Example : Cross vault

Building a cross vault with RV2 constraining the amount of forces in the edges.

In a cross vault the force distribution is more complex than in a barrel vault. Since you will manually constrain the forces in the vault, we use a square base to reduce the complexity and get help from the symmetry.

1) Create pattern: "From Lines":

A cross vault is supported in four points at the corners. The form diagram, that represents the horizontal projection of how the forces flow in the structure, should allow forces to end up in the corners. For this reason, in this case, our form diagram should look like the one in the picture below, with two diagonals added to the square pattern.

  • To build this form diagram, let's first create a square surface with the Rhino command "Rectangular Plane: Corner to Corner", typing in two times the length of the side.

  • Go to Create Pattern in the RV2 menu and use "from surface" to create the pattern. At this point, you will see a square pattern without diagonals.

  • Select the pattern and type the Rhino command "Ungroup" and press enter.

  • In Rhino, create a new layer called "Lines".

  • In Rhino, type "SelLine" and move all the selected lines to the layer "Lines" you just created.

  • Now type "SelPt" and delete all the points in the viewport.

  • In Rhino, select the command "Polyline" and draw the two diagonals from one corner to the other.

  • The edges in the form diagram should always be split at the point where they cross each other. For this reason, type the command "Split", select the two diagonals, press "Enter", and then select all the lines of the square pattern and press "Enter" again.

  • At the end of the previous step, you will see a pattern similar to the one in the picture above, with single lines.

  • Now you can click on Create pattern in the RV2 menu and use "from Lines".

2) Define boundary conditions:

  • Supports:

    • The supports of the cross vault are in the four corners (RV2>Define Boundary Conditions>Identify Supports>Select>Manual).

  • Openings:

    • No need to assign openings. This step has to be skipped and the openings on the four sides will be automatically defined by RV2.

      • Explanation: since we have to constrain the forces to be 0 along the edges perpendicular to the openings, if we use the sag function (even only 5%), it will try to put some forces in that edges.

2) Form Diagram: "Create Form Diagram".

3) Constrain force values in the edges of the form diagram:

In a cross vault, one possible way for the forces to flow within the structure is along the arches spanning between the diagonals and then from the diagonals directly to the supports. So, the edges perpendicular to the openings will carry zero forces. The arches on the boundaries will carry half of the load compared to the arches in the middle of the vault, due to the different tributary areas, as explained in the barrel vault example. So we will set the force for the arches on the boundaries equal to 1.0 and for the rest equal to 2.0. To avoid the collapse of the edges at the end of the diagonals, in the middle of the vault, we fix their force value equal to 1.0.

  • Select edges perpendicular to the openings (RV2>ModifyFormDiagram>ModifyEdgesAttributes>Manual):

    • In the pop-up window assign 0.000001 as hmax and hmin of the edges.

  • Select all the edges on the boundaries (RV2>ModifyFormDiagram>ModifyEdgesAttributes>Continuous):

    • In the pop-up window assign 1.0 as hmax and hmin of the edges.

  • Select the edges parallel to the openings and spanning between the diagonals, leaving the edges on the boundaries (RV2>ModifyFormDiagram>ModifyEdgesAttributes>Manual):

    • In the pop-up window assign 2.0 as hmax and hmin of the edges.

  • Select the four edges at the end of the diagonals meeting in the center of the vault (RV2>ModifyFormDiagram>ModifyEdgesAttributes>Manual):

    • In the pop-up window assign 1.0 as hmax and hmin of the edges.

4) Force Diagram: "Create Force Diagram".

5) run Horizontal Equilibrium:

  • Click on Horizontal Equilibrium>alpha>form100>enter

    • If equilibrium is not reached, change the number of iterations and run again.

6) run Vertical Equilibrium

  • Click on Vertical Equilibrium>target_height

    • Input as target height half of the length of the side with an opening.

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