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Section 6 – Analysis to Design
7.1 EXAMPLE 1 – THREE-SPAN, PRECAST CONCRETE I-GIRDER BRIDGE (CONTINUOUS FOR LIVE LOAD)
7.1.3.1 Step 1 – Develop Non-Composite Model
7.1.3.1.1 Step 1a – Define Geometry for Girders and Diaphragms
The girder geometry is defined by creating surface elements along the girder centerline. Four elements in cross-section are used to represent the girder webs while beam elements are used to represent the girder flanges at the top and bottom of the web. When defining the girders, nodes should be defined at the locations of the intermediate diaphragms to facilitate connections. The intermediate diaphragms are defined by beam elements connecting the points at midspan of each girder.
In a prestressed concrete girder bridge, the flanges represent a significant portion of the total girder depth; the vertical location of the supports varies depending on how the web is modeled. In this example, the shell elements defining the web are modeled to extend the full depth of the section, as recommended in Section 3.7.2.1.2 of the manual, and illustrated in Figure 23.
Figure 23 - Modeling of Web Depth
The girder web surface elements utilize thick shells, a type of surface element capable of capturing shear deformations and membrane forces further discussed in Section 3.7.1.3. Thick beam elements, a beam element capable of capturing shear deformations, are used for the girder flanges and intermediate diaphragms and further discussed in Section 3.7.1.2.
additional effort to extract force effects.
Figure 24 - Non-Composite 3D FEA Model
7.1.3.1.2 Step 1b – Define Cross-Section Properties
The cross-section properties for the girder top and bottom flanges must be defined and the element located such that the section properties of the entire girder cross-section are correct. Figure 25 shows the dimensions for the top and bottom flanges; the section properties and centroid location were determined and are listed in Table 17. Because the flange beam elements are connected at the top and bottom of the web, offsets shown in the table are used to locate the flange beam elements at the centroid of each flange so they are at the correct height relative to the girder web.
Bottom Flange (beam)
Top Flange (beam)
Intermediate Diaphragm (beam) Web Elements (surface)
DRAFT
Figure 25 - Girder Flanges for 3D FEA Model Table 17 - Girder Flange Section Properties
Section Property Top Flange Bottom Flange
Cross-section Area (A) (ft2) 1.736 1.813
Strong Axis Moment of Inertia (Iyy) (ft4) 0.075 0.230
Weak Axis Moment of Inertia (Izz) (ft4) 1.848 0.976
Torsion Constant (Jxx) (ft4) 0.139 0.195
Shear Area in y direction (Avy) (ft2) 0.084 0.039
Shear Area in z direction (Avz) (ft2) 1.544 1.615
Offset in z direction (Rz) (ft) 0.323 -0.568
The intermediate diaphragms are 10 inches wide by 46 inches deep. In this example the diaphragm is connected at the top flange of the girder and offset such that it is in the correct location, but the member could be modeled at the correct location and attached via constraint equations or rigid links. The offset is equal to the distance between the top of the modeled girder web and the centroid of the diaphragm. The section properties for these intermediate diaphragms (which are modeled with beam elements) are the same as the previous section, except for the eccentricity (offset), which is now offset from the top of the web rather than centroid of the girder, a distance of 2.3125 feet.
The only geometric property needed for the girder web is the thickness, 8 inches.
DRAFT
software library. If the latter is used, the properties should be verified to conform with design specifications. This example utilizes material properties shown in Table 18. Based on the available information, the girders utilize concrete with a 28-day compressive strength of 8 ksi while the intermediate diaphragms utilize concrete with a 28-day compressive strength of 3.5 ksi.
Table 18 - Concrete Material Properties
Material Property Girder Concrete (8 ksi) Int. Diaphragm Concrete (3.5 ksi)
Modulus of Elasticity (ksf) 765,216 490,307
Poisson’s Ratio 0.2 0.2
Unit Weight (k/ft3) 0.153 0.150
Thermal Expansion
Coefficient (ft/ft/oF) 6.0E-6 6.0E-6
7.1.3.1.4 Step 1d – Define Support Conditions
The same support conditions as the 2D model are used in this model, but are located at the actual elevation along the bottom flange.
