Step 1 – Create Non-Composite Dead Load Model

7.1.2.1.1 Step 1a – Define Girder and Intermediate Diaphragm Locations

From Figure 1 and

Figure 2, the location of the girders and intermediate diaphragms as well as the non-composite span length can be found. Assuming that the origin is located at the left end of girder B4 in Figure 1, the coordinates for the girder ends and intermediate diaphragm locations are shown in Table 10. Each girder is defined by two segments, one from x = 0 ft to x = 56.625 ft and the second from x = 56.625 ft to x = 113.25 ft. Since the software being used has elements that allow offsets to be defined, as the coordinates in Table 10 indicate, the nodes of all components lie on the same horizontal plane; in the physical structure, the component centroids do not all lie on the same plane.

Table 10 - Coordinates for Girder and Intermediate Diaphragm Ends

Component Start End

x (ft) y (ft) z (ft) x (ft) y (ft) z (ft) B1 0 34.5 0 113.25 34.5 0 B2 0 23.0 0 113.25 23.0 0 B3 0 11.5 0 113.25 11.5 0 B4 0 0 0 113.25 0 0 Int. Diaphragm B1-B2 56.625 23.0 0 56.625 34.5 0 Int. Diaphragm B2-B3 56.625 11.5 0 56.625 23.0 0 Int. Diaphragm B3-B4 56.625 0 0 56.625 11.5 0

Beam elements are used for modeling both the girders and intermediate diaphragms. In the software used, both Timoshenko and Euler beam elements are available. The Timoshenko formulation, which is capable of capturing shear deformations, is used. However, because shear deformations are insignificant in these members, Euler beam elements would be acceptable as well, though there is a negligible computational penalty for utilizing the more complete formulation.

For the girders, each line is divided into five elements. This is done such that nodes are located at the girder tenth-points. For the intermediate diaphragms, three elements per line are specified.

The intermediate diaphragm is defined as a rectangle that is 46 inches deep and 10 inches wide. Table 11 shows the intermediate diaphragm section properties as calculated by the analysis software.

Because the centroid of the diaphragms is at a different height than the girder centroid a vertical offset is required. The centroid of the girder is 36.375 inches from the bottom of the girder while the centroid of the intermediate diaphragm is 44 inches from the bottom of the girder; therefore the intermediate diaphragm cross-section must be shifted up by 7.625 inches (or 0.635 ft) (see Figure 14). In the software used, this is entered as a negative value.

Table 11 - Intermediate Diaphragm Section Properties

Section Property Value

Cross-section Area (A) (ft2_{) 3.194}

Strong Axis Moment of Inertia (Iyy) (ft4) 3.912

Weak Axis Moment of Inertia (Izz) (ft4) 0.185

Torsion Constant (Jxx) (ft4) 0.638

Shear Area in y direction (Avy) (ft2) 2.662

Shear Area in z direction (Avz) (ft2) 2.662

Offset in z direction (Rz) (ft) -0.635

Figure 14 - Diaphragm Eccentricity

7.1.2.1.3 Step 1c – Define Material Properties for Girders and Intermediate Diaphragms

Material properties can be calculated by hand and input into the analysis software or defined using materials already included in the software. If specifying material properties directly from analysis software it should be verified that they conform to the design specifications. 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. This example utilizes material properties shown in Table 12 calculated in accordance with Equation 5 and Equation 6 in Section 7.1.1.1.2.

Intermediate Diaphragm

Unit Weight (k/ft3_{) 0.153 0.150}

Thermal Expansion

Coefficient (ft/ft/oF) 6.0E-6 6.0E-6

7.1.2.1.4 Step 1d – Define Support Conditions

The beams are simply supported before the continuity diaphragms at the piers are placed. One end of each girder is restrained in the vertical and transverse directions only. At the other end, each girder is restrained vertically, transversely, and longitudinally. In this model the supports are located vertically at the centroid of the girders. For dead load, modeling the supports at the girder centroid, rather than below the bottom of the girder should not have any effect on the results.

7.1.2.1.5 Step 1e – Define Non-Composite Loads

The non-composite dead loads in addition to self-weight (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 line loads.

7.1.2.1.6 Step 1f – Define Load Cases

The next step is to define different load cases. Since all loads in this model are non-composite, they are combined into a single load case.

7.1.2.1.7 Step 1g – Ensure Correct Attributes Are Assigned to Components

After defining the geometry, member properties, material properties, support conditions, and loads, these attributes must be assigned to the appropriate geometry within the model. The elements defining the girder are assigned the properties associated with the girders and the elements defining the intermediate diaphragms are assigned the properties associated with the intermediate diaphragms. 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

o Dead loads (including self-weight, SIP forms, haunches, and deck slab)  Intermediate diaphragms

7.1.2.1.8 Step 1h – Run Analysis and Verify Results using Simplified Methods

The next step is to run the analysis. Errors may exist such that the analysis cannot be finished, these must be corrected. Errors may also exist that do not prevent the analysis from finishing but may provide erroneous results. These errors are typically more difficult to detect. Viewing the deflected shape should always be the first step as it can often reveal obvious problems. The analysis results can be verified using a variety of simplified methods. In this particular case, for a simply supported bridge, midspan moment or end reaction can be calculated using textbook equations. The results from the equations should be very similar to the results from the analysis. In this example, the reaction and midspan moment due to girder self-weight are verified.

From Section 7.1.2.1.2 and 7.1.2.1.3, the girder area (A) is 7.41 ft2 _{and the unit weight of the}

concrete is 0.153 k/ft3_{. The weight per linear foot of the girder, w, is therefore 1.134 k/ft.}

The midspan moment is calculated using the equation, M = wL2_{/8 = 1818.02 k-ft. The reaction}

is calculated using the equation, R = wL/2 = 64.21 k.

Textbook Equation Analysis Software % Difference

Midspan Moment (k-ft) 1818.02 1818.02 0.00%

End Reaction (k) 64.21 64.22 0.02%

Therefore, the model is producing reasonable results.

7.1.2.1.9 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. For the non-composite analysis, the moments in the beam elements can be used directly.