43

8.76

9.54

9.83

10.07

10.30

10.54

10.73

10.92

11.10

11.27

11.42

11.96

12.60

13.25

27”

3.79

4.12

4.72

5.31

5.86

6.36

6.84

7.29

7.70

8.08

8.43

8.76

9.55

9.83

10.08

10.31

10.54

10.74

10.93

11.11

11.28

11.43

11.97

12.61

13.26

24”

3.79

4.13

4.73

5.31

5.86

6.36

6.84

7.29

7.71

8.09

8.44

8.77

9.56

9.84

10.09

10.32

10.55

10.75

10.94

11.12

11.29

11.44

11.98

12.63

13.27

21”

3.79

4.13

4.73

5.31

5.86

6.36

6.85

7.30

7.71

8.10

8.44

8.78

9.57

9.85

10.10

10.33

10.57

10.77

10.95

11.14

11.31

11.46

12.00

12.64

13.29

18”

3.80

4.13

4.73

5.32

5.87

6.37

6.86

7.31

7.72

8.11

8.46

8.79

9.58

9.86

10.11

10.35

10.58

10.78

10.97

11.15

11.33

11.48

12.02

12.67

13.32

 

Longitudinal rib spacing = 18”

Transverse rib spacing (in.)

36”

3.79

4.12

4.72

5.31

5.86

6.35

6.84

7.29

7.70

8.08

8.43

8.76

9.55

9.83

10.08

10.31

10.54

10.74

10.92

11.11

11.27

11.43

11.96

12.61

13.25

33”

3.79

4.12

4.72

5.31

5.86

6.36

6.84

7.29

7.70

8.08

8.43

8.76

9.55

9.83

10.08

10.31

10.55

10.75

10.93

11.11

11.28

11.43

11.97

12.62

13.26

30”

3.79

4.13

4.73

5.31

5.86

6.36

6.84

7.29

7.71

8.09

8.44

8.77

9.56

9.84

10.09

10.32

10.55

10.75

10.94

11.12

11.29

11.44

11.98

12.62

13.27

27”

3.79

4.13

4.73

5.31

5.86

6.36

6.85

7.29

7.71

8.09

8.44

8.77

9.56

9.84

10.09

10.33

10.56

10.76

10.94

11.13

11.30

11.45

11.99

12.64

13.28

24”

3.79

4.13

4.73

5.32

5.87

6.37

6.85

7.30

7.72

8.10

8.45

8.78

9.57

9.85

10.10

10.33

10.57

10.77

10.95

11.14

11.31

11.46

12.00

12.65

13.30

21”

3.80

4.13

4.73

5.32

5.87

6.37

6.86

7.31

7.72

8.11

8.46

8.79

9.58

9.86

10.11

10.35

10.58

10.78

10.97

11.15

11.33

11.48

12.02

12.67

13.32

18”

3.80

4.14

4.74

5.33

5.88

6.38

6.86

7.31

7.73

8.12

8.47

8.80

9.59

9.88

10.13

10.36

10.60

10.80

10.99

11.17

11.35

11.50

12.04

12.69

13.34

 

FLEXURAL BEHAVIOR OF UHPC MEMBERS

This section presents the basic steps in estimating the flexural resistance of UHPC members with rectangular and T-beam cross-sections containing mild steel reinforcement as the primary flexural reinforcement. The flexural behavior of UHPC members can be obtained using the strain compatibility and equilibrium equations at the section level, similar to the practice used for conventional reinforced concrete members. However, differences in the UHPC material behavior should be taken into account appropriately during the strength estimation.

Rectangular Cross-Section Behavior

The cracking moment of the section can be estimated using the cross-section properties. Ignoring the contribution of the mild steel reinforcement toward the moment of inertia calculations, figure 26 provides a conservative estimate for the cracking moment of a rectangular section:

Equation. M subscript cr is equal to 0.2 times b time h square

Figure 26. Equation. Cracking moment relationship for a rectangular section.

where h = height of the section and b = width of the section.

The strain and stress distribution along the cross-section is shown in figure 27a.

Figure 27. Diagrams. Strain and stress distribution along the cross-section at cracking and ultimate limit states. Illustration: Strain and stress distribution along the rectangular cross-section at cracking and ultimate limit states. Part A shows the linear sttrain and stress distribution across the rectangualr section depth. Part B shows the linear strain along the depth and a non linear stress along the depth.

Figure 27. Diagrams. Strain and stress distribution along the cross-section at cracking and ultimate limit states.

The nominal moment capacity for the UHPC beam is estimated when either the extreme compression or tension strain reaches its corresponding limiting value, using the strain compatibility and equilibrium conditions.

At the ultimate limit state, the UHPC compressive force is approximated by a triangular stress distribution and estimated using the equation in figure 28

Figure 28. Equation. UHPC compression force.

Figure 28. Equation. UHPC compression force.

where c = neutral axis depth and b = width of the beam. The resultant compressive force, C, acts at one-third of the neutral axis depth below the extreme compression fiber of the beam.

A uniform tensile stress of 1.2 ksi may be assumed to act from the neutral axis to the extreme tension fiber when estimating the UHPC tensile force, as shown in figure 27b and given by the equation in figure 29:

Figure 29. Equation. UHPC tension force.