7.1.3.1.5 Step 1e – Define Non-Composite Loads
Self-weight of members defined in the model, the girders and intermediate diaphragms, is applied using a body force.
The other non-composite dead loads (SIP forms, haunches, and deck slab) are applied as uniform line loads to each girder. The values previously calculated in Sections 7.1.1.3.2 through 7.1.1.3.4 for the SIP form weight, haunch weight, and deck slab weight are applied to the interior and exterior girders as uniformly distributed loads.
7.1.3.1.6 Step 1f – Define Load Cases
The next step is to define load cases. Since all loads in this model are non-composite they are combined into a single load case. If the results of this example were used for design, the loads could still be combined into a single load case because they carry the same load factors. In Section 7.1.3.1.8, the results are verified using a simplified method. To facilitate checking of the model it may be desirable to create a separate load case containing only one of the loads (e.g. girder self-weight or stay-in-place forms).
7.1.3.1.7 Step 1g – Ensure Correct Attributes Are Assigned to Components The attributes assigned to each of the different components as shown below:
Body force acceleration (gravity) o Flanges
Thick beam elements Flange cross-sections
Concrete material properties, in this example, f’c = 8 ksi Body force acceleration (gravity)
Other non-composite loads (to top flange only) Intermediate Diaphragms
o Thick beam elements o Geometric cross-section
o Concrete material properties, in this example, f’c = 3.5 ksi o Body force acceleration (gravity)
7.1.3.1.8 Step 1h – Run Analysis and Verify Results
The next step is to run the analysis. The analysis results can be verified using a variety of simplified methods. Using the equations shown in Section 7.1.2.1.8 to calculate the midspan moment and the weight per linear foot from Section 7.1.1.3.1, the moment is 1789.2 k-ft. Determining the moment at midspan, by hand, from the 3D analysis model is more involved due to using multiple beam and shell elements. The required element nodal forces and moments required to determine the design moment in the girder are shown in Figure 26. The shell element nodal “forces,” indicated as Ny in Figure 26, are equal to the stress at the node
multiplied by the element thickness. A sample moment calculation is shown below Figure 26.
Figure 26 - Element Force Effects in Non-Composite Concrete Girder
0 in. 6.8125 in. 17.9375 in. 35.875 in. 53.8125 in. 71.75 in. 67.875 in. Bottom of Web
Bottom Flange Centroid Top of Web
Top Flange Centroid Node at ¼ Web Depth
Node at ½ Web Depth
Node at ¾ Web Depth
Ny0= -98.389 k/ft Nyt0.25= -47.0619 k/ft Nyb0.25= -49.0812 k/ft Nyt0.5= 2.09229 k/ft Nyb0.5= 0.0824927 k/ft Nyt0.75= 51.2594 k/ft Nyb0.75= 49.2392 k/ft Ny1= 100.576 k/ft Fxt= -225.092 k Myt= -3.54449 k-ft Fxb= 218.577 k Myb= -11.2596 k-ft
DRAFT
Nyb0.25 = -49.081 k/ft (upper node of second shell element)
Nyt0.50 = 2.092 k/ft (lower node of second shell element
Nyb0.50 = 0.082 k/ft (upper node of third shell element)
Nyt0.75 = 51.259 k/ft (lower node of third shell element)
Nyb0.75 = 49.239 k/ft (upper node of bottom shell element)
Ny1 = 100.576 k/ft (lower node of bottom shell element)
Beam Elements
Fxt = -225.092 k (axial force in top flange beam element)
Myt = -3.544 k-ft (bending moment in top flange beam element)
Fxb = 218.577 k (axial force in bottom flange beam element)
Myb = -11.260 k-ft (bending moment in bottom flange beam element)
Nodal Force Lever Arms from Bottom of Web: Ey0 = 71.75 in. Ext = 67.875 in. Eyb/t0.25 = 53.8125 in. Eyb/t0.50 = 35.875 in. Eyb/t0.75 = 17.9375 in. Exb = 6.8125 in. Ey1 = 0 in.
