Seismic Analysis

Eriksson Culvert follows the seismic analysis procedure outlined in FHWA-NHI-10-034 chapter 13. Below is an example calculation of this analysis, as well as the corresponding Eriksson Culvert file.

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Design Parameters

Estimate the Free-Field Ground Strain

(Step 1 of FHWA-NHI-10-034, chapter 13.5.1)

The maximum free-field ground shear strain, 𝛾_max, can be estimated by using the following equation,

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where 𝜏_max is the maximum earthquake induced shear stress

For this example,

Per WSDOT BDM section 8.3.4B, the ground acceleration (PGA) can be reduced according to the depth of fill. The ground motion attenuation is shown in table 1 (below). For a fill of 16', the ratio of ground motion to motion at buried structure is 1.0.

Calculate the Free-Field Relative Displacement

Determine the Racking Coefficient

(Step 2 of FHWA-NHI-10-034, chapter 13.5.1)

A 2D frame analysis is used to determine the racking stiffness by applying a lateral unit force at the top, left node of the culvert. The structural racking stiffness is defined as the ratio of force to displacement.

With a point load of 1 kip at the top of the wall, the deflection and racking stiffness are:

Calculate the Flexibility Ratio

(Step 3 of FHWA-NHI-10-034, chapter 13.5.1)

The flexibility ratio is defined as,

where w is the culvert design width and h is the culvert design height.

Calculate the Racking Ratio

(Step 4 of FHWA-NHI-10-034, chapter 13.5.1)

For a non-slip interface:

For a full-slip interface:

For this example, we have a full-slip interface; so the racking ratio is calculated as:

Determine the Racking Deformation

(Step 5 of FHWA-NHI-10-034, chapter 13.5.1)

The racking deformation is defined as,

Therefore it is calculated to be,

Racking Induced Internal Forces

(Step 6 of FHWA-NHI-10-034, chapter 13.5.1)

The internal forces can be determined through analyzing a 2D frame model with a displacement of 0.273”; though due to the formulation of FEM, it is easier to apply this as an equivalent point load of 8.06 kips.

Eriksson Culvert calculates and stores these internal forces (axial, shear, and moments) at tenth points along all members in the structure for later use in the load combination equations.

Note: Since it is necessary to capture worst-case results for all members in the culvert, Eriksson Culvert performs this analysis twice, once with a positive point load on the top left node and again with a negative point load on the top right node.

Vertical Forces due to Vertical Acceleration

(Step 7 of FHWA-NHI-10-034, chapter 13.5.1)

The vertical seismic force is determined by psudo-static loading, per section 13.5.1.3.

The vertical seismic coefficient can be equivalent to 2/3 PGA = 0.28

The dead load of 30' fill is 16' * 0.130 kcf = 2.08 klf

The dead load of the top slab = 1.17' * 0.16 = 0.187 klf

Summing these forces gives a total dead load of 2.27 klf

From WSDOT BDM section 8.3.4B, the ground motion attenuation ration of 1.0 is applied since the depth of fill is less than 20'.

Now the vertical force is calculated as,

For a rigid foundation model, an equivalent upward force is also applied to the bottom slab.

Eriksson Culvert calculates and stores these internal forces (axial, shear, and moments) at tenth points along all members in the structure for later use in the load combination equations.

Combining Forces from the Racking and Vertical Analyses

(Step 8 of FHWA-NHI-10-034, chapter 13.5.1)

Eriksson Culvert uses the internal forces from both analyses to develop unfactored EQ min/max envelopes for every member in the structure. The considered cases to develop these envelopes are:

1. Results from the racking analysis

2. Results from the vertical analysis

3. Racking + vertical analysis results

Users have the option of determining how ‘(3) Racking + vertical analysis results’ are calculated, by either of the following combinations:

1. 1.0 / 1.0, which includes:

   1. 1.0 H + 1.0 V

   2. -1.0 H - 1.0 V

   3. -1.0 H + 1.0 V

   4. 1.0 H - 1.0 V

2. 0.3 / 1.0, which includes:

   1. 0.3 H + 1.0 V

   2. -0.3 H - 1.0 V

   3. -0.3 H + 1.0 V

   4. 0.3 H - 1.0 V

   5. 1.0 H + 0.3 V

   6. -1.0 H - 0.3 V

   7. -1.0 H + 0.3 V

   8. 1.0 H - 0.3 V

These unfactored forces are then applied in the LRFD extreme event I load combination:

Results in Eriksson Culvert

The following seismic specific results are found in the Eriksson Culvert text report:

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