Figure 29. Equation. UHPC tension force.

where h = height of the beam.

From the force equilibrium at the ultimate limit state:

Figure 30. Equation. Force equilibrium equation for the rectangular section.

Figure 30. Equation. Force equilibrium equation for the rectangular section.

The procedure to determine the nominal moment capacity of a beam usually requires an iterative process and depends on the assumed method of failure (compression or tension control). However, by combining the strain compatibility and equilibrium equations at the ultimate limit state, the equation in figure 31 may be used to determine the controlling limit state.

Figure 32. Equation. Neutral axis depth and moment capacity for compression limit state.

Figure 31. Equation. Conditions for controlling limit states.

When the compression limit state controls, the neutral axis depth and corresponding moment capacity may be obtained using the equation shown in figure 32.

Figure 32. Equation. Neutral axis depth and moment capacity for compression limit state.

Figure 32. Equation. Neutral axis depth and moment capacity for compression limit state.

When the tension limit state controls, the neutral axis depth and the corresponding moment capacity may be obtained using the equation presented in figure 33:

Figure 33. Equation. Neutral axis depth and moment capacity for tension limit state.

Figure 33. Equation. Neutral axis depth and moment capacity for tension limit state.

Note that the expression estimating the neutral axis depth requires solving a quadratic equation.

T-Beam Section Behavior

The cracking moment of the T-beam section can be estimated similar to that as the rectangular section. Due to the unsymmetrical cross-section, the values of the positive cracking moment (web in tension) and negative cracking moment (flange in tension) differ significantly from each other. The cracking moment values for the T-beam can be obtained using the equation in figure 34:

Figure 34. Equation. Positive and negative cracking moment for T-beam.

Figure 34. Equation. Positive and negative cracking moment for T-beam.

The stress profile along the T-beam cross-section at cracking is shown in figure 35.

Figure 35. Diagrams. Stress profile for estimating the positive cracking moment of a T-shaped UHPC beam. Illustration showing strain and stress profiles for estimating the positive nominal moment of a t-shaped UHPC beam

Figure 35. Diagrams. Stress profile for estimating the positive cracking moment of a T-shaped UHPC beam.

Similar to the rectangular cross-section, the nominal moment capacity depends on whether the extreme compression or tension strain reaches its corresponding limiting values. However, with typical T-beam sections used in bridge decks, the flange width is expected to be very large compared to the overall depth of the section. Hence, for practical purposes, the extreme tensile strain in the web section may be assumed to reach its limiting value first when estimating the positive moment (web in tension). In addition, the neutral axis depth will be within the flange section. The corresponding strain and stress distribution across the section under positive moment conditions is shown in figure 36.

Figure 36. Diagrams. Strain and stress profiles for estimating the positive nominal moment of a T-shaped UHPC beam.

Figure 36. Diagrams. Strain and stress profiles for estimating the positive nominal moment of a T-shaped UHPC beam.

 

The concrete compressive force, C, is given by the equation in figure 37:

Figure 37. Equation. Concrete compression force.

Figure 37. Equation. Concrete compression force.

The tensile force from UHPC in tension may be obtained using the equation in figure 38:

Figure 38. Equation. UHPC tension force.

Figure 38. Equation. UHPC tension force.

From force equilibrium:

Figure 39. Equation. Force equilibrium

Figure 39. Equation. Force equilibrium

Using the equation in figure 39, neutral axis depth and positive moment capacity can be obtained by taking moments about the neutral axis.

In the case of the ultimate limit state under negative moment conditions (flange in tension), the limiting values for the strain limits can be either tension or compression, depending on the dimensions of the cross-section. Hence, the nominal negative moment capacity should be equal to the smaller of the estimated moment capacities obtained, assuming the two strain limits as controlling parameters independently. The strain and stress distributions across the section for the two negative moment conditions are shown in figure 40.

Figure 40. Diagrams. Strain and stress profiles for estimating the negative nominal moment capacity of a T-shaped UHPC beam. Illustration shows strain and stress profiles for estimating the negative nominal moment capacity of a T-shaped UHPC beam

Figure 40. Diagrams. Strain and stress profiles for estimating the negative nominal moment capacity of a T-shaped UHPC beam.

Using the equations presented in table 15, table 16, and the equilibrium condition, the neutral axis depth can be found for both cases. Once the neutral axis depths for both limit cases (c1 and c2) are estimated, the moment capacity for the flange in tension can be obtained by taking the moment of the different forces from the top of the flange. The locations of the resultant forces from the top flange are given in tables 15 and 16.

Table 15. T-beam internal forces and their location at ultimate limit state.

Case 1 ( Compressive Strain Limit State Controls)
Forces Force Location from Top
equation 1 for force location equation 2 for force location
equation 3 for force location d1
equation 4 for force location d2
equation 5 for force location equation 5 for force location

 

Table 16. T-beam internal forces and their location at ultimate limit state.