Web Element height:
he = 17.9375 in.
Sum Moments about Bottom of Web:
M = -(Ny0 × he/2× Ey0 + Fxt × Ext + Myt + (Nyt0.25 + Nyb0.25)/2 × he × Eyb/t0.25 + (Nyt0.50
+ Nyb0.50)/2 × he × Eyb/t0.50 + (Nyt0.75 + Nyb0.75)/2 × he × Eyb/t0.75 + Fxb × Exb + Myb
+ Ny1 × he/2 × Ey1)
= -(-98.389k/ft × (17.9375in/2)/(12in/ft)× 71.75in + -225.092k × 67.875in + - 3.544 k-ft × 12in/ft + (-47.062k/ft + -49.081k/ft)/2 × 17.9375in/(12in/ft) × 53.8125in + (2.092k/ft + 0.082k/ft)/2 × 17.9375in/(12in/ft) × 35.875in + (51.259k/ft + 49.239k/ft)/2 × 17.9375in/(12in/ft) × 17.9375in + 218.577k × 6.8125in + -11.260k-ft × 12in/ft + 100.576k/ft × (17.9375in/2)/(12in/ft) × 0in) = 21704.05 k-in = 1808.67 k-ft
OK, Moment from 3D model is within 1% of calculated value using equation. Model is providing reasonable results. (Note: If more elements were used, the difference would be even smaller). The moment from the software’s internal numerical integration utility is 1808.53 k-ft.
Furthermore, axial loads are summed to verify there is no net axial force on the section.
DRAFT
Sum Axial Loads:
P = -(Ny0 × he/2+ Fxt + (Nyt0.25 + Nyb0.25)/2 × he + (Nyt0.50 + Nyb0.50)/2 × he + (Nyt0.75 +
Nyb0.75)/2 × he + Fxb + Ny1 × he/2) = -(-98.389k/ft × (17.9375in/2)/(12in/ft) + -225.092k + (-47.0619k/ft + - 49.0812k/ft)/2 × 17.9375in/(12in/ft) + (2.09229k/ft + 0.0824927k/ft)/2 × 17.9375in/(12in/ft) + (51.2594k/ft + 49.2392k/ft)/2 × 17.9375in/(12in/ft) × + 218.577k + 100.576k/ft × (17.9375in/2)/(12in/ft)) = 4.88 k
OK, Net axial force from 3D is approximately zero. It is expected that due to the 3D nature of the model that this net axial force should be small (for straight bridges), but nonzero. This force arises when behavior of the girders, which do not precisely conform to typical idealized assumptions (where girders act independently), is captured.
Check reactions due to stay-in-place forms. The weight per linear foot of girder for the stay-in- place forms is 0.17 k/ft for the interior girders and 0.086 k/ft for the exterior girders.
L = 113.25 ft
R = 2 × (0.17 k/ft + 0.086 k/ft) × 113.25 ft = 57.984 k Reactions from Model:
Location Line 1 (B1) Line 2 (B2) Line 3 (B3) Line 4 (B4) Total Left Support 6.003 8.493 8.493 6.003 28.992 Right Support 6.001 8.495 8.495 6.001 28.992 Total 57.984
OK, model is providing reasonable results 7.1.3.1.9 Step 1i – Extract Required Results from Analysis Software
After verifying that the results from the analysis are reasonable, the results of interest can be extracted and input into a spreadsheet for further use. The analysis results required are typically moments and shears which are not readily available due to using multiple beam and/or shell elements. The moments can be determined as described in Section 6.2.1 and shown above, by integrating the shell stresses on the cross-section and summing the results. Some FEA software provides a utility that will integrate stresses over a cross-section automatically. Note that at the nodes where the deck is connected the forces will appear to have a discontinuity due to the node also behaving as a shear connector.
