The Owachomo bridge, Natural Bridges National Monument, Utah

From the AAA Tour Book:

The Owachomo natural bridge is in a late stage of erosion, and is approaching the day when weather and gravity will bring it crashing to the canyon floor.

Span: 180 ft

Width at center: 27 ft

Depth at center: 9 ft

Assignment

It is desired to estimate the maximum tensile stress in the bridge, and to compare it to the tensile strength s_f  of the rock.  The tensile strength of  the rock of the Owachomo bridge is not available.  Assume an average value for rock as

s_f = 10 MPa

To obtain an estimate of the maximum tensile stress with the means at our disposal, the bridge is modeled as a fixed-ended beam with a stepped cross-section, subjected to its dead weight. Two models, a simple and a more refined one, are proposed:

Model 1:  Segment BC has a 27 x 9 ft cross section

                Segments AB and CD have a 27 x 18 ft cross section

Model 2:  Segment BC has a 27 x 9 ft cross section

                Segments B'B and CC' have a 27 x 13 ft cross section               

                Segments AB' and C'D have a 27 x 22 ft cross section

Assume the specific weight g  of the rock to be

g = 2.7 Mg/m³

Conversion factors:

1 kg/m³ = 0.062428 lb/ft³

1 MPa = 145 psi

a) Obtain the maximum tensile stresses in Model 1, and compare it to the tensile strength of the rock.

b) Obtain the maximum tensile stresses in Model 2.  Does the more refined model change significantly the answer?