Case 2 ( Tension Strain Limit State Controls)
Forces Force Location from Top
Case 1 equation 2 for force location Case 2 equation 2 for force location
Case 2 equation 3 for force location d1
Case 2 equation 4 for force location d2
Case 2 equation 5 for force location Case 2 equation 6 for force location

WAFFLE DECK PANEL CAPACITY ESTIMATION

The waffle deck panel can be assumed to act like a series of T-beams to resist the dead loads and traffic loads. Moment capacity of the waffle deck panel in the positive and negative bending directions can be estimated using a transverse strip along the deck panel (see figure 41). Depending on the positive and negative bending directions, the equivalent strip width can be arrived at using the AASHTO LRFD Bridge Design Specification 4.6.2.1.3 (see figure 20 and table 4). (27)

Figure 41. Diagrams. Cross-section of an equivalent strip for positive bending. Illustration: Part A shows the equivalnet strip for +ve and negative bending. Part B shows the cross-section of an equivalent strip for positive and negative bending

Figure 41. Diagrams. Cross-section of an equivalent strip for positive bending.

As shown in figure 41b, the equivalent strip width contains a number of ribs depending on the girder span and rib spacing in the waffle deck panel. The cross-section of a typical equivalent strip is shown for positive and negative moment locations. The cross-section of the transverse strip can be further divided into a combination of T-beams with a cross-section, as shown in figure 41c. The flange width for positive bending (bf+ve) or negative bending (bf -ve) can be estimated using the equation presented in figure 42:

Figure 42. Equation. Flange width of equivalent T-beam for positive and negative bending.Equation. b subscript f superscript positive is equal to W subscript ts superscript positive divided by sum of one and intervalue of divident of W subscript ts superscript positive and S subscript tr

Figure 42. Equation. Flange width of equivalent T-beam for positive and negative bending.

where Wts+ve and Wts-ve = equivalent strip width for positive moment and negative moment regions, respectively, and Str = transverse rib spacing in the waffle panel. The flange width values for different girder and transverse rib spacing are provided in table 17.

Table 17. Equivalent flange width for the T-section for different girder and rib spacing.

Transverse Rib Spacing right pointing arrow

Flange Width for Positive Bending
( bf+ve)

Flange Width for Negative Bending
( bf-ve)

36 in.

33 in.

30 in.

27 in.

24 in.

21 in.

18 in.

36 in.

33 in.

30 in.

27 in.

24 in.

21 in.

18 in.

Girder Spacing (ft-in.)     ↑

4’-0”

26.20

26.20

26.20

26.20

17.47

17.47

17.47

30.00

30.00

20.00

20.00

20.00

20.00

15.00

4’-3”

27.03

27.03

27.03

18.02

18.02

18.02

13.51

30.38

30.38

20.25

20.25

20.25

20.25

15.19

4’-6”

27.85

27.85

27.85

18.57

18.57

18.57

13.93

30.75

30.75

20.50

20.50

20.50

20.50

15.38

4’-9”

28.68

28.68

28.68

19.12

19.12

19.12

14.34

31.13

31.13

20.75

20.75

20.75

20.75

15.56

5’-0”

29.50

29.50

29.50

19.67

19.67

19.67

14.75

31.50

31.50

21.00

21.00

21.00

15.75

15.75

5’-3”

30.33

30.33

20.22

20.22

20.22

20.22

15.16

31.88

31.88

21.25

21.25

21.25

15.94

15.94

5’-6”

31.15

31.15

20.77

20.77

20.77

20.77

15.58

32.25

32.25

21.50

21.50

21.50

16.13

16.13

5’-9”

31.98

31.98

21.32

21.32

21.32

15.99

15.99

32.63

32.63

21.75

21.75

21.75

16.31

16.31

6’-0”

32.80

32.80

21.87

21.87

21.87

16.40

16.40

33.00

22.00

22.00

22.00

22.00

16.50

16.50

6’-3”

33.63

22.42

22.42

22.42

22.42

16.81

16.81

33.38

22.25

22.25

22.25

22.25

16.69

16.69

6’-6”

34.45

22.97

22.97

22.97

22.97

17.23

17.23

33.75

22.50

22.50

22.50

22.50

16.88

16.88

6’-9”

35.28

23.52

23.52

23.52

23.52

17.64

17.64

34.13

22.75

22.75

22.75

22.75

17.06

17.06

7’-0”

24.07

24.07

24.07

24.07

18.05

18.05

14.44

34.50

23.00

23.00

23.00

23.00

17.25

17.25

7’-3”

24.62

24.62

24.62

24.62

18.46

18.46

14.77

34.88

23.25

23.25

23.25

23.25

17.44

17.44

7’-6”

25.17

25.17

25.17

25.17

18.88

18.88

15.10

35.25

23.50

23.50

23.50

23.50

17.63

17.63

7’-9”

25.72

25.72

25.72

25.72

19.29

19.29

15.43

35.63

23.75

23.75

23.75

23.75

17.81

17.81

8’-0”

26.27

26.27

26.27

26.27

19.70

19.70

15.76

24.00

24.00

24.00

24.00

18.00

18.00

14.40

8’-3”

26.82

26.82

26.82

26.82

20.11

20.11

16.09

24.25

24.25

24.25

24.25

18.19

18.19

14.55

8’-6”