7.1.3.2 Step 2 – Create Composite Dead Load Model
Step 2 is a continuation of the model created in Step 1 above. As the girders are made composite and become continuous over piers, the span lengths change from 113’-3” between centerline of bearings for all three simple spans in the non-composite model to 114’-3” for Spans 1 and 3 and 115’-3” between centerline of abutments and piers for Span 2 in the continuous composite model.
from midspan of Spans 1 and 3, modeling them at the tenth-point midspan node should not noticeably affect accuracy of results (Span 1 is shown in Figure 27):
Figure 27 - Surface Definitions for Span 1 of Composite Model
The deck slab surface elements are defined using the points along the top edges of the previously defined girder webs. The slab geometry in the 3D finite element analysis is similar to that shown in Section 7.1.2.2.1 for the PEB analysis. Thick shell elements are used to model the deck slab. Elements that do not include through-thickness shear deformation can be used; the decrease in accuracy is not significant.
7.1.3.2.2 Step 2b – Define Cross-Sections for Girders, Diaphragms, and Concrete Deck Slab The pier diaphragms are 30 inches wide and approximately 7’-2¼” deep and are connected to the concrete deck slab. The abutment diaphragms are 48 inches wide and approximately 6’-6” deep and are connected to the concrete deck slab. These diaphragms are connected to the concrete slab locations rather than at the centroid of girders for convenience, as no new nodes or lines are required. The cross-section properties for all diaphragms are shown in Table 19. Similarly to Section 7.1.2.2.2, vertical offsets are made as necessary to correctly locate members relative to the centroid of the girder web. The offsets are different than those in the 2D example as components are connected at different points in space (for example, the deck sits at the top of the girder in the 3D model, instead of at the girder centroid).
Table 19 - Abutment, Intermediate, and Pier Diaphragm Section Properties
Section Property Abutment Intermediate Pier
Cross-section Area (A) (ft2) 26.000 3.194 17.979
Strong Axis Moment of Inertia (Iyy) (ft4) 91.542 3.912 77.490
Weak Axis Moment of Inertia (Izz) (ft4) 34.667 0.185 9.364
Torsion Constant (Jxx) (ft4) 85.549 0.638 29.263
Shear Area in y direction (Avy) (ft2) 21.667 2.662 14.983
Shear Area in z direction (Avz) (ft2) 21.667 2.662 14.983
Offset in z direction (Rz) (ft) 2.773 2.313 3.135
The concrete deck slab is modeled with the 8” design thickness to neglect stiffness contributions of the integral wearing surface. An eccentricity of 0.781 ft, which includes haunches, is applied. 7.1.3.2.3 Step 2c – Define Material Properties for Girders, Diaphragms, and Deck Slabs
4 @ 11.425’ = 45.7’ 10.925’ 5 @ 11.425’ = 57.125’ 0.5’
Table 20 - Concrete Material Properties
Material Property 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/ft/oF) 6.0E-6
7.1.3.2.4 Step 2d – Define Support Conditions
The supports conditions used in the 3D composite dead load model are the same as those used in the 2D composite dead load model.
7.1.3.2.5 Step 2e – Define Loads Applied to Composite Structure
The FWS load is defined as a uniform load distributed over the bridge width. The magnitude of the FWS load is reduced from 0.030 ksf to 0.028 ksf to account for the fact that the FWS is being spread over the full bridge width instead of just between barriers.
The weight of the barrier, determined above in Section 7.1.1.3.6, is applied as a uniform line load.
7.1.3.2.6 Step 2f – Define Load Cases for Composite Dead Loads
Define load cases for the future wearing surface and barrier dead loads. Defining separate load cases is important for composite dead loads; in the design process, wearing surface loads and component dead loads have different load factors.
7.1.3.2.7 Step 2g – Ensure Correct Attributes Are Assigned to Components The attributes assigned to each of the different components as shown below:
Girders o Web
Thick shell surface elements Geometric surface cross-section
Concrete material properties, in this example, f’c = 8 ksi o Flanges
Thick beam elements
Flange cross-section properties
Concrete material properties, in this example, f’c = 8 ksi Abutment, Intermediate, and Pier Diaphragms
o Thick beam elements
o Geometric cross-section properties
o Concrete material properties, in this example, f’c = 3.5 ksi Concrete Deck Slab
o Thick shell surface elements o Geometric surface cross-section
o Concrete material properties, in this example, f’c = 4 ksi o FWS and barrier loading