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Section 6 – Analysis to Design
7.1 EXAMPLE 1 – THREE-SPAN, PRECAST CONCRETE I-GIRDER BRIDGE (CONTINUOUS FOR LIVE LOAD)
7.1.2.2 Step 2 – Create Composite Dead Load Model
This will be a continuation of the model created in Step 1 above. The bridge is made continuous for composite loads by the pier diaphragms and deck slab. As the deck concrete hardens, including concrete which is placed between ends of adjacent girders, the composite girders will span between the centerline of piers rather than between bearings. As a result, the continuous span lengths increase to 114’-3”, 115’-3”, and 114’-3”; the intermediate diaphragms are located at the center of the simple spans not at the center of the continuous spans.
7.1.2.2.1 Step 2a – Define Girder, Diaphragm, and Concrete Deck Slab Location
From Figure 1 and
Figure 2, the location of the girders, diaphragms (abutment, intermediate, and pier), concrete deck slab, barrier width, and composite span lengths can be found. Assuming that the origin is located at the lower, left corner of the deck slab in Figure 1, the coordinates for the girder supports and diaphragms are shown in Table 13. The pier diaphragms are used to make the girders continuous for composite loads.
Abut. 1 Abut. 1 Diaphragm B2-B3 Abut. 1 Diaphragm B3-B4 0 0 15.93754.4375 0 0 0 0 27.4375 15.9375 0 0 Span 1 B1 0 38.9375 0 114.25 38.9375 0 B2 0 27.4375 0 114.25 27.4375 0 B3 0 15.9375 0 114.25 15.9375 0 B4 0 4.4375 0 114.25 4.4375 0 Int. Diaphragm B1-B2 57.125 27.4375 0 57.125 38.9375 0 Int. Diaphragm B2-B3 57.125 15.9375 0 57.125 27.4375 0 Int. Diaphragm B3-B4 57.125 4.4375 0 57.125 15.9375 0 Pier 1 P1 Diaphragm B1-B2 114.25 27.4375 0 114.25 38.9375 0 P1 Diaphragm B2-B3 114.25 15.9375 0 114.25 27.4375 0 P1 Diaphragm B3-B4 114.25 4.4375 0 114.25 15.9375 0 Span 2 B5 114.25 38.9375 0 229.5 38.9375 0 B6 114.25 27.4375 0 229.5 27.4375 0 B7 114.25 15.9375 0 229.5 15.9375 0 B8 114.25 4.4375 0 229.5 4.4375 0 Int. Diaphragm B5-B6 171.875 27.4375 0 171.875 38.9375 0 Int. Diaphragm B6-B7 171.875 15.9375 0 171.875 27.4375 0 Int. Diaphragm B7-B8 171.875 4.4375 0 171.875 15.9375 0 Pier 2 P2 Diaphragm B1-B2 229.5 27.4375 0 229.5 38.9375 0 P2 Diaphragm B2-B3 229.5 15.9375 0 229.5 27.4375 0 P2 Diaphragm B3-B4 229.5 4.4375 0 229.5 15.9375 0 Span 3 B9 229.5 38.9375 0 343.75 38.9375 0 B10 229.5 27.4375 0 343.75 27.4375 0 B11 229.5 15.9375 0 343.75 15.9375 0 B12 229.5 4.4375 0 343.75 4.4375 0 Int. Diaphragm B9-B10 286.625 27.4375 0 286.625 38.9375 0 Int. Diaphragm B10-B11 286.625 15.9375 0 286.625 27.4375 0 Int. Diaphragm B11-B12 286.625 4.4375 0 286.625 15.9375 0 Abut. 2 Abut. 2 Diaphragm B1-B2 343.75 27.4375 0 343.75 38.9375 0 Abut. 2 Diaphragm B2-B3 343.75 15.9375 0 343.75 27.4375 0 Abut. 2 Diaphragm B3-B4 343.75 4.4375 0 343.75 15.9375 0
Consideration should be given to node location when defining the geometry. While it is desirable to model all components at their actual locations, sometimes models can be greatly simplified without compromising the results by modeling components in slightly shifted locations, where a node is already present for some other reason. As always, judgment should be exercised.
location. Combining these nodes makes meeting aspect ratio recommendations much easier, as otherwise a 6 inch wide element between these two locations would have been required. Using the same assumptions as in Table 13, the coordinates for the corners of the concrete deck slab are shown in Table 14.