27.37

27.37

27.37

20.53

20.53

20.53

16.42

24.50

24.50

24.50

24.50

18.38

18.38

14.70

8’-9”

27.92

27.92

27.92

20.94

20.94

20.94

16.75

24.75

24.75

24.75

24.75

18.56

18.56

14.85

9’-0”

28.47

28.47

28.47

21.35

21.35

17.08

17.08

25.00

25.00

25.00

25.00

18.75

18.75

15.00

9’-3”

29.02

29.02

29.02

21.76

21.76

17.41

17.41

25.25

25.25

25.25

25.25

18.94

18.94

15.15

9’-6”

29.57

29.57

29.57

22.18

22.18

17.74

17.74

25.50

25.50

25.50

25.50

19.13

19.13

15.30

9’-9”

30.12

30.12

22.59

22.59

22.59

18.07

15.06

25.75

25.75

25.75

25.75

19.31

19.31

15.45

10’-0”

30.67

30.67

23.00

23.00

23.00

18.40

15.33

26.00

26.00

26.00

26.00

19.50

19.50

15.60

 

The positive and negative bending moment capacity for the cross-section can be estimated using the strain compatibility approach described above.

Validation of Proposed Method

The proposed method was validated using the experimental data from the laboratory testing of waffle deck panel systems. The tests were performed at ISU as part of the FHWA HfL program. Figure 43 shows the cross-section ad reinforcement details of the transverse rib.

Figure 43. Diagram. Details of an equivalent transverse rib in the positive bending strip. Illustration showing details of an equivalent transverse rib in the positive bending strip for ISU test panel

Figure 43. Diagram. Details of an equivalent transverse rib in the positive bending strip.

The test specimen details are as follows:

It is clear from these calculations that the moment capacity estimated using the proposed procedure is 74 percent of the moment applied without failing the deck panel. The discrepancy between the observed and calculated values is expected, as the AASHTO recommendations for the equivalent strip width are conservative. In addition, the proposed method ignores the contribution of longitudinal ribs in the load-carrying capacity. Therefore, the proposed procedure results in a conservative estimate of the deck panel capacity and can be used confidently for the design of deck panels.

Estimation of Moment Capacity of Waffle Panel

The moment capacity of a waffle deck panel in the positive and negative bending directions for different girder spacing and transverse rib spacing configurations is estimated using the procedure outlined earlier in this chapter. In addition, the effect of varying the reinforcement in the transverse rib section is investigated by considering two different rebar sizes (#6 bar diameter = 0.75 in. and #7 bar diameter = 0.875 in.). All of the mild steel reinforcement is assumed to have yield strength of 60 ksi. The cross-sections considered with different reinforcement are shown in figure 44 and denoted by UWD6T6B and UWD6T7B.

Illustration shows the details of cross-sections considered for transverse ribs. Part  A shows the cross-section of UWD6T6B panel design; Part  B shows the cross-section of UWD6T7B panel design.

Figure 44. Diagrams. The details of cross-sections considered for transverse ribs.

The nominal moment capacities for the two cross-sections for different girder and transverse rib spacing using the strain compatibility method are presented in table 18 and table 19.

Table 18. Cracking and nominal moment capacity for UWP6T7B in kip-ft/ft.

Transverse
Rib Spacing
right pointing arrow

Cracking Moment Capacity
( +ve Bending)

Nominal Moment Capacity
( +ve Bending)

Cracking moment
( -ve Bending)

Nominal Moment Capacity
( -ve Bending)

36” 33” 30” 27” 24” 21” 18” 36” 33” 30” 27” 24” 21” 18” 36” 33” 30” 27” 24” 21” 18” 36” 33” 30” 27” 24” 21” 18”

Girder
Spacing (ft - in)

