Chapter 3 Prestressing with Post-Tensioning
8.4 Strip Method Analysis for a Multi-Cell Box Girder Superstructure
This section presents the transverse analysis of the four-cell box girder used in the preliminary design presented in chapter 5 and in example problem 1 presented in appendix C.
Chapter 8 – Transverse Analysis 191 of 369 8.4.1 The Transverse Model
The transverse analysis of the cross section shown in figure 8.9 is a two-dimensional solution that can be performed using either a 2D or 3D general-purpose frame analysis program. The three-dimensional analysis will typically require additional support conditions out of the model plane to provide needed analytical stability.
The analysis performed is a linear elastic analysis utilizing gross section properties of the concrete cross section. The top slabs, bottom slabs, webs, and overhangs are modeled as beam elements with depths equal to their representative thicknesses. The width of the beam elements “out of the plane of the page” is set equal to 1.0’. Node locations are provided at intersections of elements, changes in transverse member thickness, changes in transverse member orientation, and for convenience at locations of desired output.
Vertical support of the cross section model is with rigid supports at the bottom of the webs (Article 4.6.2.1.6 Paragraph 1). A horizontal support is placed rigidly to restrain the transverse movement of the bottom slab. A second horizontal support is provided to restrain the frame against side-sway, which better represents the torsional stiffness of the box girder with regard to transverse displacements.
Figure 8.10 shows the development of the cross section from transverse dimensions, to idealized members to the computer nodal analysis model for the cross section of Design Example 1.
Figure 8.10 – Developing the Two-Dimensional Transverse Model
Chapter 8 – Transverse Analysis 192 of 369 8.4.2 Transverse Bending Moment Results
Loads that uniformly repeat along the span can be applied directly to the analytical model without consideration for longitudinal distribution. These loads include self weight and superimposed dead loads such as barrier railing, wearing surfaces, and suspended utilities.
Figure 8.11 shows the bending moments from the self weight of the cross section. The analysis software used developed the applied loads internally based on the member cross section dimensions and the unit weight of the concrete. Figure 8.12 shows the effect of barrier railing weight 0.45 kips/foot, applied at the center of gravity of the barrier cross section. Figure 8.13 shows the bending moments resulting from the application of a 25 psf (or 25plf in the unit width model) wearing surface.
Small bending moments in webs and bottom slab members are not shown in some of the figures of this Section for clarity.
Figure 8.11 – Transverse Self Weight Moments (ft-kip/ft)
Figure 8.12 – Transverse Barrier Railing Moments (ft-kip/ft)
Figure 8.13 – Transverse Wearing Surface Moments (ft-kips/ft)
Chapter 8 – Transverse Analysis 193 of 369 Live load moments are determined by positioning both the Design Truck and Tandem wheel loads in position to produce maximum transverse moments. Figures 8.14, 8.15 and 8.16 show the results of three load cases for the Design Truck. Figure 8.14 shows the position of one truck to produce maximum negative transverse bending at the inside face of the outer webs. Figure 8.15 shows two trucks positioned to produce maximum transverse negative moment at interior webs. Figure 8.16 shows the load case producing maximum positive moment in the top slab.
Figure 8.14 – Maximum Negative Design Truck Moment in Outer Web
Figure 8.15 – Maximum Negative Design Truck Moment at Inner Web
Figure 8.16 – Maximum Positive Design Truck Moment in Top Slab
Chapter 8 – Transverse Analysis 194 of 369 8.4.3 Transverse Design Moments
Moment results for live load cases need to be magnified for impact and multiple presence factors, and then divided by the longitudinal length of the strips, W, based on equations 8.1, 8.2 and 8.3.
(Eqn. 8.8) W =45.0 10+ X =45.0 100+ =65"=5.42 ' (Eqn. 8.9) W =26.0 6.6+ S=26.0 6.6 12.25+ ( )=106.85"=8.90 ' (Eqn. 8.10) W =48.0 3.0+ S=48.0 3.0 12.25+ ( )=84.75"=7.06 '
Bending moment results from the transverse analysis of example problem 1 at critical nodes in foot-kips/foot are:
Node Type Self Weight Barrier Wearing
Surface Live Load
3 Overhang (Neg) -1.76 -1.97 -0.15 -32.00
13 Top Slab (Pos) 0.69 0.0 0.15 27.00
15 Top Slab (Neg) -1.06 0.0 -0.24 -36.42
Table 8.2 –Transverse Bending Moment Results from Frame Analysis
Live Load bending moments are magnified by vehicle impact and multi-presence factors, and divided over their associated strip width.
The negative live load moment in the overhang at the face of the web is:
(Eqn. 8.11) ( )( )
max
1.33 1.2 32 5.42 9.42 M ft k
ft
− = − = − −
It is interesting to note the magnitude of the overhang moment with regard to the moment caused by a 1 kip/foot load acting 1’ from the face of the railing as permitted by Article 3.6.1.3.4.
This moment would be approximately 1/3 of the moment predicted by the strength method:
(Eqn. 8.12) Mmax 1.33 1.2( )( 2.0) 3.19 ft k ft
− −
= − = −
The top slab maximum positive live load moment:
(Eqn. 8.13) ( )( )
max
1.33 1.2 27.00 8.90 4.84 M ft k
ft
= = −
The negative live load moment in the top slab at the face of the middle web:
(Eqn.8.14) ( )( )
max
1.33 1.0 36.42 7.06 6.86 M ft k
ft
− = − = − −
Chapter 8 – Transverse Analysis 195 of 369 The resulting factored moments for the AASHTO LRFD Strength I case are then:
(Eqn. 8.15) M3 1.25( 1.76 1.97) 1.5( .15) 1.75( 9.42) 21.37 ft k ft
= − − + − + − = − −
(Eqn. 8.16) M13 1.25 0.69 0( ) 1.5 .15( ) 1.75 4.84( ) 9.56 ft k ft
= + + + = −
(Eqn. 8.17) M15 1.25( 1.06 0) 1.5( .24) 1.75( 6.86) 13.69 ft k ft
= − − + − + − = − −
The three critical cross sections are designed for the amount of required reinforcing per foot along the length of the bridge. Reinforcing for the maximum positive and maximum negative ultimate moments is used in each slab and over each web. Additional reinforcing may be required over the outer webs in order to develop the capacity of the barrier when subjected to vehicle collision forces.