Table 14 - Coordinates of Concrete Deck Slab Corners
Corner x (ft) y (ft) z (ft)
Upper, Left 0 43.375 0
Upper, Right 343.75 43.375 0
Lower, Right 343.75 0 0
Lower, Left 0 0 0
The elements defined for the girders and intermediate diaphragms in the non-composite model are also used in the composite model. Thick shell elements, a type of surface element capable of capturing shear deformations and membrane forces, are used to model the concrete deck slab.
7.1.2.2.2 Step 2b – Define Diaphragm Cross-Sections and Concrete Deck Slab Thickness
The definition of the girder and intermediate diaphragm cross-section is described in Section 7.1.2.1.2. The difference between the definitions in the composite model and the non- composite model is the presence of the concrete deck slab and diaphragms at the abutments and piers. The nodes defining the deck slab, girders, and diaphragms all lie on the same horizontal plane in the analytical model but in the actual structure they are not at the same elevation. To more accurately model the structure, an eccentricity is applied to the concrete deck slab (see Figure 15), pier diaphragms, and abutment diaphragms to shift the centroids vertically to the correct position relative to the girders and intermediate diaphragms (see Section 7.1.2.1.2). Cross-slope of the deck is neglected in the geometry of the model as its affect on load distribution and load effects is negligible, while it can add a nontrivial amount of preprocessing time.
The pier diaphragms are 30 inches wide and 7’-2¼” deep while the abutment diaphragms are 48 inches wide and 6.5 feet deep. The cross-section properties are shown in Table 15.
Table 15 - Pier and Abutment Diaphragm Section Properties
Section Property Pier Abutment
Cross-section Area (A) (ft2) 17.979 26.000
Strong Axis Moment of Inertia (Iyy) (ft4) 77.490 91.542
Weak Axis Moment of Inertia (Izz) (ft4) 9.364 34.667
Torsion Constant (Jxx) (ft4) 29.263 85.549
Shear Area in y direction (Avy) (ft2) 14.983 21.667
Shear Area in z direction (Avz) (ft2) 14.983 21.667
Offset in z direction (Rz) (ft) 0.625 0.281
The only values required to define the concrete deck slab are the thickness and eccentricity. The deck slab thickness used in the model is the structural design thickness of 8”, which neglects stiffness contributions from the ½” integral wearing surface. To account for the weight of the omitted wearing surface thickness the material properties of the concrete deck slab will be modified in the following section. An eccentricity of 3.406 ft is applied to the deck slab.
Figure 15 - PEB Model
7.1.2.2.3 Step 2c – Define Material Properties for the Concrete Deck Slab
The properties for the deck slab concrete are added to the model. The deck slab concrete has a 28 day compressive strength of 4 ksi. The 150 pcf unit weight is increased by the ratio of actual deck slab thickness (including wearing surface) to the modeled deck slab thickness. Table 16 shows the material properties for the concrete deck.
Table 16 - Concrete Material Properties
Material Property Deck Slab Concrete (4 ksi) Modulus of Elasticity (ksf) 524,757 Poisson’s Ratio 0.2 Unit Weight (k/ft3) 0.159 Thermal Expansion Coefficient (ft/”F) 6.0E-6
While the deck slab, girder, and diaphragms are expected to have differential creep behavior (as a function of variable mix design and age) which suggests the use of a modular ratio, this effect is generally negligible and no modular ratio is applied.
7.1.2.2.4 Step 2d – Define Support Conditions
The supports at the abutments and Pier 1 restrain movement in the vertical and transverse directions only. At Pier 2, the girders are restrained vertically, transversely, and longitudinally against translation. Rotation about all three orthogonal axes is allowed at all bearing locations. Because of eccentricities applied in Section 7.1.2.2.2, the supports in this model are effectively reference plane at girder centroid
7.1.2.2.5 Step 2e – Define Dead Loads Applied to Composite Structure
The future wearing surface (FWS) load is defined as a uniform load distributed over the bridge width. To account for the difference in modeled width and actual width (which spans only between barriers), the FWS load is reduced from 0.030 ksf (spread over 40.0 ft) to 0.028 ksf (spread over 43.375 ft).