4’-0” 2.38 2.38 2.38 2.38 3.34 3.34 3.34 14.06 14.06 14.06 14.06 18.60 18.60 18.60 6.21 6.21 7.28 7.28 7.28 7.28 8.06 24.55 24.55 28.42 28.42 28.42 28.42 32.19
4’-3” 2.32 2.32 2.32 3.26 3.26 3.26 4.14 13.79 13.79 13.79 18.18 18.18 18.18 22.56 6.17 6.17 7.25 7.25 7.25 7.25 8.03 24.48 24.48 28.26 28.26 28.26 28.26 32.00
4’-6” 2.26 2.26 2.26 3.18 3.18 3.18 4.04 13.52 13.52 13.52 17.79 17.79 17.79 22.04 6.14 6.14 7.22 7.22 7.22 7.22 7.99 24.38 24.38 28.15 28.15 28.15 28.15 31.81
4’-9” 2.21 2.21 2.21 3.10 3.10 3.10 3.94 13.28 13.28 13.28 17.43 17.43 17.43 21.55 6.11 6.11 7.18 7.18 7.18 7.18 7.96 24.28 24.28 28.01 28.01 28.01 28.01 31.67
5’-0” 2.15 2.15 2.15 3.03 3.03 3.03 3.85 13.05 13.05 13.05 17.08 17.08 17.08 21.09 6.08 6.08 7.15 7.15 7.15 7.93 7.93 24.19 24.19 27.86 27.86 27.86 31.49 31.49
5’-3” 2.11 2.11 2.96 2.96 2.96 2.96 3.76 12.83 12.83 16.75 16.75 16.75 16.75 20.66 6.05 6.05 7.12 7.12 7.12 7.90 7.90 24.10 24.10 27.76 27.76 27.76 31.32 31.32
5’-6” 2.06 2.06 2.89 2.89 2.89 2.89 3.68 12.62 12.62 16.44 16.44 16.44 16.44 20.24 6.02 6.02 7.09 7.09 7.09 7.86 7.86 24.01 24.01 27.63 27.63 27.63 31.15 31.15
5’-9” 2.01 2.01 2.83 2.83 2.83 3.60 3.60 12.42 12.42 16.14 16.14 16.14 19.85 19.85 5.99 5.99 7.06 7.06 7.06 7.83 7.83 23.93 23.93 27.50 27.50 27.50 30.98 30.98
6’-0” 1.97 1.97 2.77 2.77 2.77 3.53 3.53 12.23 12.23 15.86 15.86 15.86 19.48 19.48 5.96 7.03 7.03 7.03 7.03 7.80 7.80 23.85 27.37 27.37 27.37 27.37 30.82 30.82
6’-3” 1.93 2.71 2.71 2.71 2.71 3.45 3.45 12.05 15.60 15.60 15.60 15.60 19.13 19.13 5.93 7.00 7.00 7.00 7.00 7.77 7.77 23.77 27.25 27.25 27.25 27.25 30.67 30.67
6’-6” 1.89 2.66 2.66 2.66 2.66 3.38 3.38 11.88 15.34 15.34 15.34 15.34 18.79 18.79 5.90 6.97 6.97 6.97 6.97 7.74 7.74 23.69 27.13 27.13 27.13 27.13 30.51 30.51
6’-9” 1.86 2.61 2.61 2.61 2.61 3.32 3.32 11.72 15.10 15.10 15.10 15.10 18.47 18.47 5.87 6.94 6.94 6.94 6.94 7.71 7.71 23.62 27.05 27.05 27.05 27.05 30.36 30.36
7’-0” 2.56 2.56 2.56 2.56 3.25 3.25 3.92 14.87 14.87 14.87 14.87 18.16 18.16 21.44 5.85 6.91 6.91 6.91 6.91 7.68 7.68 23.55 26.94 26.94 26.94 26.94 30.22 30.22
7’-3” 2.51 2.51 2.51 2.51 3.19 3.19 3.85 14.65 14.65 14.65 14.65 17.87 17.87 21.07 5.82 6.88 6.88 6.88 6.88 7.65 7.65 23.45 26.83 26.83 26.83 26.83 30.08 30.08
7’-6” 2.46 2.46 2.46 2.46 3.13 3.13 3.78 14.44 14.44 14.44 14.44 17.58 17.58 20.72 5.79 6.85 6.85 6.85 6.85 7.62 7.62 23.39 26.72 26.72 26.72 26.72 29.94 29.94
7’-9” 2.42 2.42 2.42 2.42 3.08 3.08 3.71 14.23 14.23 14.23 14.23 17.32 17.32 20.39 5.76 6.82 6.82 6.82 6.82 7.60 7.60 23.33 26.61 26.61 26.61 26.61 29.81 29.81
8’-0” 2.37 2.37 2.37 2.37 3.02 3.02 3.64 14.04 14.04 14.04 14.04 17.06 17.06 20.07 6.80 6.80 6.80 6.80 7.57 7.57 8.17 26.51 26.51 26.51 26.51 29.68 29.68 32.81
8’-3” 2.33 2.33 2.33 2.33 2.97 2.97 3.58 13.85 13.85 13.85 13.85 16.81 16.81 19.76 6.77 6.77 6.77 6.77 7.54 7.54 8.14 26.41 26.41 26.41 26.41 29.55 29.55 32.69
8’-6” 2.29 2.29 2.29 2.92 2.92 2.92 3.52 13.68 13.68 13.68 16.57 16.57 16.57 19.46 6.74 6.74 6.74 6.74 7.51 7.51 8.11 26.32 26.32 26.32 26.32 29.43 29.43 32.52
8’-9” 2.26 2.26 2.26 2.87 2.87 2.87 3.46 13.50 13.50 13.50 16.35 16.35 16.35 19.18 6.71 6.71 6.71 6.71 7.48 7.48 8.09 26.23 26.23 26.23 26.23 29.30 29.30 32.35
9’-0” 2.22 2.22 2.22 2.83 2.83 3.41 3.41 13.34 13.34 13.34 16.13 16.13 18.90 18.90 6.69 6.69 6.69 6.69 7.46 7.46 8.06 26.13 26.13 26.13 26.13 29.19 29.19 32.19
9’-3” 2.18 2.18 2.18 2.78 2.78 3.35 3.35 13.18 13.18 13.18 15.91 15.91 18.64 18.64 6.66 6.66 6.66 6.66 7.43 7.43 8.03 26.01 26.01 26.01 26.01 29.07 29.07 32.04
9’-6” 2.15 2.15 2.15 2.74 2.74 3.30 3.30 13.03 13.03 13.03 15.71 15.71 18.39 18.39 6.63 6.63 6.63 6.63 7.40 7.40 8.01 25.93 25.93 25.93 25.93 28.96 28.96 31.88
9’-9” 2.12 2.12 2.69 2.69 2.69 3.25 3.79 12.88 12.88 15.52 15.52 15.52 18.14 20.76 6.61 6.61 6.61 6.61 7.38 7.38 7.98 25.84 25.84 25.84 25.84 28.81 28.81 31.78
10’-0” 2.09 2.09 2.65 2.65 2.65 3.20 3.73 12.74 12.74 15.33 15.33 15.33 17.91 20.48 6.58 6.58 6.58 6.58 7.35 7.35 7.35 7.95 25.76 25.76 25.76 28.70 28.70 31.64
Note: All the moment capacity values are in Kip - ft/ft

