## Fall 2023

### Fallingwater Strengthening

Frank Lloyd Wright designed Fallingwater in the mid 1930’s to sit within a rocky hillside and cantilever out over a waterfall along Bear Run. The primary horizontal spanning members consist of reinforced concrete, while the primary vertical supports consist of stone masonry bearing walls. Materials testing at the site revealed that the concrete is quite strong for its time period, achieving a compressive strength (f’c) of 5,000 psi. The steel reinforcing in the main cantilevers of the living room level consists of two rows of eight (8) 1” square bars (a total of 16 bars located at an average depth to the bottom surface equal to 19.5”(d). The yield strength of the steel is approximately 36,000 psi (Fy).

### 1.Determine the unfactored service dead and live loads on the center main cantilever beam(indicated in the sketch of the living room floor framing) as loads per unit length (wd and wl).  建筑学代写

The loads from the living room level are as indicated in the detail sketch (hint: useful to calculate a line load for the 36 inch width of the main beam, and a separate line load delivered to the beam from the upturned joist areas to each side). The loads from the master terrace on this center beam are found to be 35 kips dead load and 5.25 kips live load and act as concentrated loads at the end of the 15 foot cantilever. Calculate a total factored maximum bending moment (Mu) and total factored maximum shear (Vu).

2.Calculate the reduced bending moment capacity (fMn) and the shear capacity (Vn) of the center beam in two ways:

a.Assuming a rectangular cross section with beam width (b) equal to 36 inches;

b.Assuming an increased effective flange width with symmetrical T-beam behavior;

i.Use beff in place of b where beff is equal to the smallest of:

1. ¼ the span for a simply supported beam or ½ the span for a cantilever beam
1. b + 2 * 8 * slab thickness
2. The beam’s overall tributary width

### 3.Does the section work for the applied loads in bending?  建筑学代写

4.Non-destructive testing found the presence of some shear reinforcement. What value of Vs (shear capacity from steel reinforcement) would be required to satisfy the following?

a.Vu< fVn and fVn= f( Vs + Vc)

5.If post-tensioning is to be used to reinforce the center cantilevered beam analyzed above:

a.Draw a section through the living room showing the approximate post-tensioning profile along the beam span (assume the back span of the post-tensioning, to the dead end anchorage, is approximately 2/3 of the cantilever span. Assume the centerline of the post-tensioning tendon is at the top of the structural beam at its high point over the foundation support, and is set close to the mid-height of the beam depth at its live end at the end of the cantilever.  建筑学代写

b.Estimate through calculations the unfactored magnitude of the post-tensioning force if the intent is to provide an inverse moment that is 80% of the unfactored dead load.

i.Assume two post-tensioning tendons are used to reinforce the beam, one along each side.

6.Write a short paragraph and describe a design principle that Frank Lloyd Wright and Viollet-le-Duc have in common, based upon reading the article by Donald Hoffmann from 1969.