The weight of the barrier, determined above in Section 7.1.1.3.6, is applied as a uniform line load.
7.1.2.2.6 Step 2f – Define Load Cases
Separate load cases are defined for the composite dead loads; in this analysis, the composite dead loads are simply added together as moment envelopes are being compared but if the force effects were to be used in design, the barrier would be a DC (component dead load) load while the future wearing surface would be a DW (wearing surface dead load) load with the appropriate load factors applied.
7.1.2.2.7 Step 2g – Ensure Correct Attributes Are Assigned to Components
After defining the geometry and elements, member properties, material properties, support conditions, and loads, these attributes must be assigned to the appropriate geometry within the model. The lines defining the girders, intermediate diaphragms, and pier diaphragms are assigned the properties for the respective component. The concrete deck slab surfaces are assigned the properties of the deck slab. Listed below are the different components and the attributes that must be assigned:
Girders
o Beam elements
o Geometric cross-section
o Concrete material properties, f’c = 8 ksi in this example Intermediate Diaphragms
o Beam elements
o Geometric cross-section
o Concrete material properties, f’c = 3.5 ksi in this example Pier Diaphragms
o Beam elements
o Geometric cross-section
o Concrete material properties, f’c = 3.5 ksi in this example Concrete Deck Slab
o Thick shell elements o Deck slab thickness
o Concrete material properties, f’c = 4 ksi in this example o Future wearing surface and barrier loading
7.1.2.2.8 Step 2h – Run Analysis and Verify Results using Simplified Methods
After running the analysis moments and reactions can be calculated using published equations for uniform loads. For a three equal span continuous beam with uniform loading, the reaction at the end support is 0.4wl and the reaction at the interior support is 1.1wl. Using tributary areas,
Location Line 1 Line 2 Line 3 Line 4 Total Abutment 1 13.07 14.76 14.76 13.07 55.66 Pier 1 35.95 40.59 40.59 35.95 153.08 Pier 2 35.95 40.59 40.59 35.95 153.08 Abutment 2 13.07 14.76 14.76 13.07 55.66 Total 417.48
Compare these to the reactions from the PEB Model:
Location Girder Line 1 Girder Line 2 Girder Line 3 Girder Line 4 Total
Abutment 1 12.16 15.63 15.63 12.16 55.58
Pier 1 33.53 43.05 43.05 33.53 153.16
Pier 2 33.53 43.05 43.05 33.53 153.16
Abutment 2 12.16 15.63 15.63 12.16 55.58
Total 417.48
The reactions are very similar, verifying that the model is providing accurate results. Additionally, for uniform loads applied to the entire structure, the total applied load applied can be calculated and should equal the sum of the reactions from the analysis model.
Check reactions for FWS load versus applied load: Weight of FWS = 0.028 ksf
Area over which FWS applied = 43’-4½” × 343’-9” = 14910.16 ft2
Total Weight of FWS = 0.028 ksf × 14910.16 ft2 = 417.48 k
The total applied FWS load is equal to the sum of the reactions from the model. Additionally, since the structure is symmetric the reactions should also be symmetric. The reactions are also symmetric as expected.
7.1.2.2.9 Step 2i – Extract Required Results from Analysis Software
Since the results from the analysis appear reasonable, the results of interest can be extracted and input into a spreadsheet for further analysis or used within the analysis and design software. The results of interest are moments, shears, and deflections in the girders. For a PEB model, the moment due to composite dead load and live load in the girder is not just the moment directly from one beam element. Due to the multiple elements that make up the composite section and eccentricity of the beam elements (plate and eccentric beam), the actual moment in the composite design girder is equal to the moment at the center of gravity of the beam element plus the moment in the deck shell elements and the axial force couple between the deck shell elements and girder beam element. Alternatively, if the analysis software