 

Table 19. Cracking and nominal moment capacity for UWP6T6B in kip-ft/ft.

Transverse
Rib Spacing
right pointing arrow

Cracking Moment Capacity
( +ve Bending)

Nominal Moment Capacity
( +ve Bending)

Cracking moment
( -ve Bending)

Nominal Moment Capacity
( -ve Bending)

36” 33” 30” 27” 24” 21” 18” 36” 33” 30” 27” 24” 21” 18” 36” 33” 30” 27” 24” 21” 18” 36” 33” 30” 27” 24” 21” 18”
Girder Spacing (ft. in.) ↑

4’-0”

2.50 2.50 2.50 2.50 3.52 3.52 3.52 12.00 12.00 12.00 12.00 15.52 15.52 15.52 6.52 6.52 7.66 7.66 7.66 7.66 8.48 24.65 24.65 28.55 28.55 28.55 28.55 32.33

4’-3”

2.44 2.44 2.44 3.43 3.43 3.43 4.36 11.79 11.79 11.79 15.19 15.19 15.19 18.59 6.49 6.49 7.63 7.63 7.63 7.63 8.45 24.57 24.57 28.42 28.42 28.42 28.42 32.17

4’-6”

2.38 2.38 2.38 3.34 3.34 3.34 4.25 11.59 11.59 11.59 14.89 14.89 14.89 18.19 6.46 6.46 7.59 7.59 7.59 7.59 8.41 24.46 24.46 28.26 28.26 28.26 28.26 31.97

4’-9”

2.32 2.32 2.32 3.26 3.26 3.26 4.15 11.39 11.39 11.39 14.61 14.61 14.61 17.81 6.42 6.42 7.56 7.56 7.56 7.56 8.38 24.39 24.39 28.14 28.14 28.14 28.14 31.83

5’-0”

2.26 2.26 2.26 3.19 3.19 3.19 4.05 11.21 11.21 11.21 14.34 14.34 14.34 17.45 6.39 6.39 7.53 7.53 7.53 8.34 8.34 24.29 24.29 27.99 27.99 27.99 31.64 31.64

5’-3”

2.21 2.21 3.11 3.11 3.11 3.11 3.96 11.04 11.04 14.08 14.08 14.08 14.08 17.11 6.36 6.36 7.49 7.49 7.49 8.31 8.31 24.19 24.19 27.88 27.88 27.88 31.45 31.45

5’-6”

2.16 2.16 3.04 3.04 3.04 3.04 3.87 10.88 10.88 13.84 13.84 13.84 13.84 16.79 6.32 6.32 7.46 7.46 7.46 8.28 8.28 24.10 24.10 27.74 27.74 27.74 31.32 31.32

5’-9”

2.12 2.12 2.98 2.98 2.98 3.79 3.79 10.73 10.73 13.61 13.61 13.61 16.49 16.49 6.29 6.29 7.43 7.43 7.43 8.24 8.24 24.04 24.04 27.60 27.60 27.60 31.14 31.14

6’-0”

2.07 2.07 2.91 2.91 2.91 3.71 3.71 10.59 10.59 13.40 13.40 13.40 16.20 16.20 6.26 7.39 7.39 7.39 7.39 8.21 8.21 23.95 27.51 27.51 27.51 27.51 30.97 30.97

6’-3”

2.03 2.85 2.85 2.85 2.85 3.63 3.63 10.45 13.19 13.19 13.19 13.19 15.92 15.92 6.23 7.36 7.36 7.36 7.36 8.18 8.18 23.86 27.37 27.37 27.37 27.37 30.81 30.81

6’-6”

1.99 2.80 2.80 2.80 2.80 3.56 3.56 10.32 12.99 12.99 12.99 12.99 15.66 15.66 6.20 7.33 7.33 7.33 7.33 8.15 8.15 23.78 27.28 27.28 27.28 27.28 30.69 30.69

6’-9”

1.95 2.74 2.74 2.74 2.74 3.49 3.49 10.19 12.81 12.81 12.81 12.81 15.41 15.41 6.17 7.30 7.30 7.30 7.30 8.12 8.12 23.70 27.16 27.16 27.16 27.16 30.53 30.53

7’-0”

2.69 2.69 2.69 2.69 3.42 3.42 4.13 12.63 12.63 12.63 12.63 15.18 15.18 17.72 6.14 7.27 7.27 7.27 7.27 8.08 8.08 23.63 27.04 27.04 27.04 27.04 30.38 30.38

7’-3”

2.64 2.64 2.64 2.64 3.36 3.36 4.05 12.46 12.46 12.46 12.46 14.95 14.95 17.43 6.11 7.24 7.24 7.24 7.24 8.05 8.05 23.55 26.92 26.92 26.92 26.92 30.23 30.23

7’-6”

2.59 2.59 2.59 2.59 3.30 3.30 3.98 12.29 12.29 12.29 12.29 14.73 14.73 17.16 6.08 7.21 7.21 7.21 7.21 8.02 8.02 23.48 26.84 26.84 26.84 26.84 30.08 30.08

7’-9”

2.54 2.54 2.54 2.54 3.24 3.24 3.90 12.13 12.13 12.13 12.13 14.52 14.52 16.90 6.05 7.18 7.18 7.18 7.18 7.99 7.99 23.39 26.73 26.73 26.73 26.73 29.94 29.94

8’-0”

2.50 2.50 2.50 2.50 3.18 3.18 3.84 11.98 11.98 11.98 11.98 14.32 14.32 16.65 7.15 7.15 7.15 7.15 7.96 7.96 8.60 26.62 26.62 26.62 26.62 29.80 29.80 32.98

8’-3”

2.45 2.45 2.45 2.45 3.13 3.13 3.77 11.84 11.84 11.84 11.84 14.13 14.13 16.42 7.12 7.12 7.12 7.12 7.93 7.93 8.57 26.52 26.52 26.52 26.52 29.71 29.71 32.80

8’-6”

2.41 2.41 2.41 3.07 3.07 3.07 3.71 11.70 11.70 11.70 13.95 13.95 13.95 16.19 7.09 7.09 7.09 7.09 7.90 7.90 8.54 26.41 26.41 26.41 26.41 29.58 29.58 32.67

8’-9”

2.37 2.37 2.37 3.02 3.02 3.02 3.65 11.57 11.57 11.57 13.77 13.77 13.77 15.97 7.06 7.06 7.06 7.06 7.88 7.88 8.51 26.31 26.31 26.31 26.31 29.45 29.45 32.50

9’-0”

2.33 2.33 2.33 2.97 2.97 3.59 3.59 11.44 11.44 11.44 13.60 13.60 15.75 15.75 7.03 7.03 7.03 7.03 7.85 7.85 8.48 26.22 26.22 26.22 26.22 29.32 29.32 32.33

9’-3”

2.30 2.30 2.30 2.93 2.93 3.53 3.53 11.32 11.32 11.32 13.44 13.44 15.55 15.55 7.00 7.00 7.00 7.00 7.82 7.82 8.45 26.16 26.16 26.16 26.16 29.20 29.20 32.21

9’-6”

2.26 2.26 2.26 2.88 2.88 3.47 3.47 11.20 11.20 11.20 13.28 13.28 15.35 15.35 6.98 6.98 6.98 6.98 7.79 7.79 8.43 26.06 26.06 26.06 26.06 29.08 29.08 32.05

9’-9”

2.23 2.23 2.84 2.84 2.84 3.42 3.98 11.09 11.09 13.13 13.13 13.13 15.16 17.20 6.95 6.95 6.95 6.95 7.76 7.76 8.40 25.97 25.97 25.97 25.97 28.96 28.96 31.90

10’-0”

2.19 2.19 2.79 2.79 2.79 3.37 3.92 10.98 10.98 12.98 12.98 12.98 14.98 16.98 6.92 6.92 6.92 6.92 7.74 7.74 8.37 25.89 25.89 25.89 25.89 28.85 28.85 31.79
Note: All the moment capacity values are in Kip-ft/ft

 

Overhang Design Loads

A review of current bridge deck designs used by several DOTs showed that the overhang length is less than half the girder spacing, with a maximum length of 4 ft 3 in. for typical bridges. As noted in previous sections, an F-shape standard concrete railing is used for designing the overhang region for collision loads. In addition, based on the suggestions from Iowa DOT designers, it is recommended to use a solid cross-section for the overhang region instead of a waffle configuration. The solid section for the overhang will help in addressing the variability in the types of railings and their capacities as used by DOTs, and it will provide adequate space to include the necessary details for attaching the railing to the precast deck. With the selected overhang length and type of railing, it is shown below that the overhang will have adequate capacity to meet the expected design demands.

The overhang region is designed for different combinations of dead, live, and collision loads for the strength-I and extreme event II limit states, as required by the AASHTO guidelines.(27) The critical regions along the deck cross-section for overhang design, sections AA, BB, and CC, are shown in figure 45.

Figure 45. Diagrams. Critical section locations for overhang design and cross-sections of the waffle deck at those locations. Illustration showing the critical section locations for overhang design and cross-sections of the waffle deck at those locations. Part B shows the recatngular solid section for the overhang; Part C shows the cross-section at sections BB and CC for two different panel designs: UWD6T6B and UWD6T7B.

Figure 45. Diagrams. Critical section locations for overhang design and cross-sections of the waffle deck at those locations.

Section AA is located at the interior face of the parapet. Sections BB and CC are located at critical locations used for negative moment design. The AASHTO LRFD Bridge Design Specification section A13.4 is used for finding the different design cases of the overhang. Accordingly, three design cases were considered for completing the overhang design.

Design Case I: Transverse and Longitudinal Collision Forces

Design case I accounts for the transverse and longitudinal forces experienced from a collision on the parapet and the dead load of the structure. This is an extreme event II limit state, and the appropriate load factors are given in table 20.

Table 20. Negative moment demands due to collision forces at critical locations.

 
Moment Demand @ Section AA
Moment Demand @ Section BB

Parapet self-weight
(wparapet)*
(see table 8)

Moment Demand @ Section AA
Moment Demand @ Section BB

Self-weight of UHPC overhang (woverhang)*

(woverhang = 0.105 k/ft for 8 in. solid panel)
Moment Demand @ Section AA
Moment Demand @ Section BB

Self-weight of wearing surface (Wws)* (see figure 22)

none
Moment Demand @ Section BB
Barrier collision moment (Mc)
13.9 k-ft/ft
Moment Demand @ Section BB
Design Demand

= 1.00 (0.482+0.105)+1.00 (13.9)  = 14.49 k-ft/ft

= 1.00 (1.81+0.84)+1.00(0.077)+1.00 (10.4)  = 13.13 k-ft/ft

It should be noted that the moment demand at section CC will be less than that at section BB, thus making the critical moment demand for the T-beam cross-section to be at section BB.

For critical sections away from the parapet edge (e.g., at sections BB and CC), the collision capacity moment of the parapet is dispersed with a distribution angle of 30 degrees. In addition, it is assumed the moments and tension forces resulting from a collision are transmitted and distributed effectively between adjacent UHPC panels.

In addition to the collision moment capacity of the railing, the transverse resistance of the railing (Rw) should be resisted by the deck panel. This force propagates through the face of the parapet, causing direct tension in the deck panel that can be characterized using a yield line failure mechanism. The tension force can be estimated using the equation in figure 46:

Figure 46. Equation. Tension force in deck panel due to collision loading. Equation. T is equal to R subscript w divided by sum of L subscript c and 2 time H.

Figure 46. Equation. Tension force in deck panel due to collision loading.

where Rw = total transverse resistance of the railing, Lc = critical length of the yield line failure, and H = height of the railing. The Rw and Lc values for standard Iowa DOT F-shaped barriers are presented in table 9.

Design Case II: Vertical Collision Forces

Design case II accounts for the vertical forces experienced from a vehicle overtopping on the railing in conjunction with the dead load of the structure. This is categorized as an extreme event II limit state, and applicable load factors are given in table 5. This design case generally produces much lower negative moments when compared to design cases I and III for overhangs with concrete railings, making this load case not critical for overhang design with concrete railings.

Design Case III: Dead and Live Loads

Design case III examines the presence of a live load represented by a truck wheel load on the overhang region without a collision force and the self-weight of the structure. This is a strength-I limit state, and applicable load factors are given in table 5.

The critical sections for this design case are sections BB and CC, which are the same as the critical locations used for deck negative bending design. As mentioned previously, the locations of the sections BB and CC are conservatively taken as 3 inches from the centerline of the girder (as the minimum width of the top flange, which is 12 inches for typical girders) to estimate the negative moment demand; and, the truck load is located at 12 inches from the barrier rail inside edge as prescribed by AASHTO requirements. The locations of the critical location and tire load are shown in figure 47.

Figure 47. Diagram. Locations of dead and live loads for design case III. Illustration showing the Locations of dead and live loads for design case III.

Figure 47. Diagram. Locations of dead and live loads for design case III.

The maximum demand at sections BB and CC under the combined dead load and live load effects for different girder and rib spacing were found to be less than the design demand estimated in design case I.

Overhang Capacity

The negative moment capacities of the deck panel at critical sections BB and CC are estimated during the deck design process and are provided in table 21.

Table 21. Moment capacity of the overhang section for deck panels with two different reinforcement configurations.

Critical Section Shape at Section AA right pointing arrow

critical section shape at Section AA - uwp6t6b

critical section shape at Section AA - uwp7t6b

Transverse Rib Spacing (in.)

Negative Moment Capacity
(kip-ft/ft)

Negative Moment Capacity
(kip-ft/ft)

36

37.64

37.61

33

37.98

38.04

30

38.51

38.46

27

39.03

39.08

24

39.74

39.78

21

40.64

40.67

18

41.83

41.86

 

The negative moment capacity of the rectangular cross-section at section AA may be estimated using the procedure described earlier in this chapter. The nominal negative moment capacity of the solid deck section for different transverse rib spacing is provided in table 21. It is clear from the table that the moment capacity of the overhang is nearly 225 percent of the demand. Hence, a designer can possibly use a taper for the overhang region between the transverse ribs to optimize the material (UHPC) usage, but this approach may be found to be labor intensive and in violation of the AASHTO LRFD 13.7.3.1.2 section requirement.

Page last modified on May 18, 2012.
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